Subgroups — C₄₁×#22471 #23798672

Order 10496.

1424 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#435	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1985	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1870	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1869	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1817	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1817	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1749	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₁	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
15	C₄×#41×C₄₁	—	2	8	no	—
16	C₂×C₄×Q₈×C₄₁	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₄₁	C₄	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₁	C₄	1	1	yes	characteristic
20	C₂×C₄×D₄×C₄₁	C₄	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₁	C₄	1	1	yes	characteristic
25	C₂×#117×C₄₁	C₄	1	1	yes	characteristic
26	C₂×#117×C₄₁	C₄	1	1	yes	characteristic
27	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#435	—	2	8	no	—
31	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
36	C₂×#97×C₄₁	C₂²	1	2	yes	—
37	C₄₁×#336	C₂²	1	1	yes	characteristic
38	C₄₁×#336	C₂²	1	1	yes	characteristic
39	C₄₁×#336	C₂²	1	1	yes	characteristic
40	C₄₁×#336	C₂²	1	2	yes	—
41	C₄₁×#336	C₂²	1	2	yes	—
42	C₄₁×#336	C₂²	1	2	yes	—
43	C₄₁×#336	C₂²	1	2	yes	—
44	C₄₁×#336	C₂²	1	1	yes	characteristic
45	C₂×#41×C₄₁	—	2	4	no	—
46	C₂×#41×C₄₁	—	2	4	no	—
47	C₂×#41×C₄₁	—	2	4	no	—
48	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₄₁	—	2	4	no	—
62	C₂×M₁₆×C₄₁	—	2	4	no	—
63	C₂×M₁₆×C₄₁	—	4	2	no	—
64	C₂²×C₈×C₄₁	—	2	4	no	—
65	C₂²×C₈×C₄₁	—	2	4	no	—
66	C₂²×C₈×C₄₁	—	4	2	no	—
67	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
71	C₄×Q₈×C₄₁	—	2	4	no	—
72	C₄×Q₈×C₄₁	—	2	4	no	—
73	C₄×D₄×C₄₁	—	2	4	no	—
74	C₄×D₄×C₄₁	—	2	4	no	—
75	C₄×D₄×C₄₁	—	2	4	no	—
76	C₄×D₄×C₄₁	—	2	4	no	—
77	C₄×D₄×C₄₁	—	2	4	no	—
78	C₄×D₄×C₄₁	D₄	1	8	yes	—
79	#117×C₄₁	—	2	4	no	—
80	#117×C₄₁	—	2	4	no	—
81	#117×C₄₁	—	2	4	no	—
82	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₄₁	—	2	4	no	—
95	C₂×C₄²×C₄₁	—	2	4	no	—
96	C₂×C₄²×C₄₁	—	2	4	no	—
97	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₄₁	—	2	4	no	—
108	C₂²⋊C₈×C₄₁	—	2	4	no	—
109	C₂²⋊C₈×C₄₁	—	2	4	no	—
110	C₂²⋊C₈×C₄₁	—	2	4	no	—
111	C₂²⋊C₈×C₄₁	—	4	2	no	—
112	#97×C₄₁	—	2	4	no	—
113	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
114	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
115	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
116	#41×C₄₁	—	2	32	no	—
117	C₂×Q₈×C₄₁	—	2	2	no	—
118	C₂×Q₈×C₄₁	—	2	2	no	—
119	C₂×Q₈×C₄₁	—	2	2	no	—
120	C₂×D₄×C₄₁	—	2	2	no	—
121	C₂×D₄×C₄₁	—	2	2	no	—
122	C₂×D₄×C₄₁	—	2	2	no	—
123	C₂×D₄×C₄₁	—	2	2	no	—
124	C₂×D₄×C₄₁	—	2	2	no	—
125	C₂×D₄×C₄₁	—	2	2	no	—
126	C₂×D₄×C₄₁	—	2	2	no	—
127	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	—
128	C₂×D₄×C₄₁	M₁₆	1	4	yes	complemented
129	C₂²×C₄×C₄₁	—	2	2	no	—
130	C₂²×C₄×C₄₁	—	2	2	no	—
131	C₂²×C₄×C₄₁	—	2	2	no	—
132	C₂²×C₄×C₄₁	—	2	2	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	2	no	—
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	2	no	—
137	C₂²×C₄×C₄₁	—	2	2	no	—
138	C₂²×C₄×C₄₁	—	2	2	no	—
139	C₂²×C₄×C₄₁	—	2	1	no	—
140	C₂²×C₄×C₄₁	—	2	1	no	—
141	C₂²×C₄×C₄₁	—	2	1	no	—
142	C₂²×C₄×C₄₁	—	2	1	no	—
143	C₂²×C₄×C₄₁	—	2	2	no	—
144	C₂²×C₄×C₄₁	—	2	2	no	—
145	C₂²×C₄×C₄₁	—	2	2	no	—
146	C₂²×C₄×C₄₁	—	2	2	no	—
147	C₂²×C₄×C₄₁	—	2	2	no	—
148	C₂²×C₄×C₄₁	—	2	2	no	—
149	C₂²×C₄×C₄₁	—	2	2	no	—
150	C₂²×C₄×C₄₁	—	2	2	no	—
151	C₂²×C₄×C₄₁	—	2	1	no	—
152	C₂²×C₄×C₄₁	—	2	1	no	—
153	C₂²×C₄×C₄₁	—	4	1	no	—
154	C₂²×C₄×C₄₁	—	4	1	no	—
155	C₂²×C₄×C₄₁	—	4	1	no	—
156	C₂²×C₄×C₄₁	M₁₆	1	4	yes	complemented
157	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
170	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
171	M₁₆×C₄₁	—	4	8	no	—
172	C₂×C₈×C₄₁	—	2	2	no	—
173	C₂×C₈×C₄₁	—	2	2	no	—
174	C₂×C₈×C₄₁	—	2	2	no	—
175	C₂×C₈×C₄₁	—	2	2	no	—
176	C₂×C₈×C₄₁	—	2	2	no	—
177	C₂×C₈×C₄₁	—	2	2	no	—
178	C₂×C₈×C₄₁	—	2	2	no	—
179	C₂×C₈×C₄₁	—	2	2	no	—
180	C₂×C₈×C₄₁	—	2	2	no	—
181	C₂×C₈×C₄₁	—	2	2	no	—
182	C₂×C₈×C₄₁	—	2	2	no	—
183	C₂×C₈×C₄₁	—	4	8	no	—
184	C₂×C₈×C₄₁	#41	1	4	yes	complemented
185	C₄⋊C₄×C₄₁	—	2	2	no	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	2	no	—
188	C₄⋊C₄×C₄₁	—	2	2	no	—
189	C₄⋊C₄×C₄₁	—	2	2	no	—
190	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
191	C₂²⋊C₄×C₄₁	—	2	2	no	—
192	C₂²⋊C₄×C₄₁	—	2	2	no	—
193	C₂²⋊C₄×C₄₁	—	2	2	no	—
194	C₂²⋊C₄×C₄₁	—	2	2	no	—
195	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
196	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
197	C₄²×C₄₁	—	2	2	no	—
198	C₄²×C₄₁	—	2	2	no	—
199	C₄²×C₄₁	—	2	2	no	—
200	C₄²×C₄₁	—	2	2	no	—
201	C₄²×C₄₁	—	2	2	no	—
202	C₄²×C₄₁	—	2	2	no	—
203	C₄²×C₄₁	—	2	2	no	—
204	C₄²×C₄₁	—	2	2	no	—
205	C₄²×C₄₁	—	2	2	no	—
206	C₄²×C₄₁	C₂×D₄	1	4	yes	—
207	C₂³×C₄₁	—	2	1	no	—
208	C₂³×C₄₁	—	2	1	no	—
209	C₂³×C₄₁	—	2	1	no	—
210	C₂³×C₄₁	—	2	1	no	—
211	C₂³×C₄₁	—	2	1	no	—
212	C₂³×C₄₁	—	2	1	no	—
213	C₂³×C₄₁	—	2	1	no	—
214	C₂³×C₄₁	—	2	1	no	—
215	C₂³×C₄₁	—	2	1	no	—
216	C₂³×C₄₁	—	2	1	no	—
217	C₂³×C₄₁	—	2	1	no	—
218	C₂³×C₄₁	—	2	1	no	—
219	C₂³×C₄₁	—	2	1	no	—
220	C₂³×C₄₁	—	2	1	no	—
221	C₂³×C₄₁	—	2	1	no	—
222	C₂³×C₄₁	—	2	1	no	—
223	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
225	C₂³×C₄₁	C₂×M₁₆	1	2	yes	—
226	C₂³×C₄₁	C₂×M₁₆	1	2	yes	—
227	C₂³×C₄₁	C₂×M₁₆	1	2	yes	—
228	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
229	Q₈×C₄₁	—	2	8	no	—
230	D₄×C₄₁	—	2	8	no	—
231	D₄×C₄₁	—	2	8	no	—
232	D₄×C₄₁	—	2	8	no	—
233	C₂×C₄×C₄₁	—	2	8	no	—
234	C₂×C₄×C₄₁	—	2	8	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	1	no	—
239	C₂×C₄×C₄₁	—	2	1	no	—
240	C₂×C₄×C₄₁	—	2	1	no	—
241	C₂×C₄×C₄₁	—	2	1	no	—
242	C₂×C₄×C₄₁	—	2	1	no	—
243	C₂×C₄×C₄₁	—	2	1	no	—
244	C₂×C₄×C₄₁	—	2	1	no	—
245	C₂×C₄×C₄₁	—	2	1	no	—
246	C₂×C₄×C₄₁	—	2	1	no	—
247	C₂×C₄×C₄₁	—	2	1	no	—
248	C₂×C₄×C₄₁	—	2	1	no	—
249	C₂×C₄×C₄₁	—	2	1	no	—
250	C₂×C₄×C₄₁	—	2	1	no	—
251	C₂×C₄×C₄₁	—	2	8	no	—
252	C₂×C₄×C₄₁	—	2	4	no	—
253	C₂×C₄×C₄₁	—	2	4	no	—
254	C₂×C₄×C₄₁	—	2	4	no	—
255	C₂×C₄×C₄₁	—	2	4	no	—
256	C₂×C₄×C₄₁	—	2	1	no	—
257	C₂×C₄×C₄₁	—	2	1	no	—
258	C₂×C₄×C₄₁	—	2	1	no	—
259	C₂×C₄×C₄₁	—	2	1	no	—
260	C₂×C₄×C₄₁	—	2	1	no	—
261	C₂×C₄×C₄₁	—	2	1	no	—
262	C₂×C₄×C₄₁	—	2	1	no	—
263	C₂×C₄×C₄₁	—	2	1	no	—
264	C₂×C₄×C₄₁	—	2	1	no	—
265	C₂×C₄×C₄₁	—	4	4	no	—
266	C₂×C₄×C₄₁	—	4	4	no	—
267	C₂×C₄×C₄₁	—	4	4	no	—
268	C₂×C₄×C₄₁	—	4	2	no	—
269	C₂×C₄×C₄₁	—	4	2	no	—
270	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
271	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
272	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
273	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
275	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
276	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
277	C₂×C₄×C₄₁	#131	1	2	yes	—
278	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
280	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
282	C₈×C₄₁	—	2	4	no	—
283	C₈×C₄₁	—	4	2	no	—
284	C₈×C₄₁	—	4	2	no	—
285	C₈×C₄₁	—	4	2	no	—
286	#22471	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂²×C₄₁	—	2	2	no	—
288	C₂²×C₄₁	—	2	2	no	—
289	C₂²×C₄₁	—	2	2	no	—
290	C₂²×C₄₁	—	2	2	no	—
291	C₂²×C₄₁	—	2	2	no	—
292	C₂²×C₄₁	—	2	2	no	—
293	C₂²×C₄₁	—	2	2	no	—
294	C₂²×C₄₁	—	2	2	no	—
295	C₂²×C₄₁	—	2	2	no	—
296	C₂²×C₄₁	—	2	2	no	—
297	C₂²×C₄₁	—	2	2	no	—
298	C₂²×C₄₁	—	2	2	no	—
299	C₂²×C₄₁	—	4	1	no	—
300	C₂²×C₄₁	—	4	1	no	—
301	C₂²×C₄₁	—	4	1	no	—
302	C₂²×C₄₁	—	4	1	no	—
303	C₂²×C₄₁	—	4	1	no	—
304	C₂²×C₄₁	—	4	1	no	—
305	C₂²×C₄₁	—	4	1	no	—
306	C₂²×C₄₁	—	4	1	no	—
307	C₂²×C₄₁	#407	1	4	yes	—
308	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
309	C₂²×C₄₁	C₄×#41	1	1	yes	characteristic
310	C₂²×C₄₁	#520	1	1	yes	characteristic
311	C₂²×C₄₁	C₂²×M₁₆	1	1	yes	characteristic
312	C₂²×C₄₁	C₂²×M₁₆	1	1	yes	characteristic
313	C₂²×C₄₁	C₂×#131	1	1	yes	characteristic
314	C₂²×C₄₁	#568	1	1	yes	characteristic
315	C₄×C₄₁	—	2	2	no	—
316	C₄×C₄₁	—	2	2	no	—
317	C₄×C₄₁	—	2	2	no	—
318	C₄×C₄₁	—	2	2	no	—
319	C₄×C₄₁	—	2	2	no	—
320	C₄×C₄₁	—	2	2	no	—
321	C₄×C₄₁	—	2	2	no	—
322	C₄×C₄₁	—	2	2	no	—
323	C₄×C₄₁	—	4	1	no	—
324	C₄×C₄₁	—	4	1	no	—
325	C₄×C₄₁	—	4	1	no	—
326	C₄×C₄₁	#546	1	2	yes	—
327	C₄×C₄₁	#547	1	2	yes	—
328	C₂×C₄×#41	C₂×C₄₁	1	1	yes	characteristic
329	C₂×#435	C₂×C₄₁	1	2	yes	complemented
330	#1985	C₂×C₄₁	1	2	yes	complemented
331	#1870	C₂×C₄₁	1	2	yes	complemented
332	#1869	C₂×C₄₁	1	2	yes	complemented
333	#1817	C₂×C₄₁	1	1	yes	characteristic, complemented
334	#1817	C₂×C₄₁	1	1	yes	characteristic, complemented
335	#1797	C₂×C₄₁	1	1	yes	characteristic, complemented
336	#1749	C₂×C₄₁	1	1	yes	characteristic, complemented
337	#1728	C₂×C₄₁	1	1	yes	characteristic, complemented
338	C₂×#336	C₂×C₄₁	1	1	yes	characteristic, complemented
339	C₂×C₄₁	—	2	4	no	—
340	C₂×C₄₁	—	4	2	no	—
341	C₂×C₄₁	—	4	2	no	—
342	C₂×C₄₁	#2339	1	1	yes	characteristic
343	C₂×C₄₁	C₂×#407	1	1	yes	characteristic
344	C₂×C₄₁	#2880	1	1	yes	characteristic
345	C₂×C₄₁	#2965	1	1	yes	characteristic
346	C₂×C₄₁	#2970	1	1	yes	characteristic
347	C₂×C₄₁	#2989	1	1	yes	characteristic
348	C₂×C₄₁	#2990	1	1	yes	characteristic
349	C₂²×#41	C₄×C₄₁	1	1	yes	characteristic
350	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
351	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
352	C₄×#41	—	2	8	no	—
353	C₂×C₄×Q₈	C₄×C₄₁	1	1	yes	characteristic
354	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
355	C₂×C₄×D₄	C₄×C₄₁	1	1	yes	characteristic
356	C₂×C₄×D₄	C₄×C₄₁	1	1	yes	characteristic
357	C₂×C₄×D₄	C₄×C₄₁	1	1	yes	characteristic
358	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
359	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
360	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
361	C₂×#117	C₄×C₄₁	1	1	yes	characteristic
362	C₂×#117	C₄×C₄₁	1	1	yes	characteristic
363	C₂×#117	C₄×C₄₁	1	1	yes	characteristic
364	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
365	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
366	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
367	#435	—	2	8	no	—
368	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
369	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
370	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
371	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
372	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
373	C₂×#97	C₂²×C₄₁	1	2	yes	—
374	#336	C₂²×C₄₁	1	1	yes	characteristic
375	#336	C₂²×C₄₁	1	1	yes	characteristic
376	#336	C₂²×C₄₁	1	1	yes	characteristic
377	#336	C₂²×C₄₁	1	2	yes	—
378	#336	C₂²×C₄₁	1	2	yes	—
379	#336	C₂²×C₄₁	1	2	yes	—
380	#336	C₂²×C₄₁	1	2	yes	—
381	#336	C₂²×C₄₁	1	1	yes	characteristic
382	C₄₁	#22471	1	1	yes	characteristic, direct product factor, Sylow
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	4	no	—
386	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
387	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
388	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
389	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
390	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
391	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
392	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
393	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
394	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
395	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
399	C₂×M₁₆	—	2	4	no	—
400	C₂×M₁₆	—	2	4	no	—
401	C₂×M₁₆	—	4	2	no	—
402	C₂²×C₈	—	2	4	no	—
403	C₂²×C₈	—	2	4	no	—
404	C₂²×C₈	—	4	2	no	—
405	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
406	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
407	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
408	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
409	C₄×Q₈	—	2	4	no	—
410	C₄×Q₈	—	2	4	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	4	no	—
416	C₄×D₄	D₄×C₄₁	1	8	yes	—
417	#117	—	2	4	no	—
418	#117	—	2	4	no	—
419	#117	—	2	4	no	—
420	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
429	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
432	C₂×C₄²	—	2	4	no	—
433	C₂×C₄²	—	2	4	no	—
434	C₂×C₄²	—	2	4	no	—
435	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
436	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
437	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
438	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
439	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
440	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
445	C₄⋊C₈	—	2	4	no	—
446	C₂²⋊C₈	—	2	4	no	—
447	C₂²⋊C₈	—	2	4	no	—
448	C₂²⋊C₈	—	2	4	no	—
449	C₂²⋊C₈	—	4	2	no	—
450	#97	—	2	4	no	—
451	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
452	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
453	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
454	#41	—	2	32	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	—
466	C₂×D₄	M₁₆×C₄₁	1	4	yes	complemented
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	4	1	no	—
492	C₂²×C₄	—	4	1	no	—
493	C₂²×C₄	—	4	1	no	—
494	C₂²×C₄	M₁₆×C₄₁	1	4	yes	complemented
495	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
496	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
497	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
498	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
499	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
500	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
501	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
502	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
503	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
504	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
505	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
507	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
508	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
509	M₁₆	—	4	8	no	—
510	C₂×C₈	—	2	2	no	—
511	C₂×C₈	—	2	2	no	—
512	C₂×C₈	—	2	2	no	—
513	C₂×C₈	—	2	2	no	—
514	C₂×C₈	—	2	2	no	—
515	C₂×C₈	—	2	2	no	—
516	C₂×C₈	—	2	2	no	—
517	C₂×C₈	—	2	2	no	—
518	C₂×C₈	—	2	2	no	—
519	C₂×C₈	—	2	2	no	—
520	C₂×C₈	—	2	2	no	—
521	C₂×C₈	—	4	8	no	—
522	C₂×C₈	#41×C₄₁	1	4	yes	complemented
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
534	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
535	C₄²	—	2	2	no	—
536	C₄²	—	2	2	no	—
537	C₄²	—	2	2	no	—
538	C₄²	—	2	2	no	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	2	no	—
541	C₄²	—	2	2	no	—
542	C₄²	—	2	2	no	—
543	C₄²	—	2	2	no	—
544	C₄²	C₂×D₄×C₄₁	1	4	yes	—
545	C₂³	—	2	1	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	2	1	no	—
560	C₂³	—	2	1	no	—
561	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
562	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
563	C₂³	C₂×M₁₆×C₄₁	1	2	yes	—
564	C₂³	C₂×M₁₆×C₄₁	1	2	yes	—
565	C₂³	C₂×M₁₆×C₄₁	1	2	yes	—
566	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
567	Q₈	—	2	8	no	—
568	D₄	—	2	8	no	—
569	D₄	—	2	8	no	—
570	D₄	—	2	8	no	—
571	C₂×C₄	—	2	8	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	8	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	—	4	4	no	—
605	C₂×C₄	—	4	4	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
609	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
610	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
611	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
612	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
613	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
614	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
615	C₂×C₄	#131×C₄₁	1	2	yes	—
616	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
617	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
618	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
619	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
620	C₈	—	2	4	no	—
621	C₈	—	4	2	no	—
622	C₈	—	4	2	no	—
623	C₈	—	4	2	no	—
624	C₂²	—	2	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	4	1	no	—
643	C₂²	—	4	1	no	—
644	C₂²	C₄₁×#407	1	4	yes	—
645	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
646	C₂²	C₄×#41×C₄₁	1	1	yes	characteristic
647	C₂²	C₄₁×#520	1	1	yes	characteristic
648	C₂²	C₂²×M₁₆×C₄₁	1	1	yes	characteristic
649	C₂²	C₂²×M₁₆×C₄₁	1	1	yes	characteristic
650	C₂²	C₂×#131×C₄₁	1	1	yes	characteristic
651	C₂²	C₄₁×#568	1	1	yes	characteristic
652	C₄	—	2	2	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	2	no	—
659	C₄	—	2	2	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	C₄₁×#546	1	2	yes	—
664	C₄	C₄₁×#547	1	2	yes	—
665	C₂	—	2	4	no	—
666	C₂	—	4	2	no	—
667	C₂	—	4	2	no	—
668	C₂	C₄₁×#2339	1	1	yes	characteristic
669	C₂	C₂×C₄₁×#407	1	1	yes	characteristic
670	C₂	C₄₁×#2880	1	1	yes	characteristic
671	C₂	C₄₁×#2965	1	1	yes	characteristic
672	C₂	C₄₁×#2970	1	1	yes	characteristic
673	C₂	C₄₁×#2989	1	1	yes	characteristic
674	C₂	C₄₁×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂²×#41×C₄₁	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄₁×#435	C₂	semidirect
3. C₄₁×#1985	C₂	semidirect
4. C₄₁×#1870	C₂	semidirect
5. C₄₁×#1869	C₂	semidirect
6. C₄₁×#1817	C₂	semidirect
7. C₄₁×#1817	C₂	semidirect
8. C₄₁×#1797	C₂	semidirect
9. C₄₁×#1749	C₂	semidirect
10. C₄₁×#1728	C₂	semidirect
11. C₂×C₄₁×#336	C₂	semidirect
128. C₂×D₄×C₄₁	M₁₆	semidirect
156. C₂²×C₄×C₄₁	M₁₆	semidirect
184. C₂×C₈×C₄₁	#41	semidirect
286. #22471	C₄₁	direct (N × H)
329. C₂×#435	C₂×C₄₁	semidirect
330. #1985	C₂×C₄₁	semidirect
331. #1870	C₂×C₄₁	semidirect
332. #1869	C₂×C₄₁	semidirect
333. #1817	C₂×C₄₁	semidirect
334. #1817	C₂×C₄₁	semidirect
335. #1797	C₂×C₄₁	semidirect
336. #1749	C₂×C₄₁	semidirect
337. #1728	C₂×C₄₁	semidirect
338. C₂×#336	C₂×C₄₁	semidirect
382. C₄₁	#22471	direct (N × H)
466. C₂×D₄	M₁₆×C₄₁	semidirect
494. C₂²×C₄	M₁₆×C₄₁	semidirect
522. C₂×C₈	#41×C₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#41×C₄₁	2
C₂×C₄₁×#435	2
C₄₁×#1985	2
C₄₁×#1870	2
C₄₁×#1869	2
C₄₁×#1817	2
C₄₁×#1797	2
C₄₁×#1749	2
C₄₁×#1728	2
C₂×C₄₁×#336	2
#22471	41

Subgroups — C41×#22471 #23798672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22471 #23798672