Subgroups — C₄₁×#17575 #23793776

Order 10496.

1058 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1520	C₂	1	4	yes	maximal
5	C₄₁×#1518	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1503	C₂	1	4	yes	maximal, complemented
7	C₄₁×#533	—	2	1	no	—
8	C₄₁×#533	—	2	1	no	—
9	C₄₁×#533	—	2	1	no	—
10	C₄₁×#533	—	2	1	no	—
11	C₄₁×#533	C₄	1	2	yes	complemented
12	C₄₁×#533	C₄	1	2	yes	complemented
13	C₄₁×#533	C₂²	1	2	yes	—
14	C₄₁×#533	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₁	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄⋊Q₈×C₄₁	C₄	1	1	yes	characteristic, complemented
17	C₂×#125×C₄₁	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₄₁	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₄₁	C₄	1	2	yes	complemented
22	C₂×#117×C₄₁	—	2	1	no	—
23	C₂×#117×C₄₁	—	2	1	no	—
24	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#431	—	2	1	no	—
26	C₄₁×#431	—	2	1	no	—
27	C₄₁×#431	C₂²	1	2	yes	—
28	C₄₁×#431	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
30	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₁	—	2	1	no	—
32	C₄₁×#384	—	2	1	no	—
33	C₄₁×#382	C₂²	1	2	yes	—
34	C₄₁×#380	—	2	1	no	—
35	C₄₁×#380	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
38	C₄₁×#328	C₂²	1	16	yes	—
39	C₂²×Q₈×C₄₁	—	2	1	no	—
40	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₁	—	2	2	no	—
45	C₄⋊Q₈×C₄₁	—	2	1	no	—
46	C₄⋊Q₈×C₄₁	—	2	1	no	—
47	C₄⋊Q₈×C₄₁	—	2	1	no	—
48	C₄⋊Q₈×C₄₁	—	2	1	no	—
49	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
50	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
51	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
52	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
53	#125×C₄₁	—	2	2	no	—
54	#125×C₄₁	—	2	2	no	—
55	#125×C₄₁	—	2	2	no	—
56	#125×C₄₁	—	2	2	no	—
57	C₂²⋊Q₈×C₄₁	—	2	2	no	—
58	C₂²⋊Q₈×C₄₁	—	2	2	no	—
59	C₂²⋊Q₈×C₄₁	—	2	2	no	—
60	C₂²⋊Q₈×C₄₁	—	2	2	no	—
61	C₂²⋊Q₈×C₄₁	—	2	2	no	—
62	C₂²⋊Q₈×C₄₁	—	2	2	no	—
63	C₂²⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
64	C₂²⋊Q₈×C₄₁	C₂³	1	4	yes	—
65	C₄×Q₈×C₄₁	—	2	2	no	—
66	C₄×Q₈×C₄₁	—	2	2	no	—
67	C₄×Q₈×C₄₁	—	2	2	no	—
68	C₄×Q₈×C₄₁	—	2	2	no	—
69	C₄×Q₈×C₄₁	—	2	2	no	—
70	C₄×Q₈×C₄₁	—	2	2	no	—
71	C₄×Q₈×C₄₁	C₂×C₄	1	4	yes	—
72	C₄×Q₈×C₄₁	C₂×C₄	1	4	yes	—
73	#117×C₄₁	—	2	2	no	—
74	#117×C₄₁	—	2	2	no	—
75	#117×C₄₁	—	2	2	no	—
76	#117×C₄₁	—	2	2	no	—
77	#117×C₄₁	—	4	4	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
85	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
89	C₂×C₄²×C₄₁	—	2	2	no	—
90	C₂×C₄²×C₄₁	—	2	2	no	—
91	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₄₁	—	2	2	no	—
104	C₄⋊C₈×C₄₁	—	2	2	no	—
105	C₂²⋊C₈×C₄₁	—	2	2	no	—
106	C₂²⋊C₈×C₄₁	C₂³	1	4	yes	—
107	C₄×C₈×C₄₁	—	2	2	no	—
108	C₄×C₈×C₄₁	—	2	2	no	—
109	#95×C₄₁	—	2	2	no	—
110	#95×C₄₁	C₂³	1	4	yes	—
111	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
112	C₂×Q₈×C₄₁	—	2	2	no	—
113	C₂×Q₈×C₄₁	—	2	2	no	—
114	C₂×Q₈×C₄₁	—	2	2	no	—
115	C₂×Q₈×C₄₁	—	4	4	no	—
116	C₂×Q₈×C₄₁	—	4	4	no	—
117	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
118	C₂²×C₄×C₄₁	—	2	2	no	—
119	C₂²×C₄×C₄₁	—	2	2	no	—
120	C₂²×C₄×C₄₁	—	2	4	no	—
121	C₂²×C₄×C₄₁	—	4	2	no	—
122	C₂²×C₄×C₄₁	—	4	2	no	—
123	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄×C₄₁	SD₁₆	1	4	yes	—
132	C₂²×C₄×C₄₁	Q₁₆	1	4	yes	—
133	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
143	C₂×C₈×C₄₁	—	2	4	no	—
144	C₂×C₈×C₄₁	—	4	4	no	—
145	C₄⋊C₄×C₄₁	—	2	2	no	—
146	C₄⋊C₄×C₄₁	—	2	2	no	—
147	C₄⋊C₄×C₄₁	—	2	2	no	—
148	C₄⋊C₄×C₄₁	—	2	2	no	—
149	C₄⋊C₄×C₄₁	—	2	2	no	—
150	C₄⋊C₄×C₄₁	—	2	2	no	—
151	C₄⋊C₄×C₄₁	—	2	4	no	—
152	C₄⋊C₄×C₄₁	—	2	4	no	—
153	C₄⋊C₄×C₄₁	—	2	4	no	—
154	C₄⋊C₄×C₄₁	—	2	2	no	—
155	C₄⋊C₄×C₄₁	—	2	4	no	—
156	C₄⋊C₄×C₄₁	—	4	4	no	—
157	C₄⋊C₄×C₄₁	—	4	4	no	—
158	C₄⋊C₄×C₄₁	—	4	4	no	—
159	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
160	C₂²⋊C₄×C₄₁	—	2	4	no	—
161	C₂²⋊C₄×C₄₁	—	2	4	no	—
162	C₂²⋊C₄×C₄₁	—	4	4	no	—
163	C₄²×C₄₁	—	2	4	no	—
164	C₄²×C₄₁	—	2	4	no	—
165	C₄²×C₄₁	—	4	4	no	—
166	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
167	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
169	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
170	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
171	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
172	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
175	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
176	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
182	C₂³×C₄₁	—	4	1	no	—
183	C₂³×C₄₁	—	4	1	no	—
184	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂³×C₄₁	C₂×SD₁₆	1	2	yes	—
187	C₂³×C₄₁	C₂×Q₁₆	1	2	yes	—
188	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₄₁	—	4	4	no	—
190	Q₈×C₄₁	—	4	4	no	—
191	Q₈×C₄₁	—	4	4	no	—
192	Q₈×C₄₁	—	4	4	no	—
193	C₂×C₄×C₄₁	—	2	2	no	—
194	C₂×C₄×C₄₁	—	2	2	no	—
195	C₂×C₄×C₄₁	—	2	2	no	—
196	C₂×C₄×C₄₁	—	2	2	no	—
197	C₂×C₄×C₄₁	—	2	2	no	—
198	C₂×C₄×C₄₁	—	2	2	no	—
199	C₂×C₄×C₄₁	—	2	4	no	—
200	C₂×C₄×C₄₁	—	2	4	no	—
201	C₂×C₄×C₄₁	—	4	1	no	—
202	C₂×C₄×C₄₁	—	4	1	no	—
203	C₂×C₄×C₄₁	—	4	1	no	—
204	C₂×C₄×C₄₁	—	4	1	no	—
205	C₂×C₄×C₄₁	—	4	1	no	—
206	C₂×C₄×C₄₁	—	4	1	no	—
207	C₂×C₄×C₄₁	—	4	4	no	—
208	C₂×C₄×C₄₁	—	4	4	no	—
209	C₂×C₄×C₄₁	—	4	4	no	—
210	C₂×C₄×C₄₁	—	4	4	no	—
211	C₂×C₄×C₄₁	—	4	4	no	—
212	C₂×C₄×C₄₁	#103	1	4	yes	—
213	C₂×C₄×C₄₁	#103	1	4	yes	—
214	C₂×C₄×C₄₁	#103	1	4	yes	—
215	C₂×C₄×C₄₁	#103	1	4	yes	—
216	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
227	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
228	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
229	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
230	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
231	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
232	C₈×C₄₁	—	4	4	no	—
233	#17575	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂²×C₄₁	—	2	1	no	—
235	C₂²×C₄₁	—	2	1	no	—
236	C₂²×C₄₁	—	2	1	no	—
237	C₂²×C₄₁	—	2	1	no	—
238	C₂²×C₄₁	—	2	1	no	—
239	C₂²×C₄₁	—	2	1	no	—
240	C₂²×C₄₁	—	4	2	no	—
241	C₂²×C₄₁	—	4	2	no	—
242	C₂²×C₄₁	C₂×#103	1	2	yes	—
243	C₂²×C₄₁	C₂×#103	1	2	yes	—
244	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
246	C₂²×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
247	C₄×C₄₁	—	2	1	no	—
248	C₄×C₄₁	—	2	1	no	—
249	C₄×C₄₁	—	2	1	no	—
250	C₄×C₄₁	—	2	1	no	—
251	C₄×C₄₁	—	2	1	no	—
252	C₄×C₄₁	—	2	1	no	—
253	C₄×C₄₁	—	4	4	no	—
254	C₄×C₄₁	—	4	4	no	—
255	C₄×C₄₁	—	8	4	no	—
256	C₄×C₄₁	#410	1	2	yes	—
257	C₄×C₄₁	C₂×#103	1	2	yes	—
258	C₄×C₄₁	C₂×#103	1	2	yes	—
259	C₄×C₄₁	C₂×#103	1	2	yes	—
260	C₄×C₄₁	C₂×#103	1	2	yes	—
261	C₄×C₄₁	#421	1	2	yes	—
262	C₂×#533	C₂×C₄₁	1	1	yes	characteristic
263	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
264	#2285	C₂×C₄₁	1	1	yes	characteristic
265	#1520	C₂×C₄₁	1	4	yes	—
266	#1518	C₂×C₄₁	1	4	yes	complemented
267	#1503	C₂×C₄₁	1	4	yes	complemented
268	C₂×C₄₁	—	2	1	no	—
269	C₂×C₄₁	—	2	1	no	—
270	C₂×C₄₁	—	2	2	no	—
271	C₂×C₄₁	—	2	2	no	—
272	C₂×C₄₁	C₂×#410	1	1	yes	characteristic
273	C₂×C₄₁	C₂²×#103	1	1	yes	characteristic
274	C₂×C₄₁	C₂×#421	1	1	yes	characteristic
275	#533	—	2	1	no	—
276	#533	—	2	1	no	—
277	#533	—	2	1	no	—
278	#533	—	2	1	no	—
279	#533	C₄×C₄₁	1	2	yes	complemented
280	#533	C₄×C₄₁	1	2	yes	complemented
281	#533	C₂²×C₄₁	1	2	yes	—
282	#533	C₂²×C₄₁	1	2	yes	—
283	C₂×C₄⋊Q₈	C₄×C₄₁	1	1	yes	characteristic, complemented
284	C₂×C₄⋊Q₈	C₄×C₄₁	1	1	yes	characteristic, complemented
285	C₂×#125	—	2	1	no	—
286	C₂×C₂²⋊Q₈	—	2	1	no	—
287	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
288	C₂×C₄×Q₈	—	2	1	no	—
289	C₂×C₄×Q₈	C₄×C₄₁	1	2	yes	complemented
290	C₂×#117	—	2	1	no	—
291	C₂×#117	—	2	1	no	—
292	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
293	#431	—	2	1	no	—
294	#431	—	2	1	no	—
295	#431	C₂²×C₄₁	1	2	yes	—
296	#431	C₂²×C₄₁	1	2	yes	—
297	C₂×C₄⋊C₈	—	2	1	no	—
298	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
299	C₂×C₄×C₈	—	2	1	no	—
300	#384	—	2	1	no	—
301	#382	C₂²×C₄₁	1	2	yes	—
302	#380	—	2	1	no	—
303	#380	C₂²×C₄₁	1	2	yes	—
304	C₄×C₄⋊C₄	—	2	1	no	—
305	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
306	#328	C₂²×C₄₁	1	16	yes	—
307	C₄₁	#17575	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²×Q₈	—	2	1	no	—
309	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
310	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
311	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
313	C₂²×C₈	—	2	2	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	—	2	1	no	—
317	C₄⋊Q₈	—	2	1	no	—
318	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
319	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
320	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
321	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
322	#125	—	2	2	no	—
323	#125	—	2	2	no	—
324	#125	—	2	2	no	—
325	#125	—	2	2	no	—
326	C₂²⋊Q₈	—	2	2	no	—
327	C₂²⋊Q₈	—	2	2	no	—
328	C₂²⋊Q₈	—	2	2	no	—
329	C₂²⋊Q₈	—	2	2	no	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
333	C₂²⋊Q₈	C₂³×C₄₁	1	4	yes	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	C₂×C₄×C₄₁	1	4	yes	—
341	C₄×Q₈	C₂×C₄×C₄₁	1	4	yes	—
342	#117	—	2	2	no	—
343	#117	—	2	2	no	—
344	#117	—	2	2	no	—
345	#117	—	2	2	no	—
346	#117	—	4	4	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	—	2	2	no	—
360	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
361	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
362	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
363	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
364	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
365	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
366	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
367	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
368	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
369	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
372	C₄⋊C₈	—	2	2	no	—
373	C₄⋊C₈	—	2	2	no	—
374	C₂²⋊C₈	—	2	2	no	—
375	C₂²⋊C₈	C₂³×C₄₁	1	4	yes	—
376	C₄×C₈	—	2	2	no	—
377	C₄×C₈	—	2	2	no	—
378	#95	—	2	2	no	—
379	#95	C₂³×C₄₁	1	4	yes	—
380	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	4	4	no	—
385	C₂×Q₈	—	4	4	no	—
386	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
393	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
394	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
395	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
396	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
397	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
398	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
399	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
400	C₂²×C₄	SD₁₆×C₄₁	1	4	yes	—
401	C₂²×C₄	Q₁₆×C₄₁	1	4	yes	—
402	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
403	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	4	4	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	4	4	no	—
426	C₄⋊C₄	—	4	4	no	—
427	C₄⋊C₄	—	4	4	no	—
428	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₄²	—	2	4	no	—
433	C₄²	—	2	4	no	—
434	C₄²	—	4	4	no	—
435	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
436	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
437	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
438	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
439	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
440	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
441	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
442	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
443	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
444	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
445	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
446	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
448	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
449	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
450	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
451	C₂³	—	4	1	no	—
452	C₂³	—	4	1	no	—
453	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
454	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
455	C₂³	C₂×SD₁₆×C₄₁	1	2	yes	—
456	C₂³	C₂×Q₁₆×C₄₁	1	2	yes	—
457	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
458	Q₈	—	4	4	no	—
459	Q₈	—	4	4	no	—
460	Q₈	—	4	4	no	—
461	Q₈	—	4	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	4	4	no	—
481	C₂×C₄	#103×C₄₁	1	4	yes	—
482	C₂×C₄	#103×C₄₁	1	4	yes	—
483	C₂×C₄	#103×C₄₁	1	4	yes	—
484	C₂×C₄	#103×C₄₁	1	4	yes	—
485	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
486	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
487	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
488	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
489	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
490	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
491	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
492	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
493	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
494	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
495	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
496	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
497	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
498	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
499	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
500	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
501	C₈	—	4	4	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	1	no	—
507	C₂²	—	2	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	C₂×#103×C₄₁	1	2	yes	—
511	C₂²	C₂×#103×C₄₁	1	2	yes	—
512	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
513	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
514	C₂²	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
515	C₄	—	2	1	no	—
516	C₄	—	2	1	no	—
517	C₄	—	2	1	no	—
518	C₄	—	2	1	no	—
519	C₄	—	2	1	no	—
520	C₄	—	2	1	no	—
521	C₄	—	4	4	no	—
522	C₄	—	4	4	no	—
523	C₄	—	8	4	no	—
524	C₄	C₄₁×#410	1	2	yes	—
525	C₄	C₂×#103×C₄₁	1	2	yes	—
526	C₄	C₂×#103×C₄₁	1	2	yes	—
527	C₄	C₂×#103×C₄₁	1	2	yes	—
528	C₄	C₂×#103×C₄₁	1	2	yes	—
529	C₄	C₄₁×#421	1	2	yes	—
530	C₂	—	2	1	no	—
531	C₂	—	2	1	no	—
532	C₂	—	2	2	no	—
533	C₂	—	2	2	no	—
534	C₂	C₂×C₄₁×#410	1	1	yes	characteristic
535	C₂	C₂²×#103×C₄₁	1	1	yes	characteristic
536	C₂	C₂×C₄₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄₁×#1518	C₂	semidirect
6. C₄₁×#1503	C₂	semidirect
11. C₄₁×#533	C₄	semidirect
12. C₄₁×#533	C₄	semidirect
15. C₂×C₄⋊Q₈×C₄₁	C₄	semidirect
16. C₂×C₄⋊Q₈×C₄₁	C₄	semidirect
21. C₂×C₄×Q₈×C₄₁	C₄	semidirect
233. #17575	C₄₁	direct (N × H)
266. #1518	C₂×C₄₁	semidirect
267. #1503	C₂×C₄₁	semidirect
279. #533	C₄×C₄₁	semidirect
280. #533	C₄×C₄₁	semidirect
283. C₂×C₄⋊Q₈	C₄×C₄₁	semidirect
284. C₂×C₄⋊Q₈	C₄×C₄₁	semidirect
289. C₂×C₄×Q₈	C₄×C₄₁	semidirect
307. C₄₁	#17575	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#533	2
C₂×C₄₁×#431	2
C₄₁×#2285	2
C₄₁×#1520	2
C₄₁×#1518	2
C₄₁×#1503	2
#17575	41

Subgroups — C41×#17575 #23793776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17575 #23793776