Subgroups — C₄₁×#34201 #23810402

Order 10496.

3118 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#117×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2363	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2362	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2354	C₂	1	4	yes	maximal, complemented
7	C₄₁×#2352	C₂	1	4	yes	maximal, complemented
8	C₄₁×#2321	C₂	1	4	yes	maximal, complemented
9	C₄₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#2311	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#123×C₄₁	C₂	1	4	yes	maximal, complemented
12	C₄₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#525	C₄	1	16	yes	complemented
17	C₂×#123×C₄₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
19	C₂×#121×C₄₁	C₂²	1	2	yes	—
20	C₂×#120×C₄₁	C₂²	1	2	yes	—
21	C₄₁×#520	—	2	8	no	—
22	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	complemented
23	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	complemented
26	C₂×#117×C₄₁	—	2	8	no	—
27	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₁	C₂²	1	4	yes	—
30	C₂×#117×C₄₁	C₂²	1	4	yes	—
31	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#117×C₄₁	C₂²	1	2	yes	complemented
33	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
36	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
37	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
38	C₄₁×#391	C₂²	1	2	yes	complemented
39	C₄₁×#390	C₂²	1	8	yes	—
40	C₄₁×#390	C₂²	1	2	yes	complemented
41	C₄₁×#389	C₂²	1	4	yes	complemented
42	C₄₁×#388	C₂²	1	8	yes	—
43	C₄₁×#388	C₂²	1	8	yes	—
44	C₄₁×#388	C₂²	1	8	yes	complemented
45	C₄₁×#387	C₂²	1	8	yes	complemented
46	C₄₁×#386	C₂²	1	4	yes	—
47	C₄₁×#386	C₂²	1	8	yes	—
48	C₄₁×#385	C₂²	1	4	yes	—
49	C₄₁×#384	C₂²	1	2	yes	complemented
50	C₄₁×#382	C₂²	1	2	yes	complemented
51	C₄₁×#381	C₂²	1	2	yes	—
52	C₄₁×#380	C₂²	1	2	yes	—
53	C₄₁×#380	C₂²	1	2	yes	—
54	C₄₁×#380	C₂²	1	4	yes	complemented
55	C₄₁×#379	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
57	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
58	C₄×C₄⋊C₄×C₄₁	C₂²	1	8	yes	complemented
59	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
60	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
61	C₄×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	complemented
62	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
63	C₂×#41×C₄₁	—	2	4	no	—
64	C₂×#41×C₄₁	—	2	4	no	—
65	C₂×#41×C₄₁	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₄₁	—	2	4	no	—
71	C₂³×C₄×C₄₁	—	2	4	no	—
72	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	complemented
73	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
79	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
81	#123×C₄₁	C₂×C₄	1	16	yes	complemented
82	C₂²⋊Q₈×C₄₁	C₂×C₄	1	16	yes	complemented
83	#121×C₄₁	C₂×C₄	1	16	yes	complemented
84	#120×C₄₁	C₂×C₄	1	16	yes	complemented
85	C₄×Q₈×C₄₁	—	2	8	no	—
86	C₄×D₄×C₄₁	—	2	16	no	—
87	C₄×D₄×C₄₁	—	2	8	no	—
88	C₄×D₄×C₄₁	C₂×C₄	1	32	yes	complemented
89	#117×C₄₁	—	2	4	no	—
90	#117×C₄₁	—	2	4	no	—
91	#117×C₄₁	—	2	16	no	—
92	#117×C₄₁	—	2	16	no	—
93	#117×C₄₁	—	2	8	no	—
94	#117×C₄₁	C₂×C₄	1	8	yes	complemented
95	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
96	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
97	C₂×C₄⋊C₄×C₄₁	D₄	1	8	yes	complemented
98	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
105	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
110	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₄₁	—	2	8	no	—
121	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
122	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
123	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
125	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
126	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
127	#95×C₄₁	C₂³	1	4	yes	—
128	#95×C₄₁	C₂³	1	4	yes	—
129	#95×C₄₁	C₂³	1	8	yes	—
130	#95×C₄₁	C₂³	1	8	yes	—
131	#95×C₄₁	C₂³	1	8	yes	—
132	C₂⁴×C₄₁	—	2	2	no	—
133	C₂⁴×C₄₁	—	2	2	no	—
134	C₂⁴×C₄₁	—	4	2	no	—
135	C₂⁴×C₄₁	—	4	2	no	—
136	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
137	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
138	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
139	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
140	#41×C₄₁	—	2	32	no	—
141	C₂×Q₈×C₄₁	—	2	2	no	—
142	C₂×Q₈×C₄₁	—	2	2	no	—
143	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
144	C₂×D₄×C₄₁	—	2	4	no	—
145	C₂×D₄×C₄₁	—	2	4	no	—
146	C₂×D₄×C₄₁	—	2	2	no	—
147	C₂×D₄×C₄₁	—	2	2	no	—
148	C₂×D₄×C₄₁	—	4	4	no	—
149	C₂×D₄×C₄₁	—	4	4	no	—
150	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
151	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₄₁	C₂²×C₄	1	16	yes	—
153	C₂²×C₄×C₄₁	—	2	2	no	—
154	C₂²×C₄×C₄₁	—	2	2	no	—
155	C₂²×C₄×C₄₁	—	2	32	no	—
156	C₂²×C₄×C₄₁	—	2	4	no	—
157	C₂²×C₄×C₄₁	—	2	4	no	—
158	C₂²×C₄×C₄₁	—	2	4	no	—
159	C₂²×C₄×C₄₁	—	2	4	no	—
160	C₂²×C₄×C₄₁	—	2	4	no	—
161	C₂²×C₄×C₄₁	—	2	4	no	—
162	C₂²×C₄×C₄₁	—	2	4	no	—
163	C₂²×C₄×C₄₁	—	2	8	no	—
164	C₂²×C₄×C₄₁	—	4	4	no	—
165	C₂²×C₄×C₄₁	—	4	4	no	—
166	C₂²×C₄×C₄₁	—	4	8	no	—
167	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
168	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
173	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₄₁	#41	1	4	yes	—
175	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₄₁	—	2	2	no	—
186	C₄⋊C₄×C₄₁	—	2	4	no	—
187	C₄⋊C₄×C₄₁	—	2	4	no	—
188	C₄⋊C₄×C₄₁	—	2	8	no	—
189	C₄⋊C₄×C₄₁	—	2	2	no	—
190	C₄⋊C₄×C₄₁	—	2	4	no	—
191	C₄⋊C₄×C₄₁	—	4	8	no	—
192	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
194	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
195	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
196	C₂²⋊C₄×C₄₁	—	2	4	no	—
197	C₂²⋊C₄×C₄₁	—	2	4	no	—
198	C₂²⋊C₄×C₄₁	—	2	16	no	—
199	C₂²⋊C₄×C₄₁	—	2	8	no	—
200	C₂²⋊C₄×C₄₁	—	4	4	no	—
201	C₂²⋊C₄×C₄₁	—	4	4	no	—
202	C₂²⋊C₄×C₄₁	—	4	8	no	—
203	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
204	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
205	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
206	C₄²×C₄₁	—	2	16	no	—
207	C₄²×C₄₁	—	2	8	no	—
208	C₄²×C₄₁	—	2	4	no	—
209	C₄²×C₄₁	—	2	4	no	—
210	C₄²×C₄₁	—	4	8	no	—
211	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
212	C₂³×C₄₁	—	2	2	no	—
213	C₂³×C₄₁	—	2	2	no	—
214	C₂³×C₄₁	—	2	8	no	—
215	C₂³×C₄₁	—	2	2	no	—
216	C₂³×C₄₁	—	2	2	no	—
217	C₂³×C₄₁	—	2	2	no	—
218	C₂³×C₄₁	—	2	2	no	—
219	C₂³×C₄₁	—	2	4	no	—
220	C₂³×C₄₁	—	2	2	no	—
221	C₂³×C₄₁	—	2	2	no	—
222	C₂³×C₄₁	—	2	4	no	—
223	C₂³×C₄₁	—	2	2	no	—
224	C₂³×C₄₁	—	2	2	no	—
225	C₂³×C₄₁	—	4	8	no	—
226	C₂³×C₄₁	—	4	4	no	—
227	C₂³×C₄₁	—	4	4	no	—
228	C₂³×C₄₁	—	4	8	no	—
229	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
230	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
231	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
232	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
233	C₂³×C₄₁	C₂×#41	1	2	yes	—
234	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
235	Q₈×C₄₁	—	2	8	no	—
236	D₄×C₄₁	—	2	16	no	—
237	D₄×C₄₁	—	2	8	no	—
238	D₄×C₄₁	—	4	16	no	—
239	C₂×C₄×C₄₁	—	2	8	no	—
240	C₂×C₄×C₄₁	—	2	2	no	—
241	C₂×C₄×C₄₁	—	2	2	no	—
242	C₂×C₄×C₄₁	—	2	2	no	—
243	C₂×C₄×C₄₁	—	2	2	no	—
244	C₂×C₄×C₄₁	—	2	4	no	—
245	C₂×C₄×C₄₁	—	2	8	no	—
246	C₂×C₄×C₄₁	—	2	4	no	—
247	C₂×C₄×C₄₁	—	2	16	no	—
248	C₂×C₄×C₄₁	—	2	16	no	—
249	C₂×C₄×C₄₁	—	2	16	no	—
250	C₂×C₄×C₄₁	—	2	2	no	—
251	C₂×C₄×C₄₁	—	2	4	no	—
252	C₂×C₄×C₄₁	—	2	2	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	4	no	—
257	C₂×C₄×C₄₁	—	2	8	no	—
258	C₂×C₄×C₄₁	—	2	4	no	—
259	C₂×C₄×C₄₁	—	4	8	no	—
260	C₂×C₄×C₄₁	—	4	8	no	—
261	C₂×C₄×C₄₁	—	8	8	no	—
262	C₂×C₄×C₄₁	C₄×D₄	1	16	yes	—
263	C₂×C₄×C₄₁	C₂³×C₄	1	8	yes	—
264	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
265	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
266	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₄₁	C₂²×D₄	1	8	yes	—
269	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
272	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
273	#34201	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₄₁	—	2	4	no	—
275	C₂²×C₄₁	—	2	2	no	—
276	C₂²×C₄₁	—	2	2	no	—
277	C₂²×C₄₁	—	2	4	no	—
278	C₂²×C₄₁	—	2	4	no	—
279	C₂²×C₄₁	—	2	4	no	—
280	C₂²×C₄₁	—	2	4	no	—
281	C₂²×C₄₁	—	2	8	no	—
282	C₂²×C₄₁	—	2	4	no	—
283	C₂²×C₄₁	—	4	16	no	—
284	C₂²×C₄₁	—	4	2	no	—
285	C₂²×C₄₁	—	4	2	no	—
286	C₂²×C₄₁	—	4	2	no	—
287	C₂²×C₄₁	—	4	2	no	—
288	C₂²×C₄₁	—	4	4	no	—
289	C₂²×C₄₁	—	4	2	no	—
290	C₂²×C₄₁	—	4	2	no	—
291	C₂²×C₄₁	C₂×C₄×D₄	1	4	yes	—
292	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
293	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
294	C₂²×C₄₁	C₂²×#41	1	1	yes	characteristic
295	C₂²×C₄₁	#585	1	2	yes	—
296	C₂²×C₄₁	#585	1	2	yes	—
297	C₄×C₄₁	—	2	4	no	—
298	C₄×C₄₁	—	2	4	no	—
299	C₄×C₄₁	—	2	8	no	—
300	C₄×C₄₁	—	2	8	no	—
301	C₄×C₄₁	—	4	2	no	—
302	C₄×C₄₁	—	4	2	no	—
303	C₄×C₄₁	—	4	4	no	—
304	C₄×C₄₁	—	4	4	no	—
305	C₄×C₄₁	C₂×C₄×D₄	1	4	yes	—
306	C₄×C₄₁	C₂×C₄×D₄	1	4	yes	—
307	C₂×#525	C₂×C₄₁	1	1	yes	characteristic
308	C₂×#520	C₂×C₄₁	1	1	yes	characteristic, complemented
309	C₂²×#117	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#2363	C₂×C₄₁	1	4	yes	complemented
311	#2362	C₂×C₄₁	1	4	yes	complemented
312	#2354	C₂×C₄₁	1	4	yes	complemented
313	#2352	C₂×C₄₁	1	4	yes	complemented
314	#2321	C₂×C₄₁	1	4	yes	complemented
315	#2317	C₂×C₄₁	1	1	yes	characteristic, complemented
316	#2311	C₂×C₄₁	1	1	yes	characteristic, complemented
317	C₄×#123	C₂×C₄₁	1	4	yes	complemented
318	#2280	C₂×C₄₁	1	1	yes	characteristic, complemented
319	#2274	C₂×C₄₁	1	1	yes	characteristic, complemented
320	C₂×C₄₁	—	2	4	no	—
321	C₂×C₄₁	—	4	4	no	—
322	C₂×C₄₁	—	4	4	no	—
323	C₂×C₄₁	C₂²×C₄×D₄	1	1	yes	characteristic
324	C₂×C₄₁	#3428	1	2	yes	—
325	C₂×C₄₁	#3485	1	1	yes	characteristic
326	C₂×C₄₁	#3487	1	1	yes	characteristic
327	C₂×C₄₁	#3493	1	2	yes	—
328	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
329	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
330	#525	C₄×C₄₁	1	16	yes	complemented
331	C₂×#123	C₂²×C₄₁	1	2	yes	—
332	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
333	C₂×#121	C₂²×C₄₁	1	2	yes	—
334	C₂×#120	C₂²×C₄₁	1	2	yes	—
335	#520	—	2	8	no	—
336	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	complemented
337	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
338	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
339	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	complemented
340	C₂×#117	—	2	8	no	—
341	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×#117	C₂²×C₄₁	1	4	yes	—
344	C₂×#117	C₂²×C₄₁	1	4	yes	—
345	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×#117	C₂²×C₄₁	1	2	yes	complemented
347	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
348	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
349	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	complemented
350	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
351	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
352	#391	C₂²×C₄₁	1	2	yes	complemented
353	#390	C₂²×C₄₁	1	8	yes	—
354	#390	C₂²×C₄₁	1	2	yes	complemented
355	#389	C₂²×C₄₁	1	4	yes	complemented
356	#388	C₂²×C₄₁	1	8	yes	—
357	#388	C₂²×C₄₁	1	8	yes	—
358	#388	C₂²×C₄₁	1	8	yes	complemented
359	#387	C₂²×C₄₁	1	8	yes	complemented
360	#386	C₂²×C₄₁	1	4	yes	—
361	#386	C₂²×C₄₁	1	8	yes	—
362	#385	C₂²×C₄₁	1	4	yes	—
363	#384	C₂²×C₄₁	1	2	yes	complemented
364	#382	C₂²×C₄₁	1	2	yes	complemented
365	#381	C₂²×C₄₁	1	2	yes	—
366	#380	C₂²×C₄₁	1	2	yes	—
367	#380	C₂²×C₄₁	1	2	yes	—
368	#380	C₂²×C₄₁	1	4	yes	complemented
369	#379	C₂²×C₄₁	1	2	yes	—
370	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
371	C₄×C₄⋊C₄	C₂²×C₄₁	1	8	yes	complemented
372	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
373	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
374	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
375	C₄×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	complemented
376	C₄₁	#34201	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
378	C₂×#41	—	2	4	no	—
379	C₂×#41	—	2	4	no	—
380	C₂×#41	C₂×C₄×C₄₁	1	8	yes	—
381	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
382	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
383	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
384	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
385	C₂³×C₄	—	2	4	no	—
386	C₂³×C₄	—	2	4	no	—
387	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	complemented
388	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
389	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
390	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
391	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
393	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
394	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
395	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
396	#123	C₂×C₄×C₄₁	1	16	yes	complemented
397	C₂²⋊Q₈	C₂×C₄×C₄₁	1	16	yes	complemented
398	#121	C₂×C₄×C₄₁	1	16	yes	complemented
399	#120	C₂×C₄×C₄₁	1	16	yes	complemented
400	C₄×Q₈	—	2	8	no	—
401	C₄×D₄	—	2	16	no	—
402	C₄×D₄	—	2	8	no	—
403	C₄×D₄	C₂×C₄×C₄₁	1	32	yes	complemented
404	#117	—	2	4	no	—
405	#117	—	2	4	no	—
406	#117	—	2	16	no	—
407	#117	—	2	16	no	—
408	#117	—	2	8	no	—
409	#117	C₂×C₄×C₄₁	1	8	yes	complemented
410	C₂×C₄⋊C₄	—	2	4	no	—
411	C₂×C₄⋊C₄	—	2	8	no	—
412	C₂×C₄⋊C₄	D₄×C₄₁	1	8	yes	complemented
413	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
417	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
420	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
422	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
424	C₂×C₂²⋊C₄	—	2	8	no	—
425	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
435	C₂×C₄²	—	2	8	no	—
436	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
437	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
438	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
439	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
440	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
441	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
442	#95	C₂³×C₄₁	1	4	yes	—
443	#95	C₂³×C₄₁	1	4	yes	—
444	#95	C₂³×C₄₁	1	8	yes	—
445	#95	C₂³×C₄₁	1	8	yes	—
446	#95	C₂³×C₄₁	1	8	yes	—
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	2	2	no	—
449	C₂⁴	—	4	2	no	—
450	C₂⁴	—	4	2	no	—
451	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
452	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
453	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
454	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
455	#41	—	2	32	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
466	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
467	C₂×D₄	C₂²×C₄×C₄₁	1	16	yes	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	32	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	8	no	—
479	C₂²×C₄	—	4	4	no	—
480	C₂²×C₄	—	4	4	no	—
481	C₂²×C₄	—	4	8	no	—
482	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
483	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
484	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
485	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
486	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
487	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
488	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
489	C₂²×C₄	#41×C₄₁	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	4	8	no	—
507	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
508	C₄⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
509	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
510	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	16	no	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	—	4	4	no	—
516	C₂²⋊C₄	—	4	4	no	—
517	C₂²⋊C₄	—	4	8	no	—
518	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
519	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
520	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
521	C₄²	—	2	16	no	—
522	C₄²	—	2	8	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	4	8	no	—
526	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	8	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	4	8	no	—
541	C₂³	—	4	4	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	4	8	no	—
544	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
545	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
546	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
547	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
548	C₂³	C₂×#41×C₄₁	1	2	yes	—
549	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
550	Q₈	—	2	8	no	—
551	D₄	—	2	16	no	—
552	D₄	—	2	8	no	—
553	D₄	—	4	16	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	16	no	—
563	C₂×C₄	—	2	16	no	—
564	C₂×C₄	—	2	16	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	4	8	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	—	8	8	no	—
577	C₂×C₄	C₄×D₄×C₄₁	1	16	yes	—
578	C₂×C₄	C₂³×C₄×C₄₁	1	8	yes	—
579	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
580	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
581	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
582	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₄₁	1	8	yes	—
584	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
586	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
587	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	2	4	no	—
594	C₂²	—	2	4	no	—
595	C₂²	—	2	8	no	—
596	C₂²	—	2	4	no	—
597	C₂²	—	4	16	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	4	no	—
605	C₂²	C₂×C₄×D₄×C₄₁	1	4	yes	—
606	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
607	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
608	C₂²	C₂²×#41×C₄₁	1	1	yes	characteristic
609	C₂²	C₄₁×#585	1	2	yes	—
610	C₂²	C₄₁×#585	1	2	yes	—
611	C₄	—	2	4	no	—
612	C₄	—	2	4	no	—
613	C₄	—	2	8	no	—
614	C₄	—	2	8	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	4	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	4	no	—
619	C₄	C₂×C₄×D₄×C₄₁	1	4	yes	—
620	C₄	C₂×C₄×D₄×C₄₁	1	4	yes	—
621	C₂	—	2	4	no	—
622	C₂	—	4	4	no	—
623	C₂	—	4	4	no	—
624	C₂	C₂²×C₄×D₄×C₄₁	1	1	yes	characteristic
625	C₂	C₄₁×#3428	1	2	yes	—
626	C₂	C₄₁×#3485	1	1	yes	characteristic
627	C₂	C₄₁×#3487	1	1	yes	characteristic
628	C₂	C₄₁×#3493	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂×C₄₁×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#520	C₂	semidirect
3. C₂²×#117×C₄₁	C₂	semidirect
4. C₄₁×#2363	C₂	semidirect
5. C₄₁×#2362	C₂	semidirect
6. C₄₁×#2354	C₂	semidirect
7. C₄₁×#2352	C₂	semidirect
8. C₄₁×#2321	C₂	semidirect
9. C₄₁×#2317	C₂	semidirect
10. C₄₁×#2311	C₂	semidirect
11. C₄×#123×C₄₁	C₂	semidirect
12. C₄₁×#2280	C₂	semidirect
13. C₄₁×#2274	C₂	semidirect
16. C₄₁×#525	C₄	semidirect
22. C₂×C₄×Q₈×C₄₁	C₂²	semidirect
25. C₂×C₄×D₄×C₄₁	C₂²	semidirect
32. C₂×#117×C₄₁	C₂²	semidirect
35. C₂²×C₄⋊C₄×C₄₁	C₂²	semidirect
38. C₄₁×#391	C₂²	semidirect
40. C₄₁×#390	C₂²	semidirect
41. C₄₁×#389	C₂²	semidirect
44. C₄₁×#388	C₂²	semidirect
45. C₄₁×#387	C₂²	semidirect
49. C₄₁×#384	C₂²	semidirect
50. C₄₁×#382	C₂²	semidirect
54. C₄₁×#380	C₂²	semidirect
56. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
57. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
58. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
61. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
72. C₂³×C₄×C₄₁	C₂×C₄	semidirect
81. #123×C₄₁	C₂×C₄	semidirect
82. C₂²⋊Q₈×C₄₁	C₂×C₄	semidirect
83. #121×C₄₁	C₂×C₄	semidirect
84. #120×C₄₁	C₂×C₄	semidirect
88. C₄×D₄×C₄₁	C₂×C₄	semidirect
94. #117×C₄₁	C₂×C₄	semidirect
97. C₂×C₄⋊C₄×C₄₁	D₄	semidirect
273. #34201	C₄₁	direct (N × H)
308. C₂×#520	C₂×C₄₁	semidirect
309. C₂²×#117	C₂×C₄₁	semidirect
310. #2363	C₂×C₄₁	semidirect
311. #2362	C₂×C₄₁	semidirect
312. #2354	C₂×C₄₁	semidirect
313. #2352	C₂×C₄₁	semidirect
314. #2321	C₂×C₄₁	semidirect
315. #2317	C₂×C₄₁	semidirect
316. #2311	C₂×C₄₁	semidirect
317. C₄×#123	C₂×C₄₁	semidirect
318. #2280	C₂×C₄₁	semidirect
319. #2274	C₂×C₄₁	semidirect
330. #525	C₄×C₄₁	semidirect
336. C₂×C₄×Q₈	C₂²×C₄₁	semidirect
339. C₂×C₄×D₄	C₂²×C₄₁	semidirect
346. C₂×#117	C₂²×C₄₁	semidirect
349. C₂²×C₄⋊C₄	C₂²×C₄₁	semidirect
352. #391	C₂²×C₄₁	semidirect
354. #390	C₂²×C₄₁	semidirect
355. #389	C₂²×C₄₁	semidirect
358. #388	C₂²×C₄₁	semidirect
359. #387	C₂²×C₄₁	semidirect
363. #384	C₂²×C₄₁	semidirect
364. #382	C₂²×C₄₁	semidirect
368. #380	C₂²×C₄₁	semidirect
370. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
371. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
372. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
375. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
376. C₄₁	#34201	direct (N × H)
387. C₂³×C₄	C₂×C₄×C₄₁	semidirect
396. #123	C₂×C₄×C₄₁	semidirect
397. C₂²⋊Q₈	C₂×C₄×C₄₁	semidirect
398. #121	C₂×C₄×C₄₁	semidirect
399. #120	C₂×C₄×C₄₁	semidirect
403. C₄×D₄	C₂×C₄×C₄₁	semidirect
409. #117	C₂×C₄×C₄₁	semidirect
412. C₂×C₄⋊C₄	D₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#525	2
C₂×C₄₁×#520	2
C₂²×#117×C₄₁	2
C₄₁×#2363	2
C₄₁×#2362	2
C₄₁×#2354	2
C₄₁×#2352	2
C₄₁×#2321	2
C₄₁×#2317	2
C₄₁×#2311	2
C₄×#123×C₄₁	2
C₄₁×#2280	2
C₄₁×#2274	2
#34201	41

Subgroups — C41×#34201 #23810402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34201 #23810402