Subgroups — C₄₁×#34358 #23810559

Order 10496.

2982 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2377	C₂	1	4	yes	maximal, complemented
3	C₄₁×#2364	C₂	1	4	yes	maximal, complemented
4	C₄₁×#2342	C₂	1	4	yes	maximal, complemented
5	D₄×C₂²⋊C₄×C₄₁	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2333	C₂	1	4	yes	maximal, complemented
7	C₄₁×#2325	C₂	1	4	yes	maximal, complemented
8	C₄₁×#2313	C₂	1	2	yes	maximal, complemented
9	C₄₁×#2308	C₂	1	2	yes	maximal, complemented
10	C₄₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₁×#381	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₄₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#121×C₄₁	C₄	1	16	yes	complemented
16	C₂×#121×C₄₁	C₂²	1	2	yes	—
17	C₂×#121×C₄₁	C₂²	1	2	yes	—
18	C₂×#121×C₄₁	C₂²	1	2	yes	—
19	C₂×#121×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₄₁	C₂²	1	8	yes	complemented
21	C₂×#117×C₄₁	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₄₁×#390	C₂²	1	8	yes	complemented
27	C₄₁×#388	C₂²	1	8	yes	complemented
28	C₄₁×#388	C₂²	1	8	yes	complemented
29	C₄₁×#387	C₂²	1	8	yes	complemented
30	C₄₁×#387	C₂²	1	8	yes	complemented
31	C₄₁×#386	C₂²	1	8	yes	complemented
32	C₄₁×#386	C₂²	1	8	yes	complemented
33	C₄₁×#386	C₂²	1	8	yes	complemented
34	C₄₁×#386	C₂²	1	8	yes	complemented
35	C₄₁×#385	C₂²	1	8	yes	complemented
36	C₄₁×#381	C₂²	1	2	yes	—
37	C₄₁×#381	C₂²	1	2	yes	—
38	C₄₁×#381	C₂²	1	2	yes	—
39	C₄₁×#381	C₂²	1	2	yes	complemented
40	C₄₁×#381	C₂²	1	2	yes	complemented
41	C₄₁×#380	C₂²	1	4	yes	—
42	C₄₁×#380	C₂²	1	2	yes	complemented
43	C₄₁×#380	C₂²	1	2	yes	complemented
44	C₄₁×#379	C₂²	1	2	yes	—
45	C₄₁×#379	C₂²	1	4	yes	—
46	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
47	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	complemented
48	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	complemented
49	C₄×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	complemented
50	C₂×#95×C₄₁	C₂²	1	2	yes	complemented
51	C₂×#95×C₄₁	C₂²	1	2	yes	complemented
52	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
53	C₂⁵×C₄₁	C₂³	1	2	yes	—
54	C₂²×D₄×C₄₁	—	4	4	no	—
55	C₂²×D₄×C₄₁	—	4	8	no	—
56	C₂²×D₄×C₄₁	C₂×C₄	1	8	yes	complemented
57	C₂²×D₄×C₄₁	C₂×C₄	1	16	yes	complemented
58	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
59	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
61	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
62	C₂³×C₄×C₄₁	—	4	4	no	—
63	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	complemented
64	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
67	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
68	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
73	#121×C₄₁	C₂×C₄	1	16	yes	complemented
74	#121×C₄₁	C₂×C₄	1	16	yes	complemented
75	#121×C₄₁	C₂×C₄	1	16	yes	complemented
76	#121×C₄₁	C₂×C₄	1	16	yes	complemented
77	C₄×D₄×C₄₁	—	4	16	no	—
78	#117×C₄₁	—	4	4	no	—
79	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
80	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₄₁	—	4	4	no	—
85	C₂×C₂²⋊C₄×C₄₁	—	4	8	no	—
86	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	16	yes	complemented
87	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
97	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
98	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
99	#95×C₄₁	C₂³	1	4	yes	—
100	#95×C₄₁	C₂³	1	4	yes	—
101	#95×C₄₁	C₂³	1	4	yes	—
102	#95×C₄₁	C₂³	1	4	yes	—
103	#95×C₄₁	C₂³	1	8	yes	—
104	#95×C₄₁	C₂³	1	8	yes	—
105	#95×C₄₁	C₂³	1	4	yes	—
106	#95×C₄₁	C₂³	1	4	yes	—
107	C₂⁴×C₄₁	—	2	4	no	—
108	C₂⁴×C₄₁	—	2	4	no	—
109	C₂⁴×C₄₁	—	2	8	no	—
110	C₂⁴×C₄₁	—	4	2	no	—
111	C₂⁴×C₄₁	—	4	4	no	—
112	C₂⁴×C₄₁	—	4	4	no	—
113	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
114	C₂⁴×C₄₁	C₂²×C₄	1	8	yes	—
115	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
116	C₂⁴×C₄₁	C₂⁴	1	4	yes	—
117	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
118	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
119	C₂×D₄×C₄₁	—	2	16	no	—
120	C₂×D₄×C₄₁	—	4	16	no	—
121	C₂×D₄×C₄₁	—	4	8	no	—
122	C₂×D₄×C₄₁	—	4	2	no	—
123	C₂×D₄×C₄₁	—	4	4	no	—
124	C₂×D₄×C₄₁	—	4	2	no	—
125	C₂×D₄×C₄₁	—	4	32	no	—
126	C₂×D₄×C₄₁	C₂²×C₄	1	16	yes	—
127	C₂×D₄×C₄₁	C₂²×C₄	1	16	yes	—
128	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
129	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
130	C₂²×C₄×C₄₁	—	2	4	no	—
131	C₂²×C₄×C₄₁	—	2	4	no	—
132	C₂²×C₄×C₄₁	—	2	8	no	—
133	C₂²×C₄×C₄₁	—	4	16	no	—
134	C₂²×C₄×C₄₁	—	4	2	no	—
135	C₂²×C₄×C₄₁	—	4	2	no	—
136	C₂²×C₄×C₄₁	—	4	2	no	—
137	C₂²×C₄×C₄₁	—	4	2	no	—
138	C₂²×C₄×C₄₁	—	4	4	no	—
139	C₂²×C₄×C₄₁	—	4	8	no	—
140	C₂²×C₄×C₄₁	—	4	4	no	—
141	C₂²×C₄×C₄₁	—	4	8	no	—
142	C₂²×C₄×C₄₁	—	4	8	no	—
143	C₂²×C₄×C₄₁	—	4	4	no	—
144	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
147	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
152	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
153	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
158	C₄⋊C₄×C₄₁	—	4	4	no	—
159	C₄⋊C₄×C₄₁	—	4	8	no	—
160	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
161	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
162	C₂²⋊C₄×C₄₁	—	2	8	no	—
163	C₂²⋊C₄×C₄₁	—	2	8	no	—
164	C₂²⋊C₄×C₄₁	—	2	8	no	—
165	C₂²⋊C₄×C₄₁	—	4	8	no	—
166	C₂²⋊C₄×C₄₁	—	4	4	no	—
167	C₂²⋊C₄×C₄₁	—	4	4	no	—
168	C₂²⋊C₄×C₄₁	—	4	8	no	—
169	C₂²⋊C₄×C₄₁	—	4	8	no	—
170	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
171	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
172	C₄²×C₄₁	—	4	4	no	—
173	C₄²×C₄₁	—	4	8	no	—
174	C₂³×C₄₁	—	2	2	no	—
175	C₂³×C₄₁	—	2	2	no	—
176	C₂³×C₄₁	—	2	2	no	—
177	C₂³×C₄₁	—	2	2	no	—
178	C₂³×C₄₁	—	2	8	no	—
179	C₂³×C₄₁	—	2	8	no	—
180	C₂³×C₄₁	—	2	2	no	—
181	C₂³×C₄₁	—	2	4	no	—
182	C₂³×C₄₁	—	2	8	no	—
183	C₂³×C₄₁	—	2	4	no	—
184	C₂³×C₄₁	—	2	2	no	—
185	C₂³×C₄₁	—	2	2	no	—
186	C₂³×C₄₁	—	2	8	no	—
187	C₂³×C₄₁	—	2	16	no	—
188	C₂³×C₄₁	—	4	4	no	—
189	C₂³×C₄₁	—	4	8	no	—
190	C₂³×C₄₁	—	4	8	no	—
191	C₂³×C₄₁	—	4	16	no	—
192	C₂³×C₄₁	—	8	16	no	—
193	C₂³×C₄₁	C₂³×C₄	1	8	yes	—
194	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
195	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
196	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
197	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
198	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
199	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
200	D₄×C₄₁	—	2	32	no	—
201	D₄×C₄₁	—	4	8	no	—
202	D₄×C₄₁	—	4	8	no	—
203	D₄×C₄₁	—	4	16	no	—
204	C₂×C₄×C₄₁	—	2	8	no	—
205	C₂×C₄×C₄₁	—	2	8	no	—
206	C₂×C₄×C₄₁	—	2	2	no	—
207	C₂×C₄×C₄₁	—	2	8	no	—
208	C₂×C₄×C₄₁	—	2	2	no	—
209	C₂×C₄×C₄₁	—	2	2	no	—
210	C₂×C₄×C₄₁	—	2	2	no	—
211	C₂×C₄×C₄₁	—	2	8	no	—
212	C₂×C₄×C₄₁	—	2	8	no	—
213	C₂×C₄×C₄₁	—	2	8	no	—
214	C₂×C₄×C₄₁	—	2	8	no	—
215	C₂×C₄×C₄₁	—	2	8	no	—
216	C₂×C₄×C₄₁	—	2	8	no	—
217	C₂×C₄×C₄₁	—	4	4	no	—
218	C₂×C₄×C₄₁	—	4	4	no	—
219	C₂×C₄×C₄₁	—	4	4	no	—
220	C₂×C₄×C₄₁	—	4	4	no	—
221	C₂×C₄×C₄₁	—	4	8	no	—
222	C₂×C₄×C₄₁	—	8	8	no	—
223	C₂×C₄×C₄₁	—	8	16	no	—
224	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₄₁	C₂³×C₄	1	8	yes	—
227	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
228	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
229	#34358	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂²×C₄₁	—	2	2	no	—
231	C₂²×C₄₁	—	2	2	no	—
232	C₂²×C₄₁	—	2	2	no	—
233	C₂²×C₄₁	—	2	2	no	—
234	C₂²×C₄₁	—	2	8	no	—
235	C₂²×C₄₁	—	2	8	no	—
236	C₂²×C₄₁	—	2	2	no	—
237	C₂²×C₄₁	—	2	8	no	—
238	C₂²×C₄₁	—	2	2	no	—
239	C₂²×C₄₁	—	2	8	no	—
240	C₂²×C₄₁	—	2	2	no	—
241	C₂²×C₄₁	—	2	8	no	—
242	C₂²×C₄₁	—	4	16	no	—
243	C₂²×C₄₁	—	4	8	no	—
244	C₂²×C₄₁	—	4	4	no	—
245	C₂²×C₄₁	—	4	8	no	—
246	C₂²×C₄₁	—	8	16	no	—
247	C₂²×C₄₁	#518	1	4	yes	—
248	C₂²×C₄₁	#518	1	4	yes	—
249	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
250	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
251	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
252	C₂²×C₄₁	#585	1	2	yes	—
253	C₄×C₄₁	—	2	2	no	—
254	C₄×C₄₁	—	2	2	no	—
255	C₄×C₄₁	—	2	2	no	—
256	C₄×C₄₁	—	2	2	no	—
257	C₄×C₄₁	—	2	8	no	—
258	C₄×C₄₁	—	4	8	no	—
259	C₄×C₄₁	—	4	8	no	—
260	C₂²×#121	C₂×C₄₁	1	1	yes	characteristic
261	#2377	C₂×C₄₁	1	4	yes	complemented
262	#2364	C₂×C₄₁	1	4	yes	complemented
263	#2342	C₂×C₄₁	1	4	yes	complemented
264	D₄×C₂²⋊C₄	C₂×C₄₁	1	4	yes	complemented
265	#2333	C₂×C₄₁	1	4	yes	complemented
266	#2325	C₂×C₄₁	1	4	yes	complemented
267	#2313	C₂×C₄₁	1	2	yes	complemented
268	#2308	C₂×C₄₁	1	2	yes	complemented
269	#2280	C₂×C₄₁	1	1	yes	characteristic, complemented
270	C₂×#381	C₂×C₄₁	1	1	yes	characteristic, complemented
271	C₂×C₄₁	—	2	4	no	—
272	C₂×C₄₁	—	2	4	no	—
273	C₂×C₄₁	—	4	2	no	—
274	C₂×C₄₁	—	4	8	no	—
275	C₂×C₄₁	C₂×#518	1	1	yes	characteristic
276	C₂×C₄₁	C₂×#518	1	1	yes	characteristic
277	C₂×C₄₁	#3428	1	1	yes	characteristic
278	C₂×C₄₁	#3534	1	4	yes	—
279	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
280	C₂³×D₄	C₂²×C₄₁	1	2	yes	—
281	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
282	C₂×#121	C₄×C₄₁	1	16	yes	complemented
283	C₂×#121	C₂²×C₄₁	1	2	yes	—
284	C₂×#121	C₂²×C₄₁	1	2	yes	—
285	C₂×#121	C₂²×C₄₁	1	2	yes	—
286	C₂×#121	C₂²×C₄₁	1	2	yes	—
287	C₂×C₄×D₄	C₂²×C₄₁	1	8	yes	complemented
288	C₂×#117	C₂²×C₄₁	1	2	yes	—
289	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
290	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
291	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
292	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
293	#390	C₂²×C₄₁	1	8	yes	complemented
294	#388	C₂²×C₄₁	1	8	yes	complemented
295	#388	C₂²×C₄₁	1	8	yes	complemented
296	#387	C₂²×C₄₁	1	8	yes	complemented
297	#387	C₂²×C₄₁	1	8	yes	complemented
298	#386	C₂²×C₄₁	1	8	yes	complemented
299	#386	C₂²×C₄₁	1	8	yes	complemented
300	#386	C₂²×C₄₁	1	8	yes	complemented
301	#386	C₂²×C₄₁	1	8	yes	complemented
302	#385	C₂²×C₄₁	1	8	yes	complemented
303	#381	C₂²×C₄₁	1	2	yes	—
304	#381	C₂²×C₄₁	1	2	yes	—
305	#381	C₂²×C₄₁	1	2	yes	—
306	#381	C₂²×C₄₁	1	2	yes	complemented
307	#381	C₂²×C₄₁	1	2	yes	complemented
308	#380	C₂²×C₄₁	1	4	yes	—
309	#380	C₂²×C₄₁	1	2	yes	complemented
310	#380	C₂²×C₄₁	1	2	yes	complemented
311	#379	C₂²×C₄₁	1	4	yes	—
312	#379	C₂²×C₄₁	1	2	yes	—
313	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	complemented
315	C₄×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	complemented
316	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	complemented
317	C₂×#95	C₂²×C₄₁	1	2	yes	complemented
318	C₂×#95	C₂²×C₄₁	1	2	yes	complemented
319	C₄₁	#34358	1	1	yes	characteristic, direct product factor, Sylow
320	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
321	C₂⁵	C₂³×C₄₁	1	2	yes	—
322	C₂²×D₄	—	4	4	no	—
323	C₂²×D₄	—	4	8	no	—
324	C₂²×D₄	C₂×C₄×C₄₁	1	8	yes	complemented
325	C₂²×D₄	C₂×C₄×C₄₁	1	16	yes	complemented
326	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
327	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
328	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
329	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
330	C₂³×C₄	—	4	4	no	—
331	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	complemented
332	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
334	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
335	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
336	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
337	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
341	#121	C₂×C₄×C₄₁	1	16	yes	complemented
342	#121	C₂×C₄×C₄₁	1	16	yes	complemented
343	#121	C₂×C₄×C₄₁	1	16	yes	complemented
344	#121	C₂×C₄×C₄₁	1	16	yes	complemented
345	C₄×D₄	—	4	16	no	—
346	#117	—	4	4	no	—
347	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
348	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
349	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
350	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
352	C₂×C₂²⋊C₄	—	4	4	no	—
353	C₂×C₂²⋊C₄	—	4	8	no	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	16	yes	complemented
355	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
365	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
366	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
367	#95	C₂³×C₄₁	1	4	yes	—
368	#95	C₂³×C₄₁	1	4	yes	—
369	#95	C₂³×C₄₁	1	4	yes	—
370	#95	C₂³×C₄₁	1	4	yes	—
371	#95	C₂³×C₄₁	1	8	yes	—
372	#95	C₂³×C₄₁	1	8	yes	—
373	#95	C₂³×C₄₁	1	4	yes	—
374	#95	C₂³×C₄₁	1	4	yes	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	2	8	no	—
378	C₂⁴	—	4	2	no	—
379	C₂⁴	—	4	4	no	—
380	C₂⁴	—	4	4	no	—
381	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
382	C₂⁴	C₂²×C₄×C₄₁	1	8	yes	—
383	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
384	C₂⁴	C₂⁴×C₄₁	1	4	yes	—
385	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
386	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
387	C₂×D₄	—	2	16	no	—
388	C₂×D₄	—	4	16	no	—
389	C₂×D₄	—	4	8	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	4	32	no	—
394	C₂×D₄	C₂²×C₄×C₄₁	1	16	yes	—
395	C₂×D₄	C₂²×C₄×C₄₁	1	16	yes	—
396	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
397	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	8	no	—
401	C₂²×C₄	—	4	16	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	—	4	2	no	—
406	C₂²×C₄	—	4	4	no	—
407	C₂²×C₄	—	4	8	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	8	no	—
411	C₂²×C₄	—	4	4	no	—
412	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
413	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
414	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
415	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
416	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
417	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
418	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
420	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
421	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
422	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
426	C₄⋊C₄	—	4	4	no	—
427	C₄⋊C₄	—	4	8	no	—
428	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
429	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	8	no	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₂²⋊C₄	—	4	8	no	—
434	C₂²⋊C₄	—	4	4	no	—
435	C₂²⋊C₄	—	4	8	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₂²⋊C₄	—	4	8	no	—
438	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
439	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
440	C₄²	—	4	4	no	—
441	C₄²	—	4	8	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	8	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	16	no	—
456	C₂³	—	4	4	no	—
457	C₂³	—	4	8	no	—
458	C₂³	—	4	8	no	—
459	C₂³	—	4	16	no	—
460	C₂³	—	8	16	no	—
461	C₂³	C₂³×C₄×C₄₁	1	8	yes	—
462	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
463	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
464	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
465	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
466	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
467	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
468	D₄	—	2	32	no	—
469	D₄	—	4	8	no	—
470	D₄	—	4	8	no	—
471	D₄	—	4	16	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	8	no	—
480	C₂×C₄	—	2	8	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	2	8	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	8	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	8	8	no	—
491	C₂×C₄	—	8	16	no	—
492	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
493	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
494	C₂×C₄	C₂³×C₄×C₄₁	1	8	yes	—
495	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
496	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	8	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	8	no	—
509	C₂²	—	4	16	no	—
510	C₂²	—	4	8	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	8	no	—
513	C₂²	—	8	16	no	—
514	C₂²	C₄₁×#518	1	4	yes	—
515	C₂²	C₄₁×#518	1	4	yes	—
516	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
517	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
518	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
519	C₂²	C₄₁×#585	1	2	yes	—
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	8	no	—
525	C₄	—	4	8	no	—
526	C₄	—	4	8	no	—
527	C₂	—	2	4	no	—
528	C₂	—	2	4	no	—
529	C₂	—	4	2	no	—
530	C₂	—	4	8	no	—
531	C₂	C₂×C₄₁×#518	1	1	yes	characteristic
532	C₂	C₂×C₄₁×#518	1	1	yes	characteristic
533	C₂	C₄₁×#3428	1	1	yes	characteristic
534	C₂	C₄₁×#3534	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂²×#121×C₄₁	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. C₄₁×#2377	C₂	semidirect
3. C₄₁×#2364	C₂	semidirect
4. C₄₁×#2342	C₂	semidirect
5. D₄×C₂²⋊C₄×C₄₁	C₂	semidirect
6. C₄₁×#2333	C₂	semidirect
7. C₄₁×#2325	C₂	semidirect
8. C₄₁×#2313	C₂	semidirect
9. C₄₁×#2308	C₂	semidirect
10. C₄₁×#2280	C₂	semidirect
11. C₂×C₄₁×#381	C₂	semidirect
15. C₂×#121×C₄₁	C₄	semidirect
20. C₂×C₄×D₄×C₄₁	C₂²	semidirect
26. C₄₁×#390	C₂²	semidirect
27. C₄₁×#388	C₂²	semidirect
28. C₄₁×#388	C₂²	semidirect
29. C₄₁×#387	C₂²	semidirect
30. C₄₁×#387	C₂²	semidirect
31. C₄₁×#386	C₂²	semidirect
32. C₄₁×#386	C₂²	semidirect
33. C₄₁×#386	C₂²	semidirect
34. C₄₁×#386	C₂²	semidirect
35. C₄₁×#385	C₂²	semidirect
39. C₄₁×#381	C₂²	semidirect
40. C₄₁×#381	C₂²	semidirect
42. C₄₁×#380	C₂²	semidirect
43. C₄₁×#380	C₂²	semidirect
47. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
50. C₂×#95×C₄₁	C₂²	semidirect
51. C₂×#95×C₄₁	C₂²	semidirect
56. C₂²×D₄×C₄₁	C₂×C₄	semidirect
57. C₂²×D₄×C₄₁	C₂×C₄	semidirect
63. C₂³×C₄×C₄₁	C₂×C₄	semidirect
73. #121×C₄₁	C₂×C₄	semidirect
74. #121×C₄₁	C₂×C₄	semidirect
75. #121×C₄₁	C₂×C₄	semidirect
76. #121×C₄₁	C₂×C₄	semidirect
79. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
86. C₂×C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
229. #34358	C₄₁	direct (N × H)
261. #2377	C₂×C₄₁	semidirect
262. #2364	C₂×C₄₁	semidirect
263. #2342	C₂×C₄₁	semidirect
264. D₄×C₂²⋊C₄	C₂×C₄₁	semidirect
265. #2333	C₂×C₄₁	semidirect
266. #2325	C₂×C₄₁	semidirect
267. #2313	C₂×C₄₁	semidirect
268. #2308	C₂×C₄₁	semidirect
269. #2280	C₂×C₄₁	semidirect
270. C₂×#381	C₂×C₄₁	semidirect
282. C₂×#121	C₄×C₄₁	semidirect
287. C₂×C₄×D₄	C₂²×C₄₁	semidirect
293. #390	C₂²×C₄₁	semidirect
294. #388	C₂²×C₄₁	semidirect
295. #388	C₂²×C₄₁	semidirect
296. #387	C₂²×C₄₁	semidirect
297. #387	C₂²×C₄₁	semidirect
298. #386	C₂²×C₄₁	semidirect
299. #386	C₂²×C₄₁	semidirect
300. #386	C₂²×C₄₁	semidirect
301. #386	C₂²×C₄₁	semidirect
302. #385	C₂²×C₄₁	semidirect
306. #381	C₂²×C₄₁	semidirect
307. #381	C₂²×C₄₁	semidirect
309. #380	C₂²×C₄₁	semidirect
310. #380	C₂²×C₄₁	semidirect
314. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
315. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
316. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
317. C₂×#95	C₂²×C₄₁	semidirect
318. C₂×#95	C₂²×C₄₁	semidirect
319. C₄₁	#34358	direct (N × H)
324. C₂²×D₄	C₂×C₄×C₄₁	semidirect
325. C₂²×D₄	C₂×C₄×C₄₁	semidirect
331. C₂³×C₄	C₂×C₄×C₄₁	semidirect
341. #121	C₂×C₄×C₄₁	semidirect
342. #121	C₂×C₄×C₄₁	semidirect
343. #121	C₂×C₄×C₄₁	semidirect
344. #121	C₂×C₄×C₄₁	semidirect
347. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
354. C₂×C₂²⋊C₄	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#121×C₄₁	2
C₄₁×#2377	2
C₄₁×#2364	2
C₄₁×#2342	2
D₄×C₂²⋊C₄×C₄₁	2
C₄₁×#2333	2
C₄₁×#2325	2
C₄₁×#2313	2
C₄₁×#2308	2
C₄₁×#2280	2
C₂×C₄₁×#381	2
#34358	41

Subgroups — C41×#34358 #23810559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34358 #23810559