Subgroups — C₄₁×#17765 #23793966

Order 10496.

964 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1561	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1557	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1552	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1547	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₄₁	—	2	1	no	—
13	C₄₁×#556	—	2	1	no	—
14	C₄₁×#555	C₄	1	2	yes	—
15	C₄₁×#554	C₄	1	2	yes	—
16	C₄₁×#553	C₄	1	4	yes	—
17	C₄₁×#548	—	2	2	no	—
18	C₄₁×#547	—	2	1	no	—
19	C₄₁×#542	—	2	2	no	—
20	C₄₁×#537	—	2	1	no	—
21	C₄₁×#533	C₂²	1	2	yes	—
22	C₄₁×#532	C₂²	1	1	yes	characteristic
23	C₄₁×#532	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#520	—	2	1	no	—
26	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#446	—	2	1	no	—
28	C₄₁×#436	—	2	1	no	—
29	C₄₁×#435	—	2	2	no	—
30	C₄₁×#432	C₂²	1	2	yes	—
31	C₄₁×#431	C₂²	1	1	yes	characteristic
32	C₄₁×#424	—	2	1	no	—
33	C₄₁×#424	—	2	2	no	—
34	C₄₁×#423	C₂²	1	1	yes	characteristic
35	C₄₁×#423	C₂²	1	2	yes	—
36	C₄₁×#423	C₂²	1	2	yes	—
37	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
40	C₄₁×#332	C₂²	1	2	yes	—
41	C₄₁×#332	C₂²	1	2	yes	—
42	C₄₁×#332	C₂²	1	1	yes	characteristic
43	C₄₁×#332	C₂²	1	1	yes	characteristic
44	C₄₁×#332	C₂²	1	2	yes	—
45	C₄₁×#332	C₂²	1	2	yes	—
46	C₄₁×#332	C₂²	1	2	yes	—
47	C₄₁×#331	C₂²	1	1	yes	characteristic
48	C₄₁×#331	C₂²	1	1	yes	characteristic
49	C₄₁×#331	C₂²	1	2	yes	—
50	C₂×#41×C₄₁	—	2	1	no	—
51	C₂×#41×C₄₁	—	2	2	no	—
52	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₄₁	—	2	1	no	—
54	#131×C₄₁	—	4	4	no	—
55	C₂×M₁₆×C₄₁	—	2	2	no	—
56	C₂×M₁₆×C₄₁	—	2	1	no	—
57	C₂×M₁₆×C₄₁	—	2	2	no	—
58	C₂²×C₈×C₄₁	—	2	2	no	—
59	C₂²×C₈×C₄₁	—	2	1	no	—
60	C₂²×C₈×C₄₁	—	2	1	no	—
61	C₄⋊Q₈×C₄₁	—	2	1	no	—
62	C₄⋊Q₈×C₄₁	—	2	1	no	—
63	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
64	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
66	#127×C₄₁	C₂³	1	1	yes	characteristic
67	#126×C₄₁	—	2	4	no	—
68	#125×C₄₁	C₂×C₄	1	4	yes	—
69	#124×C₄₁	—	2	4	no	—
70	#124×C₄₁	—	2	1	no	—
71	#124×C₄₁	—	2	2	no	—
72	#123×C₄₁	—	2	4	no	—
73	C₂²⋊Q₈×C₄₁	—	2	2	no	—
74	C₂²⋊Q₈×C₄₁	—	2	2	no	—
75	C₂²⋊Q₈×C₄₁	—	2	2	no	—
76	C₂²⋊Q₈×C₄₁	—	2	4	no	—
77	#121×C₄₁	—	2	2	no	—
78	#121×C₄₁	—	2	4	no	—
79	#121×C₄₁	—	2	2	no	—
80	#121×C₄₁	—	2	1	no	—
81	#121×C₄₁	C₂×C₄	1	2	yes	—
82	#121×C₄₁	C₂×C₄	1	4	yes	—
83	C₄×Q₈×C₄₁	—	2	2	no	—
84	C₄×Q₈×C₄₁	—	2	2	no	—
85	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
87	C₄×D₄×C₄₁	—	2	4	no	—
88	C₄×D₄×C₄₁	—	2	2	no	—
89	C₄×D₄×C₄₁	—	2	1	no	—
90	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
91	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
92	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
95	C₄×D₄×C₄₁	D₄	1	2	yes	—
96	#117×C₄₁	—	2	2	no	—
97	#117×C₄₁	C₂×C₄	1	4	yes	—
98	#117×C₄₁	D₄	1	1	yes	characteristic
99	#117×C₄₁	D₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
105	C₄⋊C₈×C₄₁	—	2	2	no	—
106	C₄⋊C₈×C₄₁	—	2	1	no	—
107	C₄⋊C₈×C₄₁	—	2	1	no	—
108	C₄⋊C₈×C₄₁	—	2	2	no	—
109	C₄⋊C₈×C₄₁	—	2	1	no	—
110	C₄⋊C₈×C₄₁	—	2	2	no	—
111	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
114	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
115	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
117	C₂²⋊C₈×C₄₁	—	2	1	no	—
118	C₂²⋊C₈×C₄₁	—	2	2	no	—
119	#97×C₄₁	—	2	2	no	—
120	C₄×C₈×C₄₁	—	2	1	no	—
121	#41×C₄₁	—	2	2	no	—
122	#41×C₄₁	—	2	2	no	—
123	#41×C₄₁	—	4	2	no	—
124	#41×C₄₁	—	4	2	no	—
125	#41×C₄₁	—	8	2	no	—
126	#41×C₄₁	—	8	2	no	—
127	C₂×Q₈×C₄₁	—	2	2	no	—
128	C₂×Q₈×C₄₁	—	2	1	no	—
129	C₂×Q₈×C₄₁	—	2	2	no	—
130	C₂×Q₈×C₄₁	—	8	1	no	—
131	C₂×Q₈×C₄₁	—	8	1	no	—
132	C₂×Q₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×D₄×C₄₁	—	2	2	no	—
134	C₂×D₄×C₄₁	—	2	4	no	—
135	C₂×D₄×C₄₁	—	2	1	no	—
136	C₂×D₄×C₄₁	—	2	1	no	—
137	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
138	C₂×D₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
139	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	2	2	no	—
142	C₂²×C₄×C₄₁	—	2	4	no	—
143	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
149	M₁₆×C₄₁	—	4	4	no	—
150	M₁₆×C₄₁	—	4	2	no	—
151	M₁₆×C₄₁	—	8	2	no	—
152	C₂×C₈×C₄₁	—	2	1	no	—
153	C₂×C₈×C₄₁	—	2	1	no	—
154	C₂×C₈×C₄₁	—	2	2	no	—
155	C₂×C₈×C₄₁	—	2	1	no	—
156	C₂×C₈×C₄₁	—	2	2	no	—
157	C₂×C₈×C₄₁	—	2	1	no	—
158	C₂×C₈×C₄₁	—	4	4	no	—
159	C₂×C₈×C₄₁	—	4	2	no	—
160	C₂×C₈×C₄₁	—	8	1	no	—
161	C₄⋊C₄×C₄₁	—	2	2	no	—
162	C₄⋊C₄×C₄₁	—	2	4	no	—
163	C₄⋊C₄×C₄₁	—	2	2	no	—
164	C₄⋊C₄×C₄₁	—	2	4	no	—
165	C₄⋊C₄×C₄₁	—	2	2	no	—
166	C₄⋊C₄×C₄₁	—	2	1	no	—
167	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
169	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
170	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
171	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
172	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
173	C₂²⋊C₄×C₄₁	—	2	4	no	—
174	C₂²⋊C₄×C₄₁	—	2	4	no	—
175	C₂²⋊C₄×C₄₁	—	2	2	no	—
176	C₂²⋊C₄×C₄₁	—	2	4	no	—
177	C₂²⋊C₄×C₄₁	—	2	2	no	—
178	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
179	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
180	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
181	C₄²×C₄₁	—	2	4	no	—
182	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
183	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₄₁	C₂×D₄	1	2	yes	—
186	C₄²×C₄₁	C₂×D₄	1	2	yes	—
187	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
189	C₂³×C₄₁	—	2	2	no	—
190	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	Q₈×C₄₁	—	2	2	no	—
193	Q₈×C₄₁	—	4	1	no	—
194	Q₈×C₄₁	—	4	1	no	—
195	Q₈×C₄₁	—	4	2	no	—
196	Q₈×C₄₁	—	4	2	no	—
197	Q₈×C₄₁	—	8	2	no	—
198	D₄×C₄₁	—	2	4	no	—
199	D₄×C₄₁	—	2	2	no	—
200	D₄×C₄₁	—	4	2	no	—
201	D₄×C₄₁	—	4	4	no	—
202	D₄×C₄₁	—	4	1	no	—
203	D₄×C₄₁	—	4	1	no	—
204	D₄×C₄₁	—	4	1	no	—
205	D₄×C₄₁	—	4	1	no	—
206	D₄×C₄₁	—	8	4	no	—
207	C₂×C₄×C₄₁	—	2	2	no	—
208	C₂×C₄×C₄₁	—	2	2	no	—
209	C₂×C₄×C₄₁	—	2	1	no	—
210	C₂×C₄×C₄₁	—	2	2	no	—
211	C₂×C₄×C₄₁	—	2	1	no	—
212	C₂×C₄×C₄₁	—	2	2	no	—
213	C₂×C₄×C₄₁	—	2	2	no	—
214	C₂×C₄×C₄₁	—	2	4	no	—
215	C₂×C₄×C₄₁	—	4	1	no	—
216	C₂×C₄×C₄₁	—	4	2	no	—
217	C₂×C₄×C₄₁	—	4	4	no	—
218	C₂×C₄×C₄₁	—	8	4	no	—
219	C₂×C₄×C₄₁	—	8	2	no	—
220	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
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	AGL(1,ℤ/8)	1	2	yes	—
227	C₂×C₄×C₄₁	#137	1	2	yes	—
228	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
229	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
231	C₈×C₄₁	—	4	1	no	—
232	C₈×C₄₁	—	4	2	no	—
233	C₈×C₄₁	—	4	1	no	—
234	C₈×C₄₁	—	4	2	no	—
235	C₈×C₄₁	—	4	1	no	—
236	C₈×C₄₁	—	4	1	no	—
237	#17765	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂²×C₄₁	—	2	1	no	—
239	C₂²×C₄₁	—	2	1	no	—
240	C₂²×C₄₁	—	2	2	no	—
241	C₂²×C₄₁	—	2	2	no	—
242	C₂²×C₄₁	—	2	2	no	—
243	C₂²×C₄₁	—	4	2	no	—
244	C₂²×C₄₁	—	4	2	no	—
245	C₂²×C₄₁	—	4	2	no	—
246	C₂²×C₄₁	#417	1	2	yes	—
247	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
248	C₄×C₄₁	—	2	1	no	—
249	C₄×C₄₁	—	2	2	no	—
250	C₄×C₄₁	—	2	1	no	—
251	C₄×C₄₁	—	2	2	no	—
252	C₄×C₄₁	—	2	2	no	—
253	C₄×C₄₁	—	2	1	no	—
254	C₄×C₄₁	—	4	2	no	—
255	C₄×C₄₁	—	4	4	no	—
256	C₄×C₄₁	#417	1	2	yes	—
257	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
259	C₄×C₄₁	#576	1	1	yes	characteristic
260	C₄×C₄₁	#577	1	1	yes	characteristic
261	#3511	C₂×C₄₁	1	1	yes	characteristic
262	#2989	C₂×C₄₁	1	1	yes	characteristic, complemented
263	#2906	C₂×C₄₁	1	1	yes	characteristic, complemented
264	#1561	C₂×C₄₁	1	2	yes	complemented
265	#1557	C₂×C₄₁	1	2	yes	complemented
266	#1552	C₂×C₄₁	1	2	yes	complemented
267	#1549	C₂×C₄₁	1	1	yes	characteristic, complemented
268	#1547	C₂×C₄₁	1	1	yes	characteristic, complemented
269	#1547	C₂×C₄₁	1	2	yes	complemented
270	#1540	C₂×C₄₁	1	1	yes	characteristic, complemented
271	C₂×#332	C₂×C₄₁	1	1	yes	characteristic, complemented
272	C₂×C₄₁	—	2	2	no	—
273	C₂×C₄₁	—	4	2	no	—
274	C₂×C₄₁	—	8	2	no	—
275	C₂×C₄₁	#2889	1	1	yes	characteristic
276	C₂×C₄₁	C₂×#417	1	1	yes	characteristic
277	C₂×C₄₁	#2897	1	1	yes	characteristic
278	C₂×#131	—	2	1	no	—
279	#556	—	2	1	no	—
280	#555	C₄×C₄₁	1	2	yes	—
281	#554	C₄×C₄₁	1	2	yes	—
282	#553	C₄×C₄₁	1	4	yes	—
283	#548	—	2	2	no	—
284	#547	—	2	1	no	—
285	#542	—	2	2	no	—
286	#537	—	2	1	no	—
287	#533	C₂²×C₄₁	1	2	yes	—
288	#532	C₂²×C₄₁	1	1	yes	characteristic
289	#532	C₂²×C₄₁	1	2	yes	—
290	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
291	#520	—	2	1	no	—
292	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
293	#446	—	2	1	no	—
294	#436	—	2	1	no	—
295	#435	—	2	2	no	—
296	#432	C₂²×C₄₁	1	2	yes	—
297	#431	C₂²×C₄₁	1	1	yes	characteristic
298	#424	—	2	1	no	—
299	#424	—	2	2	no	—
300	#423	C₂²×C₄₁	1	1	yes	characteristic
301	#423	C₂²×C₄₁	1	2	yes	—
302	#423	C₂²×C₄₁	1	2	yes	—
303	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
304	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
305	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
306	#332	C₂²×C₄₁	1	2	yes	—
307	#332	C₂²×C₄₁	1	2	yes	—
308	#332	C₂²×C₄₁	1	1	yes	characteristic
309	#332	C₂²×C₄₁	1	1	yes	characteristic
310	#332	C₂²×C₄₁	1	2	yes	—
311	#332	C₂²×C₄₁	1	2	yes	—
312	#332	C₂²×C₄₁	1	2	yes	—
313	#331	C₂²×C₄₁	1	1	yes	characteristic
314	#331	C₂²×C₄₁	1	1	yes	characteristic
315	#331	C₂²×C₄₁	1	2	yes	—
316	C₄₁	#17765	1	1	yes	characteristic, direct product factor, Sylow
317	C₂×#41	—	2	1	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
320	C₂²×Q₈	—	2	1	no	—
321	#131	—	4	4	no	—
322	C₂×M₁₆	—	2	2	no	—
323	C₂×M₁₆	—	2	1	no	—
324	C₂×M₁₆	—	2	2	no	—
325	C₂²×C₈	—	2	2	no	—
326	C₂²×C₈	—	2	1	no	—
327	C₂²×C₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	—	2	1	no	—
330	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
331	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
332	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
333	#127	C₂³×C₄₁	1	1	yes	characteristic
334	#126	—	2	4	no	—
335	#125	C₂×C₄×C₄₁	1	4	yes	—
336	#124	—	2	4	no	—
337	#124	—	2	1	no	—
338	#124	—	2	2	no	—
339	#123	—	2	4	no	—
340	C₂²⋊Q₈	—	2	2	no	—
341	C₂²⋊Q₈	—	2	2	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	#121	—	2	2	no	—
345	#121	—	2	4	no	—
346	#121	—	2	2	no	—
347	#121	—	2	1	no	—
348	#121	C₂×C₄×C₄₁	1	2	yes	—
349	#121	C₂×C₄×C₄₁	1	4	yes	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
353	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
354	C₄×D₄	—	2	4	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	1	no	—
357	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
358	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
359	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
360	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
361	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
362	C₄×D₄	D₄×C₄₁	1	2	yes	—
363	#117	—	2	2	no	—
364	#117	C₂×C₄×C₄₁	1	4	yes	—
365	#117	D₄×C₄₁	1	1	yes	characteristic
366	#117	D₄×C₄₁	1	2	yes	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
370	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
372	C₄⋊C₈	—	2	2	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	2	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	2	no	—
378	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
381	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
382	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
383	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
384	C₂²⋊C₈	—	2	1	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	#97	—	2	2	no	—
387	C₄×C₈	—	2	1	no	—
388	#41	—	2	2	no	—
389	#41	—	2	2	no	—
390	#41	—	4	2	no	—
391	#41	—	4	2	no	—
392	#41	—	8	2	no	—
393	#41	—	8	2	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	8	1	no	—
398	C₂×Q₈	—	8	1	no	—
399	C₂×Q₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
405	C₂×D₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
406	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
411	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
416	M₁₆	—	4	4	no	—
417	M₁₆	—	4	2	no	—
418	M₁₆	—	8	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	4	4	no	—
426	C₂×C₈	—	4	2	no	—
427	C₂×C₈	—	8	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
435	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
436	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
437	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
438	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
439	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
440	C₂²⋊C₄	—	2	4	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	2	no	—
445	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
446	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
447	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
448	C₄²	—	2	4	no	—
449	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
450	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
451	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
452	C₄²	C₂×D₄×C₄₁	1	2	yes	—
453	C₄²	C₂×D₄×C₄₁	1	2	yes	—
454	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
455	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
456	C₂³	—	2	2	no	—
457	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
458	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
459	Q₈	—	2	2	no	—
460	Q₈	—	4	1	no	—
461	Q₈	—	4	1	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	8	2	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	2	no	—
467	D₄	—	4	2	no	—
468	D₄	—	4	4	no	—
469	D₄	—	4	1	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	1	no	—
472	D₄	—	4	1	no	—
473	D₄	—	8	4	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	8	4	no	—
486	C₂×C₄	—	8	2	no	—
487	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
488	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
489	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
490	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
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	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
494	C₂×C₄	#137×C₄₁	1	2	yes	—
495	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
498	C₈	—	4	1	no	—
499	C₈	—	4	2	no	—
500	C₈	—	4	1	no	—
501	C₈	—	4	2	no	—
502	C₈	—	4	1	no	—
503	C₈	—	4	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	C₄₁×#417	1	2	yes	—
513	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	1	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	1	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	4	no	—
522	C₄	C₄₁×#417	1	2	yes	—
523	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
524	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
525	C₄	C₄₁×#576	1	1	yes	characteristic
526	C₄	C₄₁×#577	1	1	yes	characteristic
527	C₂	—	2	2	no	—
528	C₂	—	4	2	no	—
529	C₂	—	8	2	no	—
530	C₂	C₄₁×#2889	1	1	yes	characteristic
531	C₂	C₂×C₄₁×#417	1	1	yes	characteristic
532	C₂	C₄₁×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2989	C₂	semidirect
3. C₄₁×#2906	C₂	semidirect
4. C₄₁×#1561	C₂	semidirect
5. C₄₁×#1557	C₂	semidirect
6. C₄₁×#1552	C₂	semidirect
7. C₄₁×#1549	C₂	semidirect
8. C₄₁×#1547	C₂	semidirect
9. C₄₁×#1547	C₂	semidirect
10. C₄₁×#1540	C₂	semidirect
11. C₂×C₄₁×#332	C₂	semidirect
237. #17765	C₄₁	direct (N × H)
262. #2989	C₂×C₄₁	semidirect
263. #2906	C₂×C₄₁	semidirect
264. #1561	C₂×C₄₁	semidirect
265. #1557	C₂×C₄₁	semidirect
266. #1552	C₂×C₄₁	semidirect
267. #1549	C₂×C₄₁	semidirect
268. #1547	C₂×C₄₁	semidirect
269. #1547	C₂×C₄₁	semidirect
270. #1540	C₂×C₄₁	semidirect
271. C₂×#332	C₂×C₄₁	semidirect
316. C₄₁	#17765	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3511	2
C₄₁×#2989	2
C₄₁×#2906	2
C₄₁×#1561	2
C₄₁×#1557	2
C₄₁×#1552	2
C₄₁×#1549	2
C₄₁×#1547	2
C₄₁×#1540	2
C₂×C₄₁×#332	2
#17765	41

Subgroups — C41×#17765 #23793966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17765 #23793966