Subgroups — C₄₁×#61412 #23837613

Order 10496.

1886 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3510	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3403	C₂	1	4	yes	maximal, complemented
3	C₄₁×#3402	C₂	1	4	yes	maximal, complemented
4	C₄₁×#3394	C₂	1	8	yes	maximal, complemented
5	C₄₁×#3154	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#3044	C₂	1	2	yes	maximal, complemented
8	C₄₁×#3041	C₂	1	2	yes	maximal, complemented
9	C₄₁×#3040	C₂	1	2	yes	maximal, complemented
10	C₄₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#2949	C₂	1	2	yes	maximal, complemented
12	C₄×#135×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#137×C₄₁	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
15	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
18	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#559	C₂²	1	8	yes	complemented
20	C₄₁×#556	C₂²	1	8	yes	complemented
21	C₄₁×#551	—	2	8	no	—
22	C₄₁×#542	—	2	4	no	—
23	C₄₁×#536	C₂²	1	2	yes	—
24	C₄₁×#535	C₂²	1	2	yes	—
25	C₄₁×#533	C₂²	1	1	yes	characteristic
26	C₄₁×#529	—	2	4	no	—
27	C₂×#124×C₄₁	C₂²	1	2	yes	—
28	C₄₁×#519	C₂²	1	2	yes	—
29	C₄₁×#518	C₂²	1	2	yes	—
30	C₄×#41×C₄₁	C₂²	1	2	yes	—
31	C₄₁×#497	C₂²	1	4	yes	complemented
32	C₄₁×#496	C₂²	1	4	yes	complemented
33	C₄₁×#495	C₂²	1	4	yes	—
34	C₄₁×#495	C₂²	1	4	yes	—
35	C₄₁×#495	C₂²	1	4	yes	complemented
36	C₄₁×#492	C₂²	1	4	yes	complemented
37	C₄₁×#464	C₂²	1	8	yes	complemented
38	C₄₁×#464	C₂²	1	8	yes	complemented
39	C₄₁×#463	C₂²	1	8	yes	—
40	C₄₁×#463	C₂²	1	8	yes	complemented
41	C₄₁×#462	C₂²	1	8	yes	complemented
42	C₄₁×#459	C₂²	1	8	yes	—
43	C₄₁×#446	C₂²	1	4	yes	complemented
44	C₈×Q₈×C₄₁	C₂²	1	4	yes	complemented
45	C₄₁×#440	C₂²	1	8	yes	complemented
46	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	complemented
47	C₄×SD₁₆×C₄₁	C₂²	1	4	yes	complemented
48	C₄×SD₁₆×C₄₁	C₂²	1	8	yes	complemented
49	C₄×D₈×C₄₁	C₂²	1	2	yes	complemented
50	C₄₁×#427	C₂²	1	1	yes	characteristic
51	C₄₁×#424	C₂²	1	2	yes	—
52	C₄₁×#423	C₂²	1	1	yes	characteristic
53	C₄₁×#419	C₂²	1	2	yes	—
54	C₄₁×#418	C₂²	1	2	yes	—
55	C₄₁×#416	C₂²	1	2	yes	—
56	C₄₁×#405	C₂²	1	2	yes	—
57	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
58	C₂×#41×C₄₁	C₂³	1	2	yes	—
59	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
61	#137×C₄₁	—	4	4	no	—
62	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
63	#135×C₄₁	—	2	4	no	—
64	C₂×Q₁₆×C₄₁	—	4	2	no	—
65	C₂×Q₁₆×C₄₁	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
67	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
68	C₂×SD₁₆×C₄₁	—	2	8	no	—
69	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
70	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
71	C₂×SD₁₆×C₄₁	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
74	C₂×D₈×C₄₁	—	4	2	no	—
75	C₂×D₈×C₄₁	C₂³	1	1	yes	characteristic
76	C₂×D₈×C₄₁	C₂³	1	4	yes	—
77	C₂×D₈×C₄₁	C₂³	1	4	yes	—
78	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
79	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
82	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
83	#127×C₄₁	C₂³	1	4	yes	—
84	#126×C₄₁	—	2	8	no	—
85	#126×C₄₁	—	2	8	no	—
86	#125×C₄₁	D₄	1	2	yes	—
87	#124×C₄₁	—	2	8	no	—
88	#124×C₄₁	—	2	2	no	—
89	#124×C₄₁	—	2	2	no	—
90	#124×C₄₁	C₂³	1	8	yes	—
91	#124×C₄₁	C₂³	1	4	yes	—
92	#124×C₄₁	C₂³	1	4	yes	—
93	#123×C₄₁	—	2	8	no	—
94	#123×C₄₁	—	2	8	no	—
95	#123×C₄₁	—	2	4	no	—
96	C₂²⋊Q₈×C₄₁	—	2	8	no	—
97	C₂²⋊Q₈×C₄₁	—	2	4	no	—
98	C₂²⋊Q₈×C₄₁	D₄	1	4	yes	—
99	#121×C₄₁	—	2	4	no	—
100	#120×C₄₁	—	2	4	no	—
101	C₄×Q₈×C₄₁	—	2	4	no	—
102	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
103	C₄×Q₈×C₄₁	C₂³	1	8	yes	—
104	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
105	C₄×D₄×C₄₁	—	2	4	no	—
106	C₄×D₄×C₄₁	—	2	4	no	—
107	C₄×D₄×C₄₁	—	2	8	no	—
108	C₄×D₄×C₄₁	C₂³	1	4	yes	—
109	C₄×D₄×C₄₁	C₂³	1	8	yes	—
110	#117×C₄₁	—	2	8	no	—
111	#117×C₄₁	—	2	4	no	—
112	#117×C₄₁	—	2	2	no	—
113	#117×C₄₁	C₂³	1	2	yes	—
114	#117×C₄₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
116	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
117	C₂×C₄²×C₄₁	—	2	2	no	—
118	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
119	#107×C₄₁	C₂³	1	2	yes	—
120	#106×C₄₁	C₂³	1	2	yes	—
121	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
122	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
123	#103×C₄₁	C₂³	1	4	yes	—
124	#103×C₄₁	C₂³	1	8	yes	—
125	#102×C₄₁	C₂³	1	4	yes	—
126	#102×C₄₁	C₂³	1	8	yes	—
127	#97×C₄₁	C₂³	1	4	yes	—
128	C₄×C₈×C₄₁	C₂³	1	2	yes	—
129	C₄×C₈×C₄₁	C₂³	1	2	yes	—
130	C₄×C₈×C₄₁	C₂³	1	4	yes	—
131	C₂⁴×C₄₁	—	2	2	no	—
132	#41×C₄₁	—	2	4	no	—
133	#41×C₄₁	—	2	4	no	—
134	C₂×Q₈×C₄₁	—	2	2	no	—
135	C₂×Q₈×C₄₁	—	2	4	no	—
136	C₂×Q₈×C₄₁	—	2	4	no	—
137	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
138	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
140	C₂×D₄×C₄₁	—	2	2	no	—
141	C₂×D₄×C₄₁	—	2	2	no	—
142	C₂×D₄×C₄₁	—	2	8	no	—
143	C₂×D₄×C₄₁	—	2	4	no	—
144	C₂×D₄×C₄₁	—	2	4	no	—
145	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
146	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₄₁	—	2	2	no	—
148	C₂²×C₄×C₄₁	—	2	2	no	—
149	C₂²×C₄×C₄₁	—	2	4	no	—
150	C₂²×C₄×C₄₁	—	2	8	no	—
151	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
153	Q₁₆×C₄₁	—	2	2	no	—
154	Q₁₆×C₄₁	—	4	4	no	—
155	Q₁₆×C₄₁	—	4	4	no	—
156	SD₁₆×C₄₁	—	2	8	no	—
157	SD₁₆×C₄₁	—	2	8	no	—
158	SD₁₆×C₄₁	—	2	4	no	—
159	SD₁₆×C₄₁	—	4	4	no	—
160	SD₁₆×C₄₁	—	4	4	no	—
161	D₈×C₄₁	—	2	2	no	—
162	D₈×C₄₁	—	4	4	no	—
163	D₈×C₄₁	—	4	4	no	—
164	M₁₆×C₄₁	—	2	4	no	—
165	C₂×C₈×C₄₁	—	2	2	no	—
166	C₂×C₈×C₄₁	#41	1	4	yes	—
167	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
168	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
169	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
170	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
171	C₄⋊C₄×C₄₁	—	2	4	no	—
172	C₄⋊C₄×C₄₁	—	2	8	no	—
173	C₄⋊C₄×C₄₁	—	2	8	no	—
174	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
175	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
177	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
178	C₂²⋊C₄×C₄₁	—	2	2	no	—
179	C₂²⋊C₄×C₄₁	—	2	8	no	—
180	C₂²⋊C₄×C₄₁	—	2	8	no	—
181	C₂²⋊C₄×C₄₁	—	2	8	no	—
182	C₂²⋊C₄×C₄₁	—	2	4	no	—
183	C₂²⋊C₄×C₄₁	—	2	8	no	—
184	C₂²⋊C₄×C₄₁	—	2	8	no	—
185	C₂²⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
186	C₄²×C₄₁	—	2	4	no	—
187	C₄²×C₄₁	—	2	2	no	—
188	C₄²×C₄₁	—	2	8	no	—
189	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₄₁	C₂⁴	1	2	yes	—
192	C₄²×C₄₁	C₂⁴	1	4	yes	—
193	C₂³×C₄₁	—	2	4	no	—
194	C₂³×C₄₁	—	2	2	no	—
195	C₂³×C₄₁	—	4	4	no	—
196	C₂³×C₄₁	—	8	4	no	—
197	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₄₁	—	2	4	no	—
199	Q₈×C₄₁	—	2	2	no	—
200	Q₈×C₄₁	—	2	2	no	—
201	Q₈×C₄₁	—	2	4	no	—
202	Q₈×C₄₁	—	4	4	no	—
203	Q₈×C₄₁	—	4	4	no	—
204	D₄×C₄₁	—	2	2	no	—
205	D₄×C₄₁	—	2	4	no	—
206	D₄×C₄₁	—	2	4	no	—
207	D₄×C₄₁	—	2	2	no	—
208	D₄×C₄₁	—	4	4	no	—
209	D₄×C₄₁	—	4	4	no	—
210	D₄×C₄₁	—	8	8	no	—
211	C₂×C₄×C₄₁	—	2	4	no	—
212	C₂×C₄×C₄₁	—	2	4	no	—
213	C₂×C₄×C₄₁	—	2	2	no	—
214	C₂×C₄×C₄₁	—	4	2	no	—
215	C₂×C₄×C₄₁	—	4	4	no	—
216	C₂×C₄×C₄₁	—	4	4	no	—
217	C₂×C₄×C₄₁	—	8	4	no	—
218	C₂×C₄×C₄₁	—	8	8	no	—
219	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₄₁	C₂²×D₄	1	8	yes	—
221	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
224	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
225	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
226	C₈×C₄₁	—	2	2	no	—
227	C₈×C₄₁	—	2	4	no	—
228	C₈×C₄₁	C₂×#41	1	2	yes	—
229	C₈×C₄₁	C₂×#41	1	2	yes	—
230	#61412	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂²×C₄₁	—	2	2	no	—
232	C₂²×C₄₁	—	4	4	no	—
233	C₂²×C₄₁	—	4	8	no	—
234	C₂²×C₄₁	—	4	2	no	—
235	C₂²×C₄₁	—	4	4	no	—
236	C₂²×C₄₁	—	16	4	no	—
237	C₂²×C₄₁	#550	1	2	yes	—
238	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₄₁	—	2	2	no	—
240	C₄×C₄₁	—	2	2	no	—
241	C₄×C₄₁	—	2	8	no	—
242	C₄×C₄₁	—	4	4	no	—
243	C₄×C₄₁	—	4	2	no	—
244	C₄×C₄₁	—	4	4	no	—
245	C₄×C₄₁	#550	1	2	yes	—
246	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
247	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
248	#3510	C₂×C₄₁	1	2	yes	complemented
249	#3403	C₂×C₄₁	1	4	yes	complemented
250	#3402	C₂×C₄₁	1	4	yes	complemented
251	#3394	C₂×C₄₁	1	8	yes	complemented
252	#3154	C₂×C₄₁	1	2	yes	complemented
253	C₂×#495	C₂×C₄₁	1	1	yes	characteristic, complemented
254	#3044	C₂×C₄₁	1	2	yes	complemented
255	#3041	C₂×C₄₁	1	2	yes	complemented
256	#3040	C₂×C₄₁	1	2	yes	complemented
257	#2960	C₂×C₄₁	1	1	yes	characteristic, complemented
258	#2949	C₂×C₄₁	1	2	yes	complemented
259	C₄×#135	C₂×C₄₁	1	1	yes	characteristic, complemented
260	C₂×C₄₁	—	2	2	no	—
261	C₂×C₄₁	—	8	2	no	—
262	C₂×C₄₁	—	8	4	no	—
263	C₂×C₄₁	C₂×#550	1	1	yes	characteristic
264	C₂×C₄₁	#3584	1	2	yes	—
265	C₂×#137	C₂²×C₄₁	1	2	yes	—
266	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
267	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
268	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
269	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
270	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
271	#559	C₂²×C₄₁	1	8	yes	complemented
272	#556	C₂²×C₄₁	1	8	yes	complemented
273	#551	—	2	8	no	—
274	#542	—	2	4	no	—
275	#536	C₂²×C₄₁	1	2	yes	—
276	#535	C₂²×C₄₁	1	2	yes	—
277	#533	C₂²×C₄₁	1	1	yes	characteristic
278	#529	—	2	4	no	—
279	C₂×#124	C₂²×C₄₁	1	2	yes	—
280	#519	C₂²×C₄₁	1	2	yes	—
281	#518	C₂²×C₄₁	1	2	yes	—
282	C₄×#41	C₂²×C₄₁	1	2	yes	—
283	#497	C₂²×C₄₁	1	4	yes	complemented
284	#496	C₂²×C₄₁	1	4	yes	complemented
285	#495	C₂²×C₄₁	1	4	yes	—
286	#495	C₂²×C₄₁	1	4	yes	—
287	#495	C₂²×C₄₁	1	4	yes	complemented
288	#492	C₂²×C₄₁	1	4	yes	complemented
289	#464	C₂²×C₄₁	1	8	yes	complemented
290	#464	C₂²×C₄₁	1	8	yes	complemented
291	#463	C₂²×C₄₁	1	8	yes	—
292	#463	C₂²×C₄₁	1	8	yes	complemented
293	#462	C₂²×C₄₁	1	8	yes	complemented
294	#459	C₂²×C₄₁	1	8	yes	—
295	#446	C₂²×C₄₁	1	4	yes	complemented
296	C₈×Q₈	C₂²×C₄₁	1	4	yes	complemented
297	#440	C₂²×C₄₁	1	8	yes	complemented
298	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	complemented
299	C₄×SD₁₆	C₂²×C₄₁	1	4	yes	complemented
300	C₄×SD₁₆	C₂²×C₄₁	1	8	yes	complemented
301	C₄×D₈	C₂²×C₄₁	1	2	yes	complemented
302	#427	C₂²×C₄₁	1	1	yes	characteristic
303	#424	C₂²×C₄₁	1	2	yes	—
304	#423	C₂²×C₄₁	1	1	yes	characteristic
305	#419	C₂²×C₄₁	1	2	yes	—
306	#418	C₂²×C₄₁	1	2	yes	—
307	#416	C₂²×C₄₁	1	2	yes	—
308	#405	C₂²×C₄₁	1	2	yes	—
309	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
310	C₄₁	#61412	1	1	yes	characteristic, direct product factor, Sylow
311	C₂×#41	C₂³×C₄₁	1	2	yes	—
312	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
313	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
314	#137	—	4	4	no	—
315	AGL(1,ℤ/8)	—	4	4	no	—
316	#135	—	2	4	no	—
317	C₂×Q₁₆	—	4	2	no	—
318	C₂×Q₁₆	C₂³×C₄₁	1	1	yes	characteristic
319	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
320	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
321	C₂×SD₁₆	—	2	8	no	—
322	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
323	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
324	C₂×SD₁₆	C₂³×C₄₁	1	2	yes	—
325	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
326	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
327	C₂×D₈	—	4	2	no	—
328	C₂×D₈	C₂³×C₄₁	1	1	yes	characteristic
329	C₂×D₈	C₂³×C₄₁	1	4	yes	—
330	C₂×D₈	C₂³×C₄₁	1	4	yes	—
331	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
332	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
334	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
335	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
336	#127	C₂³×C₄₁	1	4	yes	—
337	#126	—	2	8	no	—
338	#126	—	2	8	no	—
339	#125	D₄×C₄₁	1	2	yes	—
340	#124	—	2	8	no	—
341	#124	—	2	2	no	—
342	#124	—	2	2	no	—
343	#124	C₂³×C₄₁	1	8	yes	—
344	#124	C₂³×C₄₁	1	4	yes	—
345	#124	C₂³×C₄₁	1	4	yes	—
346	#123	—	2	8	no	—
347	#123	—	2	8	no	—
348	#123	—	2	4	no	—
349	C₂²⋊Q₈	—	2	8	no	—
350	C₂²⋊Q₈	—	2	4	no	—
351	C₂²⋊Q₈	D₄×C₄₁	1	4	yes	—
352	#121	—	2	4	no	—
353	#120	—	2	4	no	—
354	C₄×Q₈	—	2	4	no	—
355	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
356	C₄×Q₈	C₂³×C₄₁	1	8	yes	—
357	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
358	C₄×D₄	—	2	4	no	—
359	C₄×D₄	—	2	4	no	—
360	C₄×D₄	—	2	8	no	—
361	C₄×D₄	C₂³×C₄₁	1	4	yes	—
362	C₄×D₄	C₂³×C₄₁	1	8	yes	—
363	#117	—	2	8	no	—
364	#117	—	2	4	no	—
365	#117	—	2	2	no	—
366	#117	C₂³×C₄₁	1	2	yes	—
367	#117	C₂³×C₄₁	1	2	yes	—
368	C₂×C₂²⋊C₄	—	2	4	no	—
369	C₂×C₂²⋊C₄	—	2	4	no	—
370	C₂×C₄²	—	2	2	no	—
371	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
372	#107	C₂³×C₄₁	1	2	yes	—
373	#106	C₂³×C₄₁	1	2	yes	—
374	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
375	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
376	#103	C₂³×C₄₁	1	4	yes	—
377	#103	C₂³×C₄₁	1	8	yes	—
378	#102	C₂³×C₄₁	1	4	yes	—
379	#102	C₂³×C₄₁	1	8	yes	—
380	#97	C₂³×C₄₁	1	4	yes	—
381	C₄×C₈	C₂³×C₄₁	1	2	yes	—
382	C₄×C₈	C₂³×C₄₁	1	2	yes	—
383	C₄×C₈	C₂³×C₄₁	1	4	yes	—
384	C₂⁴	—	2	2	no	—
385	#41	—	2	4	no	—
386	#41	—	2	4	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
391	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
392	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
399	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
406	Q₁₆	—	2	2	no	—
407	Q₁₆	—	4	4	no	—
408	Q₁₆	—	4	4	no	—
409	SD₁₆	—	2	8	no	—
410	SD₁₆	—	2	8	no	—
411	SD₁₆	—	2	4	no	—
412	SD₁₆	—	4	4	no	—
413	SD₁₆	—	4	4	no	—
414	D₈	—	2	2	no	—
415	D₈	—	4	4	no	—
416	D₈	—	4	4	no	—
417	M₁₆	—	2	4	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	#41×C₄₁	1	4	yes	—
420	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
421	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
422	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
423	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	8	no	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
428	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
429	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
430	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₂²⋊C₄	—	2	8	no	—
434	C₂²⋊C₄	—	2	8	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
439	C₄²	—	2	4	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	8	no	—
442	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
443	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
444	C₄²	C₂⁴×C₄₁	1	2	yes	—
445	C₄²	C₂⁴×C₄₁	1	4	yes	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	4	no	—
449	C₂³	—	8	4	no	—
450	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
451	Q₈	—	2	4	no	—
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	2	4	no	—
455	Q₈	—	4	4	no	—
456	Q₈	—	4	4	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	4	no	—
459	D₄	—	2	4	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	4	no	—
462	D₄	—	4	4	no	—
463	D₄	—	8	8	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	8	4	no	—
471	C₂×C₄	—	8	8	no	—
472	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₄₁	1	8	yes	—
474	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
475	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
477	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
478	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
479	C₈	—	2	2	no	—
480	C₈	—	2	4	no	—
481	C₈	C₂×#41×C₄₁	1	2	yes	—
482	C₈	C₂×#41×C₄₁	1	2	yes	—
483	C₂²	—	2	2	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	8	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	16	4	no	—
489	C₂²	C₄₁×#550	1	2	yes	—
490	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	8	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	4	no	—
497	C₄	C₄₁×#550	1	2	yes	—
498	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
499	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
500	C₂	—	2	2	no	—
501	C₂	—	8	4	no	—
502	C₂	—	8	2	no	—
503	C₂	C₂×C₄₁×#550	1	1	yes	characteristic
504	C₂	C₄₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3510	C₂	semidirect
2. C₄₁×#3403	C₂	semidirect
3. C₄₁×#3402	C₂	semidirect
4. C₄₁×#3394	C₂	semidirect
5. C₄₁×#3154	C₂	semidirect
6. C₂×C₄₁×#495	C₂	semidirect
7. C₄₁×#3044	C₂	semidirect
8. C₄₁×#3041	C₂	semidirect
9. C₄₁×#3040	C₂	semidirect
10. C₄₁×#2960	C₂	semidirect
11. C₄₁×#2949	C₂	semidirect
12. C₄×#135×C₄₁	C₂	semidirect
19. C₄₁×#559	C₂²	semidirect
20. C₄₁×#556	C₂²	semidirect
31. C₄₁×#497	C₂²	semidirect
32. C₄₁×#496	C₂²	semidirect
35. C₄₁×#495	C₂²	semidirect
36. C₄₁×#492	C₂²	semidirect
37. C₄₁×#464	C₂²	semidirect
38. C₄₁×#464	C₂²	semidirect
40. C₄₁×#463	C₂²	semidirect
41. C₄₁×#462	C₂²	semidirect
43. C₄₁×#446	C₂²	semidirect
44. C₈×Q₈×C₄₁	C₂²	semidirect
45. C₄₁×#440	C₂²	semidirect
46. C₄×Q₁₆×C₄₁	C₂²	semidirect
47. C₄×SD₁₆×C₄₁	C₂²	semidirect
48. C₄×SD₁₆×C₄₁	C₂²	semidirect
49. C₄×D₈×C₄₁	C₂²	semidirect
230. #61412	C₄₁	direct (N × H)
248. #3510	C₂×C₄₁	semidirect
249. #3403	C₂×C₄₁	semidirect
250. #3402	C₂×C₄₁	semidirect
251. #3394	C₂×C₄₁	semidirect
252. #3154	C₂×C₄₁	semidirect
253. C₂×#495	C₂×C₄₁	semidirect
254. #3044	C₂×C₄₁	semidirect
255. #3041	C₂×C₄₁	semidirect
256. #3040	C₂×C₄₁	semidirect
257. #2960	C₂×C₄₁	semidirect
258. #2949	C₂×C₄₁	semidirect
259. C₄×#135	C₂×C₄₁	semidirect
271. #559	C₂²×C₄₁	semidirect
272. #556	C₂²×C₄₁	semidirect
283. #497	C₂²×C₄₁	semidirect
284. #496	C₂²×C₄₁	semidirect
287. #495	C₂²×C₄₁	semidirect
288. #492	C₂²×C₄₁	semidirect
289. #464	C₂²×C₄₁	semidirect
290. #464	C₂²×C₄₁	semidirect
292. #463	C₂²×C₄₁	semidirect
293. #462	C₂²×C₄₁	semidirect
295. #446	C₂²×C₄₁	semidirect
296. C₈×Q₈	C₂²×C₄₁	semidirect
297. #440	C₂²×C₄₁	semidirect
298. C₄×Q₁₆	C₂²×C₄₁	semidirect
299. C₄×SD₁₆	C₂²×C₄₁	semidirect
300. C₄×SD₁₆	C₂²×C₄₁	semidirect
301. C₄×D₈	C₂²×C₄₁	semidirect
310. C₄₁	#61412	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×#3510	2
C₄₁×#3403	2
C₄₁×#3402	2
C₄₁×#3394	2
C₄₁×#3154	2
C₂×C₄₁×#495	2
C₄₁×#3044	2
C₄₁×#3041	2
C₄₁×#3040	2
C₄₁×#2960	2
C₄₁×#2949	2
C₄×#135×C₄₁	2
#61412	41

Subgroups — C41×#61412 #23837613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#61412 #23837613