Subgroups — C₄₁×#60915 #23837116

Order 10496.

1790 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3455	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3263	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3262	C₂	1	2	yes	maximal, complemented
4	C₄₁×#3261	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#3260	C₂	1	2	yes	maximal, complemented
6	C₄₁×#3259	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#3258	C₂	1	4	yes	maximal, complemented
8	C₄₁×#3257	C₂	1	4	yes	maximal, complemented
9	C₄₁×#3076	C₂	1	4	yes	maximal, complemented
10	C₄₁×#3054	C₂	1	2	yes	maximal, complemented
11	C₄₁×#3050	C₂	1	2	yes	maximal, complemented
12	C₄₁×#3035	C₂	1	4	yes	maximal, complemented
13	C₂×C₄₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#135×C₄₁	C₂²	1	2	yes	—
15	C₂×#135×C₄₁	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#542	C₂²	1	4	yes	—
19	C₄₁×#542	C₂²	1	4	yes	—
20	C₄₁×#541	C₂²	1	2	yes	—
21	C₄₁×#541	C₂²	1	2	yes	—
22	C₄₁×#540	C₂²	1	2	yes	—
23	C₄₁×#540	C₂²	1	2	yes	—
24	C₄₁×#539	—	2	4	no	—
25	C₄₁×#538	—	2	4	no	—
26	C₄₁×#533	C₂²	1	2	yes	—
27	C₄₁×#532	C₂²	1	2	yes	—
28	C₄₁×#529	—	2	4	no	—
29	C₄₁×#529	—	2	4	no	—
30	C₄₁×#525	C₂²	1	4	yes	—
31	C₄×#41×C₄₁	C₂²	1	4	yes	—
32	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
33	C₄₁×#479	C₂²	1	4	yes	complemented
34	C₄₁×#479	C₂²	1	4	yes	complemented
35	C₄₁×#478	C₂²	1	4	yes	complemented
36	C₄₁×#478	C₂²	1	4	yes	complemented
37	C₄₁×#477	C₂²	1	4	yes	complemented
38	C₄₁×#476	C₂²	1	4	yes	complemented
39	C₄₁×#475	C₂²	1	4	yes	complemented
40	C₄₁×#474	C₂²	1	4	yes	complemented
41	C₄₁×#470	C₂²	1	2	yes	—
42	C₄₁×#470	C₂²	1	2	yes	—
43	C₄₁×#469	C₂²	1	2	yes	—
44	C₄₁×#469	C₂²	1	2	yes	—
45	C₄₁×#468	C₂²	1	4	yes	—
46	C₄₁×#468	C₂²	1	4	yes	—
47	C₄₁×#467	C₂²	1	2	yes	—
48	C₄₁×#467	C₂²	1	2	yes	—
49	C₄₁×#466	C₂²	1	2	yes	—
50	C₄₁×#466	C₂²	1	2	yes	—
51	C₄₁×#465	C₂²	1	4	yes	—
52	C₄₁×#465	C₂²	1	4	yes	—
53	C₄₁×#464	C₂²	1	4	yes	complemented
54	C₄₁×#464	C₂²	1	4	yes	complemented
55	C₄₁×#463	C₂²	1	4	yes	complemented
56	C₄₁×#463	C₂²	1	4	yes	complemented
57	C₄₁×#462	C₂²	1	4	yes	complemented
58	C₄₁×#461	C₂²	1	4	yes	complemented
59	C₄₁×#460	C₂²	1	4	yes	complemented
60	C₄₁×#459	C₂²	1	4	yes	complemented
61	C₄₁×#427	C₂²	1	2	yes	—
62	C₄₁×#427	C₂²	1	2	yes	—
63	C₄₁×#424	C₂²	1	4	yes	—
64	C₄₁×#424	C₂²	1	1	yes	characteristic
65	C₄₁×#424	C₂²	1	1	yes	characteristic
66	C₄₁×#424	C₂²	1	1	yes	characteristic
67	C₄₁×#424	C₂²	1	1	yes	characteristic
68	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
69	C₄₁×#416	C₂²	1	4	yes	—
70	C₄₁×#416	C₂²	1	4	yes	—
71	C₂×#41×C₄₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
73	#135×C₄₁	—	4	4	no	—
74	#135×C₄₁	—	4	4	no	—
75	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
76	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
77	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
79	C₂×D₈×C₄₁	C₂³	1	2	yes	—
80	C₂×D₈×C₄₁	C₂³	1	2	yes	—
81	C₂×M₁₆×C₄₁	—	4	2	no	—
82	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
83	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
84	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₄₁	—	4	2	no	—
86	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
87	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
88	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
90	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
92	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
93	#127×C₄₁	C₂³	1	2	yes	—
94	#126×C₄₁	—	2	4	no	—
95	#126×C₄₁	—	2	4	no	—
96	#126×C₄₁	—	2	4	no	—
97	#126×C₄₁	—	2	4	no	—
98	#125×C₄₁	—	2	4	no	—
99	#125×C₄₁	—	2	4	no	—
100	#125×C₄₁	C₂³	1	2	yes	—
101	#125×C₄₁	C₂³	1	2	yes	—
102	#124×C₄₁	—	2	4	no	—
103	#124×C₄₁	—	2	4	no	—
104	#124×C₄₁	C₂³	1	4	yes	—
105	#123×C₄₁	—	2	4	no	—
106	#123×C₄₁	—	2	4	no	—
107	#123×C₄₁	—	2	4	no	—
108	#123×C₄₁	—	2	4	no	—
109	#123×C₄₁	—	2	4	no	—
110	C₂²⋊Q₈×C₄₁	—	2	4	no	—
111	C₂²⋊Q₈×C₄₁	—	2	4	no	—
112	C₂²⋊Q₈×C₄₁	—	2	4	no	—
113	C₂²⋊Q₈×C₄₁	C₂³	1	4	yes	—
114	C₂²⋊Q₈×C₄₁	C₂³	1	4	yes	—
115	#121×C₄₁	—	2	4	no	—
116	#121×C₄₁	—	2	4	no	—
117	#121×C₄₁	—	2	4	no	—
118	#121×C₄₁	C₂³	1	4	yes	—
119	#121×C₄₁	C₂³	1	4	yes	—
120	#120×C₄₁	—	2	4	no	—
121	C₄×Q₈×C₄₁	—	2	4	no	—
122	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
123	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
124	C₄×D₄×C₄₁	—	2	4	no	—
125	C₄×D₄×C₄₁	—	2	4	no	—
126	C₄×D₄×C₄₁	—	2	4	no	—
127	C₄×D₄×C₄₁	—	2	4	no	—
128	C₄×D₄×C₄₁	—	2	4	no	—
129	C₄×D₄×C₄₁	C₂³	1	4	yes	—
130	C₄×D₄×C₄₁	C₂³	1	4	yes	—
131	#117×C₄₁	—	2	4	no	—
132	#117×C₄₁	D₄	1	2	yes	—
133	#117×C₄₁	D₄	1	2	yes	—
134	#117×C₄₁	C₂³	1	4	yes	—
135	#117×C₄₁	C₂³	1	1	yes	characteristic
136	#117×C₄₁	C₂³	1	2	yes	—
137	#117×C₄₁	C₂³	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
139	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
140	C₂×C₄²×C₄₁	—	2	4	no	—
141	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
142	#107×C₄₁	C₂³	1	2	yes	—
143	#107×C₄₁	C₂³	1	2	yes	—
144	#107×C₄₁	C₂³	1	4	yes	—
145	#106×C₄₁	C₂³	1	4	yes	—
146	#106×C₄₁	C₂³	1	2	yes	—
147	#106×C₄₁	C₂³	1	2	yes	—
148	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
149	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
150	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
151	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
152	#103×C₄₁	C₂³	1	4	yes	—
153	#103×C₄₁	C₂³	1	4	yes	—
154	#103×C₄₁	C₂³	1	4	yes	—
155	#103×C₄₁	C₂³	1	4	yes	—
156	#102×C₄₁	C₂³	1	4	yes	—
157	#102×C₄₁	C₂³	1	4	yes	—
158	#102×C₄₁	C₂³	1	4	yes	—
159	#102×C₄₁	C₂³	1	4	yes	—
160	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
161	#41×C₄₁	—	2	8	no	—
162	#41×C₄₁	—	2	8	no	—
163	C₂×Q₈×C₄₁	—	2	4	no	—
164	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
165	C₂×D₄×C₄₁	—	2	4	no	—
166	C₂×D₄×C₄₁	—	2	4	no	—
167	C₂×D₄×C₄₁	—	2	4	no	—
168	C₂×D₄×C₄₁	—	2	4	no	—
169	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₄₁	—	2	2	no	—
171	C₂²×C₄×C₄₁	—	2	2	no	—
172	C₂²×C₄×C₄₁	—	2	4	no	—
173	C₂²×C₄×C₄₁	—	2	4	no	—
174	C₂²×C₄×C₄₁	—	2	4	no	—
175	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₄₁	—	4	2	no	—
183	Q₁₆×C₄₁	—	4	2	no	—
184	SD₁₆×C₄₁	—	4	4	no	—
185	SD₁₆×C₄₁	—	4	4	no	—
186	D₈×C₄₁	—	4	2	no	—
187	D₈×C₄₁	—	4	2	no	—
188	M₁₆×C₄₁	—	4	1	no	—
189	M₁₆×C₄₁	—	4	2	no	—
190	M₁₆×C₄₁	—	4	1	no	—
191	C₂×C₈×C₄₁	—	2	4	no	—
192	C₂×C₈×C₄₁	—	2	4	no	—
193	C₂×C₈×C₄₁	—	4	1	no	—
194	C₂×C₈×C₄₁	—	4	2	no	—
195	C₂×C₈×C₄₁	—	4	1	no	—
196	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
197	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
198	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
199	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
200	C₄⋊C₄×C₄₁	—	2	4	no	—
201	C₄⋊C₄×C₄₁	—	2	4	no	—
202	C₄⋊C₄×C₄₁	—	2	4	no	—
203	C₄⋊C₄×C₄₁	—	2	4	no	—
204	C₄⋊C₄×C₄₁	—	2	4	no	—
205	C₄⋊C₄×C₄₁	—	2	4	no	—
206	C₄⋊C₄×C₄₁	—	2	4	no	—
207	C₄⋊C₄×C₄₁	—	2	4	no	—
208	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
210	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
211	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
212	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
213	C₂²⋊C₄×C₄₁	—	2	4	no	—
214	C₂²⋊C₄×C₄₁	—	2	4	no	—
215	C₂²⋊C₄×C₄₁	—	2	4	no	—
216	C₂²⋊C₄×C₄₁	—	2	4	no	—
217	C₂²⋊C₄×C₄₁	—	2	4	no	—
218	C₂²⋊C₄×C₄₁	—	2	4	no	—
219	C₂²⋊C₄×C₄₁	—	2	4	no	—
220	C₂²⋊C₄×C₄₁	—	2	4	no	—
221	C₂²⋊C₄×C₄₁	—	2	4	no	—
222	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
223	C₂²⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
224	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
225	C₄²×C₄₁	—	2	4	no	—
226	C₄²×C₄₁	—	2	4	no	—
227	C₄²×C₄₁	—	2	4	no	—
228	C₄²×C₄₁	C₂×D₄	1	4	yes	—
229	C₄²×C₄₁	C₂⁴	1	2	yes	—
230	C₄²×C₄₁	C₂⁴	1	2	yes	—
231	C₂³×C₄₁	—	2	2	no	—
232	C₂³×C₄₁	—	2	4	no	—
233	C₂³×C₄₁	—	4	2	no	—
234	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
235	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
236	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
237	Q₈×C₄₁	—	2	4	no	—
238	Q₈×C₄₁	—	2	4	no	—
239	Q₈×C₄₁	—	8	4	no	—
240	D₄×C₄₁	—	2	4	no	—
241	D₄×C₄₁	—	2	4	no	—
242	D₄×C₄₁	—	4	8	no	—
243	D₄×C₄₁	—	8	4	no	—
244	D₄×C₄₁	—	8	4	no	—
245	D₄×C₄₁	—	8	4	no	—
246	C₂×C₄×C₄₁	—	2	2	no	—
247	C₂×C₄×C₄₁	—	2	2	no	—
248	C₂×C₄×C₄₁	—	2	2	no	—
249	C₂×C₄×C₄₁	—	2	1	no	—
250	C₂×C₄×C₄₁	—	2	1	no	—
251	C₂×C₄×C₄₁	—	2	1	no	—
252	C₂×C₄×C₄₁	—	2	1	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	—	4	4	no	—
257	C₂×C₄×C₄₁	—	4	8	no	—
258	C₂×C₄×C₄₁	—	8	4	no	—
259	C₂×C₄×C₄₁	—	8	4	no	—
260	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
261	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
262	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
266	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
267	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
268	C₈×C₄₁	—	2	2	no	—
269	C₈×C₄₁	—	2	2	no	—
270	C₈×C₄₁	—	2	2	no	—
271	C₈×C₄₁	—	2	2	no	—
272	#60915	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂²×C₄₁	—	2	2	no	—
274	C₂²×C₄₁	—	2	2	no	—
275	C₂²×C₄₁	—	2	2	no	—
276	C₂²×C₄₁	—	2	1	no	—
277	C₂²×C₄₁	—	2	1	no	—
278	C₂²×C₄₁	—	4	4	no	—
279	C₂²×C₄₁	—	4	4	no	—
280	C₂²×C₄₁	—	4	4	no	—
281	C₂²×C₄₁	—	4	4	no	—
282	C₂²×C₄₁	#537	1	2	yes	—
283	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
284	C₄×C₄₁	—	2	4	no	—
285	C₄×C₄₁	—	2	4	no	—
286	C₄×C₄₁	—	2	1	no	—
287	C₄×C₄₁	—	2	1	no	—
288	C₄×C₄₁	—	4	4	no	—
289	C₄×C₄₁	—	4	4	no	—
290	C₄×C₄₁	—	4	4	no	—
291	C₄×C₄₁	#537	1	2	yes	—
292	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
293	C₄×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
294	#3455	C₂×C₄₁	1	2	yes	complemented
295	#3263	C₂×C₄₁	1	2	yes	complemented
296	#3262	C₂×C₄₁	1	2	yes	complemented
297	#3261	C₂×C₄₁	1	1	yes	characteristic, complemented
298	#3260	C₂×C₄₁	1	2	yes	complemented
299	#3259	C₂×C₄₁	1	1	yes	characteristic, complemented
300	#3258	C₂×C₄₁	1	4	yes	complemented
301	#3257	C₂×C₄₁	1	4	yes	complemented
302	#3076	C₂×C₄₁	1	4	yes	complemented
303	#3054	C₂×C₄₁	1	2	yes	complemented
304	#3050	C₂×C₄₁	1	2	yes	complemented
305	#3035	C₂×C₄₁	1	4	yes	complemented
306	C₂×#424	C₂×C₄₁	1	1	yes	characteristic, complemented
307	C₂×C₄₁	—	2	2	no	—
308	C₂×C₄₁	—	4	1	no	—
309	C₂×C₄₁	—	4	1	no	—
310	C₂×C₄₁	—	8	4	no	—
311	C₂×C₄₁	C₂×#537	1	1	yes	characteristic
312	C₂×C₄₁	#3584	1	2	yes	—
313	C₂×#135	C₂²×C₄₁	1	2	yes	—
314	C₂×#135	C₂²×C₄₁	1	2	yes	—
315	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
316	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
317	#542	C₂²×C₄₁	1	4	yes	—
318	#542	C₂²×C₄₁	1	4	yes	—
319	#541	C₂²×C₄₁	1	2	yes	—
320	#541	C₂²×C₄₁	1	2	yes	—
321	#540	C₂²×C₄₁	1	2	yes	—
322	#540	C₂²×C₄₁	1	2	yes	—
323	#539	—	2	4	no	—
324	#538	—	2	4	no	—
325	#533	C₂²×C₄₁	1	2	yes	—
326	#532	C₂²×C₄₁	1	2	yes	—
327	#529	—	2	4	no	—
328	#529	—	2	4	no	—
329	#525	C₂²×C₄₁	1	4	yes	—
330	C₄×#41	C₂²×C₄₁	1	4	yes	—
331	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
332	#479	C₂²×C₄₁	1	4	yes	complemented
333	#479	C₂²×C₄₁	1	4	yes	complemented
334	#478	C₂²×C₄₁	1	4	yes	complemented
335	#478	C₂²×C₄₁	1	4	yes	complemented
336	#477	C₂²×C₄₁	1	4	yes	complemented
337	#476	C₂²×C₄₁	1	4	yes	complemented
338	#475	C₂²×C₄₁	1	4	yes	complemented
339	#474	C₂²×C₄₁	1	4	yes	complemented
340	#470	C₂²×C₄₁	1	2	yes	—
341	#470	C₂²×C₄₁	1	2	yes	—
342	#469	C₂²×C₄₁	1	2	yes	—
343	#469	C₂²×C₄₁	1	2	yes	—
344	#468	C₂²×C₄₁	1	4	yes	—
345	#468	C₂²×C₄₁	1	4	yes	—
346	#467	C₂²×C₄₁	1	2	yes	—
347	#467	C₂²×C₄₁	1	2	yes	—
348	#466	C₂²×C₄₁	1	2	yes	—
349	#466	C₂²×C₄₁	1	2	yes	—
350	#465	C₂²×C₄₁	1	4	yes	—
351	#465	C₂²×C₄₁	1	4	yes	—
352	#464	C₂²×C₄₁	1	4	yes	complemented
353	#464	C₂²×C₄₁	1	4	yes	complemented
354	#463	C₂²×C₄₁	1	4	yes	complemented
355	#463	C₂²×C₄₁	1	4	yes	complemented
356	#462	C₂²×C₄₁	1	4	yes	complemented
357	#461	C₂²×C₄₁	1	4	yes	complemented
358	#460	C₂²×C₄₁	1	4	yes	complemented
359	#459	C₂²×C₄₁	1	4	yes	complemented
360	#427	C₂²×C₄₁	1	2	yes	—
361	#427	C₂²×C₄₁	1	2	yes	—
362	#424	C₂²×C₄₁	1	4	yes	—
363	#424	C₂²×C₄₁	1	1	yes	characteristic
364	#424	C₂²×C₄₁	1	1	yes	characteristic
365	#424	C₂²×C₄₁	1	1	yes	characteristic
366	#424	C₂²×C₄₁	1	1	yes	characteristic
367	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
368	#416	C₂²×C₄₁	1	4	yes	—
369	#416	C₂²×C₄₁	1	4	yes	—
370	C₄₁	#60915	1	1	yes	characteristic, direct product factor, Sylow
371	C₂×#41	C₂³×C₄₁	1	4	yes	—
372	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
373	#135	—	4	4	no	—
374	#135	—	4	4	no	—
375	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
376	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
377	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
378	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
379	C₂×D₈	C₂³×C₄₁	1	2	yes	—
380	C₂×D₈	C₂³×C₄₁	1	2	yes	—
381	C₂×M₁₆	—	4	2	no	—
382	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
383	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
384	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
385	C₂²×C₈	—	4	2	no	—
386	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
387	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
388	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
390	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
391	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
392	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
393	#127	C₂³×C₄₁	1	2	yes	—
394	#126	—	2	4	no	—
395	#126	—	2	4	no	—
396	#126	—	2	4	no	—
397	#126	—	2	4	no	—
398	#125	—	2	4	no	—
399	#125	—	2	4	no	—
400	#125	C₂³×C₄₁	1	2	yes	—
401	#125	C₂³×C₄₁	1	2	yes	—
402	#124	—	2	4	no	—
403	#124	—	2	4	no	—
404	#124	C₂³×C₄₁	1	4	yes	—
405	#123	—	2	4	no	—
406	#123	—	2	4	no	—
407	#123	—	2	4	no	—
408	#123	—	2	4	no	—
409	#123	—	2	4	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₂²⋊Q₈	—	2	4	no	—
412	C₂²⋊Q₈	—	2	4	no	—
413	C₂²⋊Q₈	C₂³×C₄₁	1	4	yes	—
414	C₂²⋊Q₈	C₂³×C₄₁	1	4	yes	—
415	#121	—	2	4	no	—
416	#121	—	2	4	no	—
417	#121	—	2	4	no	—
418	#121	C₂³×C₄₁	1	4	yes	—
419	#121	C₂³×C₄₁	1	4	yes	—
420	#120	—	2	4	no	—
421	C₄×Q₈	—	2	4	no	—
422	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
423	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
424	C₄×D₄	—	2	4	no	—
425	C₄×D₄	—	2	4	no	—
426	C₄×D₄	—	2	4	no	—
427	C₄×D₄	—	2	4	no	—
428	C₄×D₄	—	2	4	no	—
429	C₄×D₄	C₂³×C₄₁	1	4	yes	—
430	C₄×D₄	C₂³×C₄₁	1	4	yes	—
431	#117	—	2	4	no	—
432	#117	D₄×C₄₁	1	2	yes	—
433	#117	D₄×C₄₁	1	2	yes	—
434	#117	C₂³×C₄₁	1	4	yes	—
435	#117	C₂³×C₄₁	1	1	yes	characteristic
436	#117	C₂³×C₄₁	1	2	yes	—
437	#117	C₂³×C₄₁	1	1	yes	characteristic
438	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
439	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
440	C₂×C₄²	—	2	4	no	—
441	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
442	#107	C₂³×C₄₁	1	2	yes	—
443	#107	C₂³×C₄₁	1	2	yes	—
444	#107	C₂³×C₄₁	1	4	yes	—
445	#106	C₂³×C₄₁	1	4	yes	—
446	#106	C₂³×C₄₁	1	2	yes	—
447	#106	C₂³×C₄₁	1	2	yes	—
448	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
449	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
450	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
451	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
452	#103	C₂³×C₄₁	1	4	yes	—
453	#103	C₂³×C₄₁	1	4	yes	—
454	#103	C₂³×C₄₁	1	4	yes	—
455	#103	C₂³×C₄₁	1	4	yes	—
456	#102	C₂³×C₄₁	1	4	yes	—
457	#102	C₂³×C₄₁	1	4	yes	—
458	#102	C₂³×C₄₁	1	4	yes	—
459	#102	C₂³×C₄₁	1	4	yes	—
460	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
461	#41	—	2	8	no	—
462	#41	—	2	8	no	—
463	C₂×Q₈	—	2	4	no	—
464	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
482	Q₁₆	—	4	2	no	—
483	Q₁₆	—	4	2	no	—
484	SD₁₆	—	4	4	no	—
485	SD₁₆	—	4	4	no	—
486	D₈	—	4	2	no	—
487	D₈	—	4	2	no	—
488	M₁₆	—	4	1	no	—
489	M₁₆	—	4	2	no	—
490	M₁₆	—	4	1	no	—
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	—	2	4	no	—
493	C₂×C₈	—	4	1	no	—
494	C₂×C₈	—	4	2	no	—
495	C₂×C₈	—	4	1	no	—
496	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
497	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
498	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
499	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
509	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
510	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
511	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
512	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	4	no	—
522	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
523	C₂²⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
524	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	4	no	—
527	C₄²	—	2	4	no	—
528	C₄²	C₂×D₄×C₄₁	1	4	yes	—
529	C₄²	C₂⁴×C₄₁	1	2	yes	—
530	C₄²	C₂⁴×C₄₁	1	2	yes	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	4	2	no	—
534	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
535	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
536	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
537	Q₈	—	2	4	no	—
538	Q₈	—	2	4	no	—
539	Q₈	—	8	4	no	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	4	8	no	—
543	D₄	—	8	4	no	—
544	D₄	—	8	4	no	—
545	D₄	—	8	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	—	8	4	no	—
559	C₂×C₄	—	8	4	no	—
560	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
561	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
562	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
565	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
566	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
567	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	C₄₁×#537	1	2	yes	—
582	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
583	C₄	—	2	4	no	—
584	C₄	—	2	4	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	4	no	—
588	C₄	—	4	4	no	—
589	C₄	—	4	4	no	—
590	C₄	C₄₁×#537	1	2	yes	—
591	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
592	C₄	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
593	C₂	—	2	2	no	—
594	C₂	—	4	1	no	—
595	C₂	—	4	1	no	—
596	C₂	—	8	4	no	—
597	C₂	C₂×C₄₁×#537	1	1	yes	characteristic
598	C₂	C₄₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
socle	C₂²×C₄₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3455	C₂	semidirect
2. C₄₁×#3263	C₂	semidirect
3. C₄₁×#3262	C₂	semidirect
4. C₄₁×#3261	C₂	semidirect
5. C₄₁×#3260	C₂	semidirect
6. C₄₁×#3259	C₂	semidirect
7. C₄₁×#3258	C₂	semidirect
8. C₄₁×#3257	C₂	semidirect
9. C₄₁×#3076	C₂	semidirect
10. C₄₁×#3054	C₂	semidirect
11. C₄₁×#3050	C₂	semidirect
12. C₄₁×#3035	C₂	semidirect
13. C₂×C₄₁×#424	C₂	semidirect
33. C₄₁×#479	C₂²	semidirect
34. C₄₁×#479	C₂²	semidirect
35. C₄₁×#478	C₂²	semidirect
36. C₄₁×#478	C₂²	semidirect
37. C₄₁×#477	C₂²	semidirect
38. C₄₁×#476	C₂²	semidirect
39. C₄₁×#475	C₂²	semidirect
40. C₄₁×#474	C₂²	semidirect
53. C₄₁×#464	C₂²	semidirect
54. C₄₁×#464	C₂²	semidirect
55. C₄₁×#463	C₂²	semidirect
56. C₄₁×#463	C₂²	semidirect
57. C₄₁×#462	C₂²	semidirect
58. C₄₁×#461	C₂²	semidirect
59. C₄₁×#460	C₂²	semidirect
60. C₄₁×#459	C₂²	semidirect
272. #60915	C₄₁	direct (N × H)
294. #3455	C₂×C₄₁	semidirect
295. #3263	C₂×C₄₁	semidirect
296. #3262	C₂×C₄₁	semidirect
297. #3261	C₂×C₄₁	semidirect
298. #3260	C₂×C₄₁	semidirect
299. #3259	C₂×C₄₁	semidirect
300. #3258	C₂×C₄₁	semidirect
301. #3257	C₂×C₄₁	semidirect
302. #3076	C₂×C₄₁	semidirect
303. #3054	C₂×C₄₁	semidirect
304. #3050	C₂×C₄₁	semidirect
305. #3035	C₂×C₄₁	semidirect
306. C₂×#424	C₂×C₄₁	semidirect
332. #479	C₂²×C₄₁	semidirect
333. #479	C₂²×C₄₁	semidirect
334. #478	C₂²×C₄₁	semidirect
335. #478	C₂²×C₄₁	semidirect
336. #477	C₂²×C₄₁	semidirect
337. #476	C₂²×C₄₁	semidirect
338. #475	C₂²×C₄₁	semidirect
339. #474	C₂²×C₄₁	semidirect
352. #464	C₂²×C₄₁	semidirect
353. #464	C₂²×C₄₁	semidirect
354. #463	C₂²×C₄₁	semidirect
355. #463	C₂²×C₄₁	semidirect
356. #462	C₂²×C₄₁	semidirect
357. #461	C₂²×C₄₁	semidirect
358. #460	C₂²×C₄₁	semidirect
359. #459	C₂²×C₄₁	semidirect
370. C₄₁	#60915	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×#3455	2
C₄₁×#3263	2
C₄₁×#3262	2
C₄₁×#3261	2
C₄₁×#3260	2
C₄₁×#3259	2
C₄₁×#3258	2
C₄₁×#3257	2
C₄₁×#3076	2
C₄₁×#3054	2
C₄₁×#3050	2
C₄₁×#3035	2
C₂×C₄₁×#424	2
#60915	41

Subgroups — C41×#60915 #23837116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#60915 #23837116