Subgroups — C₄₁×#34191 #23810392

Order 10496.

2470 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2355	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2352	C₂	1	4	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₄₁	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2346	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2343	C₂	1	4	yes	maximal, complemented
7	C₄₁×#2339	C₂	1	2	yes	maximal
8	C₄₁×#2320	C₂	1	2	yes	maximal, complemented
9	C₄₁×#2318	C₂	1	4	yes	maximal, complemented
10	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#2283	C₂	1	2	yes	maximal, complemented
12	C₄₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄₁×#382	C₂	1	2	yes	maximal, complemented
14	C₄₁×#2275	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₄₁	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
24	C₂×#117×C₄₁	C₂²	1	2	yes	—
25	C₂×#117×C₄₁	C₂²	1	2	yes	—
26	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
28	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#391	C₂²	1	4	yes	—
31	C₄₁×#389	C₂²	1	4	yes	—
32	C₄₁×#388	C₂²	1	8	yes	—
33	C₄₁×#387	C₂²	1	8	yes	—
34	C₄₁×#387	C₂²	1	8	yes	—
35	C₄₁×#386	C₂²	1	8	yes	—
36	C₄₁×#385	C₂²	1	8	yes	—
37	C₄₁×#384	C₂²	1	2	yes	—
38	C₄₁×#383	C₂²	1	4	yes	—
39	C₄₁×#383	C₂²	1	2	yes	—
40	C₄₁×#382	C₂²	1	1	yes	characteristic
41	C₄₁×#382	C₂²	1	2	yes	—
42	C₄₁×#382	C₂²	1	4	yes	—
43	C₄₁×#382	C₂²	1	4	yes	—
44	C₄₁×#382	C₂²	1	4	yes	—
45	C₄₁×#382	C₂²	1	1	yes	characteristic
46	C₄₁×#381	C₂²	1	2	yes	—
47	C₄₁×#381	C₂²	1	4	yes	—
48	C₄₁×#380	C₂²	1	4	yes	—
49	C₄₁×#380	C₂²	1	2	yes	—
50	C₄₁×#380	C₂²	1	2	yes	—
51	C₄₁×#380	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
57	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
58	C₄×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
59	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
60	C₄₁×#376	C₂²	1	2	yes	—
61	C₄₁×#376	C₂²	1	4	yes	—
62	C₂×#95×C₄₁	C₂²	1	4	yes	—
63	C₂×#95×C₄₁	C₂²	1	4	yes	—
64	C₂×#41×C₄₁	—	2	8	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	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
74	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₄₁	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₄₁	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₄₁	—	2	16	no	—
79	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₄₁	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₄₁	C₂×C₄	1	16	yes	complemented
83	#117×C₄₁	—	4	4	no	—
84	#117×C₄₁	C₂×C₄	1	16	yes	complemented
85	#117×C₄₁	C₂×C₄	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₄₁	—	4	8	no	—
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	complemented
104	C₂×C₄²×C₄₁	C₂×C₄	1	16	yes	complemented
105	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
106	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
108	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
113	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
115	#95×C₄₁	C₂³	1	2	yes	—
116	#95×C₄₁	C₂³	1	2	yes	—
117	#95×C₄₁	C₂³	1	4	yes	—
118	#95×C₄₁	C₂³	1	4	yes	—
119	#95×C₄₁	C₂³	1	8	yes	—
120	#95×C₄₁	C₂³	1	8	yes	—
121	#95×C₄₁	C₂³	1	2	yes	—
122	#95×C₄₁	C₂³	1	4	yes	—
123	#95×C₄₁	C₂³	1	2	yes	—
124	#95×C₄₁	C₂³	1	8	yes	—
125	#95×C₄₁	C₂³	1	4	yes	—
126	#95×C₄₁	C₂³	1	8	yes	—
127	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
128	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
129	#41×C₄₁	—	2	32	no	—
130	C₂×Q₈×C₄₁	—	2	4	no	—
131	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₄₁	—	2	8	no	—
133	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
134	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₄₁	#41	1	8	yes	—
136	C₂²×C₄×C₄₁	—	2	8	no	—
137	C₂²×C₄×C₄₁	—	2	4	no	—
138	C₂²×C₄×C₄₁	—	2	4	no	—
139	C₂²×C₄×C₄₁	—	2	2	no	—
140	C₂²×C₄×C₄₁	—	2	4	no	—
141	C₂²×C₄×C₄₁	—	2	2	no	—
142	C₂²×C₄×C₄₁	—	2	16	no	—
143	C₂²×C₄×C₄₁	—	4	4	no	—
144	C₂²×C₄×C₄₁	—	4	2	no	—
145	C₂²×C₄×C₄₁	—	4	8	no	—
146	C₂²×C₄×C₄₁	—	4	8	no	—
147	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
148	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
165	C₄⋊C₄×C₄₁	—	2	8	no	—
166	C₄⋊C₄×C₄₁	—	2	4	no	—
167	C₄⋊C₄×C₄₁	—	2	8	no	—
168	C₄⋊C₄×C₄₁	—	2	4	no	—
169	C₄⋊C₄×C₄₁	—	4	8	no	—
170	C₄⋊C₄×C₄₁	—	4	4	no	—
171	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
172	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
173	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
174	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
175	C₂²⋊C₄×C₄₁	—	2	16	no	—
176	C₂²⋊C₄×C₄₁	—	4	4	no	—
177	C₂²⋊C₄×C₄₁	—	4	4	no	—
178	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
179	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
180	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
181	C₄²×C₄₁	—	2	8	no	—
182	C₄²×C₄₁	—	4	8	no	—
183	C₄²×C₄₁	—	4	4	no	—
184	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
185	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
186	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
187	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
188	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
189	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
190	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
191	C₂³×C₄₁	—	2	2	no	—
192	C₂³×C₄₁	—	2	2	no	—
193	C₂³×C₄₁	—	2	4	no	—
194	C₂³×C₄₁	—	2	2	no	—
195	C₂³×C₄₁	—	2	1	no	—
196	C₂³×C₄₁	—	2	1	no	—
197	C₂³×C₄₁	—	2	2	no	—
198	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
199	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
200	C₂³×C₄₁	C₂×#41	1	8	yes	—
201	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
202	Q₈×C₄₁	—	2	8	no	—
203	D₄×C₄₁	—	2	16	no	—
204	D₄×C₄₁	#117	1	16	yes	complemented
205	C₂×C₄×C₄₁	—	2	16	no	—
206	C₂×C₄×C₄₁	—	2	2	no	—
207	C₂×C₄×C₄₁	—	2	4	no	—
208	C₂×C₄×C₄₁	—	2	1	no	—
209	C₂×C₄×C₄₁	—	2	2	no	—
210	C₂×C₄×C₄₁	—	2	1	no	—
211	C₂×C₄×C₄₁	—	2	2	no	—
212	C₂×C₄×C₄₁	—	2	4	no	—
213	C₂×C₄×C₄₁	—	2	2	no	—
214	C₂×C₄×C₄₁	—	2	2	no	—
215	C₂×C₄×C₄₁	—	2	8	no	—
216	C₂×C₄×C₄₁	—	2	16	no	—
217	C₂×C₄×C₄₁	—	2	8	no	—
218	C₂×C₄×C₄₁	—	2	4	no	—
219	C₂×C₄×C₄₁	—	2	4	no	—
220	C₂×C₄×C₄₁	—	2	8	no	—
221	C₂×C₄×C₄₁	—	2	8	no	—
222	C₂×C₄×C₄₁	—	2	4	no	—
223	C₂×C₄×C₄₁	—	2	4	no	—
224	C₂×C₄×C₄₁	—	2	4	no	—
225	C₂×C₄×C₄₁	—	2	8	no	—
226	C₂×C₄×C₄₁	—	2	4	no	—
227	C₂×C₄×C₄₁	—	4	8	no	—
228	C₂×C₄×C₄₁	—	4	4	no	—
229	C₂×C₄×C₄₁	—	8	16	no	—
230	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
231	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
232	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
233	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
234	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
236	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
239	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
240	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
241	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
242	#34191	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂²×C₄₁	—	2	4	no	—
244	C₂²×C₄₁	—	2	4	no	—
245	C₂²×C₄₁	—	2	8	no	—
246	C₂²×C₄₁	—	2	8	no	—
247	C₂²×C₄₁	—	2	2	no	—
248	C₂²×C₄₁	—	2	2	no	—
249	C₂²×C₄₁	—	2	2	no	—
250	C₂²×C₄₁	—	4	2	no	—
251	C₂²×C₄₁	—	4	2	no	—
252	C₂²×C₄₁	C₂×#117	1	8	yes	—
253	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
254	C₂²×C₄₁	C₂²×#41	1	2	yes	—
255	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
257	C₂²×C₄₁	#585	1	2	yes	—
258	C₄×C₄₁	—	2	2	no	—
259	C₄×C₄₁	—	2	4	no	—
260	C₄×C₄₁	—	2	2	no	—
261	C₄×C₄₁	—	2	4	no	—
262	C₄×C₄₁	—	2	2	no	—
263	C₄×C₄₁	—	2	4	no	—
264	C₄×C₄₁	—	2	2	no	—
265	C₄×C₄₁	—	4	8	no	—
266	C₄×C₄₁	—	4	4	no	—
267	C₄×C₄₁	—	4	4	no	—
268	C₄×C₄₁	C₂×#117	1	4	yes	—
269	C₄×C₄₁	#520	1	4	yes	—
270	C₂×C₄×#41	C₂×C₄₁	1	1	yes	characteristic
271	#2355	C₂×C₄₁	1	2	yes	complemented
272	#2352	C₂×C₄₁	1	4	yes	complemented
273	D₄×C₄⋊C₄	C₂×C₄₁	1	2	yes	complemented
274	#2346	C₂×C₄₁	1	2	yes	complemented
275	#2343	C₂×C₄₁	1	4	yes	complemented
276	#2339	C₂×C₄₁	1	2	yes	—
277	#2320	C₂×C₄₁	1	2	yes	complemented
278	#2318	C₂×C₄₁	1	4	yes	complemented
279	#2285	C₂×C₄₁	1	1	yes	characteristic, complemented
280	#2283	C₂×C₄₁	1	1	yes	characteristic, complemented
281	#2283	C₂×C₄₁	1	2	yes	complemented
282	C₂×#382	C₂×C₄₁	1	2	yes	complemented
283	#2275	C₂×C₄₁	1	2	yes	complemented
284	C₂×C₄₁	—	2	8	no	—
285	C₂×C₄₁	—	4	2	no	—
286	C₂×C₄₁	C₂²×#117	1	1	yes	characteristic
287	C₂×C₄₁	C₂×#520	1	1	yes	characteristic
288	C₂×C₄₁	#3428	1	1	yes	characteristic
289	C₂×C₄₁	#3514	1	2	yes	—
290	C₂×C₄₁	#3515	1	2	yes	—
291	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
292	C₄×#41	C₄×C₄₁	1	16	yes	complemented
293	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
294	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
296	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
297	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
299	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
300	C₂×#117	C₂²×C₄₁	1	2	yes	—
301	C₂×#117	C₂²×C₄₁	1	2	yes	—
302	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
304	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
305	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
306	#391	C₂²×C₄₁	1	4	yes	—
307	#389	C₂²×C₄₁	1	4	yes	—
308	#388	C₂²×C₄₁	1	8	yes	—
309	#387	C₂²×C₄₁	1	8	yes	—
310	#387	C₂²×C₄₁	1	8	yes	—
311	#386	C₂²×C₄₁	1	8	yes	—
312	#385	C₂²×C₄₁	1	8	yes	—
313	#384	C₂²×C₄₁	1	2	yes	—
314	#383	C₂²×C₄₁	1	4	yes	—
315	#383	C₂²×C₄₁	1	2	yes	—
316	#382	C₂²×C₄₁	1	4	yes	—
317	#382	C₂²×C₄₁	1	4	yes	—
318	#382	C₂²×C₄₁	1	4	yes	—
319	#382	C₂²×C₄₁	1	1	yes	characteristic
320	#382	C₂²×C₄₁	1	2	yes	—
321	#382	C₂²×C₄₁	1	1	yes	characteristic
322	#381	C₂²×C₄₁	1	2	yes	—
323	#381	C₂²×C₄₁	1	4	yes	—
324	#380	C₂²×C₄₁	1	2	yes	—
325	#380	C₂²×C₄₁	1	2	yes	—
326	#380	C₂²×C₄₁	1	4	yes	—
327	#380	C₂²×C₄₁	1	2	yes	—
328	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
330	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
331	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
334	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
335	C₄×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
336	#376	C₂²×C₄₁	1	2	yes	—
337	#376	C₂²×C₄₁	1	4	yes	—
338	C₂×#95	C₂²×C₄₁	1	4	yes	—
339	C₂×#95	C₂²×C₄₁	1	4	yes	—
340	C₄₁	#34191	1	1	yes	characteristic, direct product factor, Sylow
341	C₂×#41	—	2	8	no	—
342	C₂×#41	C₂×C₄×C₄₁	1	8	yes	—
343	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
344	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
345	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
347	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
349	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
351	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
352	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
353	C₄×Q₈	C₂×C₄×C₄₁	1	8	yes	complemented
354	C₄×Q₈	C₂×C₄×C₄₁	1	8	yes	complemented
355	C₄×D₄	—	2	16	no	—
356	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
357	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
358	C₄×D₄	C₂×C₄×C₄₁	1	16	yes	complemented
359	C₄×D₄	C₂×C₄×C₄₁	1	16	yes	complemented
360	#117	—	4	4	no	—
361	#117	C₂×C₄×C₄₁	1	16	yes	complemented
362	#117	C₂×C₄×C₄₁	1	8	yes	complemented
363	C₂×C₄⋊C₄	—	4	8	no	—
364	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
369	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
380	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	complemented
381	C₂×C₄²	C₂×C₄×C₄₁	1	16	yes	complemented
382	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
383	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
384	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
385	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
386	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
389	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
391	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
392	#95	C₂³×C₄₁	1	2	yes	—
393	#95	C₂³×C₄₁	1	2	yes	—
394	#95	C₂³×C₄₁	1	4	yes	—
395	#95	C₂³×C₄₁	1	4	yes	—
396	#95	C₂³×C₄₁	1	8	yes	—
397	#95	C₂³×C₄₁	1	8	yes	—
398	#95	C₂³×C₄₁	1	2	yes	—
399	#95	C₂³×C₄₁	1	4	yes	—
400	#95	C₂³×C₄₁	1	2	yes	—
401	#95	C₂³×C₄₁	1	8	yes	—
402	#95	C₂³×C₄₁	1	4	yes	—
403	#95	C₂³×C₄₁	1	8	yes	—
404	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
405	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
406	#41	—	2	32	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
411	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
412	C₂×D₄	#41×C₄₁	1	8	yes	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	16	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	8	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
425	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
426	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
427	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
428	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
429	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
432	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
433	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	4	8	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
449	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
450	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
451	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₂²⋊C₄	—	4	4	no	—
454	C₂²⋊C₄	—	4	4	no	—
455	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
456	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
457	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
458	C₄²	—	2	8	no	—
459	C₄²	—	4	4	no	—
460	C₄²	—	4	8	no	—
461	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
462	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
463	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
464	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
465	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
466	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
467	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	2	no	—
475	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
476	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
477	C₂³	C₂×#41×C₄₁	1	8	yes	—
478	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
479	Q₈	—	2	8	no	—
480	D₄	—	2	16	no	—
481	D₄	#117×C₄₁	1	16	yes	complemented
482	C₂×C₄	—	2	16	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	16	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	8	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	8	16	no	—
507	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
508	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
509	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
510	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
511	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
512	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
513	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
514	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
515	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
516	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
517	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
518	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	2	8	no	—
522	C₂²	—	2	8	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	C₂×#117×C₄₁	1	8	yes	—
529	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
530	C₂²	C₂²×#41×C₄₁	1	2	yes	—
531	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
532	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
533	C₂²	C₄₁×#585	1	2	yes	—
534	C₄	—	2	2	no	—
535	C₄	—	2	4	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	2	4	no	—
540	C₄	—	2	2	no	—
541	C₄	—	4	4	no	—
542	C₄	—	4	8	no	—
543	C₄	—	4	4	no	—
544	C₄	C₂×#117×C₄₁	1	4	yes	—
545	C₄	C₄₁×#520	1	4	yes	—
546	C₂	—	2	8	no	—
547	C₂	—	4	2	no	—
548	C₂	C₂²×#117×C₄₁	1	1	yes	characteristic
549	C₂	C₂×C₄₁×#520	1	1	yes	characteristic
550	C₂	C₄₁×#3428	1	1	yes	characteristic
551	C₂	C₄₁×#3514	1	2	yes	—
552	C₂	C₄₁×#3515	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂²×#41×C₄₁	C₂²
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×#2355	C₂	semidirect
3. C₄₁×#2352	C₂	semidirect
4. D₄×C₄⋊C₄×C₄₁	C₂	semidirect
5. C₄₁×#2346	C₂	semidirect
6. C₄₁×#2343	C₂	semidirect
8. C₄₁×#2320	C₂	semidirect
9. C₄₁×#2318	C₂	semidirect
10. C₄₁×#2285	C₂	semidirect
11. C₄₁×#2283	C₂	semidirect
12. C₄₁×#2283	C₂	semidirect
13. C₂×C₄₁×#382	C₂	semidirect
14. C₄₁×#2275	C₂	semidirect
16. C₄×#41×C₄₁	C₄	semidirect
76. C₄×Q₈×C₄₁	C₂×C₄	semidirect
77. C₄×Q₈×C₄₁	C₂×C₄	semidirect
81. C₄×D₄×C₄₁	C₂×C₄	semidirect
82. C₄×D₄×C₄₁	C₂×C₄	semidirect
84. #117×C₄₁	C₂×C₄	semidirect
85. #117×C₄₁	C₂×C₄	semidirect
103. C₂×C₄²×C₄₁	C₂×C₄	semidirect
104. C₂×C₄²×C₄₁	C₂×C₄	semidirect
204. D₄×C₄₁	#117	semidirect
242. #34191	C₄₁	direct (N × H)
271. #2355	C₂×C₄₁	semidirect
272. #2352	C₂×C₄₁	semidirect
273. D₄×C₄⋊C₄	C₂×C₄₁	semidirect
274. #2346	C₂×C₄₁	semidirect
275. #2343	C₂×C₄₁	semidirect
277. #2320	C₂×C₄₁	semidirect
278. #2318	C₂×C₄₁	semidirect
279. #2285	C₂×C₄₁	semidirect
280. #2283	C₂×C₄₁	semidirect
281. #2283	C₂×C₄₁	semidirect
282. C₂×#382	C₂×C₄₁	semidirect
283. #2275	C₂×C₄₁	semidirect
292. C₄×#41	C₄×C₄₁	semidirect
340. C₄₁	#34191	direct (N × H)
353. C₄×Q₈	C₂×C₄×C₄₁	semidirect
354. C₄×Q₈	C₂×C₄×C₄₁	semidirect
358. C₄×D₄	C₂×C₄×C₄₁	semidirect
359. C₄×D₄	C₂×C₄×C₄₁	semidirect
361. #117	C₂×C₄×C₄₁	semidirect
362. #117	C₂×C₄×C₄₁	semidirect
380. C₂×C₄²	C₂×C₄×C₄₁	semidirect
381. C₂×C₄²	C₂×C₄×C₄₁	semidirect
481. D₄	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₄₁	2
C₄₁×#2355	2
C₄₁×#2352	2
D₄×C₄⋊C₄×C₄₁	2
C₄₁×#2346	2
C₄₁×#2343	2
C₄₁×#2339	2
C₄₁×#2320	2
C₄₁×#2318	2
C₄₁×#2285	2
C₄₁×#2283	2
C₂×C₄₁×#382	2
C₄₁×#2275	2
#34191	41

Subgroups — C41×#34191 #23810392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34191 #23810392