Subgroups — C₄₁×#20774 #23796975

Order 10496.

818 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1840	C₂	1	4	yes	maximal, complemented
3	C₄₁×#1839	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1729	C₂	1	2	yes	maximal
5	C₄₁×#1729	C₂	1	2	yes	maximal
6	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₁×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
15	C₄×C₂²⋊C₄×C₄₁	C₄	1	4	yes	—
16	C₄₁×#376	C₂²	1	2	yes	—
17	C₄³×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#336	C₂²	1	2	yes	—
19	C₄₁×#336	C₂²	1	2	yes	—
20	C₄₁×#336	C₂²	1	8	yes	—
21	C₄₁×#336	C₂²	1	8	yes	—
22	C₄₁×#336	C₂²	1	2	yes	—
23	C₄₁×#336	C₂²	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₁	—	2	4	no	—
27	C₂×M₁₆×C₄₁	—	2	4	no	—
28	C₂×M₁₆×C₄₁	—	2	4	no	—
29	C₂²×C₈×C₄₁	—	2	4	no	—
30	C₂²×C₈×C₄₁	—	2	4	no	—
31	C₂²×C₈×C₄₁	—	2	4	no	—
32	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
34	#117×C₄₁	—	2	2	no	—
35	#117×C₄₁	—	2	4	no	—
36	#117×C₄₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₁	—	2	4	no	—
43	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
50	#95×C₄₁	C₂×C₄	1	4	yes	—
51	#95×C₄₁	C₂×C₄	1	4	yes	—
52	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	4	no	—
54	C₂²×C₄×C₄₁	—	2	4	no	—
55	C₂²×C₄×C₄₁	—	4	2	no	—
56	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	#41	1	4	yes	—
63	C₂²×C₄×C₄₁	#41	1	2	yes	—
64	C₂²×C₄×C₄₁	#41	1	4	yes	—
65	C₂²×C₄×C₄₁	#41	1	2	yes	—
66	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₄₁	—	4	8	no	—
68	C₂×C₈×C₄₁	—	2	4	no	—
69	C₂×C₈×C₄₁	—	2	4	no	—
70	C₂×C₈×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	—	2	4	no	—
74	C₂×C₈×C₄₁	—	4	8	no	—
75	C₄⋊C₄×C₄₁	—	2	2	no	—
76	C₄⋊C₄×C₄₁	—	2	2	no	—
77	C₄⋊C₄×C₄₁	—	2	4	no	—
78	C₂²⋊C₄×C₄₁	—	2	4	no	—
79	C₂²⋊C₄×C₄₁	—	2	4	no	—
80	C₄²×C₄₁	—	2	4	no	—
81	C₄²×C₄₁	—	2	8	no	—
82	C₄²×C₄₁	—	2	2	no	—
83	C₄²×C₄₁	—	2	2	no	—
84	C₄²×C₄₁	—	2	4	no	—
85	C₄²×C₄₁	—	2	2	no	—
86	C₄²×C₄₁	—	2	2	no	—
87	C₄²×C₄₁	—	4	4	no	—
88	C₂³×C₄₁	—	2	2	no	—
89	C₂³×C₄₁	—	2	2	no	—
90	C₂³×C₄₁	#117	1	2	yes	—
91	C₂³×C₄₁	#117	1	2	yes	—
92	C₂³×C₄₁	#117	1	2	yes	—
93	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	4	no	—
100	C₂×C₄×C₄₁	—	2	4	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	4	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	4	8	no	—
107	C₂×C₄×C₄₁	#117	1	2	yes	—
108	C₂×C₄×C₄₁	#117	1	2	yes	—
109	C₂×C₄×C₄₁	#117	1	2	yes	—
110	C₂×C₄×C₄₁	#126	1	8	yes	—
111	C₂×C₄×C₄₁	#126	1	8	yes	—
112	C₂×C₄×C₄₁	#131	1	4	yes	—
113	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
114	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
115	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
117	C₈×C₄₁	—	4	4	no	—
118	C₈×C₄₁	—	4	4	no	—
119	#20774	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₄₁	—	2	2	no	—
121	C₂²×C₄₁	—	2	2	no	—
122	C₂²×C₄₁	—	2	4	no	—
123	C₂²×C₄₁	—	2	4	no	—
124	C₂²×C₄₁	#383	1	4	yes	—
125	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
126	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
127	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
128	C₂²×C₄₁	C₂×#131	1	2	yes	—
129	C₂²×C₄₁	#568	1	2	yes	—
130	C₄×C₄₁	—	2	4	no	—
131	C₄×C₄₁	—	4	2	no	—
132	C₄×C₄₁	—	4	2	no	—
133	C₄×C₄₁	—	4	2	no	—
134	C₄×C₄₁	—	4	4	no	—
135	C₄×C₄₁	#383	1	4	yes	—
136	C₄×C₄₁	C₂×#126	1	2	yes	—
137	C₄×C₄₁	C₂×#126	1	2	yes	—
138	C₄×#117	C₂×C₄₁	1	1	yes	characteristic
139	#1840	C₂×C₄₁	1	4	yes	complemented
140	#1839	C₂×C₄₁	1	4	yes	complemented
141	#1729	C₂×C₄₁	1	2	yes	—
142	#1729	C₂×C₄₁	1	2	yes	—
143	#1728	C₂×C₄₁	1	1	yes	characteristic
144	C₂×#336	C₂×C₄₁	1	1	yes	characteristic
145	C₂×C₄₁	—	2	4	no	—
146	C₂×C₄₁	C₂×#383	1	1	yes	characteristic
147	C₂×C₄₁	#2922	1	1	yes	characteristic
148	C₂×C₄₁	#2923	1	1	yes	characteristic
149	C₂×C₄₁	#2924	1	2	yes	—
150	C₂×C₄₁	#2925	1	2	yes	—
151	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
152	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
153	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
154	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
155	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
156	C₄×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
157	C₄×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
158	C₄×C₂²⋊C₄	C₄×C₄₁	1	4	yes	—
159	#376	C₂²×C₄₁	1	2	yes	—
160	C₄³	C₂²×C₄₁	1	2	yes	—
161	#336	C₂²×C₄₁	1	2	yes	—
162	#336	C₂²×C₄₁	1	2	yes	—
163	#336	C₂²×C₄₁	1	8	yes	—
164	#336	C₂²×C₄₁	1	8	yes	—
165	#336	C₂²×C₄₁	1	2	yes	—
166	#336	C₂²×C₄₁	1	2	yes	—
167	C₄₁	#20774	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
169	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
177	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
178	#117	—	2	2	no	—
179	#117	—	2	4	no	—
180	#117	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
188	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
189	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
190	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
191	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
192	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
194	#95	C₂×C₄×C₄₁	1	4	yes	—
195	#95	C₂×C₄×C₄₁	1	4	yes	—
196	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	#41×C₄₁	1	4	yes	—
207	C₂²×C₄	#41×C₄₁	1	2	yes	—
208	C₂²×C₄	#41×C₄₁	1	4	yes	—
209	C₂²×C₄	#41×C₄₁	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
211	M₁₆	—	4	8	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	8	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	#117×C₄₁	1	2	yes	—
235	C₂³	#117×C₄₁	1	2	yes	—
236	C₂³	#117×C₄₁	1	2	yes	—
237	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	#117×C₄₁	1	2	yes	—
252	C₂×C₄	#117×C₄₁	1	2	yes	—
253	C₂×C₄	#117×C₄₁	1	2	yes	—
254	C₂×C₄	#126×C₄₁	1	8	yes	—
255	C₂×C₄	#126×C₄₁	1	8	yes	—
256	C₂×C₄	#131×C₄₁	1	4	yes	—
257	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
258	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
259	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
261	C₈	—	4	4	no	—
262	C₈	—	4	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	C₄₁×#383	1	4	yes	—
268	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
269	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
270	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
271	C₂²	C₂×#131×C₄₁	1	2	yes	—
272	C₂²	C₄₁×#568	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	4	no	—
278	C₄	C₄₁×#383	1	4	yes	—
279	C₄	C₂×#126×C₄₁	1	2	yes	—
280	C₄	C₂×#126×C₄₁	1	2	yes	—
281	C₂	—	2	4	no	—
282	C₂	C₂×C₄₁×#383	1	1	yes	characteristic
283	C₂	C₄₁×#2922	1	1	yes	characteristic
284	C₂	C₄₁×#2923	1	1	yes	characteristic
285	C₂	C₄₁×#2924	1	2	yes	—
286	C₂	C₄₁×#2925	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1840	C₂	semidirect
3. C₄₁×#1839	C₂	semidirect
119. #20774	C₄₁	direct (N × H)
139. #1840	C₂×C₄₁	semidirect
140. #1839	C₂×C₄₁	semidirect
167. C₄₁	#20774	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#117×C₄₁	2
C₄₁×#1840	2
C₄₁×#1839	2
C₄₁×#1729	2
C₄₁×#1728	2
C₂×C₄₁×#336	2
#20774	41

Subgroups — C41×#20774 #23796975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20774 #23796975