Subgroups — C₄₁×#21429 #23797630

Order 10496.

1010 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#541	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2921	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2873	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1998	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1966	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1901	C₂	1	4	yes	maximal
7	C₄×#101×C₄₁	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#541	—	2	4	no	—
10	C₄₁×#541	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#126×C₄₁	C₄	1	4	yes	—
13	C₂×#125×C₄₁	C₄	1	4	yes	—
14	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
17	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#433	—	2	4	no	—
19	C₄₁×#423	C₂²	1	2	yes	—
20	C₂×#101×C₄₁	C₂²	1	8	yes	—
21	C₄₁×#407	C₂²	1	2	yes	—
22	C₄₁×#405	—	2	4	no	—
23	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
24	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
25	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
26	C₄₁×#343	C₂²	1	8	yes	—
27	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₄₁	—	2	2	no	—
30	C₂×M₁₆×C₄₁	—	2	8	no	—
31	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
32	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₄₁	—	2	2	no	—
34	C₄⋊Q₈×C₄₁	—	2	2	no	—
35	C₄⋊Q₈×C₄₁	—	2	2	no	—
36	#126×C₄₁	—	2	16	no	—
37	#125×C₄₁	—	2	16	no	—
38	#125×C₄₁	—	2	2	no	—
39	#125×C₄₁	—	2	2	no	—
40	C₂²⋊Q₈×C₄₁	—	2	4	no	—
41	C₂²⋊Q₈×C₄₁	—	2	4	no	—
42	C₄×Q₈×C₄₁	—	2	4	no	—
43	C₄×Q₈×C₄₁	—	2	4	no	—
44	#117×C₄₁	D₄	1	4	yes	—
45	#117×C₄₁	D₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₁	—	2	4	no	—
58	#101×C₄₁	—	2	16	no	—
59	C₂²⋊C₈×C₄₁	—	2	4	no	—
60	#97×C₄₁	—	2	2	no	—
61	#97×C₄₁	—	2	2	no	—
62	#97×C₄₁	—	2	2	no	—
63	C₄×C₈×C₄₁	—	2	2	no	—
64	C₄×C₈×C₄₁	—	2	2	no	—
65	C₄×C₈×C₄₁	—	2	2	no	—
66	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₁	—	2	2	no	—
68	C₂×Q₈×C₄₁	—	2	2	no	—
69	C₂×Q₈×C₄₁	—	2	8	no	—
70	C₂²×C₄×C₄₁	—	2	4	no	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₄₁	—	2	8	no	—
80	M₁₆×C₄₁	—	2	4	no	—
81	M₁₆×C₄₁	—	2	8	no	—
82	M₁₆×C₄₁	#41	1	8	yes	—
83	C₂×C₈×C₄₁	—	2	8	no	—
84	C₂×C₈×C₄₁	—	2	4	no	—
85	C₂×C₈×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	2	8	no	—
87	C₄⋊C₄×C₄₁	—	2	8	no	—
88	C₄⋊C₄×C₄₁	—	2	8	no	—
89	C₄⋊C₄×C₄₁	—	2	8	no	—
90	C₄⋊C₄×C₄₁	—	2	2	no	—
91	C₄⋊C₄×C₄₁	—	2	2	no	—
92	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
93	C₂²⋊C₄×C₄₁	—	2	8	no	—
94	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
95	C₄²×C₄₁	—	2	8	no	—
96	C₄²×C₄₁	C₂×D₄	1	2	yes	—
97	C₄²×C₄₁	C₂×D₄	1	2	yes	—
98	C₄²×C₄₁	C₂×D₄	1	2	yes	—
99	C₄²×C₄₁	C₂×D₄	1	2	yes	—
100	C₂³×C₄₁	—	2	2	no	—
101	C₂³×C₄₁	—	4	2	no	—
102	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
104	Q₈×C₄₁	—	4	4	no	—
105	Q₈×C₄₁	—	4	4	no	—
106	C₂×C₄×C₄₁	—	2	2	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	8	no	—
113	C₂×C₄×C₄₁	—	4	2	no	—
114	C₂×C₄×C₄₁	—	4	2	no	—
115	C₂×C₄×C₄₁	—	4	4	no	—
116	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
122	C₈×C₄₁	—	2	8	no	—
123	C₈×C₄₁	—	4	4	no	—
124	#21429	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₁	—	2	2	no	—
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	2	2	no	—
128	C₂²×C₄₁	—	8	2	no	—
129	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₁	C₄×#41	1	2	yes	—
131	C₄×C₄₁	—	2	4	no	—
132	C₄×C₄₁	—	2	4	no	—
133	C₄×C₄₁	—	2	2	no	—
134	C₄×C₄₁	—	4	4	no	—
135	C₄×C₄₁	—	4	4	no	—
136	C₄×C₄₁	#525	1	2	yes	—
137	C₄×C₄₁	#532	1	2	yes	—
138	C₂×#541	C₂×C₄₁	1	1	yes	characteristic
139	#2921	C₂×C₄₁	1	1	yes	characteristic
140	#2873	C₂×C₄₁	1	1	yes	characteristic
141	#1998	C₂×C₄₁	1	2	yes	complemented
142	#1966	C₂×C₄₁	1	2	yes	complemented
143	#1901	C₂×C₄₁	1	4	yes	—
144	C₄×#101	C₂×C₄₁	1	4	yes	complemented
145	C₂×C₄₁	—	2	2	no	—
146	C₂×C₄₁	—	4	2	no	—
147	C₂×C₄₁	#2297	1	1	yes	characteristic
148	C₂×C₄₁	#2889	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
150	#541	—	2	4	no	—
151	#541	—	2	4	no	—
152	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
153	C₂×#126	C₄×C₄₁	1	4	yes	—
154	C₂×#125	C₄×C₄₁	1	4	yes	—
155	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
157	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
158	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
159	#433	—	2	4	no	—
160	#423	C₂²×C₄₁	1	2	yes	—
161	C₂×#101	C₂²×C₄₁	1	8	yes	—
162	#407	C₂²×C₄₁	1	2	yes	—
163	#405	—	2	4	no	—
164	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
165	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
166	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
167	#343	C₂²×C₄₁	1	8	yes	—
168	C₄₁	#21429	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
170	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	8	no	—
173	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
174	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	#126	—	2	16	no	—
179	#125	—	2	16	no	—
180	#125	—	2	2	no	—
181	#125	—	2	2	no	—
182	C₂²⋊Q₈	—	2	4	no	—
183	C₂²⋊Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	4	no	—
186	#117	D₄×C₄₁	1	4	yes	—
187	#117	D₄×C₄₁	1	4	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
197	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
199	C₄⋊C₈	—	2	4	no	—
200	#101	—	2	16	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	#97	—	2	2	no	—
203	#97	—	2	2	no	—
204	#97	—	2	2	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	2	2	no	—
208	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
221	M₁₆	—	2	8	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	8	no	—
224	M₁₆	#41×C₄₁	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
237	C₄²	—	2	8	no	—
238	C₄²	C₂×D₄×C₄₁	1	2	yes	—
239	C₄²	C₂×D₄×C₄₁	1	2	yes	—
240	C₄²	C₂×D₄×C₄₁	1	2	yes	—
241	C₄²	C₂×D₄×C₄₁	1	2	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
245	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
246	Q₈	—	4	4	no	—
247	Q₈	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
261	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	4	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	8	2	no	—
270	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
271	C₂²	C₄×#41×C₄₁	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	4	no	—
277	C₄	C₄₁×#525	1	2	yes	—
278	C₄	C₄₁×#532	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	C₄₁×#2297	1	1	yes	characteristic
282	C₂	C₄₁×#2889	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1998	C₂	semidirect
5. C₄₁×#1966	C₂	semidirect
7. C₄×#101×C₄₁	C₂	semidirect
124. #21429	C₄₁	direct (N × H)
141. #1998	C₂×C₄₁	semidirect
142. #1966	C₂×C₄₁	semidirect
144. C₄×#101	C₂×C₄₁	semidirect
168. C₄₁	#21429	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#541	2
C₄₁×#2921	2
C₄₁×#2873	2
C₄₁×#1998	2
C₄₁×#1966	2
C₄₁×#1901	2
C₄×#101×C₄₁	2
#21429	41

Subgroups — C41×#21429 #23797630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21429 #23797630