Subgroups — C₂×C₄₁×#3255 #23837106

Order 10496.

1866 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#533	C₂	1	2	yes	maximal
2	C₄₁×#3255	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄₁×#475	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₁×#470	C₂	1	4	yes	maximal
5	C₂×C₄₁×#462	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₁×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#533	—	2	8	no	—
10	C₄₁×#533	C₂²	1	16	yes	—
11	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
13	C₂×#125×C₄₁	—	2	2	no	—
14	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
15	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	4	yes	—
16	C₂×C₄×Q₈×C₄₁	—	2	2	no	—
17	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
18	C₂×#117×C₄₁	—	2	2	no	—
19	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#475	C₂²	1	32	yes	complemented
21	C₄₁×#470	C₂²	1	32	yes	—
22	C₄₁×#462	C₂²	1	32	yes	complemented
23	C₂×#106×C₄₁	C₂²	1	4	yes	—
24	C₄₁×#423	C₂²	1	8	yes	—
25	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
26	C₂×#103×C₄₁	C₂²	1	8	yes	—
27	C₂²×Q₈×C₄₁	—	2	2	no	—
28	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
29	C₂³×C₄×C₄₁	D₄	1	2	yes	—
30	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂×Q₁₆×C₄₁	—	4	8	no	—
32	C₂×Q₁₆×C₄₁	C₂³	1	16	yes	—
33	C₂×M₁₆×C₄₁	—	4	4	no	—
34	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
35	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
36	C₄⋊Q₈×C₄₁	—	2	4	no	—
37	C₄⋊Q₈×C₄₁	—	2	4	no	—
38	C₄⋊Q₈×C₄₁	C₂³	1	8	yes	—
39	C₄⋊Q₈×C₄₁	C₂³	1	8	yes	—
40	#125×C₄₁	—	2	16	no	—
41	C₂²⋊Q₈×C₄₁	—	2	16	no	—
42	C₂²⋊Q₈×C₄₁	—	2	8	no	—
43	C₂²⋊Q₈×C₄₁	C₂³	1	16	yes	—
44	C₄×Q₈×C₄₁	—	2	16	no	—
45	C₄×Q₈×C₄₁	—	2	8	no	—
46	C₄×Q₈×C₄₁	C₂³	1	16	yes	—
47	#117×C₄₁	—	2	16	no	—
48	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
52	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
55	C₂×C₄²×C₄₁	—	2	4	no	—
56	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₁	D₄	1	4	yes	—
59	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
61	#106×C₄₁	C₂³	1	16	yes	—
62	C₄⋊C₈×C₄₁	C₂³	1	16	yes	—
63	#103×C₄₁	C₂³	1	32	yes	—
64	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₄₁	—	2	4	no	—
66	C₂×Q₈×C₄₁	—	2	8	no	—
67	C₂×Q₈×C₄₁	—	2	8	no	—
68	C₂×Q₈×C₄₁	—	2	8	no	—
69	C₂×Q₈×C₄₁	—	4	8	no	—
70	C₂×Q₈×C₄₁	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₄₁	—	2	8	no	—
72	C₂²×C₄×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	2	4	no	—
75	C₂²×C₄×C₄₁	—	2	4	no	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
83	Q₁₆×C₄₁	—	4	16	no	—
84	M₁₆×C₄₁	—	4	8	no	—
85	C₂×C₈×C₄₁	—	2	8	no	—
86	C₂×C₈×C₄₁	C₂⁴	1	8	yes	—
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	16	no	—
91	C₄⋊C₄×C₄₁	—	2	16	no	—
92	C₄⋊C₄×C₄₁	—	2	4	no	—
93	C₄⋊C₄×C₄₁	C₂⁴	1	8	yes	—
94	C₂²⋊C₄×C₄₁	—	2	16	no	—
95	C₄²×C₄₁	—	2	16	no	—
96	C₄²×C₄₁	C₂×D₄	1	4	yes	—
97	C₄²×C₄₁	C₂×D₄	1	4	yes	—
98	C₄²×C₄₁	C₂×D₄	1	4	yes	—
99	C₄²×C₄₁	C₂⁴	1	4	yes	—
100	C₂³×C₄₁	—	2	2	no	—
101	C₂³×C₄₁	—	2	4	no	—
102	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
103	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₄₁	—	2	8	no	—
105	Q₈×C₄₁	—	2	8	no	—
106	Q₈×C₄₁	—	4	8	no	—
107	Q₈×C₄₁	—	4	16	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	16	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	8	no	—
115	C₂×C₄×C₄₁	—	2	8	no	—
116	C₂×C₄×C₄₁	—	4	8	no	—
117	C₂×C₄×C₄₁	—	4	8	no	—
118	C₂×C₄×C₄₁	#137	1	8	yes	—
119	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
123	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
124	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
125	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
126	C₈×C₄₁	—	2	8	no	—
127	C₂×#3255	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₄₁	—	2	2	no	—
129	C₂²×C₄₁	—	2	4	no	—
130	C₂²×C₄₁	—	4	4	no	—
131	C₂²×C₄₁	#537	1	2	yes	—
132	C₂²×C₄₁	C₂×#137	1	4	yes	—
133	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₄₁	—	2	2	no	—
135	C₄×C₄₁	—	2	4	no	—
136	C₄×C₄₁	—	4	8	no	—
137	C₄×C₄₁	—	4	8	no	—
138	C₄×C₄₁	#537	1	2	yes	—
139	C₄×C₄₁	C₂×#137	1	8	yes	—
140	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	C₄×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×#533	C₂×C₄₁	1	2	yes	—
143	#3255	C₂×C₄₁	1	16	yes	direct product factor
144	C₂×#475	C₂×C₄₁	1	4	yes	complemented
145	C₂×#470	C₂×C₄₁	1	4	yes	—
146	C₂×#462	C₂×C₄₁	1	4	yes	complemented
147	C₂×#423	C₂×C₄₁	1	1	yes	characteristic
148	C₂×C₄₁	—	4	2	no	—
149	C₂×C₄₁	#3255	1	4	yes	direct product factor
150	C₂×C₄₁	C₂×#537	1	1	yes	characteristic
151	C₂×C₄₁	C₂²×#137	1	2	yes	—
152	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
153	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
154	#533	—	2	8	no	—
155	#533	C₂²×C₄₁	1	16	yes	—
156	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
157	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
158	C₂×#125	—	2	2	no	—
159	C₂×C₂²⋊Q₈	—	2	2	no	—
160	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	4	yes	—
161	C₂×C₄×Q₈	—	2	2	no	—
162	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
163	C₂×#117	—	2	2	no	—
164	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
165	#475	C₂²×C₄₁	1	32	yes	complemented
166	#470	C₂²×C₄₁	1	32	yes	—
167	#462	C₂²×C₄₁	1	32	yes	complemented
168	C₂×#106	C₂²×C₄₁	1	4	yes	—
169	#423	C₂²×C₄₁	1	8	yes	—
170	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
171	C₂×#103	C₂²×C₄₁	1	8	yes	—
172	C₄₁	C₂×#3255	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×Q₈	—	2	2	no	—
174	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
175	C₂³×C₄	D₄×C₄₁	1	2	yes	—
176	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
177	C₂×Q₁₆	—	4	8	no	—
178	C₂×Q₁₆	C₂³×C₄₁	1	16	yes	—
179	C₂×M₁₆	—	4	4	no	—
180	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
181	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	2	4	no	—
184	C₄⋊Q₈	C₂³×C₄₁	1	8	yes	—
185	C₄⋊Q₈	C₂³×C₄₁	1	8	yes	—
186	#125	—	2	16	no	—
187	C₂²⋊Q₈	—	2	16	no	—
188	C₂²⋊Q₈	—	2	8	no	—
189	C₂²⋊Q₈	C₂³×C₄₁	1	16	yes	—
190	C₄×Q₈	—	2	16	no	—
191	C₄×Q₈	—	2	8	no	—
192	C₄×Q₈	C₂³×C₄₁	1	16	yes	—
193	#117	—	2	16	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₄₁	1	4	yes	—
205	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
206	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
207	#106	C₂³×C₄₁	1	16	yes	—
208	C₄⋊C₈	C₂³×C₄₁	1	16	yes	—
209	#103	C₂³×C₄₁	1	32	yes	—
210	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	2	8	no	—
213	C₂×Q₈	—	2	8	no	—
214	C₂×Q₈	—	2	8	no	—
215	C₂×Q₈	—	4	8	no	—
216	C₂×Q₈	C₂⁴×C₄₁	1	8	yes	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
226	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
229	Q₁₆	—	4	16	no	—
230	M₁₆	—	4	8	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	C₂⁴×C₄₁	1	8	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	C₂⁴×C₄₁	1	8	yes	—
240	C₂²⋊C₄	—	2	16	no	—
241	C₄²	—	2	16	no	—
242	C₄²	C₂×D₄×C₄₁	1	4	yes	—
243	C₄²	C₂×D₄×C₄₁	1	4	yes	—
244	C₄²	C₂×D₄×C₄₁	1	4	yes	—
245	C₄²	C₂⁴×C₄₁	1	4	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	4	no	—
248	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
249	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
250	Q₈	—	2	8	no	—
251	Q₈	—	2	8	no	—
252	Q₈	—	4	8	no	—
253	Q₈	—	4	16	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	#137×C₄₁	1	8	yes	—
265	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
266	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
267	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
269	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
270	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
271	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₄₁×#537	1	2	yes	—
277	C₂²	C₂×#137×C₄₁	1	4	yes	—
278	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	8	no	—
282	C₄	—	4	8	no	—
283	C₄	C₄₁×#537	1	2	yes	—
284	C₄	C₂×#137×C₄₁	1	8	yes	—
285	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
286	C₄	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
287	C₂	—	4	2	no	—
288	C₂	C₄₁×#3255	1	4	yes	direct product factor
289	C₂	C₂×C₄₁×#537	1	1	yes	characteristic
290	C₂	C₂²×#137×C₄₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#3255	C₂	direct (N × H)
3. C₂×C₄₁×#475	C₂	semidirect
5. C₂×C₄₁×#462	C₂	semidirect
20. C₄₁×#475	C₂²	semidirect
22. C₄₁×#462	C₂²	semidirect
127. C₂×#3255	C₄₁	direct (N × H)
143. #3255	C₂×C₄₁	direct (N × H)
144. C₂×#475	C₂×C₄₁	semidirect
146. C₂×#462	C₂×C₄₁	semidirect
149. C₂×C₄₁	#3255	direct (N × H)
165. #475	C₂²×C₄₁	semidirect
167. #462	C₂²×C₄₁	semidirect
172. C₄₁	C₂×#3255	direct (N × H)
288. C₂	C₄₁×#3255	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×#533	2
C₄₁×#3255	2
C₂×C₄₁×#475	2
C₂×C₄₁×#470	2
C₂×C₄₁×#462	2
C₂×C₄₁×#423	2
C₂×#3255	41

Subgroups — C2×C41×#3255 #23837106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3255 #23837106