Subgroups — C₂²×#10622820 #15617204

Order 2816.

7214 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#4318	C₂	1	2	yes	maximal, complemented
2	C₂×#10622820	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂²⋊(C₁₁⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂⁵×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁶×C₁₁	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₁₁	—	2	16	no	—
7	C₂⁴×C₄×C₁₁	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₁	C₄	1	12	yes	—
9	C₂⁴×C₄×C₁₁	C₄	1	2	yes	—
10	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₁₁	C₂²	1	12	yes	—
12	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²×#4318	—	2	16	no	—
14	C₂²×#4318	C₂²	1	24	yes	complemented
15	C₂³×C₁₁⋊C₈	C₂²	1	4	yes	complemented
16	#10622820	C₂²	1	64	yes	direct product factor
17	C₂×C₂²⋊(C₁₁⋊C₈)	C₂²	1	48	yes	complemented
18	C₂⁵×C₁₁	—	2	8	no	—
19	C₂⁵×C₁₁	—	2	16	no	—
20	C₂⁵×C₁₁	C₂×C₄	1	12	yes	—
21	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂⁵×C₁₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₁	—	2	8	no	—
24	C₂³×C₄×C₁₁	—	2	96	no	—
25	C₂³×C₄×C₁₁	—	2	16	no	—
26	C₂³×C₄×C₁₁	—	2	16	no	—
27	C₂³×C₄×C₁₁	—	2	96	no	—
28	C₂³×C₄×C₁₁	C₂×C₄	1	12	yes	—
29	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₁	C₂×C₄	1	48	yes	—
31	C₂³×C₄×C₁₁	C₂×C₄	1	12	yes	—
32	C₂³×C₄×C₁₁	C₂×C₄	1	24	yes	—
33	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
34	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₁	D₄	1	16	yes	—
36	C₂³×C₄×C₁₁	C₂³	1	12	yes	—
37	C₂³×C₄×C₁₁	C₂³	1	16	yes	—
38	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
39	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
40	C₂×#4318	—	2	96	no	—
41	C₂×#4318	C₂³	1	32	yes	complemented
42	C₂²×C₁₁⋊C₈	—	2	16	no	—
43	C₂²×C₁₁⋊C₈	C₂³	1	24	yes	complemented
44	C₂²⋊(C₁₁⋊C₈)	C₂³	1	64	yes	complemented
45	C₂²×#407	—	11	1	no	maximal, Sylow
46	C₂⁴×C₁₁	—	2	128	no	—
47	C₂⁴×C₁₁	—	2	48	no	—
48	C₂⁴×C₁₁	—	2	8	no	—
49	C₂⁴×C₁₁	—	2	8	no	—
50	C₂⁴×C₁₁	—	2	8	no	—
51	C₂⁴×C₁₁	—	2	96	no	—
52	C₂⁴×C₁₁	C₂²⋊C₄	1	8	yes	—
53	C₂⁴×C₁₁	M₁₆	1	16	yes	complemented
54	C₂⁴×C₁₁	C₂²×C₄	1	16	yes	—
55	C₂⁴×C₁₁	C₂²×C₄	1	6	yes	—
56	C₂⁴×C₁₁	C₂²×C₄	1	12	yes	—
57	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	—	2	48	no	—
59	C₂²×C₄×C₁₁	—	2	8	no	—
60	C₂²×C₄×C₁₁	—	2	192	no	—
61	C₂²×C₄×C₁₁	—	2	48	no	—
62	C₂²×C₄×C₁₁	—	2	96	no	—
63	C₂²×C₄×C₁₁	—	2	48	no	—
64	C₂²×C₄×C₁₁	—	2	64	no	—
65	C₂²×C₄×C₁₁	C₂²⋊C₄	1	48	yes	—
66	C₂²×C₄×C₁₁	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₁₁	C₂²×C₄	1	16	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	32	yes	—
71	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
73	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₁₁	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₁₁	C₂×D₄	1	48	yes	—
76	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
77	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
78	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
79	#4318	—	2	64	no	—
80	C₂×C₁₁⋊C₈	—	2	48	no	—
81	C₂×C₁₁⋊C₈	C₂⁴	1	16	yes	complemented
82	C₂⁵×C₄	D₁₁	1	1	yes	characteristic
83	C₂³×M₁₆	—	11	2	no	—
84	C₂×#407	—	11	24	no	—
85	C₂²×C₂²⋊C₈	—	11	4	no	—
86	C₂³×C₁₁	—	2	384	no	—
87	C₂³×C₁₁	—	2	32	no	—
88	C₂³×C₁₁	—	2	24	no	—
89	C₂³×C₁₁	—	2	24	no	—
90	C₂³×C₁₁	—	2	24	no	—
91	C₂³×C₁₁	—	2	64	no	—
92	C₂³×C₁₁	—	2	32	no	—
93	C₂³×C₁₁	—	2	64	no	—
94	C₂³×C₁₁	C₂×C₂²⋊C₄	1	24	yes	—
95	C₂³×C₁₁	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂³×C₁₁	C₂×M₁₆	1	8	yes	—
97	C₂³×C₁₁	C₂×M₁₆	1	48	yes	complemented
98	C₂³×C₁₁	C₂³×C₄	1	4	yes	—
99	C₂³×C₁₁	C₂³×C₄	1	8	yes	—
100	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
101	C₂×C₄×C₁₁	—	2	32	no	—
102	C₂×C₄×C₁₁	—	2	24	no	—
103	C₂×C₄×C₁₁	—	2	48	no	—
104	C₂×C₄×C₁₁	—	2	64	no	—
105	C₂×C₄×C₁₁	—	2	16	no	—
106	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	48	yes	—
107	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	12	yes	—
108	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	24	yes	—
109	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
111	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
112	C₂×C₄×C₁₁	C₂²×D₄	1	12	yes	—
113	C₂×C₄×C₁₁	C₂²×D₄	1	16	yes	—
114	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
115	C₁₁⋊C₈	—	2	16	no	—
116	C₂⁶	Dic₁₁	1	1	yes	characteristic
117	C₂⁴×C₄	—	2	16	no	—
118	C₂⁴×C₄	Dic₁₁	1	1	yes	characteristic
119	C₂⁴×C₄	Dic₁₁	1	12	yes	—
120	C₂⁴×C₄	Dic₁₁	1	2	yes	—
121	C₂⁴×C₄	C₂×D₁₁	1	12	yes	—
122	C₂⁴×C₄	C₂×D₁₁	1	2	yes	—
123	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
124	C₂²×M₁₆	—	11	24	no	—
125	C₂²×M₁₆	—	22	16	no	—
126	C₂³×C₈	—	11	4	no	—
127	#407	—	11	64	no	—
128	C₂×C₂²⋊C₈	—	11	48	no	—
129	C₂²×C₁₁	—	2	128	no	—
130	C₂²×C₁₁	—	2	48	no	—
131	C₂²×C₁₁	—	2	96	no	—
132	C₂²×C₁₁	—	2	8	no	—
133	C₂²×C₁₁	—	2	8	no	—
134	C₂²×C₁₁	—	2	8	no	—
135	C₂²×C₁₁	#407	1	16	yes	complemented
136	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	8	yes	—
137	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	6	yes	—
138	C₂²×C₁₁	C₂²×M₁₆	1	12	yes	—
139	C₂²×C₁₁	C₂²×M₁₆	1	16	yes	complemented
140	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
141	C₄×C₁₁	—	2	8	no	—
142	C₄×C₁₁	C₂²×C₂²⋊C₄	1	6	yes	—
143	C₄×C₁₁	C₂²×C₂²⋊C₄	1	8	yes	—
144	C₄×C₁₁	C₂³×D₄	1	2	yes	—
145	C₂⁵	—	2	16	no	—
146	C₂⁵	—	2	8	no	—
147	C₂⁵	C₂×Dic₁₁	1	12	yes	—
148	C₂⁵	C₂×Dic₁₁	1	2	yes	—
149	C₂⁵	C₂×Dic₁₁	1	1	yes	characteristic
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	2	96	no	—
152	C₂³×C₄	—	2	16	no	—
153	C₂³×C₄	—	2	16	no	—
154	C₂³×C₄	—	2	96	no	—
155	C₂³×C₄	C₂×Dic₁₁	1	12	yes	—
156	C₂³×C₄	C₂×Dic₁₁	1	2	yes	—
157	C₂³×C₄	C₂×Dic₁₁	1	48	yes	—
158	C₂³×C₄	C₂×Dic₁₁	1	24	yes	—
159	C₂³×C₄	C₂×Dic₁₁	1	12	yes	—
160	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₁₁	1	6	yes	—
162	C₂³×C₄	C₂²⋊D₁₁	1	16	yes	—
163	C₂³×C₄	C₂²×D₁₁	1	12	yes	—
164	C₂³×C₄	C₂²×D₁₁	1	16	yes	—
165	C₂³×C₄	C₂²×D₁₁	1	6	yes	—
166	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
167	C₂×M₁₆	—	11	32	no	—
168	C₂×M₁₆	—	22	96	no	—
169	C₂²×C₈	—	11	24	no	—
170	C₂²×C₈	—	22	16	no	—
171	C₂²⋊C₈	—	11	64	no	—
172	C₂×C₁₁	—	2	8	no	—
173	C₂×C₁₁	—	2	16	no	—
174	C₂×C₁₁	C₂×#407	1	12	yes	complemented
175	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₁₁	C₂³×M₁₆	1	2	yes	—
177	C₂⁴	—	2	128	no	—
178	C₂⁴	—	2	8	no	—
179	C₂⁴	—	2	8	no	—
180	C₂⁴	—	2	96	no	—
181	C₂⁴	—	2	48	no	—
182	C₂⁴	—	2	8	no	—
183	C₂⁴	#4318	1	16	yes	complemented
184	C₂⁴	C₂²⋊Dic₁₁	1	8	yes	—
185	C₂⁴	C₂²×Dic₁₁	1	16	yes	—
186	C₂⁴	C₂²×Dic₁₁	1	12	yes	—
187	C₂⁴	C₂²×Dic₁₁	1	6	yes	—
188	C₂⁴	C₂²×Dic₁₁	1	1	yes	characteristic
189	C₂²×C₄	—	2	48	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	192	no	—
192	C₂²×C₄	—	2	96	no	—
193	C₂²×C₄	—	2	48	no	—
194	C₂²×C₄	—	2	48	no	—
195	C₂²×C₄	—	2	64	no	—
196	C₂²×C₄	C₂²⋊Dic₁₁	1	48	yes	—
197	C₂²×C₄	C₂²⋊Dic₁₁	1	8	yes	—
198	C₂²×C₄	C₂²×Dic₁₁	1	16	yes	—
199	C₂²×C₄	C₂²×Dic₁₁	1	12	yes	—
200	C₂²×C₄	C₂²×Dic₁₁	1	24	yes	—
201	C₂²×C₄	C₂²×Dic₁₁	1	32	yes	—
202	C₂²×C₄	C₂²×Dic₁₁	1	12	yes	—
203	C₂²×C₄	C₂²×Dic₁₁	1	6	yes	—
204	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₁₁	1	8	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₁₁	1	48	yes	—
207	C₂²×C₄	C₂³×D₁₁	1	8	yes	—
208	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
209	C₂²×C₄	C₂³×D₁₁	1	4	yes	—
210	M₁₆	—	22	64	no	—
211	C₂×C₈	—	11	16	no	—
212	C₂×C₈	—	22	48	no	—
213	C₁₁	C₂²×#407	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	2	384	no	—
215	C₂³	—	2	24	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	64	no	—
218	C₂³	—	2	64	no	—
219	C₂³	—	2	32	no	—
220	C₂³	—	2	32	no	—
221	C₂³	—	2	24	no	—
222	C₂³	C₂×#4318	1	8	yes	—
223	C₂³	C₂×#4318	1	48	yes	complemented
224	C₂³	C₂×C₂²⋊Dic₁₁	1	24	yes	—
225	C₂³	C₂×C₂²⋊Dic₁₁	1	4	yes	—
226	C₂³	C₂³×Dic₁₁	1	8	yes	—
227	C₂³	C₂³×Dic₁₁	1	4	yes	—
228	C₂³	C₂³×Dic₁₁	1	3	yes	—
229	C₂×C₄	—	2	32	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	2	48	no	—
232	C₂×C₄	—	2	64	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	48	yes	—
235	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	24	yes	—
236	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	12	yes	—
237	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
238	C₂×C₄	C₂³×Dic₁₁	1	4	yes	—
239	C₂×C₄	C₂³×Dic₁₁	1	3	yes	—
240	C₂×C₄	C₂²×C₂²⋊D₁₁	1	12	yes	—
241	C₂×C₄	C₂²×C₂²⋊D₁₁	1	16	yes	—
242	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
243	C₈	—	22	16	no	—
244	C₂²	—	2	128	no	—
245	C₂²	—	2	96	no	—
246	C₂²	—	2	48	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	2	8	no	—
250	C₂²	#10622820	1	16	yes	direct product factor
251	C₂²	C₂²×#4318	1	12	yes	—
252	C₂²	C₂²×#4318	1	16	yes	complemented
253	C₂²	C₂²×C₂²⋊Dic₁₁	1	8	yes	—
254	C₂²	C₂²×C₂²⋊Dic₁₁	1	6	yes	—
255	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
256	C₄	—	2	8	no	—
257	C₄	C₂²×C₂²⋊Dic₁₁	1	6	yes	—
258	C₄	C₂²×C₂²⋊Dic₁₁	1	8	yes	—
259	C₄	C₂³×C₂²⋊D₁₁	1	2	yes	—
260	C₂	—	2	16	no	—
261	C₂	—	2	8	no	—
262	C₂	C₂×#10622820	1	12	yes	direct product factor
263	C₂	C₂³×C₂²⋊Dic₁₁	1	1	yes	characteristic
264	C₂	C₂³×#4318	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂⁵×C₄×C₁₁	C₂
socle	C₂⁴×C₁₁	C₂²×C₄
prime radical	C₂⁶×C₁₁	C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂⁵×C₄	D₁₁
hypocenter	C₁₁	C₂²×#407
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#4318	C₂	semidirect
2. C₂×#10622820	C₂	direct (N × H)
3. C₂²×C₂²⋊(C₁₁⋊C₈)	C₂	semidirect
14. C₂²×#4318	C₂²	semidirect
15. C₂³×C₁₁⋊C₈	C₂²	semidirect
16. #10622820	C₂²	direct (N × H)
17. C₂×C₂²⋊(C₁₁⋊C₈)	C₂²	semidirect
41. C₂×#4318	C₂³	semidirect
43. C₂²×C₁₁⋊C₈	C₂³	semidirect
44. C₂²⋊(C₁₁⋊C₈)	C₂³	semidirect
53. C₂⁴×C₁₁	M₁₆	semidirect
81. C₂×C₁₁⋊C₈	C₂⁴	semidirect
97. C₂³×C₁₁	C₂×M₁₆	semidirect
135. C₂²×C₁₁	#407	semidirect
139. C₂²×C₁₁	C₂²×M₁₆	semidirect
174. C₂×C₁₁	C₂×#407	semidirect
183. C₂⁴	#4318	semidirect
213. C₁₁	C₂²×#407	semidirect
223. C₂³	C₂×#4318	semidirect
250. C₂²	#10622820	direct (N × H)
252. C₂²	C₂²×#4318	semidirect
262. C₂	C₂×#10622820	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#4318	2
C₂×#10622820	2
C₂²×C₂²⋊(C₁₁⋊C₈)	2
C₂⁵×C₄×C₁₁	2
C₂²×#407	11

Subgroups — C22×#10622820 #15617204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622820 #15617204