Subgroups — C₂²×#10623014 #15660495

Order 2816.

9310 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10623014	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#66977	C₂	1	4	yes	maximal, complemented
3	C₂²×#66931	C₂	1	8	yes	maximal, complemented
4	C₂³×C₂²⋊Dic₁₁	C₂	1	2	yes	maximal
5	C₂³×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
8	C₂⁴×Dic₁₁	C₂²	1	4	yes	—
9	C₂³×Dic₂₂	C₂²	1	4	yes	complemented
10	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	24	yes	—
11	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₁₁	C₂²	1	48	yes	—
13	C₂²×C₂²⋊Dic₁₁	C₂²	1	4	yes	—
14	C₂²×C₂²⋊Dic₁₁	C₂²	1	4	yes	—
15	C₂²×C₄⋊Dic₁₁	C₂²	1	4	yes	complemented
16	C₂×#66977	C₂²	1	96	yes	complemented
17	C₂²×Dic₁₁⋊C₄	C₂²	1	8	yes	complemented
18	#10623014	C₂²	1	256	yes	direct product factor
19	C₂×#66931	C₂²	1	192	yes	complemented
20	C₂⁵×C₁₁	—	4	8	no	—
21	C₂⁵×C₁₁	Q₈	1	16	yes	complemented
22	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
23	C₂⁵×C₁₁	C₂³	1	2	yes	—
24	C₂⁵×C₁₁	C₂³	1	12	yes	—
25	C₂³×C₄×C₁₁	—	4	8	no	—
26	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₁	C₂³	1	24	yes	—
28	C₂³×Dic₁₁	—	4	16	no	—
29	C₂³×Dic₁₁	C₂³	1	8	yes	—
30	C₂³×Dic₁₁	C₂³	1	48	yes	—
31	C₂²×Dic₂₂	—	4	16	no	—
32	C₂²×Dic₂₂	C₂³	1	48	yes	complemented
33	C₂×C₁₁×C₂²⋊C₄	C₂³	1	48	yes	—
34	C₂×C₁₁×C₂²⋊C₄	C₂³	1	64	yes	—
35	C₂×C₂²⋊Dic₁₁	C₂³	1	48	yes	—
36	C₂×C₂²⋊Dic₁₁	C₂³	1	128	yes	—
37	C₂×C₂²⋊Dic₁₁	C₂³	1	48	yes	—
38	C₂×C₄⋊Dic₁₁	C₂³	1	48	yes	complemented
39	#66977	C₂³	1	256	yes	complemented
40	C₂×Dic₁₁⋊C₄	C₂³	1	96	yes	complemented
41	#66931	C₂³	1	512	yes	complemented
42	C₂²×#543	—	11	1	no	maximal, Sylow
43	C₂⁴×C₁₁	—	2	16	no	—
44	C₂⁴×C₁₁	—	2	8	no	—
45	C₂⁴×C₁₁	—	4	64	no	—
46	C₂⁴×C₁₁	—	4	48	no	—
47	C₂⁴×C₁₁	C₂×Q₈	1	8	yes	—
48	C₂⁴×C₁₁	C₂×Q₈	1	16	yes	—
49	C₂⁴×C₁₁	C₂×Q₈	1	96	yes	complemented
50	C₂⁴×C₁₁	C₂⁴	1	6	yes	—
51	C₂⁴×C₁₁	C₂⁴	1	12	yes	—
52	C₂⁴×C₁₁	C₂⁴	1	16	yes	—
53	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	—	2	16	no	—
55	C₂²×C₄×C₁₁	—	4	48	no	—
56	C₂²×C₄×C₁₁	C₂⁴	1	24	yes	—
57	C₂²×C₄×C₁₁	C₂⁴	1	32	yes	—
58	C₂²×Dic₁₁	—	2	32	no	—
59	C₂²×Dic₁₁	—	4	96	no	—
60	C₂²×Dic₁₁	C₂⁴	1	48	yes	—
61	C₂²×Dic₁₁	C₂⁴	1	64	yes	—
62	C₂×Dic₂₂	—	4	96	no	—
63	C₂×Dic₂₂	C₂⁴	1	64	yes	complemented
64	C₁₁×C₂²⋊C₄	C₂⁴	1	64	yes	—
65	C₂²⋊Dic₁₁	C₂⁴	1	64	yes	—
66	C₂²⋊Dic₁₁	C₂⁴	1	64	yes	—
67	C₄⋊Dic₁₁	C₂⁴	1	64	yes	complemented
68	Dic₁₁⋊C₄	C₂⁴	1	128	yes	complemented
69	C₂×#543	—	11	48	no	—
70	C₂²×C₂²⋊Q₈	—	11	4	no	—
71	C₂²×C₂²⋊Q₈	—	11	8	no	—
72	C₂³×C₂²⋊C₄	—	11	2	no	—
73	C₂³×C₂²⋊C₄	D₁₁	1	1	yes	characteristic
74	C₂³×C₁₁	—	2	64	no	—
75	C₂³×C₁₁	—	2	32	no	—
76	C₂³×C₁₁	—	2	48	no	—
77	C₂³×C₁₁	—	2	24	no	—
78	C₂³×C₁₁	—	4	192	no	—
79	C₂³×C₁₁	—	4	32	no	—
80	C₂³×C₁₁	C₂²×Q₈	1	24	yes	—
81	C₂³×C₁₁	C₂²×Q₈	1	48	yes	—
82	C₂³×C₁₁	C₂²×Q₈	1	4	yes	—
83	C₂³×C₁₁	C₂²×Q₈	1	64	yes	complemented
84	C₂³×C₁₁	H(5,𝔽₂)	1	8	yes	—
85	C₂³×C₁₁	C₂⁵	1	4	yes	—
86	C₂³×C₁₁	C₂⁵	1	8	yes	—
87	C₂³×C₁₁	C₂⁵	1	3	yes	—
88	C₂×C₄×C₁₁	—	2	48	no	—
89	C₂×C₄×C₁₁	—	4	32	no	—
90	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
91	C₂×Dic₁₁	—	2	96	no	—
92	C₂×Dic₁₁	—	4	64	no	—
93	C₂×Dic₁₁	C₂⁵	1	32	yes	—
94	Dic₂₂	—	4	64	no	—
95	C₂⁶	C₂×D₁₁	1	1	yes	characteristic
96	C₂³×Q₈	—	11	4	no	—
97	C₂⁴×C₄	—	11	4	no	—
98	C₂⁴×C₄	C₂×D₁₁	1	2	yes	—
99	#543	—	11	256	no	—
100	C₂×C₂²⋊Q₈	—	11	96	no	—
101	C₂×C₂²⋊Q₈	—	11	192	no	—
102	C₂²×C₄⋊C₄	—	11	8	no	—
103	C₂²×C₄⋊C₄	—	11	4	no	—
104	C₂²×C₂²⋊C₄	—	11	4	no	—
105	C₂²×C₂²⋊C₄	—	11	4	no	—
106	C₂²×C₂²⋊C₄	—	11	48	no	—
107	C₂²×C₂²⋊C₄	C₂×D₁₁	1	4	yes	—
108	C₂²×C₂²⋊C₄	C₂×D₁₁	1	24	yes	—
109	C₂²×C₁₁	—	2	8	no	—
110	C₂²×C₁₁	—	2	16	no	—
111	C₂²×C₁₁	—	2	96	no	—
112	C₂²×C₁₁	—	2	48	no	—
113	C₂²×C₁₁	—	4	64	no	—
114	C₂²×C₁₁	#543	1	16	yes	complemented
115	C₂²×C₁₁	C₂³×Q₈	1	8	yes	—
116	C₂²×C₁₁	C₂³×Q₈	1	16	yes	—
117	C₂²×C₁₁	C₂³×Q₈	1	6	yes	—
118	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	12	yes	—
119	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
120	C₄×C₁₁	—	2	16	no	—
121	Dic₁₁	—	2	32	no	—
122	C₂⁵	—	4	8	no	—
123	C₂⁵	Dic₂₂	1	16	yes	complemented
124	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
125	C₂⁵	C₂²×D₁₁	1	2	yes	—
126	C₂⁵	C₂²×D₁₁	1	12	yes	—
127	C₂²×Q₈	—	11	48	no	—
128	C₂²×Q₈	—	44	16	no	—
129	C₂³×C₄	—	4	8	no	—
130	C₂³×C₄	—	11	8	no	—
131	C₂³×C₄	—	11	48	no	—
132	C₂³×C₄	—	44	16	no	—
133	C₂³×C₄	C₂²×D₁₁	1	4	yes	—
134	C₂³×C₄	C₂²×D₁₁	1	24	yes	—
135	C₂²⋊Q₈	—	11	256	no	—
136	C₂²⋊Q₈	—	11	512	no	—
137	C₂×C₄⋊C₄	—	11	96	no	—
138	C₂×C₄⋊C₄	—	11	48	no	—
139	C₂×C₂²⋊C₄	—	11	48	no	—
140	C₂×C₂²⋊C₄	—	11	48	no	—
141	C₂×C₂²⋊C₄	—	11	128	no	—
142	C₂×C₂²⋊C₄	C₂²×D₁₁	1	48	yes	—
143	C₂×C₂²⋊C₄	C₂²×D₁₁	1	64	yes	—
144	C₂×C₁₁	—	2	8	no	—
145	C₂×C₁₁	—	2	16	no	—
146	C₂×C₁₁	C₂×#543	1	12	yes	complemented
147	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
148	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	2	yes	—
149	C₂⁴	—	2	8	no	—
150	C₂⁴	—	2	16	no	—
151	C₂⁴	—	4	48	no	—
152	C₂⁴	—	4	64	no	—
153	C₂⁴	C₂×Dic₂₂	1	8	yes	—
154	C₂⁴	C₂×Dic₂₂	1	16	yes	—
155	C₂⁴	C₂×Dic₂₂	1	96	yes	complemented
156	C₂⁴	C₂³×D₁₁	1	6	yes	—
157	C₂⁴	C₂³×D₁₁	1	12	yes	—
158	C₂⁴	C₂³×D₁₁	1	16	yes	—
159	C₂⁴	C₂³×D₁₁	1	1	yes	characteristic
160	C₂×Q₈	—	11	64	no	—
161	C₂×Q₈	—	44	96	no	—
162	C₂²×C₄	—	2	16	no	—
163	C₂²×C₄	—	4	48	no	—
164	C₂²×C₄	—	11	48	no	—
165	C₂²×C₄	—	11	64	no	—
166	C₂²×C₄	—	22	32	no	—
167	C₂²×C₄	—	44	96	no	—
168	C₂²×C₄	C₂³×D₁₁	1	24	yes	—
169	C₂²×C₄	C₂³×D₁₁	1	32	yes	—
170	C₄⋊C₄	—	11	128	no	—
171	C₄⋊C₄	—	11	64	no	—
172	C₂²⋊C₄	—	11	64	no	—
173	C₂²⋊C₄	—	11	64	no	—
174	C₂²⋊C₄	C₂³×D₁₁	1	64	yes	—
175	C₁₁	C₂²×#543	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	32	no	—
177	C₂³	—	2	64	no	—
178	C₂³	—	2	24	no	—
179	C₂³	—	2	48	no	—
180	C₂³	—	4	192	no	—
181	C₂³	—	4	32	no	—
182	C₂³	C₂²×Dic₂₂	1	24	yes	—
183	C₂³	C₂²×Dic₂₂	1	48	yes	—
184	C₂³	C₂²×Dic₂₂	1	4	yes	—
185	C₂³	C₂²×Dic₂₂	1	64	yes	complemented
186	C₂³	#67037	1	8	yes	—
187	C₂³	C₂⁴×D₁₁	1	4	yes	—
188	C₂³	C₂⁴×D₁₁	1	8	yes	—
189	C₂³	C₂⁴×D₁₁	1	3	yes	—
190	Q₈	—	44	64	no	—
191	C₂×C₄	—	2	48	no	—
192	C₂×C₄	—	4	32	no	—
193	C₂×C₄	—	11	32	no	—
194	C₂×C₄	—	22	96	no	—
195	C₂×C₄	—	44	64	no	—
196	C₂×C₄	C₂⁴×D₁₁	1	16	yes	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	16	no	—
199	C₂²	—	2	48	no	—
200	C₂²	—	2	96	no	—
201	C₂²	—	4	64	no	—
202	C₂²	#10623014	1	16	yes	direct product factor
203	C₂²	C₂³×Dic₂₂	1	8	yes	—
204	C₂²	C₂³×Dic₂₂	1	16	yes	—
205	C₂²	C₂³×Dic₂₂	1	6	yes	—
206	C₂²	C₂×#67037	1	12	yes	—
207	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
208	C₄	—	2	16	no	—
209	C₄	—	22	32	no	—
210	C₂	—	2	8	no	—
211	C₂	—	2	16	no	—
212	C₂	C₂×#10623014	1	12	yes	direct product factor
213	C₂	C₂⁴×Dic₂₂	1	1	yes	characteristic
214	C₂	C₂²×#67037	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂³×C₁₁×C₂²⋊C₄	C₂
socle	C₂⁴×C₁₁	C₂⁴
prime radical	C₂⁶×C₁₁	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	C₂²×#543
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10623014	C₂	direct (N × H)
2. C₂²×#66977	C₂	semidirect
3. C₂²×#66931	C₂	semidirect
9. C₂³×Dic₂₂	C₂²	semidirect
15. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
16. C₂×#66977	C₂²	semidirect
17. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
18. #10623014	C₂²	direct (N × H)
19. C₂×#66931	C₂²	semidirect
21. C₂⁵×C₁₁	Q₈	semidirect
32. C₂²×Dic₂₂	C₂³	semidirect
38. C₂×C₄⋊Dic₁₁	C₂³	semidirect
39. #66977	C₂³	semidirect
40. C₂×Dic₁₁⋊C₄	C₂³	semidirect
41. #66931	C₂³	semidirect
49. C₂⁴×C₁₁	C₂×Q₈	semidirect
63. C₂×Dic₂₂	C₂⁴	semidirect
67. C₄⋊Dic₁₁	C₂⁴	semidirect
68. Dic₁₁⋊C₄	C₂⁴	semidirect
83. C₂³×C₁₁	C₂²×Q₈	semidirect
114. C₂²×C₁₁	#543	semidirect
123. C₂⁵	Dic₂₂	semidirect
146. C₂×C₁₁	C₂×#543	semidirect
155. C₂⁴	C₂×Dic₂₂	semidirect
175. C₁₁	C₂²×#543	semidirect
185. C₂³	C₂²×Dic₂₂	semidirect
202. C₂²	#10623014	direct (N × H)
212. C₂	C₂×#10623014	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×#10623014	2
C₂²×#66977	2
C₂²×#66931	2
C₂³×C₂²⋊Dic₁₁	2
C₂³×C₁₁×C₂²⋊C₄	2
C₂²×#543	11

Subgroups — C22×#10623014 #15660495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10623014 #15660495