Subgroups — C₂²×#10623101 #15660819

Order 2816.

7414 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10623101	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#66922	C₂	1	3	yes	maximal
3	C₂²×#66943	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₁₁	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₂₂	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₂	C₂²	1	3	yes	—
9	C₂×C₄×Q₈×C₁₁	C₂²	1	24	yes	—
10	C₂²×C₁₁×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₁	C₂²	1	3	yes	—
12	C₂×Q₈×Dic₁₁	C₂²	1	48	yes	—
13	C₂²×C₄⋊Dic₁₁	C₂²	1	6	yes	—
14	C₂²×C₄⋊Dic₁₁	C₂²	1	6	yes	—
15	C₂²×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₁⋊C₄	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₁₁	C₂²	1	6	yes	—
18	#10623101	C₂²	1	256	yes	direct product factor
19	C₂×#66943	C₂²	1	144	yes	—
20	C₂×#66922	C₂²	1	72	yes	—
21	C₂×C₄×Dic₂₂	C₂²	1	72	yes	—
22	C₂²×Q₈×C₁₁	Q₈	1	16	yes	complemented
23	C₂²×Q₈×C₁₁	C₂³	1	12	yes	—
24	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
25	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
26	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₁	C₂³	1	2	yes	—
28	C₂³×Dic₁₁	C₂³	1	6	yes	—
29	C₂²×Dic₂₂	—	2	24	no	—
30	C₂²×Dic₂₂	C₂³	1	36	yes	—
31	C₄×Q₈×C₁₁	C₂³	1	64	yes	—
32	C₂×C₁₁×C₄⋊C₄	C₂³	1	36	yes	—
33	C₂×C₄²×C₁₁	C₂³	1	36	yes	—
34	Q₈×Dic₁₁	C₂³	1	128	yes	—
35	C₂×C₄⋊Dic₁₁	C₂³	1	12	yes	—
36	C₂×C₄⋊Dic₁₁	C₂³	1	72	yes	—
37	C₂×C₄⋊Dic₁₁	C₂³	1	72	yes	—
38	C₂×Dic₁₁⋊C₄	C₂³	1	72	yes	—
39	C₂×C₄×Dic₁₁	C₂³	1	72	yes	—
40	#66943	C₂³	1	384	yes	—
41	#66922	C₂³	1	192	yes	—
42	C₄×Dic₂₂	C₂³	1	192	yes	—
43	C₂²×#557	—	11	1	no	maximal, Sylow
44	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₁₁	C₂×Q₈	1	96	yes	complemented
46	C₂×Q₈×C₁₁	C₂⁴	1	16	yes	—
47	C₂²×C₄×C₁₁	—	2	12	no	—
48	C₂²×C₄×C₁₁	C₂×Q₈	1	24	yes	—
49	C₂²×C₄×C₁₁	#41	1	8	yes	—
50	C₂²×C₄×C₁₁	C₂⁴	1	18	yes	—
51	C₂²×C₄×C₁₁	C₂⁴	1	18	yes	—
52	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
53	C₂²×Dic₁₁	—	2	24	no	—
54	C₂²×Dic₁₁	—	2	8	no	—
55	C₂²×Dic₁₁	C₂⁴	1	12	yes	—
56	C₂²×Dic₁₁	C₂⁴	1	36	yes	—
57	C₂×Dic₂₂	—	2	144	no	—
58	C₂×Dic₂₂	C₂⁴	1	48	yes	—
59	C₁₁×C₄⋊C₄	C₂⁴	1	48	yes	—
60	C₄²×C₁₁	C₂⁴	1	48	yes	—
61	C₄⋊Dic₁₁	C₂⁴	1	16	yes	—
62	C₄⋊Dic₁₁	C₂⁴	1	96	yes	—
63	C₄⋊Dic₁₁	C₂⁴	1	96	yes	—
64	Dic₁₁⋊C₄	C₂⁴	1	96	yes	—
65	C₄×Dic₁₁	C₂⁴	1	96	yes	—
66	C₂×#557	—	11	48	no	—
67	C₂²×C₄⋊Q₈	—	11	3	no	—
68	C₂²×#125	—	11	6	no	—
69	C₂²×C₄×Q₈	—	11	3	no	—
70	C₂²×C₄×Q₈	—	11	2	no	—
71	C₂²×C₄×Q₈	D₁₁	1	1	yes	characteristic
72	C₂³×C₁₁	C₂²×Q₈	1	4	yes	—
73	C₂³×C₁₁	C₂×#41	1	4	yes	—
74	C₂³×C₁₁	2₋¹⁺⁴	1	4	yes	—
75	C₂³×C₁₁	C₂⁵	1	3	yes	—
76	Q₈×C₁₁	C₂²×Q₈	1	64	yes	complemented
77	C₂×C₄×C₁₁	—	2	36	no	—
78	C₂×C₄×C₁₁	C₂²×Q₈	1	72	yes	—
79	C₂×C₄×C₁₁	C₂×#41	1	24	yes	—
80	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
81	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
82	C₂×C₄×C₁₁	C₂⁵	1	4	yes	—
83	C₂×Dic₁₁	—	2	72	no	—
84	C₂×Dic₁₁	—	2	24	no	—
85	C₂×Dic₁₁	C₂⁵	1	8	yes	—
86	C₂×Dic₁₁	C₂⁵	1	24	yes	—
87	Dic₂₂	—	2	96	no	—
88	C₂³×Q₈	—	11	3	no	—
89	C₂³×Q₈	C₂×D₁₁	1	1	yes	characteristic
90	#557	—	11	256	no	—
91	C₂×C₄⋊Q₈	—	11	72	no	—
92	C₂×#125	—	11	144	no	—
93	C₂×C₄×Q₈	—	11	48	no	—
94	C₂×C₄×Q₈	—	11	72	no	—
95	C₂×C₄×Q₈	C₂×D₁₁	1	24	yes	—
96	C₂²×C₄⋊C₄	—	11	6	no	—
97	C₂²×C₄⋊C₄	—	11	1	no	—
98	C₂²×C₄⋊C₄	—	11	6	no	—
99	C₂²×C₄⋊C₄	—	11	6	no	—
100	C₂²×C₄⋊C₄	C₂×D₁₁	1	3	yes	—
101	C₂²×C₄²	—	11	6	no	—
102	C₂²×C₄²	C₂×D₁₁	1	3	yes	—
103	C₂²×C₁₁	#557	1	16	yes	complemented
104	C₂²×C₁₁	C₂³×Q₈	1	6	yes	—
105	C₂²×C₁₁	C₂²×#41	1	6	yes	—
106	C₂²×C₁₁	C₂×2₋¹⁺⁴	1	6	yes	—
107	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
108	C₄×C₁₁	—	2	12	no	—
109	C₄×C₁₁	C₂³×Q₈	1	24	yes	—
110	C₄×C₁₁	C₂²×#41	1	8	yes	—
111	Dic₁₁	—	2	24	no	—
112	Dic₁₁	—	2	8	no	—
113	C₂²×Q₈	—	11	36	no	—
114	C₂²×Q₈	—	22	24	no	—
115	C₂²×Q₈	Dic₂₂	1	16	yes	complemented
116	C₂²×Q₈	C₂²×D₁₁	1	12	yes	—
117	C₂³×C₄	—	11	6	no	—
118	C₂³×C₄	—	11	2	no	—
119	C₂³×C₄	C₂²×D₁₁	1	3	yes	—
120	C₂³×C₄	C₂²×D₁₁	1	3	yes	—
121	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
122	C₄⋊Q₈	—	11	192	no	—
123	#125	—	11	384	no	—
124	C₄×Q₈	—	11	128	no	—
125	C₄×Q₈	—	11	192	no	—
126	C₄×Q₈	C₂²×D₁₁	1	64	yes	—
127	C₂×C₄⋊C₄	—	11	12	no	—
128	C₂×C₄⋊C₄	—	11	72	no	—
129	C₂×C₄⋊C₄	—	11	72	no	—
130	C₂×C₄⋊C₄	—	11	72	no	—
131	C₂×C₄⋊C₄	C₂²×D₁₁	1	36	yes	—
132	C₂×C₄²	—	11	72	no	—
133	C₂×C₄²	C₂²×D₁₁	1	36	yes	—
134	C₂×C₁₁	C₂×#557	1	12	yes	complemented
135	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
136	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
137	C₂×C₁₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
138	C₂⁴	C₂³×D₁₁	1	1	yes	characteristic
139	C₂×Q₈	—	11	48	no	—
140	C₂×Q₈	—	22	144	no	—
141	C₂×Q₈	C₂×Dic₂₂	1	96	yes	complemented
142	C₂×Q₈	C₂³×D₁₁	1	16	yes	—
143	C₂²×C₄	—	2	12	no	—
144	C₂²×C₄	—	11	12	no	—
145	C₂²×C₄	—	11	36	no	—
146	C₂²×C₄	—	22	24	no	—
147	C₂²×C₄	—	22	8	no	—
148	C₂²×C₄	C₂×Dic₂₂	1	24	yes	—
149	C₂²×C₄	#4343	1	8	yes	—
150	C₂²×C₄	C₂³×D₁₁	1	18	yes	—
151	C₂²×C₄	C₂³×D₁₁	1	18	yes	—
152	C₂²×C₄	C₂³×D₁₁	1	6	yes	—
153	C₄⋊C₄	—	11	16	no	—
154	C₄⋊C₄	—	11	96	no	—
155	C₄⋊C₄	—	11	96	no	—
156	C₄⋊C₄	—	11	96	no	—
157	C₄⋊C₄	C₂³×D₁₁	1	48	yes	—
158	C₄²	—	11	96	no	—
159	C₄²	C₂³×D₁₁	1	48	yes	—
160	C₁₁	C₂²×#557	1	1	yes	characteristic, complemented, Sylow
161	C₂³	C₂²×Dic₂₂	1	4	yes	—
162	C₂³	C₂×#4343	1	4	yes	—
163	C₂³	#67043	1	4	yes	—
164	C₂³	C₂⁴×D₁₁	1	3	yes	—
165	Q₈	—	22	96	no	—
166	Q₈	C₂²×Dic₂₂	1	64	yes	complemented
167	C₂×C₄	—	2	36	no	—
168	C₂×C₄	—	11	8	no	—
169	C₂×C₄	—	11	24	no	—
170	C₂×C₄	—	22	72	no	—
171	C₂×C₄	—	22	24	no	—
172	C₂×C₄	C₂²×Dic₂₂	1	72	yes	—
173	C₂×C₄	C₂×#4343	1	24	yes	—
174	C₂×C₄	C₂⁴×D₁₁	1	12	yes	—
175	C₂×C₄	C₂⁴×D₁₁	1	12	yes	—
176	C₂×C₄	C₂⁴×D₁₁	1	4	yes	—
177	C₂²	#10623101	1	16	yes	direct product factor
178	C₂²	C₂³×Dic₂₂	1	6	yes	—
179	C₂²	C₂²×#4343	1	6	yes	—
180	C₂²	C₂×#67043	1	6	yes	—
181	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
182	C₄	—	2	12	no	—
183	C₄	—	22	24	no	—
184	C₄	—	22	8	no	—
185	C₄	C₂³×Dic₂₂	1	24	yes	—
186	C₄	C₂²×#4343	1	8	yes	—
187	C₂	C₂×#10623101	1	12	yes	direct product factor
188	C₂	C₂⁴×Dic₂₂	1	1	yes	characteristic
189	C₂	C₂³×#4343	1	1	yes	characteristic
190	C₂	C₂²×#67043	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂²×C₄×Q₈×C₁₁	C₂
socle, prime radical	C₂⁴×C₁₁	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₁₁
hypocenter	C₁₁	C₂²×#557
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#10623101	C₂	direct (N × H)
18. #10623101	C₂²	direct (N × H)
22. C₂²×Q₈×C₁₁	Q₈	semidirect
45. C₂×Q₈×C₁₁	C₂×Q₈	semidirect
76. Q₈×C₁₁	C₂²×Q₈	semidirect
103. C₂²×C₁₁	#557	semidirect
115. C₂²×Q₈	Dic₂₂	semidirect
134. C₂×C₁₁	C₂×#557	semidirect
141. C₂×Q₈	C₂×Dic₂₂	semidirect
160. C₁₁	C₂²×#557	semidirect
166. Q₈	C₂²×Dic₂₂	semidirect
177. C₂²	#10623101	direct (N × H)
187. C₂	C₂×#10623101	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×#10623101	2
C₂²×#66922	2
C₂²×#66943	2
C₂²×Q₈×Dic₁₁	2
C₂²×C₄×Dic₂₂	2
C₂²×C₄×Q₈×C₁₁	2
C₂²×#557	11

Subgroups — C22×#10623101 #15660819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10623101 #15660819