Subgroups — C₂²×#10622824 #14912037

Order 2816.

8062 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	C₂³×C₂²⋊Dic₁₁	C₂	1	2	yes	maximal
2	C₂×#10622824	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₁₁⋊#95	C₂	1	4	yes	maximal, complemented
4	C₂⁵×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₁₁	—	2	16	no	—
7	C₂⁴×C₄×C₁₁	C₄	1	16	yes	complemented
8	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₁	C₂²	1	12	yes	—
10	C₂⁴×Dic₁₁	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₁₁	—	2	16	no	—
12	C₂²×C₂²⋊Dic₁₁	C₂²	1	24	yes	—
13	#10622824	C₂²	1	64	yes	direct product factor
14	C₂×C₁₁⋊#95	C₂²	1	48	yes	complemented
15	C₂⁵×C₁₁	—	2	16	no	—
16	C₂⁵×C₁₁	C₂×C₄	1	8	yes	—
17	C₂⁵×C₁₁	D₄	1	16	yes	—
18	C₂⁵×C₁₁	C₂³	1	6	yes	—
19	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₁	—	2	128	no	—
21	C₂³×C₄×C₁₁	—	2	16	no	—
22	C₂³×C₄×C₁₁	—	2	96	no	—
23	C₂³×C₄×C₁₁	C₂×C₄	1	16	yes	—
24	C₂³×C₄×C₁₁	C₂×C₄	1	96	yes	complemented
25	C₂³×C₄×C₁₁	C₂³	1	12	yes	—
26	C₂³×C₄×C₁₁	C₂³	1	16	yes	—
27	C₂³×Dic₁₁	—	2	16	no	—
28	C₂³×Dic₁₁	—	2	32	no	—
29	C₂³×Dic₁₁	C₂³	1	24	yes	—
30	C₂×C₂²⋊Dic₁₁	—	2	96	no	—
31	C₂×C₂²⋊Dic₁₁	C₂³	1	32	yes	—
32	C₁₁⋊#95	C₂³	1	64	yes	complemented
33	C₂²×#381	—	11	1	no	maximal, Sylow
34	C₂⁴×C₁₁	—	2	64	no	—
35	C₂⁴×C₁₁	—	2	128	no	—
36	C₂⁴×C₁₁	—	2	48	no	—
37	C₂⁴×C₁₁	C₂²⋊C₄	1	64	yes	—
38	C₂⁴×C₁₁	C₂²×C₄	1	24	yes	—
39	C₂⁴×C₁₁	C₂²×C₄	1	4	yes	—
40	C₂⁴×C₁₁	C₂×D₄	1	48	yes	—
41	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
42	C₂⁴×C₁₁	#41	1	16	yes	—
43	C₂⁴×C₁₁	C₂⁴	1	4	yes	—
44	C₂⁴×C₁₁	C₂⁴	1	3	yes	—
45	C₂²×C₄×C₁₁	—	2	64	no	—
46	C₂²×C₄×C₁₁	—	2	384	no	—
47	C₂²×C₄×C₁₁	—	2	48	no	—
48	C₂²×C₄×C₁₁	—	2	64	no	—
49	C₂²×C₄×C₁₁	C₂²×C₄	1	48	yes	—
50	C₂²×C₄×C₁₁	C₂²×C₄	1	64	yes	complemented
51	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
52	C₂²×Dic₁₁	—	2	48	no	—
53	C₂²×Dic₁₁	—	2	96	no	—
54	C₂²×Dic₁₁	—	4	64	no	—
55	C₂²×Dic₁₁	C₂⁴	1	16	yes	—
56	C₂²⋊Dic₁₁	—	2	64	no	—
57	C₂⁵×C₄	D₁₁	1	1	yes	characteristic
58	C₂³×C₂²⋊C₄	—	11	2	no	—
59	C₂×#381	—	11	24	no	—
60	C₂²×#95	—	11	4	no	—
61	C₂³×C₁₁	—	2	96	no	—
62	C₂³×C₁₁	—	2	192	no	—
63	C₂³×C₁₁	—	2	16	no	—
64	C₂³×C₁₁	—	2	256	no	—
65	C₂³×C₁₁	C₂×C₂²⋊C₄	1	96	yes	—
66	C₂³×C₁₁	C₂×C₂²⋊C₄	1	16	yes	—
67	C₂³×C₁₁	#117	1	32	yes	—
68	C₂³×C₁₁	#123	1	64	yes	—
69	C₂³×C₁₁	C₂³×C₄	1	8	yes	—
70	C₂³×C₁₁	C₂³×C₄	1	6	yes	—
71	C₂³×C₁₁	C₂²×D₄	1	16	yes	—
72	C₂³×C₁₁	C₂²×D₄	1	12	yes	—
73	C₂³×C₁₁	C₂×#41	1	24	yes	—
74	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
75	C₂×C₄×C₁₁	—	2	96	no	—
76	C₂×C₄×C₁₁	—	2	128	no	—
77	C₂×C₄×C₁₁	—	2	16	no	—
78	C₂×C₄×C₁₁	C₂³×C₄	1	16	yes	—
79	C₂×Dic₁₁	—	2	16	no	—
80	C₂×Dic₁₁	—	2	32	no	—
81	C₂×Dic₁₁	—	4	96	no	—
82	C₂⁶	C₂×D₁₁	1	1	yes	characteristic
83	C₂⁴×C₄	—	2	16	no	—
84	C₂⁴×C₄	—	11	4	no	—
85	C₂⁴×C₄	Dic₁₁	1	16	yes	complemented
86	C₂⁴×C₄	C₂×D₁₁	1	2	yes	—
87	C₂⁴×C₄	C₂×D₁₁	1	12	yes	—
88	C₂²×C₂²⋊C₄	—	11	24	no	—
89	C₂²×C₂²⋊C₄	—	22	16	no	—
90	#381	—	11	64	no	—
91	C₂×#95	—	11	48	no	—
92	C₂²×C₁₁	—	2	16	no	—
93	C₂²×C₁₁	—	2	32	no	—
94	C₂²×C₁₁	—	2	192	no	—
95	C₂²×C₁₁	#381	1	64	yes	complemented
96	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	16	yes	—
97	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	12	yes	—
98	C₂²×C₁₁	C₂×#117	1	24	yes	—
99	C₂²×C₁₁	C₂×#123	1	48	yes	—
100	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
101	C₂²×C₁₁	C₂³×D₄	1	2	yes	—
102	C₂²×C₁₁	C₂²×#41	1	4	yes	—
103	C₄×C₁₁	—	2	16	no	—
104	Dic₁₁	—	4	16	no	—
105	C₂⁵	—	2	16	no	—
106	C₂⁵	C₂×Dic₁₁	1	8	yes	—
107	C₂⁵	C₂²⋊D₁₁	1	16	yes	—
108	C₂⁵	C₂²×D₁₁	1	6	yes	—
109	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
110	C₂³×C₄	—	2	128	no	—
111	C₂³×C₄	—	2	16	no	—
112	C₂³×C₄	—	2	96	no	—
113	C₂³×C₄	—	11	24	no	—
114	C₂³×C₄	—	22	32	no	—
115	C₂³×C₄	—	22	16	no	—
116	C₂³×C₄	C₂×Dic₁₁	1	16	yes	—
117	C₂³×C₄	C₂×Dic₁₁	1	96	yes	complemented
118	C₂³×C₄	C₂²×D₁₁	1	12	yes	—
119	C₂³×C₄	C₂²×D₁₁	1	16	yes	—
120	C₂×C₂²⋊C₄	—	11	32	no	—
121	C₂×C₂²⋊C₄	—	22	96	no	—
122	#95	—	11	64	no	—
123	C₂×C₁₁	—	2	16	no	—
124	C₂×C₁₁	C₂×#381	1	24	yes	complemented
125	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₁₁	C₂²×#117	1	2	yes	—
127	C₂×C₁₁	C₂²×#123	1	4	yes	—
128	C₂⁴	—	2	64	no	—
129	C₂⁴	—	2	128	no	—
130	C₂⁴	—	2	48	no	—
131	C₂⁴	C₂²⋊Dic₁₁	1	64	yes	—
132	C₂⁴	#4339	1	16	yes	—
133	C₂⁴	C₂²×Dic₁₁	1	24	yes	—
134	C₂⁴	C₂²×Dic₁₁	1	4	yes	—
135	C₂⁴	C₂×C₂²⋊D₁₁	1	48	yes	—
136	C₂⁴	C₂×C₂²⋊D₁₁	1	8	yes	—
137	C₂⁴	C₂³×D₁₁	1	4	yes	—
138	C₂⁴	C₂³×D₁₁	1	3	yes	—
139	C₂²×C₄	—	2	64	no	—
140	C₂²×C₄	—	2	384	no	—
141	C₂²×C₄	—	2	48	no	—
142	C₂²×C₄	—	2	64	no	—
143	C₂²×C₄	—	11	16	no	—
144	C₂²×C₄	—	22	96	no	—
145	C₂²×C₄	—	22	48	no	—
146	C₂²×C₄	—	44	64	no	—
147	C₂²×C₄	C₂²×Dic₁₁	1	48	yes	—
148	C₂²×C₄	C₂²×Dic₁₁	1	64	yes	complemented
149	C₂²×C₄	C₂³×D₁₁	1	8	yes	—
150	C₂²⋊C₄	—	22	64	no	—
151	C₁₁	C₂²×#381	1	1	yes	characteristic, complemented, Sylow
152	C₂³	—	2	96	no	—
153	C₂³	—	2	192	no	—
154	C₂³	—	2	256	no	—
155	C₂³	—	2	16	no	—
156	C₂³	#66979	1	32	yes	—
157	C₂³	#66982	1	64	yes	—
158	C₂³	C₂×C₂²⋊Dic₁₁	1	96	yes	—
159	C₂³	C₂×C₂²⋊Dic₁₁	1	16	yes	—
160	C₂³	C₂×#4339	1	24	yes	—
161	C₂³	C₂³×Dic₁₁	1	8	yes	—
162	C₂³	C₂³×Dic₁₁	1	6	yes	—
163	C₂³	C₂²×C₂²⋊D₁₁	1	16	yes	—
164	C₂³	C₂²×C₂²⋊D₁₁	1	12	yes	—
165	C₂³	C₂⁴×D₁₁	1	1	yes	characteristic
166	C₂×C₄	—	2	96	no	—
167	C₂×C₄	—	2	128	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	22	32	no	—
170	C₂×C₄	—	22	16	no	—
171	C₂×C₄	—	44	96	no	—
172	C₂×C₄	C₂³×Dic₁₁	1	16	yes	—
173	C₂²	—	2	192	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	2	32	no	—
176	C₂²	#10622824	1	64	yes	direct product factor
177	C₂²	C₂×#66979	1	24	yes	—
178	C₂²	C₂×#66982	1	48	yes	—
179	C₂²	C₂²×C₂²⋊Dic₁₁	1	16	yes	—
180	C₂²	C₂²×C₂²⋊Dic₁₁	1	12	yes	—
181	C₂²	C₂²×#4339	1	4	yes	—
182	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
183	C₂²	C₂³×C₂²⋊D₁₁	1	2	yes	—
184	C₄	—	2	16	no	—
185	C₄	—	44	16	no	—
186	C₂	—	2	16	no	—
187	C₂	C₂×#10622824	1	24	yes	direct product factor
188	C₂	C₂³×C₂²⋊Dic₁₁	1	1	yes	characteristic
189	C₂	C₂²×#66979	1	2	yes	—
190	C₂	C₂²×#66982	1	4	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₁
Fitting	C₂⁵×C₄×C₁₁	C₂
socle	C₂⁵×C₁₁	C₂³
prime radical	C₂⁶×C₁₁	C₂²
cosocle	C₂³×C₁₁	C₂⁵
hypercenter	C₂⁵×C₄	D₁₁
hypocenter	C₁₁	C₂²×#381
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#10622824	C₂	direct (N × H)
3. C₂²×C₁₁⋊#95	C₂	semidirect
7. C₂⁴×C₄×C₁₁	C₄	semidirect
13. #10622824	C₂²	direct (N × H)
14. C₂×C₁₁⋊#95	C₂²	semidirect
24. C₂³×C₄×C₁₁	C₂×C₄	semidirect
32. C₁₁⋊#95	C₂³	semidirect
50. C₂²×C₄×C₁₁	C₂²×C₄	semidirect
85. C₂⁴×C₄	Dic₁₁	semidirect
95. C₂²×C₁₁	#381	semidirect
117. C₂³×C₄	C₂×Dic₁₁	semidirect
124. C₂×C₁₁	C₂×#381	semidirect
148. C₂²×C₄	C₂²×Dic₁₁	semidirect
151. C₁₁	C₂²×#381	semidirect
176. C₂²	#10622824	direct (N × H)
187. C₂	C₂×#10622824	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×C₂²⋊Dic₁₁	2
C₂×#10622824	2
C₂²×C₁₁⋊#95	2
C₂⁵×C₄×C₁₁	2
C₂²×#381	11

Subgroups — C22×#10622824 #14912037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622824 #14912037