Subgroups — #14670208

Order 2816.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622143	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622128	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#13068774	C₂	1	2	yes	maximal
5	C₂×C₁₁×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#107	C₂²	1	2	yes	—
8	C₂×#66882	C₂²	1	2	yes	—
9	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#10622589	—	2	2	no	—
12	C₂×#66871	—	2	2	no	—
13	C₁₁×#367	—	2	2	no	—
14	#10622143	—	2	2	no	—
15	C₁₆×Dic₁₁	C₄	1	4	yes	complemented
16	#10622128	—	2	4	no	—
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊Dic₁₁	—	2	2	no	—
20	C₂×Dic₁₁⋊C₄	—	4	4	no	—
21	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
22	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
25	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
26	C₁₁×#107	—	2	2	no	—
27	C₁₁×#107	—	2	2	no	—
28	#66882	—	2	2	no	—
29	#66882	—	2	2	no	—
30	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₁	Q₈	1	2	yes	—
32	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
34	C₂×C₁₁⋊C₁₆	Q₈	1	2	yes	—
35	#66871	—	4	4	no	—
36	#12906	—	11	1	no	maximal, Sylow
37	C₂²×C₄×C₁₁	—	4	2	no	—
38	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₁	—	4	2	no	—
40	C₂²×Dic₁₁	D₈	1	2	yes	—
41	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₁₁×C₄⋊C₄	—	4	2	no	—
48	C₁₁×C₄⋊C₄	—	4	2	no	—
49	C₄⋊Dic₁₁	—	4	2	no	—
50	C₄⋊Dic₁₁	—	4	2	no	—
51	Dic₁₁⋊C₄	—	8	8	no	—
52	C₄×Dic₁₁	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₁₁	Q₁₆	1	2	yes	—
54	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₁₁⋊C₈	D₈	1	2	yes	—
56	C₂×C₁₁⋊C₈	Q₁₆	1	2	yes	—
57	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	complemented
58	C₁₁⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
59	C₂×#367	—	11	1	no	—
60	C₂×#367	—	11	2	no	—
61	C₂×#367	D₁₁	1	1	yes	characteristic
62	C₂×C₄×C₁₆	—	11	1	no	—
63	#1845	—	11	2	no	—
64	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₁	—	8	2	no	—
66	C₂×C₄×C₁₁	—	8	4	no	—
67	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₁₁	—	8	2	no	—
74	C₂×Dic₁₁	—	8	4	no	—
75	C₂×Dic₁₁	#107	1	4	yes	—
76	C₂×Dic₁₁	SD₃₂	1	8	yes	—
77	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₁₁⋊C₈	#107	1	4	yes	—
82	C₂²×C₁₆	—	11	1	no	—
83	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
84	C₂×#107	—	11	2	no	—
85	C₂×#107	—	22	2	no	—
86	C₂×#107	C₂×D₁₁	1	2	yes	—
87	C₂×C₄×C₈	—	11	1	no	—
88	#384	—	22	2	no	—
89	#367	—	2	2	no	—
90	#367	—	22	4	no	—
91	#367	—	22	2	no	—
92	C₄×C₁₆	—	11	4	no	—
93	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
94	C₂²×C₁₁	#493	1	1	yes	characteristic
95	C₂²×C₁₁	#500	1	1	yes	characteristic
96	C₂²×C₁₁	C₂×SD₃₂	1	4	yes	—
97	C₄×C₁₁	—	16	2	no	—
98	C₄×C₁₁	#384	1	1	yes	characteristic
99	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
100	C₄×C₁₁	#494	1	1	yes	characteristic
101	C₄×C₁₁	#500	1	1	yes	characteristic
102	Dic₁₁	—	16	2	no	—
103	Dic₁₁	#367	1	8	yes	complemented
104	C₂²×C₈	—	11	1	no	—
105	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	22	2	no	—
108	C₂×C₄⋊C₄	—	44	4	no	—
109	C₂×C₄²	—	11	1	no	—
110	C₂×C₁₆	—	11	2	no	—
111	C₂×C₁₆	—	11	2	no	—
112	C₂×C₁₆	—	11	2	no	—
113	C₂×C₁₆	Dic₂₂	1	2	yes	—
114	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
115	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
116	#107	—	2	2	no	—
117	#107	—	2	2	no	—
118	#107	—	22	2	no	—
119	#107	—	22	2	no	—
120	#107	—	44	4	no	—
121	C₄×C₈	—	11	2	no	—
122	C₄×C₈	—	11	2	no	—
123	C₂×C₁₁	#1845	1	1	yes	characteristic
124	C₂×C₁₁	C₂×#367	1	2	yes	—
125	C₂×C₁₁	#2248	1	2	yes	—
126	C₂×C₁₁	#2254	1	2	yes	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	11	2	no	—
129	C₂²×C₄	—	44	2	no	—
130	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
131	C₂×C₈	—	11	2	no	—
132	C₂×C₈	—	11	2	no	—
133	C₂×C₈	—	11	2	no	—
134	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
136	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
137	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	44	2	no	—
143	C₄⋊C₄	—	44	2	no	—
144	C₄⋊C₄	—	88	8	no	—
145	C₄²	—	11	2	no	—
146	C₄²	—	11	2	no	—
147	C₁₆	—	11	4	no	—
148	C₁₆	Dic₁₁⋊C₄	1	4	yes	complemented
149	C₁₁	#12906	1	1	yes	characteristic, complemented, Sylow
150	C₂³	D₈×D₁₁	1	1	yes	characteristic
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	11	4	no	—
154	C₂×C₄	—	11	8	no	—
155	C₂×C₄	—	88	2	no	—
156	C₂×C₄	—	88	4	no	—
157	C₂×C₄	#66941	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
159	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
160	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
162	C₂×C₄	#66993	1	1	yes	characteristic
163	C₈	—	11	4	no	—
164	C₈	#66941	1	1	yes	characteristic
165	C₈	C₄⋊C₄×D₁₁	1	1	yes	characteristic
166	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
167	C₈	#66997	1	1	yes	characteristic
168	C₂²	#10622487	1	1	yes	characteristic
169	C₂²	D₁₁×#107	1	1	yes	characteristic
170	C₂²	D₁₁×SD₃₂	1	4	yes	—
171	C₂²	#10622764	1	1	yes	characteristic
172	C₄	—	11	8	no	—
173	C₄	—	16	2	no	—
174	C₄	—	176	2	no	—
175	C₄	#10622487	1	1	yes	characteristic
176	C₄	D₁₁×#107	1	1	yes	characteristic
177	C₄	#10622589	1	1	yes	characteristic
178	C₄	#10622794	1	1	yes	characteristic
179	C₂	#13068774	1	1	yes	characteristic
180	C₂	D₁₁×#367	1	2	yes	—
181	C₂	#13070870	1	2	yes	—
182	C₂	#13070897	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₁₁
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#367	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#367	D₁₁
hypocenter	C₁₁	#12906
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. C₁₆×Dic₁₁	C₄	semidirect
57. C₁₁×C₁₆	C₄⋊C₄	semidirect
58. C₁₁⋊C₁₆	C₄⋊C₄	semidirect
103. Dic₁₁	#367	semidirect
148. C₁₆	Dic₁₁⋊C₄	semidirect
149. C₁₁	#12906	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622143	2
C₂×#10622128	2
C₂×C₁₆×Dic₁₁	2
#13068774	2
C₂×C₁₁×#367	2
#12906	11