Subgroups — #14670205

Order 2816.

374 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622142	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622127	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×#366	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×#366	—	2	2	no	—
14	#10622142	—	2	2	no	—
15	C₁₆×Dic₁₁	C₄	1	4	yes	complemented
16	#10622127	—	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	#12905	—	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×#366	—	11	1	no	—
60	C₂×#366	—	11	2	no	—
61	C₂×#366	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	D₁₆	1	4	yes	—
77	C₂×Dic₁₁	Q₃₂	1	4	yes	—
78	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₁₁⋊C₈	#107	1	4	yes	—
83	C₂²×C₁₆	—	11	1	no	—
84	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
85	C₂×#107	—	11	2	no	—
86	C₂×#107	—	22	2	no	—
87	C₂×#107	C₂×D₁₁	1	2	yes	—
88	C₂×C₄×C₈	—	11	1	no	—
89	#384	—	22	2	no	—
90	#366	—	2	2	no	—
91	#366	—	22	4	no	—
92	#366	—	22	2	no	—
93	C₄×C₁₆	—	11	4	no	—
94	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
95	C₂²×C₁₁	#493	1	1	yes	characteristic
96	C₂²×C₁₁	#500	1	1	yes	characteristic
97	C₂²×C₁₁	C₂×D₁₆	1	2	yes	—
98	C₂²×C₁₁	C₂×Q₃₂	1	2	yes	—
99	C₄×C₁₁	—	16	2	no	—
100	C₄×C₁₁	#384	1	1	yes	characteristic
101	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
102	C₄×C₁₁	#494	1	1	yes	characteristic
103	C₄×C₁₁	#500	1	1	yes	characteristic
104	Dic₁₁	—	16	2	no	—
105	Dic₁₁	#366	1	8	yes	complemented
106	C₂²×C₈	—	11	1	no	—
107	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	22	2	no	—
110	C₂×C₄⋊C₄	—	44	4	no	—
111	C₂×C₄²	—	11	1	no	—
112	C₂×C₁₆	—	11	2	no	—
113	C₂×C₁₆	—	11	2	no	—
114	C₂×C₁₆	—	11	2	no	—
115	C₂×C₁₆	Dic₂₂	1	2	yes	—
116	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
117	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	—	22	2	no	—
121	#107	—	22	2	no	—
122	#107	—	44	4	no	—
123	C₄×C₈	—	11	2	no	—
124	C₄×C₈	—	11	2	no	—
125	C₂×C₁₁	#1845	1	1	yes	characteristic
126	C₂×C₁₁	C₂×#366	1	2	yes	—
127	C₂×C₁₁	#2246	1	1	yes	characteristic
128	C₂×C₁₁	#2247	1	1	yes	characteristic
129	C₂×C₁₁	#2252	1	2	yes	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	11	2	no	—
132	C₂²×C₄	—	44	2	no	—
133	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
134	C₂×C₈	—	11	2	no	—
135	C₂×C₈	—	11	2	no	—
136	C₂×C₈	—	11	2	no	—
137	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
139	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	44	2	no	—
146	C₄⋊C₄	—	44	2	no	—
147	C₄⋊C₄	—	88	8	no	—
148	C₄²	—	11	2	no	—
149	C₄²	—	11	2	no	—
150	C₁₆	—	11	4	no	—
151	C₁₆	Dic₁₁⋊C₄	1	4	yes	complemented
152	C₁₁	#12905	1	1	yes	characteristic, complemented, Sylow
153	C₂³	D₈×D₁₁	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	11	4	no	—
157	C₂×C₄	—	11	4	no	—
158	C₂×C₄	—	11	4	no	—
159	C₂×C₄	—	88	2	no	—
160	C₂×C₄	—	88	4	no	—
161	C₂×C₄	#66941	1	1	yes	characteristic
162	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
163	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
164	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
166	C₂×C₄	#66993	1	1	yes	characteristic
167	C₈	—	11	4	no	—
168	C₈	#66941	1	1	yes	characteristic
169	C₈	C₄⋊C₄×D₁₁	1	1	yes	characteristic
170	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
171	C₈	#66997	1	1	yes	characteristic
172	C₂²	#10622487	1	1	yes	characteristic
173	C₂²	D₁₁×#107	1	1	yes	characteristic
174	C₂²	D₁₁×D₁₆	1	2	yes	—
175	C₂²	D₁₁×Q₃₂	1	2	yes	—
176	C₂²	#10622764	1	1	yes	characteristic
177	C₄	—	11	8	no	—
178	C₄	—	16	2	no	—
179	C₄	—	176	2	no	—
180	C₄	#10622487	1	1	yes	characteristic
181	C₄	D₁₁×#107	1	1	yes	characteristic
182	C₄	#10622589	1	1	yes	characteristic
183	C₄	#10622794	1	1	yes	characteristic
184	C₂	#13068774	1	1	yes	characteristic
185	C₂	D₁₁×#366	1	2	yes	—
186	C₂	#13070864	1	1	yes	characteristic
187	C₂	#13070867	1	1	yes	characteristic
188	C₂	#13070890	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₁₁
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#366	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#366	D₁₁
hypocenter	C₁₁	#12905
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

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
105. Dic₁₁	#366	semidirect
151. C₁₆	Dic₁₁⋊C₄	semidirect
152. C₁₁	#12905	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622142	2
C₂×#10622127	2
C₂×C₁₆×Dic₁₁	2
#13068774	2
C₂×C₁₁×#366	2
#12905	11