Subgroups — #14670186

Order 2816.

370 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁⋊C₁₆	C₂	1	4	yes	maximal
3	#13068302	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×#66879	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
9	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
10	#10622545	C₄	1	1	yes	characteristic
11	C₂×#66867	C₄	1	1	yes	characteristic
12	C₁₁×C₄⋊C₁₆	—	2	2	no	—
13	Dic₁₁⋊C₁₆	—	2	8	no	—
14	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
15	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
19	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₁×C₁₆	—	2	2	no	—
21	C₂×C₁₁×C₁₆	—	2	4	no	—
22	C₄×C₈×C₁₁	D₄	1	2	yes	—
23	C₄×C₈×C₁₁	Q₈	1	2	yes	—
24	C₁₁⋊#95	C₈	1	2	yes	—
25	C₁₁⋊#95	C₈	1	2	yes	—
26	#66879	—	2	4	no	—
27	C₈×Dic₁₁	D₄	1	4	yes	—
28	C₈×Dic₁₁	Q₈	1	4	yes	—
29	C₂×C₁₁⋊C₁₆	—	2	2	no	—
30	C₂×C₁₁⋊C₁₆	—	2	4	no	—
31	#66867	—	2	2	no	—
32	#12899	—	11	1	no	maximal, Sylow
33	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₁	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₁₁	—	2	2	no	—
38	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
40	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
43	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₁	M₁₆	1	2	yes	—
45	C₄×Dic₁₁	C₄⋊C₄	1	4	yes	—
46	C₄×Dic₁₁	M₁₆	1	4	yes	—
47	C₂×C₁₁⋊C₈	—	2	2	no	—
48	C₂×C₁₁⋊C₈	—	2	2	no	—
49	C₂×C₁₁⋊C₈	C₄⋊C₄	1	4	yes	—
50	C₁₁×C₁₆	—	4	2	no	—
51	C₁₁⋊C₁₆	—	4	2	no	—
52	C₂×C₄⋊C₁₆	—	11	4	no	—
53	C₂×C₄⋊C₁₆	—	11	1	no	—
54	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic
55	#1726	—	11	1	no	—
56	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₁	—	2	2	no	—
58	C₂×C₄×C₁₁	—	2	2	no	—
59	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
65	C₂×Dic₁₁	—	2	4	no	—
66	C₂×Dic₁₁	—	2	4	no	—
67	C₂×Dic₁₁	C₄⋊C₈	1	8	yes	—
68	C₈×C₁₁	—	2	2	no	—
69	C₈×C₁₁	#127	1	1	yes	characteristic
70	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
71	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₁₁⋊C₈	—	2	4	no	—
73	C₂²×C₁₆	—	11	2	no	—
74	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
75	C₂×#97	—	11	2	no	—
76	C₂×#97	—	11	1	no	—
77	C₂×C₄×C₈	—	11	2	no	—
78	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
79	#376	—	11	1	no	—
80	C₄⋊C₁₆	—	2	2	no	—
81	C₄⋊C₁₆	—	22	8	no	—
82	C₄⋊C₁₆	—	22	2	no	—
83	C₂²×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
84	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₁₁	#504	1	2	yes	—
86	C₂²×C₁₁	#504	1	2	yes	—
87	C₄×C₁₁	—	2	2	no	—
88	C₄×C₁₁	—	2	2	no	—
89	C₄×C₁₁	#384	1	1	yes	characteristic
90	C₄×C₁₁	#423	1	1	yes	characteristic
91	C₄×C₁₁	#423	1	2	yes	—
92	Dic₁₁	—	2	8	no	—
93	C₂²×C₈	—	11	2	no	—
94	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
96	C₂×C₄²	—	11	2	no	—
97	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	22	4	no	—
101	C₂×C₁₆	—	22	2	no	—
102	#97	—	22	4	no	—
103	#97	—	22	2	no	—
104	C₄×C₈	—	11	4	no	—
105	C₄×C₈	—	11	4	no	—
106	C₄×C₈	Dic₂₂	1	2	yes	—
107	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
108	#95	—	11	2	no	—
109	#95	—	11	2	no	—
110	C₂×C₁₁	#1842	1	1	yes	characteristic
111	C₂×C₁₁	#2151	1	4	yes	—
112	C₂×C₁₁	#2151	1	1	yes	characteristic
113	C₂×C₁₁	#2151	1	1	yes	characteristic
114	C₂²×C₄	—	11	4	no	—
115	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
116	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	11	4	no	—
120	C₂×C₈	—	22	2	no	—
121	C₂×C₈	—	22	2	no	—
122	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
123	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
124	C₂×C₈	D₄×D₁₁	1	2	yes	—
125	C₂×C₈	Q₈×D₁₁	1	2	yes	—
126	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
127	C₄²	—	11	4	no	—
128	C₄²	—	11	4	no	—
129	C₄²	#4313	1	2	yes	—
130	C₄²	Dic₁₁⋊C₄	1	2	yes	—
131	C₁₆	—	4	2	no	—
132	C₁₆	—	44	2	no	—
133	C₁₁	#12899	1	1	yes	characteristic, complemented, Sylow
134	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	11	8	no	—
138	C₂×C₄	—	22	4	no	—
139	C₂×C₄	—	22	4	no	—
140	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
141	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
142	C₂×C₄	C₄⋊C₄×D₁₁	1	2	yes	—
143	C₂×C₄	C₂×#4313	1	1	yes	characteristic
144	C₂×C₄	M₁₆×D₁₁	1	2	yes	—
145	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	22	4	no	—
148	C₈	#66941	1	2	yes	—
149	C₈	#66993	1	1	yes	characteristic
150	C₈	#66997	1	1	yes	characteristic
151	C₂²	D₁₁×C₄⋊C₈	1	2	yes	—
152	C₂²	#10622514	1	2	yes	—
153	C₂²	#10622520	1	2	yes	—
154	C₂²	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	22	8	no	—
158	C₄	#10622450	1	2	yes	—
159	C₄	#10622589	1	1	yes	characteristic
160	C₄	#10622731	1	1	yes	characteristic
161	C₂	#13068767	1	1	yes	characteristic
162	C₂	#13070373	1	4	yes	—
163	C₂	#13070374	1	1	yes	characteristic
164	C₂	#13070376	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator, socle, prime radical	C₂³×C₁₁	C₂²×C₈
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	#12899
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
133. C₁₁	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×Dic₁₁⋊C₁₆	2
#13068302	2
C₂×C₁₁×C₄⋊C₁₆	2
#12899	11