Subgroups — #14670210

Order 2816.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 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	4	yes	maximal
2	#13068776	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×#66882	—	2	2	no	—
7	C₂×#66882	C₂²	1	4	yes	—
8	#10622548	—	2	2	no	—
9	C₄×C₁₁×C₁₆	C₄	1	4	yes	complemented
10	#10622143	—	2	8	no	—
11	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₁	—	2	4	no	—
15	C₂×C₄⋊Dic₁₁	—	4	4	no	—
16	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₁₁×C₁₆	D₄	1	4	yes	—
18	C₂×C₁₁×C₁₆	Q₈	1	4	yes	—
19	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₁	Q₈	1	2	yes	—
21	#66882	—	2	8	no	—
22	#66882	—	4	4	no	—
23	#12906	—	11	1	no	maximal, Sylow
24	C₂²×C₄×C₁₁	D₈	1	2	yes	—
25	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
26	C₂²×Dic₁₁	—	4	4	no	—
27	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₁₁	D₈	1	2	yes	—
29	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
30	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
32	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
34	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
35	C₄²×C₁₁	Q₁₆	1	2	yes	—
36	C₄⋊Dic₁₁	—	4	4	no	—
37	C₄⋊Dic₁₁	—	4	4	no	—
38	C₄⋊Dic₁₁	—	8	8	no	—
39	C₁₁×C₁₆	C₄⋊C₄	1	8	yes	complemented
40	C₂×#367	—	11	4	no	—
41	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic
42	#1845	—	11	2	no	—
43	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄×C₁₁	#107	1	4	yes	—
45	C₂×C₄×C₁₁	SD₃₂	1	8	yes	—
46	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×Dic₁₁	—	8	4	no	—
53	C₂×Dic₁₁	—	8	8	no	—
54	C₈×C₁₁	#107	1	4	yes	—
55	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
56	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
57	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
58	C₂×#107	—	11	4	no	—
59	C₂×#107	—	22	2	no	—
60	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
61	#384	—	22	2	no	—
62	#367	—	22	8	no	—
63	C₄×C₁₆	Dic₁₁	1	4	yes	complemented
64	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
65	C₂²×C₁₁	#493	1	1	yes	characteristic
66	C₂²×C₁₁	#500	1	1	yes	characteristic
67	C₂²×C₁₁	C₂×SD₃₂	1	4	yes	—
68	C₄×C₁₁	#367	1	8	yes	complemented
69	C₄×C₁₁	#384	1	1	yes	characteristic
70	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
71	C₄×C₁₁	#494	1	1	yes	characteristic
72	C₄×C₁₁	#500	1	1	yes	characteristic
73	Dic₁₁	—	16	4	no	—
74	C₂²×C₈	D₄₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	22	4	no	—
77	C₂×C₄⋊C₄	—	44	4	no	—
78	C₂×C₄²	D₄₄	1	1	yes	characteristic
79	C₂×C₁₆	Dic₂₂	1	4	yes	—
80	C₂×C₁₆	D₄₄	1	4	yes	—
81	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	—
82	#107	—	22	8	no	—
83	#107	—	44	4	no	—
84	C₄×C₈	Dic₂₂	1	2	yes	—
85	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
86	C₂×C₁₁	#1845	1	1	yes	characteristic
87	C₂×C₁₁	C₂×#367	1	2	yes	—
88	C₂×C₁₁	#2248	1	2	yes	—
89	C₂×C₁₁	#2254	1	2	yes	—
90	C₂²×C₄	—	44	4	no	—
91	C₂²×C₄	D₈₈	1	2	yes	—
92	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
93	C₂×C₈	D₈₈	1	2	yes	—
94	C₂×C₈	Dic₄₄	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
96	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
97	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
98	C₂×C₈	C₂×D₄₄	1	2	yes	—
99	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
100	C₄⋊C₄	—	44	4	no	—
101	C₄⋊C₄	—	44	4	no	—
102	C₄⋊C₄	—	88	8	no	—
103	C₄²	Dic₄₄	1	2	yes	—
104	C₄²	C₄⋊Dic₁₁	1	2	yes	—
105	C₁₆	C₄⋊Dic₁₁	1	8	yes	complemented
106	C₁₁	#12906	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂×D₈₈	1	1	yes	characteristic
108	C₂×C₄	—	88	8	no	—
109	C₂×C₄	—	88	4	no	—
110	C₂×C₄	#66864	1	8	yes	—
111	C₂×C₄	#66882	1	4	yes	—
112	C₂×C₄	#66922	1	1	yes	characteristic
113	C₂×C₄	#66927	1	1	yes	characteristic
114	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
118	C₈	#66882	1	4	yes	—
119	C₈	#66922	1	2	yes	—
120	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
121	C₂²	#10622295	1	1	yes	characteristic
122	C₂²	#10622308	1	1	yes	characteristic
123	C₂²	C₂×#66864	1	4	yes	—
124	C₂²	C₂×#66882	1	1	yes	characteristic
125	C₄	—	176	4	no	—
126	C₄	#10622143	1	8	yes	complemented
127	C₄	#10622295	1	1	yes	characteristic
128	C₄	#10622312	1	1	yes	characteristic
129	C₄	#10622548	1	1	yes	characteristic
130	C₄	C₂×#66882	1	1	yes	characteristic
131	C₂	#13068776	1	1	yes	characteristic
132	C₂	C₂×#10622143	1	2	yes	—
133	C₂	#13070872	1	2	yes	—
134	C₂	#13070898	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#12906
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₁×C₁₆	C₄	semidirect
39. C₁₁×C₁₆	C₄⋊C₄	semidirect
63. C₄×C₁₆	Dic₁₁	semidirect
68. C₄×C₁₁	#367	semidirect
105. C₁₆	C₄⋊Dic₁₁	semidirect
106. C₁₁	#12906	semidirect
126. C₄	#10622143	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
#13068776	2
C₂×C₄×C₁₁×C₁₆	2
#12906	11