Subgroups — #14670222

Order 2816.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622144	C₂	1	4	yes	maximal
2	#13068788	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×#66883	—	2	2	no	—
7	C₂×#66883	C₂²	1	4	yes	—
8	#10622537	—	2	2	no	—
9	C₄×C₁₁×C₁₆	C₄	1	2	yes	—
10	C₄×C₁₁×C₁₆	C₄	1	2	yes	—
11	#10622144	—	2	8	no	—
12	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
15	C₂×#4318	—	2	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	1	yes	characteristic
20	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
22	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
23	#66883	—	2	8	no	—
24	#66883	—	4	4	no	—
25	C₄⋊(C₁₁⋊C₈)	—	4	4	no	—
26	#12909	—	11	1	no	maximal, Sylow
27	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₁	D₈	1	2	yes	—
32	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
33	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
37	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₁	D₈	1	2	yes	—
40	#4318	—	4	4	no	—
41	#4318	—	4	4	no	—
42	C₂×C₁₁⋊C₈	—	4	4	no	—
43	C₁₁×C₁₆	C₄⋊C₄	1	8	yes	—
44	C₂×#368	—	11	4	no	—
45	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic
46	#1848	—	11	2	no	—
47	C₂³×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₁	—	2	4	no	—
49	C₂×C₄×C₁₁	#107	1	2	yes	—
50	C₂×C₄×C₁₁	#107	1	2	yes	—
51	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
53	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₁₁	#107	1	2	yes	—
58	C₈×C₁₁	#107	1	2	yes	—
59	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
60	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
61	C₁₁⋊C₈	—	8	4	no	—
62	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
63	C₂×#108	—	11	4	no	—
64	C₂×#108	—	22	2	no	—
65	#423	—	22	2	no	—
66	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
67	#368	—	22	8	no	—
68	C₄×C₁₆	Dic₁₁	1	2	yes	—
69	C₄×C₁₆	Dic₁₁	1	2	yes	—
70	C₂²×C₁₁	—	2	2	no	—
71	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
72	C₂²×C₁₁	#494	1	1	yes	characteristic
73	C₂²×C₁₁	#500	1	1	yes	characteristic
74	C₄×C₁₁	—	2	2	no	—
75	C₄×C₁₁	#368	1	4	yes	complemented
76	C₄×C₁₁	#384	1	1	yes	characteristic
77	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
78	C₄×C₁₁	#493	1	1	yes	characteristic
79	C₄×C₁₁	#500	1	1	yes	characteristic
80	C₂×M₁₆	—	22	4	no	—
81	C₂²×C₈	D₄₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
83	C₂×C₄²	D₄₄	1	1	yes	characteristic
84	C₂×C₁₆	Dic₂₂	1	4	yes	—
85	C₂×C₁₆	D₄₄	1	4	yes	—
86	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	—
87	#108	—	22	8	no	—
88	#108	—	44	4	no	—
89	C₄⋊C₈	—	44	4	no	—
90	C₄×C₈	Dic₂₂	1	1	yes	characteristic
91	C₄×C₈	Dic₂₂	1	1	yes	characteristic
92	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
93	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
94	C₂×C₁₁	—	2	1	no	—
95	C₂×C₁₁	—	2	1	no	—
96	C₂×C₁₁	#1845	1	1	yes	characteristic
97	C₂×C₁₁	C₂×#368	1	2	yes	—
98	C₂²×C₄	Dic₄₄	1	2	yes	—
99	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
100	M₁₆	—	44	4	no	—
101	M₁₆	—	44	4	no	—
102	C₂×C₈	—	44	4	no	—
103	C₂×C₈	D₈₈	1	2	yes	—
104	C₂×C₈	Dic₄₄	1	2	yes	—
105	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
106	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
107	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
108	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
109	C₂×C₈	C₂×D₄₄	1	2	yes	—
110	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
111	C₄²	D₈₈	1	2	yes	—
112	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
113	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
114	C₁₆	C₄⋊Dic₁₁	1	8	yes	—
115	C₁₁	#12909	1	1	yes	characteristic, complemented, Sylow
116	C₂³	C₂×Dic₄₄	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#66882	1	2	yes	—
119	C₂×C₄	#66882	1	2	yes	—
120	C₂×C₄	#66922	1	1	yes	characteristic
121	C₂×C₄	#66927	1	1	yes	characteristic
122	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
124	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
126	C₈	—	88	4	no	—
127	C₈	#66882	1	2	yes	—
128	C₈	#66882	1	2	yes	—
129	C₈	#66922	1	2	yes	—
130	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	#10622295	1	1	yes	characteristic
133	C₂²	#10622312	1	1	yes	characteristic
134	C₂²	C₂×#66882	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	#10622144	1	4	yes	complemented
137	C₄	#10622295	1	1	yes	characteristic
138	C₄	#10622308	1	1	yes	characteristic
139	C₄	#10622548	1	1	yes	characteristic
140	C₄	C₂×#66882	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#13068776	1	1	yes	characteristic
144	C₂	C₂×#10622144	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle	C₂²×C₁₁	C₂×#107
prime radical	C₂³×C₁₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#12909
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
75. C₄×C₁₁	#368	semidirect
115. C₁₁	#12909	semidirect
136. C₄	#10622144	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622144	2
#13068788	2
C₂×C₄×C₁₁×C₁₆	2
#12909	11