Subgroups — #14670214

Order 2816.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 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	2	yes	maximal
2	C₂×#10622142	C₂	1	2	yes	maximal
3	#13068780	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×#66882	C₂²	1	4	yes	—
9	C₂×#66881	—	2	2	no	—
10	#10622549	—	2	2	no	—
11	C₄×C₁₁×C₁₆	C₄	1	4	yes	complemented
12	#10622143	—	2	4	no	—
13	#10622142	—	2	4	no	—
14	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₁	—	2	4	no	—
18	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
21	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
22	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
23	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
24	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₁	Q₈	1	2	yes	—
26	C₁₁⋊#95	—	4	4	no	—
27	#66882	—	2	4	no	—
28	#66882	—	2	4	no	—
29	#66881	—	4	4	no	—
30	#12907	—	11	1	no	maximal, Sylow
31	C₂²×C₄×C₁₁	D₈	1	2	yes	—
32	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₁	—	4	4	no	—
34	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₁	D₈	1	2	yes	—
36	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
37	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₁	Q₁₆	1	2	yes	—
45	C₄⋊Dic₁₁	—	4	4	no	—
46	C₄⋊Dic₁₁	—	4	4	no	—
47	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	—
48	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	—
49	C₂×#367	—	11	2	no	—
50	C₂×#366	—	11	2	no	—
51	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic
52	#1846	—	11	2	no	—
53	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₁	—	2	4	no	—
55	C₂×C₄×C₁₁	#107	1	4	yes	—
56	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×Dic₁₁	—	8	4	no	—
63	C₂×Dic₁₁	—	8	8	no	—
64	C₈×C₁₁	#107	1	4	yes	—
65	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
71	C₂×#107	—	11	4	no	—
72	C₂×#106	—	22	2	no	—
73	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
74	#382	—	22	2	no	—
75	#367	—	22	4	no	—
76	#366	—	22	4	no	—
77	C₄×C₁₆	Dic₁₁	1	4	yes	complemented
78	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₁	#493	1	1	yes	characteristic
80	C₂²×C₁₁	#500	1	1	yes	characteristic
81	C₂²×C₁₁	#508	1	4	yes	—
82	C₄×C₁₁	—	2	4	no	—
83	C₄×C₁₁	#384	1	1	yes	characteristic
84	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
85	C₄×C₁₁	#494	1	1	yes	characteristic
86	C₄×C₁₁	#500	1	1	yes	characteristic
87	Dic₁₁	—	16	4	no	—
88	C₂²×C₈	D₄₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	22	4	no	—
91	C₂×C₄²	D₄₄	1	1	yes	characteristic
92	C₂×C₁₆	Dic₂₂	1	2	yes	—
93	C₂×C₁₆	Dic₂₂	1	2	yes	—
94	C₂×C₁₆	D₄₄	1	2	yes	—
95	C₂×C₁₆	D₄₄	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
97	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
98	#107	—	22	4	no	—
99	#107	—	22	4	no	—
100	#106	—	44	4	no	—
101	C₄×C₈	Dic₂₂	1	2	yes	—
102	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
103	#95	—	44	4	no	—
104	C₂×C₁₁	#1845	1	1	yes	characteristic
105	C₂×C₁₁	#2158	1	2	yes	—
106	C₂×C₁₁	#2249	1	2	yes	—
107	C₂×C₁₁	#2253	1	2	yes	—
108	C₂²×C₄	—	44	4	no	—
109	C₂²×C₄	D₈₈	1	2	yes	—
110	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
111	C₂×C₈	D₈₈	1	2	yes	—
112	C₂×C₈	Dic₄₄	1	2	yes	—
113	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
114	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
115	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
117	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
119	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
120	C₄⋊C₄	—	44	4	no	—
121	C₄⋊C₄	—	44	4	no	—
122	C₄²	Dic₄₄	1	2	yes	—
123	C₄²	C₄⋊Dic₁₁	1	2	yes	—
124	C₁₆	C₄⋊Dic₁₁	1	4	yes	—
125	C₁₆	C₄⋊Dic₁₁	1	4	yes	—
126	C₁₁	#12907	1	1	yes	characteristic, complemented, Sylow
127	C₂³	C₂×D₈₈	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	88	8	no	—
130	C₂×C₄	—	88	4	no	—
131	C₂×C₄	#66882	1	4	yes	—
132	C₂×C₄	#66922	1	1	yes	characteristic
133	C₂×C₄	#66927	1	1	yes	characteristic
134	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
138	C₈	#66882	1	4	yes	—
139	C₈	#66922	1	1	yes	characteristic
140	C₈	#66922	1	1	yes	characteristic
141	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
142	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
143	C₂²	#10622295	1	1	yes	characteristic
144	C₂²	#10622308	1	1	yes	characteristic
145	C₂²	#10622517	1	4	yes	—
146	C₂²	C₂×#66882	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	176	4	no	—
149	C₄	#10622295	1	1	yes	characteristic
150	C₄	#10622312	1	1	yes	characteristic
151	C₄	#10622548	1	1	yes	characteristic
152	C₄	C₂×#66882	1	1	yes	characteristic
153	C₂	#13068776	1	1	yes	characteristic
154	C₂	#13070400	1	2	yes	—
155	C₂	#13070877	1	2	yes	—
156	C₂	#13070895	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	#12907
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
11. C₄×C₁₁×C₁₆	C₄	semidirect
77. C₄×C₁₆	Dic₁₁	semidirect
126. C₁₁	#12907	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×#10622142	2
#13068780	2
C₂×C₄×C₁₁×C₁₆	2
#12907	11