Subgroups — #14754158

Order 2816.

1350 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622537	C₂	1	6	yes	maximal
2	#13068768	C₂	1	8	yes	maximal, complemented
3	C₂×C₄³×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₁	C₄	1	4	yes	—
5	C₂²×C₄²×C₁₁	C₂²	1	3	yes	—
6	C₂²×#4318	C₂²	1	4	yes	—
7	C₄³×C₁₁	C₄	1	3	yes	—
8	C₄³×C₁₁	C₄	1	1	yes	characteristic
9	C₄³×C₁₁	C₂²	1	4	yes	—
10	#10622537	—	2	24	no	—
11	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	24	yes	—
12	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₁	—	2	16	no	—
15	C₂×C₄²×C₁₁	C₂×C₄	1	12	yes	—
16	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
17	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
18	C₂×C₄²×C₁₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₁₁	D₄	1	12	yes	—
20	C₂×C₄²×C₁₁	Q₈	1	12	yes	—
21	C₂×C₄²×C₁₁	C₂³	1	6	yes	—
22	C₂×#4318	—	2	24	no	—
23	C₂²×C₁₁⋊C₈	C₂³	1	8	yes	—
24	C₄⋊(C₁₁⋊C₈)	—	2	48	no	—
25	#20791	—	11	1	no	maximal, Sylow
26	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₁	—	2	24	no	—
28	C₂²×C₄×C₁₁	C₄⋊C₄	1	24	yes	—
29	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
30	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
32	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
33	C₂²×C₄×C₁₁	C₂×Q₈	1	6	yes	—
34	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
35	C₄²×C₁₁	—	2	24	no	—
36	C₄²×C₁₁	C₄⋊C₄	1	12	yes	—
37	C₄²×C₁₁	C₄⋊C₄	1	12	yes	—
38	C₄²×C₁₁	M₁₆	1	16	yes	complemented
39	C₄²×C₁₁	C₂×D₄	1	12	yes	—
40	C₄²×C₁₁	C₂×Q₈	1	12	yes	—
41	#4318	—	4	16	no	—
42	C₂×C₁₁⋊C₈	—	2	24	no	—
43	C₂×#423	—	11	6	no	—
44	C₂×C₄³	D₁₁	1	1	yes	characteristic
45	#1842	—	11	8	no	—
46	C₂³×C₁₁	—	2	4	no	—
47	C₂³×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
48	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₁	—	2	12	no	—
50	C₂×C₄×C₁₁	—	2	48	no	—
51	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	12	yes	—
52	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
53	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	12	yes	—
54	C₂×C₄×C₁₁	#127	1	4	yes	—
55	C₂×C₄×C₁₁	C₄⋊Q₈	1	12	yes	—
56	C₂×C₄×C₁₁	C₂×M₁₆	1	24	yes	—
57	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
58	C₂×C₄×C₁₁	C₂²×Q₈	1	3	yes	—
59	C₁₁⋊C₈	—	4	8	no	—
60	C₂²×M₁₆	—	11	4	no	—
61	C₂²×C₄²	Dic₁₁	1	4	yes	—
62	C₂²×C₄²	C₂×D₁₁	1	3	yes	—
63	#423	—	22	24	no	—
64	C₂×C₄⋊C₈	—	11	24	no	—
65	C₄³	Dic₁₁	1	3	yes	—
66	C₄³	Dic₁₁	1	1	yes	characteristic
67	C₄³	C₂×D₁₁	1	4	yes	—
68	C₂²×C₁₁	—	2	12	no	—
69	C₂²×C₁₁	#384	1	4	yes	—
70	C₂²×C₁₁	C₂²×C₄⋊C₄	1	3	yes	—
71	C₂²×C₁₁	C₂²×M₁₆	1	4	yes	—
72	C₄×C₁₁	—	2	12	no	—
73	C₄×C₁₁	#384	1	4	yes	—
74	C₄×C₁₁	#423	1	24	yes	complemented
75	C₄×C₁₁	C₂×#127	1	1	yes	characteristic
76	C₄×C₁₁	C₂×C₄⋊Q₈	1	3	yes	—
77	C₂³×C₄	C₂×Dic₁₁	1	6	yes	—
78	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
79	C₂×M₁₆	—	22	24	no	—
80	C₂²×C₈	—	11	8	no	—
81	C₂×C₄²	—	2	16	no	—
82	C₂×C₄²	Dic₂₂	1	12	yes	—
83	C₂×C₄²	D₄₄	1	12	yes	—
84	C₂×C₄²	C₂×Dic₁₁	1	12	yes	—
85	C₂×C₄²	C₂×Dic₁₁	1	4	yes	—
86	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
87	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
88	C₂×C₄²	C₂²×D₁₁	1	6	yes	—
89	C₄⋊C₈	—	22	48	no	—
90	C₂×C₁₁	—	2	4	no	—
91	C₂×C₁₁	C₂×#384	1	1	yes	characteristic
92	C₂×C₁₁	C₂×#423	1	6	yes	—
93	C₂⁴	C₂²×Dic₁₁	1	1	yes	characteristic
94	C₂²×C₄	—	2	24	no	—
95	C₂²×C₄	C₄⋊Dic₁₁	1	24	yes	—
96	C₂²×C₄	C₂×Dic₂₂	1	6	yes	—
97	C₂²×C₄	C₂×D₄₄	1	6	yes	—
98	C₂²×C₄	C₂²×Dic₁₁	1	6	yes	—
99	C₂²×C₄	C₂²×Dic₁₁	1	6	yes	—
100	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
101	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
102	M₁₆	—	44	16	no	—
103	C₂×C₈	—	22	24	no	—
104	C₄²	—	2	24	no	—
105	C₄²	#4318	1	16	yes	complemented
106	C₄²	C₄⋊Dic₁₁	1	12	yes	—
107	C₄²	C₄⋊Dic₁₁	1	12	yes	—
108	C₄²	C₂×Dic₂₂	1	12	yes	—
109	C₄²	C₂×D₄₄	1	12	yes	—
110	C₁₁	#20791	1	1	yes	characteristic, complemented, Sylow
111	C₂³	—	2	4	no	—
112	C₂³	C₂×C₄⋊Dic₁₁	1	6	yes	—
113	C₂³	C₂³×Dic₁₁	1	1	yes	characteristic
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	#66922	1	12	yes	—
117	C₂×C₄	#66927	1	4	yes	—
118	C₂×C₄	C₂×#4318	1	24	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₁₁	1	12	yes	—
120	C₂×C₄	C₂×C₄⋊Dic₁₁	1	12	yes	—
121	C₂×C₄	C₂×C₄⋊Dic₁₁	1	6	yes	—
122	C₂×C₄	C₂²×Dic₂₂	1	3	yes	—
123	C₂×C₄	C₂²×D₄₄	1	3	yes	—
124	C₈	—	44	8	no	—
125	C₂²	—	2	12	no	—
126	C₂²	#10622548	1	4	yes	—
127	C₂²	C₂²×#4318	1	4	yes	—
128	C₂²	C₂²×C₄⋊Dic₁₁	1	3	yes	—
129	C₄	—	2	12	no	—
130	C₄	#10622537	1	24	yes	complemented
131	C₄	#10622548	1	4	yes	—
132	C₄	C₂×#66922	1	3	yes	—
133	C₄	C₂×#66927	1	1	yes	characteristic
134	C₂	—	2	4	no	—
135	C₂	C₂×#10622548	1	1	yes	characteristic
136	C₂	C₂×#10622537	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₁
commutator, socle	C₂³×C₁₁	C₂³×C₄
Fitting	C₂×C₄³×C₁₁	C₂
prime radical	C₂⁴×C₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₄³	D₁₁
hypocenter	C₁₁	#20791
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #13068768	C₂	semidirect
38. C₄²×C₁₁	M₁₆	semidirect
74. C₄×C₁₁	#423	semidirect
105. C₄²	#4318	semidirect
110. C₁₁	#20791	semidirect
130. C₄	#10622537	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622537	2
#13068768	2
C₂×C₄³×C₁₁	2
#20791	11