Subgroups — #28334565

Order 13696.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812045	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133802	—	2	2	no	—
4	#16812045	—	2	2	no	—
5	#16812045	C₄	1	4	yes	—
6	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₁₀₇	—	2	2	no	—
8	C₂×C₁₆×C₁₀₇	C₄	1	2	yes	—
9	#13133802	—	2	4	no	—
10	#13133802	—	4	2	no	—
11	C₂×C₁₀₇⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
16	C₂×C₈×C₁₀₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₀₇	—	2	2	no	—
18	C₂³×C₁₀₇	Q₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₀₇	SD₁₆	1	2	yes	—
23	C₈×C₁₀₇	C₄²	1	1	yes	characteristic
24	C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
26	C₁₀₇⋊C₈	—	4	4	no	—
27	C₂²×C₁₀₇	—	4	1	no	—
28	C₂²×C₁₀₇	#103	1	2	yes	—
29	C₂²×C₁₀₇	#107	1	1	yes	characteristic
30	C₄×C₁₀₇	#95	1	1	yes	characteristic
31	C₄×C₁₀₇	#102	1	2	yes	—
32	C₄×C₁₀₇	#106	1	1	yes	characteristic
33	C₂×C₁₀₇	—	2	2	no	—
34	C₂×C₁₀₇	—	2	1	no	—
35	C₂×C₁₀₇	#340	1	1	yes	characteristic
36	#1385	—	107	1	no	maximal, Sylow
37	C₁₀₇	#1385	1	1	yes	characteristic, complemented, Sylow
38	C₂×M₃₂	D₁₀₇	1	1	yes	characteristic
39	C₂×#108	—	107	2	no	—
40	C₂×M₁₆	—	214	2	no	—
41	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	Dic₁₀₇	1	2	yes	—
44	#108	—	107	4	no	—
45	#108	—	214	2	no	—
46	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
47	M₁₆	—	428	2	no	—
48	M₁₆	—	214	4	no	—
49	C₂×C₈	—	428	2	no	—
50	C₂×C₈	Dic₂₁₄	1	1	yes	characteristic
51	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
52	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₄₂₈	1	1	yes	characteristic
56	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₁₀₇	1	1	yes	characteristic
58	C₂×C₄	D₈₅₆	1	1	yes	characteristic
59	C₂×C₄	#13133807	1	2	yes	—
60	C₈	—	428	4	no	—
61	C₈	C₄×Dic₁₀₇	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
63	C₈	Dic₁₀₇⋊C₄	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#16812034	1	2	yes	—
66	C₂²	#16812043	1	1	yes	characteristic
67	C₄	C₁₀₇⋊#95	1	1	yes	characteristic
68	C₄	#16812031	1	2	yes	—
69	C₄	#16812041	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#20292390	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16812041
commutator	C₈×C₁₀₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₀₇
Fitting	C₂×M₃₂×C₁₀₇	C₂
socle	C₂×C₁₀₇	#340
prime radical	C₂³×C₁₀₇	Q₁₆
cosocle	C₂²×C₈×C₁₀₇	C₂²
hypercenter	C₂×M₃₂	D₁₀₇
hypocenter	C₁₀₇	#1385
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
37. C₁₀₇	#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812045	2
C₂×M₃₂×C₁₀₇	2
#1385	107