Subgroups — #28334509

Order 13696.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293007	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₀₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₈	—	2	1	no	—
5	C₂×Dic₄₂₈	C₄	1	2	yes	—
6	#16812109	—	2	1	no	—
7	C₂×C₁₀₇⋊C₁₆	—	2	1	no	—
8	C₄⋊C₈×C₁₀₇	—	2	1	no	—
9	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁₄	—	2	2	no	—
11	Dic₄₂₈	—	4	2	no	—
12	Dic₄₂₈	—	4	1	no	—
13	Dic₄₂₈	—	4	1	no	—
14	C₄⋊Dic₁₀₇	—	4	1	no	—
15	C₄⋊Dic₁₀₇	—	4	1	no	—
16	C₁₀₇⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₀₇	—	4	1	no	—
18	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
24	C₂×Dic₁₀₇	—	4	2	no	—
25	Dic₂₁₄	—	4	2	no	—
26	Dic₂₁₄	—	4	2	no	—
27	C₈×C₁₀₇	—	2	1	no	—
28	C₈×C₁₀₇	—	8	1	no	—
29	C₈×C₁₀₇	D₈	1	1	yes	characteristic
30	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₀₇	#102	1	1	yes	characteristic
32	C₄×C₁₀₇	—	2	1	no	—
33	C₄×C₁₀₇	#100	1	1	yes	characteristic
34	C₄×C₁₀₇	#102	1	1	yes	characteristic
35	C₄×C₁₀₇	SD₃₂	1	1	yes	characteristic
36	C₄×C₁₀₇	Q₃₂	1	1	yes	characteristic
37	Dic₁₀₇	—	8	2	no	—
38	C₂×C₁₀₇	#331	1	1	yes	characteristic
39	C₂×C₁₀₇	#358	1	1	yes	characteristic
40	C₂×C₁₀₇	#362	1	1	yes	characteristic
41	#1362	—	107	1	no	maximal, Sylow
42	C₁₀₇	#1362	1	1	yes	characteristic, complemented, Sylow
43	#494	—	107	1	no	—
44	C₄⋊C₁₆	—	107	1	no	—
45	#335	D₁₀₇	1	1	yes	characteristic
46	C₂×Q₁₆	—	107	2	no	—
47	C₂×Q₁₆	—	214	1	no	—
48	C₄⋊Q₈	—	214	1	no	—
49	C₂×C₁₆	—	214	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
52	C₂×Q₈	—	214	2	no	—
53	Q₁₆	—	428	2	no	—
54	Q₁₆	—	428	1	no	—
55	Q₁₆	—	428	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	C₄×D₁₀₇	1	1	yes	characteristic
58	C₂×C₈	D₄₂₈	1	1	yes	characteristic
59	C₄⋊C₄	—	428	1	no	—
60	C₄⋊C₄	—	428	1	no	—
61	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
62	C₁₆	—	428	1	no	—
63	Q₈	—	428	2	no	—
64	Q₈	—	428	2	no	—
65	C₂×C₄	—	428	2	no	—
66	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
67	C₂×C₄	#13133803	1	1	yes	characteristic
68	C₂×C₄	#13133805	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	8	1	no	—
71	C₈	D₈₅₆	1	1	yes	characteristic
72	C₈	#13133807	1	1	yes	characteristic
73	C₂²	#16812029	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	856	2	no	—
76	C₄	#16812025	1	1	yes	characteristic
77	C₄	#16812031	1	1	yes	characteristic
78	C₄	#16812053	1	1	yes	characteristic
79	C₄	#16812055	1	1	yes	characteristic
80	C₂	#20292370	1	1	yes	characteristic
81	C₂	#20292430	1	1	yes	characteristic
82	C₂	#20292442	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812029
commutator	C₂×C₈×C₁₀₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₇
Fitting	#335×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	#102
cosocle	C₄×C₈×C₁₀₇	C₂²
hypercenter	#335	D₁₀₇
hypocenter	C₁₀₇	#1362
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
42. C₁₀₇	#1362	semidirect

Maximal subgroups

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

subgroup	index
#20293007	2
C₄⋊(C₁₀₇⋊C₁₆)	2
#335×C₁₀₇	2
#1362	107