Subgroups — #28338332

Order 13696.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₀₇	C₂	1	2	yes	maximal
2	#20292455	C₂	1	1	yes	maximal, characteristic
3	#20292431	C₂	1	2	yes	maximal
4	#20292402	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₇	—	2	2	no	—
7	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812043	C₂²	1	2	yes	—
9	#16812033	—	2	2	no	—
10	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₀₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₇	—	2	2	no	—
16	C₄⋊Dic₁₀₇	—	4	2	no	—
17	C₄×Dic₁₀₇	—	4	2	no	—
18	C₄×Dic₁₀₇	D₄	1	1	yes	characteristic
19	C₁₀₇⋊C₁₆	—	2	1	no	—
20	C₂×Q₈×C₁₀₇	—	2	2	no	—
21	Q₁₆×C₁₀₇	—	2	2	no	—
22	Q₁₆×C₁₀₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
25	Q₈×C₁₀₇	—	2	4	no	—
26	Q₈×C₁₀₇	—	4	2	no	—
27	C₂×C₄×C₁₀₇	—	4	2	no	—
28	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₇	—	4	2	no	—
30	C₂×Dic₁₀₇	D₈	1	2	yes	—
31	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₀₇	#41	1	1	yes	characteristic
33	C₁₀₇⋊C₈	—	2	1	no	—
34	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₀₇	—	4	4	no	—
36	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₀₇	#135	1	1	yes	characteristic
38	Dic₁₀₇	—	2	2	no	—
39	Dic₁₀₇	—	8	2	no	—
40	C₂×C₁₀₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₀₇	#510	1	2	yes	—
42	#2176	—	107	1	no	maximal, Sylow
43	C₁₀₇	#2176	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	D₁₀₇	1	1	yes	characteristic
45	C₄×Q₁₆	—	107	2	no	—
46	#367	—	107	1	no	—
47	#358	—	107	2	no	—
48	#346	—	107	1	no	—
49	C₂×Q₁₆	C₂×D₁₀₇	1	2	yes	—
50	C₄×Q₈	—	214	2	no	—
51	Q₃₂	Dic₁₀₇	1	4	yes	complemented
52	C₂×C₁₆	—	107	1	no	—
53	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
54	#107	—	107	2	no	—
55	#103	—	214	2	no	—
56	C₄×C₈	—	107	1	no	—
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×Dic₁₀₇	1	4	yes	—
60	C₂×C₈	—	107	1	no	—
61	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
62	C₄⋊C₄	—	428	2	no	—
63	C₄⋊C₄	—	214	2	no	—
64	C₄²	—	107	1	no	—
65	C₄²	—	428	2	no	—
66	C₁₆	—	214	1	no	—
67	C₁₆	C₂×Dic₁₀₇	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	107	2	no	—
72	C₂×C₄	—	428	2	no	—
73	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
74	C₈	—	214	1	no	—
75	C₈	C₂²×Dic₁₀₇	1	1	yes	characteristic
76	C₈	#13133817	1	1	yes	characteristic
77	C₂²	D₈×D₁₀₇	1	1	yes	characteristic
78	C₄	—	4	4	no	—
79	C₄	—	856	2	no	—
80	C₄	—	214	2	no	—
81	C₄	D₄×Dic₁₀₇	1	1	yes	characteristic
82	C₄	#16812128	1	1	yes	characteristic
83	C₂	D₈×Dic₁₀₇	1	1	yes	characteristic
84	C₂	#20293069	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₀₇
commutator	C₈×C₁₀₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₂×Q₃₂×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	C₂×Q₃₂	D₁₀₇
hypocenter	C₁₀₇	#2176
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
12. Q₃₂×C₁₀₇	C₄	semidirect
43. C₁₀₇	#2176	semidirect
51. Q₃₂	Dic₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₀₇	2
#20292455	2
#20292431	2
#20292402	2
C₂×Q₃₂×C₁₀₇	2
#2176	107