Subgroups — #28338712

Order 13696.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812055	C₂	1	2	yes	maximal
3	#20293005	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₈	C₂²	1	2	yes	—
7	C₂×#13133808	—	2	2	no	—
8	#16812107	—	2	2	no	—
9	Dic₈₅₆	—	2	2	no	—
10	#16812055	—	2	4	no	—
11	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
14	Q₃₂×C₁₀₇	—	2	2	no	—
15	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₄	—	2	2	no	—
17	Dic₄₂₈	—	2	2	no	—
18	Dic₄₂₈	—	2	2	no	—
19	#13133808	—	4	4	no	—
20	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
21	Dic₁₀₇⋊C₄	—	4	4	no	—
22	C₄×Dic₁₀₇	D₄	1	1	yes	characteristic
23	C₁₀₇⋊C₁₆	D₄	1	2	yes	—
24	C₂×Q₈×C₁₀₇	—	2	2	no	—
25	Q₁₆×C₁₀₇	—	2	2	no	—
26	Q₁₆×C₁₀₇	—	2	2	no	—
27	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₀₇	D₄	1	2	yes	—
29	Q₈×C₁₀₇	—	4	2	no	—
30	Q₈×C₁₀₇	—	4	2	no	—
31	C₂×C₄×C₁₀₇	—	4	2	no	—
32	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₀₇	—	4	2	no	—
34	C₂×Dic₁₀₇	D₈	1	2	yes	—
35	Dic₂₁₄	—	4	2	no	—
36	Dic₂₁₄	—	4	2	no	—
37	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₁₀₇⋊C₈	D₈	1	2	yes	—
40	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
41	C₄×C₁₀₇	—	8	2	no	—
42	C₄×C₁₀₇	#127	1	1	yes	characteristic
43	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₀₇	—	8	2	no	—
45	Dic₁₀₇	Q₃₂	1	4	yes	complemented
46	C₂×C₁₀₇	#493	1	1	yes	characteristic
47	C₂×C₁₀₇	C₂×Q₃₂	1	2	yes	—
48	#2247	—	107	1	no	maximal, Sylow
49	C₁₀₇	#2247	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₃₂	—	107	2	no	—
51	C₂×Q₃₂	—	107	1	no	—
52	C₂×Q₃₂	D₁₀₇	1	1	yes	characteristic
53	#494	—	107	2	no	—
54	C₄×C₁₆	—	107	1	no	—
55	C₂×Q₁₆	—	107	2	no	—
56	C₂×Q₁₆	—	214	2	no	—
57	C₂×Q₁₆	C₂×D₁₀₇	1	2	yes	—
58	C₄⋊Q₈	—	214	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	214	4	no	—
61	Q₃₂	—	214	2	no	—
62	C₂×C₁₆	—	107	1	no	—
63	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
64	C₄×C₈	—	107	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	214	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	428	4	no	—
70	Q₁₆	—	214	2	no	—
71	Q₁₆	—	214	2	no	—
72	C₂×C₈	—	107	1	no	—
73	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
74	C₄⋊C₄	—	428	4	no	—
75	C₄²	—	107	1	no	—
76	C₁₆	—	107	2	no	—
77	C₁₆	C₂²⋊D₁₀₇	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	428	2	no	—
81	Q₈	—	428	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	107	2	no	—
84	C₂×C₄	—	428	2	no	—
85	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
86	C₈	—	107	2	no	—
87	C₈	D₄×D₁₀₇	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
89	C₂²	D₈×D₁₀₇	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	856	2	no	—
92	C₄	—	107	4	no	—
93	C₄	#16812105	1	1	yes	characteristic
94	C₄	D₈×D₁₀₇	1	1	yes	characteristic
95	C₂	#20293002	1	1	yes	characteristic
96	C₂	Q₃₂×D₁₀₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₀₇
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₂×Q₃₂×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₁₀₇
hypocenter	C₁₀₇	#2247
LCS	G ▷ C₂×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
45. Dic₁₀₇	Q₃₂	semidirect
49. C₁₀₇	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₅₆	2
C₂×#16812055	2
#20293005	2
C₁₆×Dic₁₀₇	2
C₂×Q₃₂×C₁₀₇	2
#2247	107