Subgroups — #28336618

Order 13696.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812041	C₂	1	4	yes	maximal
2	#20292497	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₁₀₇	—	2	4	no	—
5	C₂×C₄⋊Dic₁₀₇	C₂²	1	4	yes	—
6	#16812041	—	2	8	no	—
7	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₀₇	C₄	1	4	yes	complemented
10	C₂²×Dic₁₀₇	—	2	4	no	—
11	C₄⋊Dic₁₀₇	—	2	8	no	—
12	C₄⋊Dic₁₀₇	—	4	8	no	—
13	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
14	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₀₇	D₄	1	4	yes	—
17	C₂×C₈×C₁₀₇	Q₈	1	4	yes	—
18	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₄²×C₁₀₇	Q₈	1	2	yes	—
20	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₁₀₇	SD₁₆	1	8	yes	—
23	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
26	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×Dic₁₀₇	—	4	4	no	—
28	C₂×Dic₁₀₇	—	4	8	no	—
29	C₈×C₁₀₇	C₄⋊C₄	1	8	yes	complemented
30	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₁₀₇	#127	1	1	yes	characteristic
32	C₂²×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₂²×C₁₀₇	C₂×SD₁₆	1	4	yes	—
34	C₄×C₁₀₇	#106	1	8	yes	complemented
35	C₄×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
36	C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
37	Dic₁₀₇	—	8	4	no	—
38	C₂×C₁₀₇	#384	1	1	yes	characteristic
39	C₂×C₁₀₇	C₂×#106	1	2	yes	—
40	C₂×C₁₀₇	#492	1	2	yes	—
41	C₂×C₁₀₇	#498	1	2	yes	—
42	#1844	—	107	1	no	maximal, Sylow
43	C₁₀₇	#1844	1	1	yes	characteristic, complemented, Sylow
44	C₂×#106	—	107	4	no	—
45	C₂×C₄×C₈	D₁₀₇	1	1	yes	characteristic
46	#384	—	107	2	no	—
47	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
48	C₂×C₄⋊C₄	—	107	4	no	—
49	C₂×C₄⋊C₄	—	214	4	no	—
50	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
51	#106	—	214	8	no	—
52	C₄×C₈	Dic₁₀₇	1	4	yes	complemented
53	C₂²×C₄	—	214	4	no	—
54	C₂²×C₄	D₄₂₈	1	2	yes	—
55	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
56	C₂×C₈	Dic₂₁₄	1	4	yes	—
57	C₂×C₈	D₄₂₈	1	4	yes	—
58	C₂×C₈	C₂×Dic₁₀₇	1	4	yes	—
59	C₄⋊C₄	—	428	8	no	—
60	C₄⋊C₄	—	214	8	no	—
61	C₄²	Dic₂₁₄	1	2	yes	—
62	C₄²	C₂×Dic₁₀₇	1	2	yes	—
63	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
64	C₂×C₄	—	428	4	no	—
65	C₂×C₄	—	428	8	no	—
66	C₂×C₄	C₄⋊Dic₁₀₇	1	4	yes	—
67	C₂×C₄	#13133807	1	8	yes	—
68	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
70	C₂×C₄	C₂×Dic₂₁₄	1	2	yes	—
71	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
72	C₈	C₄⋊Dic₁₀₇	1	8	yes	complemented
73	C₂²	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
74	C₂²	#16812106	1	1	yes	characteristic
75	C₂²	#16812109	1	1	yes	characteristic
76	C₂²	C₂×#13133807	1	4	yes	—
77	C₄	—	856	4	no	—
78	C₄	#16812041	1	8	yes	complemented
79	C₄	C₂×C₄⋊Dic₁₀₇	1	2	yes	—
80	C₄	#16812109	1	2	yes	—
81	C₂	#20292497	1	1	yes	characteristic
82	C₂	C₂×#16812041	1	2	yes	—
83	C₂	#20293000	1	2	yes	—
84	C₂	#20293025	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₈
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×C₄×C₈×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₇
hypocenter	C₁₀₇	#1844
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₁₀₇	C₄	semidirect
29. C₈×C₁₀₇	C₄⋊C₄	semidirect
34. C₄×C₁₀₇	#106	semidirect
43. C₁₀₇	#1844	semidirect
52. C₄×C₈	Dic₁₀₇	semidirect
72. C₈	C₄⋊Dic₁₀₇	semidirect
78. C₄	#16812041	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812041	2
#20292497	2
C₂×C₄×C₈×C₁₀₇	2
#1844	107