Subgroups — #28334448

Order 13696.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20292885	C₂	1	1	yes	maximal, characteristic
2	#20292409	C₂	1	1	yes	maximal, characteristic
3	H(3,ℤ/4)×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₈	C₄	1	1	yes	characteristic, complemented
5	#16812088	—	2	1	no	—
6	#16812049	—	2	1	no	—
7	#16812041	C₄	1	1	yes	characteristic, complemented
8	#16812034	—	2	1	no	—
9	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
10	#117×C₁₀₇	—	2	1	no	—
11	C₂×Dic₂₁₄	—	2	1	no	—
12	Dic₄₂₈	—	4	1	no	—
13	C₄⋊Dic₁₀₇	—	2	1	no	—
14	Dic₁₀₇⋊C₄	—	4	1	no	—
15	C₂²⋊Dic₁₀₇	—	4	1	no	—
16	C₁₀₇⋊C₁₆	—	4	1	no	—
17	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
18	M₁₆×C₁₀₇	—	4	1	no	—
19	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₇	—	4	1	no	—
22	C₂²⋊C₄×C₁₀₇	—	4	1	no	—
23	C₄²×C₁₀₇	—	2	2	no	—
24	C₂³×C₁₀₇	D₈	1	1	yes	characteristic
25	C₂×C₄×C₁₀₇	—	2	1	no	—
26	C₂×C₄×C₁₀₇	—	2	1	no	—
27	C₂×C₄×C₁₀₇	—	4	2	no	—
28	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
30	C₂×Dic₁₀₇	—	4	1	no	—
31	C₂×Dic₁₀₇	—	4	1	no	—
32	Dic₂₁₄	—	4	1	no	—
33	Dic₂₁₄	—	4	1	no	—
34	C₈×C₁₀₇	—	2	1	no	—
35	C₈×C₁₀₇	—	2	1	no	—
36	C₂²×C₁₀₇	—	2	1	no	—
37	C₂²×C₁₀₇	—	4	1	no	—
38	C₂²×C₁₀₇	#102	1	1	yes	characteristic
39	C₄×C₁₀₇	—	2	1	no	—
40	C₄×C₁₀₇	—	8	2	no	—
41	C₄×C₁₀₇	#99	1	1	yes	characteristic
42	C₄×C₁₀₇	#104	1	1	yes	characteristic
43	Dic₁₀₇	—	8	1	no	—
44	Dic₁₀₇	—	8	1	no	—
45	C₂×C₁₀₇	—	2	1	no	—
46	C₂×C₁₀₇	—	4	1	no	—
47	C₂×C₁₀₇	#327	1	1	yes	characteristic
48	#1340	—	107	1	no	maximal, Sylow
49	C₁₀₇	#1340	1	1	yes	characteristic, complemented, Sylow
50	#470	—	107	1	no	—
51	#349	—	107	1	no	—
52	H(3,ℤ/4)	D₁₀₇	1	1	yes	characteristic
53	C₂×Q₁₆	—	107	1	no	—
54	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	214	1	no	—
56	#117	—	2	1	no	—
57	M₃₂	—	214	1	no	—
58	#106	—	107	1	no	—
59	#103	—	214	1	no	—
60	C₂×Q₈	—	214	1	no	—
61	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
62	Q₁₆	—	428	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₄×D₁₀₇	1	1	yes	characteristic
65	C₂×C₈	D₄₂₈	1	1	yes	characteristic
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	428	1	no	—
68	C₄⋊C₄	—	214	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	428	1	no	—
71	C₄²	—	2	2	no	—
72	C₁₆	—	428	1	no	—
73	C₂³	#13133803	1	1	yes	characteristic
74	Q₈	—	428	1	no	—
75	Q₈	—	428	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	428	1	no	—
80	C₂×C₄	—	428	1	no	—
81	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
82	C₂×C₄	#13133805	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#16812029	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	2	no	—
90	C₄	—	856	1	no	—
91	C₄	—	856	1	no	—
92	C₄	#16812023	1	1	yes	characteristic
93	C₄	#16812037	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#20292360	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292360
commutator	C₂×C₈×C₁₀₇	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₁₀₇
Fitting	H(3,ℤ/4)×C₁₀₇	C₂
socle	C₂×C₁₀₇	#327
prime radical	C₂³×C₁₀₇	D₈
cosocle	C₂×M₁₆×C₁₀₇	C₂²
hypercenter	H(3,ℤ/4)	D₁₀₇
hypocenter	C₁₀₇	#1340
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₄₂₈	C₄	semidirect
7. #16812041	C₄	semidirect
49. C₁₀₇	#1340	semidirect

Maximal subgroups

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

subgroup	index
#20292885	2
#20292409	2
H(3,ℤ/4)×C₁₀₇	2
#1340	107