Subgroups — #28336213

Order 13696.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20292382	C₂	1	4	yes	maximal
4	#376×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	#16812020	C₄	1	4	yes	complemented
8	C₄×C₁₀₇⋊C₈	C₄	1	4	yes	complemented
9	C₂×C₄²×C₁₀₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
11	#95×C₁₀₇	C₄	1	4	yes	—
12	C₂×C₁₀₇⋊C₈	—	2	4	no	—
13	C₂×C₁₀₇⋊C₈	—	2	4	no	—
14	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₀₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
19	C₄²×C₁₀₇	—	2	4	no	—
20	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
21	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
22	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₇	—	2	4	no	—
24	C₂×C₄×C₁₀₇	—	2	4	no	—
25	C₂×C₄×C₁₀₇	—	2	8	no	—
26	C₂×C₄×C₁₀₇	C₄²	1	4	yes	—
27	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	#41	1	4	yes	—
30	C₁₀₇⋊C₈	—	2	4	no	—
31	C₁₀₇⋊C₈	—	2	4	no	—
32	C₂²×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₁₀₇	#117	1	2	yes	—
34	C₂²×C₁₀₇	#131	1	4	yes	—
35	C₄×C₁₀₇	—	2	4	no	—
36	C₄×C₁₀₇	—	4	4	no	—
37	C₄×C₁₀₇	#117	1	2	yes	—
38	C₄×C₁₀₇	#117	1	2	yes	—
39	C₂×C₁₀₇	#376	1	1	yes	characteristic
40	C₂×C₁₀₇	#405	1	2	yes	—
41	C₂×C₁₀₇	#433	1	4	yes	—
42	#1746	—	107	1	no	maximal, Sylow
43	C₁₀₇	#1746	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	—	107	1	no	—
45	C₂×C₄×C₈	—	107	1	no	—
46	#376	D₁₀₇	1	1	yes	characteristic
47	#336	—	107	4	no	—
48	C₂²×C₈	—	107	2	no	—
49	C₂²×C₈	—	107	2	no	—
50	C₂×C₄²	C₂×D₁₀₇	1	2	yes	—
51	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
52	#97	—	107	4	no	—
53	C₄×C₈	—	107	4	no	—
54	#95	Dic₁₀₇	1	4	yes	—
55	C₂²×C₄	C₂×Dic₁₀₇	1	4	yes	—
56	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
58	C₂×C₈	—	107	4	no	—
59	C₂×C₈	—	107	4	no	—
60	C₂×C₈	—	214	4	no	—
61	C₂×C₈	—	214	4	no	—
62	C₄²	—	2	4	no	—
63	C₄²	C₄×D₁₀₇	1	2	yes	—
64	C₄²	C₄×D₁₀₇	1	2	yes	—
65	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	C₄×Dic₁₀₇	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
71	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
72	C₂×C₄	#13133818	1	4	yes	—
73	C₈	—	214	4	no	—
74	C₈	—	214	4	no	—
75	C₂²	C₂×C₄×Dic₁₀₇	1	1	yes	characteristic
76	C₂²	#16812068	1	2	yes	—
77	C₂²	#16812116	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	4	4	no	—
80	C₄	#16812066	1	2	yes	—
81	C₄	#16812066	1	2	yes	—
82	C₂	#20292473	1	1	yes	characteristic
83	C₂	#20292583	1	2	yes	—
84	C₂	#20292688	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₇
commutator	C₂²×C₁₀₇	C₂×C₄²
Fitting	#376×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#376	D₁₀₇
hypocenter	C₁₀₇	#1746
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
7. #16812020	C₄	semidirect
8. C₄×C₁₀₇⋊C₈	C₄	semidirect
43. C₁₀₇	#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812020	2
C₂×C₄×C₁₀₇⋊C₈	2
#20292382	2
#376×C₁₀₇	2
#1746	107