Subgroups — #28338066

Order 13696.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812049	C₂	1	2	yes	maximal
2	#20292405	C₂	1	4	yes	maximal, complemented
3	#405×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	#16812049	C₄	1	4	yes	complemented
5	#16812049	C₂²	1	4	yes	—
6	C₂×C₁₀₇⋊C₁₆	C₄	1	4	yes	complemented
7	C₂×M₁₆×C₁₀₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
9	#117×C₁₀₇	C₄	1	1	yes	characteristic
10	#97×C₁₀₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₇	C₄	1	2	yes	—
12	C₁₀₇⋊C₁₆	C₂×C₄	1	8	yes	—
13	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
14	M₁₆×C₁₀₇	—	2	2	no	—
15	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₇	—	2	1	no	—
21	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
22	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
23	C₂³×C₁₀₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₀₇	—	2	2	no	—
25	C₂×C₄×C₁₀₇	C₄²	1	2	yes	—
26	C₂×C₄×C₁₀₇	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₀₇	M₁₆	1	2	yes	—
28	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₁₀₇	—	2	2	no	—
30	C₈×C₁₀₇	C₄²	1	1	yes	characteristic
31	C₈×C₁₀₇	C₄²	1	1	yes	characteristic
32	C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
33	C₂²×C₁₀₇	—	4	1	no	—
34	C₂²×C₁₀₇	#97	1	2	yes	—
35	C₂²×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₁₀₇	—	4	2	no	—
37	C₄×C₁₀₇	#97	1	2	yes	—
38	C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₁₀₇	—	2	1	no	—
41	C₂×C₁₀₇	—	2	2	no	—
42	C₂×C₁₀₇	C₂×#97	1	1	yes	characteristic
43	#2110	—	107	1	no	maximal, Sylow
44	C₁₀₇	#2110	1	1	yes	characteristic, complemented, Sylow
45	C₂×M₃₂	—	107	2	no	—
46	#405	D₁₀₇	1	1	yes	characteristic
47	#347	—	107	4	no	—
48	C₂×M₁₆	Dic₁₀₇	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
50	#117	Dic₁₀₇	1	1	yes	characteristic
51	M₃₂	—	107	4	no	—
52	M₃₂	—	107	4	no	—
53	C₂×C₁₆	—	107	4	no	—
54	#97	C₂×D₁₀₇	1	2	yes	—
55	C₄×C₈	Dic₁₀₇	1	2	yes	—
56	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
59	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
60	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
61	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
62	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×Dic₁₀₇	1	2	yes	—
66	C₁₆	—	107	8	no	—
67	C₂³	#13133802	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄×Dic₁₀₇	1	2	yes	—
70	C₂×C₄	#13133802	1	2	yes	—
71	C₂×C₄	#13133802	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₄×Dic₁₀₇	1	1	yes	characteristic
75	C₈	C₄×Dic₁₀₇	1	1	yes	characteristic
76	C₈	C₂×C₄×D₁₀₇	1	2	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	#16812020	1	2	yes	—
79	C₂²	C₂×#13133802	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#16812020	1	2	yes	—
82	C₄	C₂×C₄×Dic₁₀₇	1	1	yes	characteristic
83	C₄	C₂×#13133802	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#16812020	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₀₇
commutator	C₄×C₁₀₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#405×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×#97
prime radical	C₂³×C₁₀₇	M₁₆
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	#405	D₁₀₇
hypocenter	C₁₀₇	#2110
LCS	G ▷ C₄×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20292405	C₂	semidirect
4. #16812049	C₄	semidirect
6. C₂×C₁₀₇⋊C₁₆	C₄	semidirect
44. C₁₀₇	#2110	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812049	2
#20292405	2
#405×C₁₀₇	2
#2110	107