Subgroups — #20295034

Order 6976.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182013	C₂	1	1	yes	maximal, characteristic
3	#16812622	C₂	1	4	yes	maximal, complemented
4	#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₈	—	2	2	no	—
6	C₂×Dic₂₁₈	—	2	1	no	—
7	C₂×Dic₂₁₈	C₄	1	4	yes	complemented
8	C₂×Dic₂₁₈	C₂²	1	2	yes	—
9	C₂²×Dic₁₀₉	—	2	1	no	—
10	#13182013	—	2	2	no	—
11	C₂×#11724784	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
15	C₄²×C₁₀₉	—	2	1	no	—
16	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
20	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₀₉	—	2	4	no	—
23	C₂×Dic₁₀₉	—	2	2	no	—
24	Dic₂₁₈	—	2	4	no	—
25	Dic₂₁₈	—	2	2	no	—
26	Dic₂₁₈	C₂×C₄	1	4	yes	—
27	#11724784	—	2	2	no	—
28	C₂²×C₁₀₉	—	2	2	no	—
29	C₂²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₀₉	—	4	2	no	—
32	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
33	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	Dic₁₀₉	—	2	4	no	—
36	C₂×C₁₀₉	—	2	2	no	—
37	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₁₀₉	#137	1	2	yes	—
39	C₁₀₉	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	109	1	no	maximal, Sylow
41	C₂²×Q₈	—	109	1	no	—
42	C₂×M₁₆	—	109	1	no	—
43	#117	D₁₀₉	1	1	yes	characteristic
44	#103	—	109	4	no	—
45	C₂×Q₈	—	109	4	no	—
46	C₂×Q₈	—	109	2	no	—
47	C₂×Q₈	—	218	2	no	—
48	C₂×Q₈	—	218	1	no	—
49	C₂²×C₄	—	218	1	no	—
50	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
51	M₁₆	—	218	2	no	—
52	C₂×C₈	—	109	2	no	—
53	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
57	Q₈	—	109	4	no	—
58	Q₈	—	218	4	no	—
59	Q₈	—	218	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	218	4	no	—
62	C₂×C₄	—	218	2	no	—
63	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
64	C₂×C₄	D₄₃₆	1	2	yes	—
65	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
67	C₈	—	218	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	#13182006	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	218	4	no	—
73	C₄	#13182006	1	2	yes	—
74	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
75	C₄	C₂×D₄₃₆	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×#13182006	1	1	yes	characteristic
78	C₂	#16812729	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#117×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#117	D₁₀₉
hypocenter	C₁₀₉	#419
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #16812622	C₂	semidirect
7. C₂×Dic₂₁₈	C₄	semidirect
39. C₁₀₉	#419	semidirect