Subgroups — #20295259

Order 6976.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812670	C₂	1	2	yes	maximal, complemented
3	#16812633	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812620	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
7	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#11724784	C₂²	1	1	yes	characteristic
9	C₈×D₁₀₉	—	2	2	no	—
10	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²⋊Dic₁₀₉	—	2	2	no	—
12	C₂×D₄×C₁₀₉	C₂²	1	2	yes	—
13	D₈×C₁₀₉	—	2	2	no	—
14	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×C₁₀₉	—	2	2	no	—
16	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
17	D₄×C₁₀₉	—	2	2	no	—
18	D₄×C₁₀₉	—	2	2	no	—
19	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂²⋊D₁₀₉	—	4	4	no	—
21	C₂×Dic₁₀₉	—	2	2	no	—
22	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₄×D₁₀₉	—	2	1	no	—
24	C₄×D₁₀₉	—	2	1	no	—
25	C₈×C₁₀₉	D₄	1	2	yes	complemented
26	#11724784	—	2	1	no	—
27	C₂²×C₁₀₉	—	4	2	no	—
28	C₂²×C₁₀₉	—	4	2	no	—
29	C₂²×C₁₀₉	—	4	2	no	—
30	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×D₁₀₉	—	2	1	no	—
32	C₂×D₁₀₉	—	2	1	no	—
33	C₂×D₁₀₉	D₈	1	2	yes	complemented
34	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₄×C₁₀₉	#41	1	1	yes	characteristic
36	Dic₁₀₉	—	2	1	no	—
37	Dic₁₀₉	—	4	2	no	—
38	C₂×C₁₀₉	—	8	2	no	—
39	C₂×C₁₀₉	#121	1	1	yes	characteristic
40	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
41	C₂×C₁₀₉	#135	1	1	yes	characteristic
42	D₁₀₉	—	2	2	no	—
43	C₁₀₉	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	109	1	no	maximal, Sylow
45	C₂×D₈	D₁₀₉	1	1	yes	characteristic, complemented
46	C₂²×C₈	—	109	1	no	—
47	#121	—	109	2	no	—
48	#107	—	109	1	no	—
49	#102	—	109	2	no	—
50	C₂×D₄	—	218	2	no	—
51	C₂×D₄	C₂×D₁₀₉	1	2	yes	—
52	C₂²×C₄	—	109	1	no	—
53	D₈	—	2	2	no	—
54	C₂×C₈	—	109	1	no	—
55	C₂×C₈	—	218	2	no	—
56	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
57	C₄⋊C₄	—	109	2	no	—
58	C₂²⋊C₄	—	218	2	no	—
59	C₂³	—	2	2	no	—
60	C₂³	—	109	1	no	—
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	—	436	4	no	—
64	C₂×C₄	—	109	1	no	—
65	C₂×C₄	—	218	2	no	—
66	C₂×C₄	—	218	1	no	—
67	C₂×C₄	—	218	1	no	—
68	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
69	C₈	—	218	1	no	—
70	C₈	C₂²⋊D₁₀₉	1	2	yes	complemented
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	109	2	no	—
75	C₂²	—	218	1	no	—
76	C₂²	—	218	1	no	—
77	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
78	C₄	—	436	2	no	—
79	C₄	—	218	1	no	—
80	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
81	C₄	#13182028	1	1	yes	characteristic
82	C₂	—	8	2	no	—
83	C₂	—	218	2	no	—
84	C₂	#16812670	1	1	yes	characteristic
85	C₂	D₈×D₁₀₉	1	1	yes	characteristic
86	C₂	#16812718	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×D₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×D₈	D₁₀₉
hypocenter	C₁₀₉	#466
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₁₀₉	C₂	semidirect
2. #16812670	C₂	semidirect
3. #16812633	C₂	semidirect
4. #16812620	C₂	semidirect
5. C₂×D₈×C₁₀₉	C₂	semidirect
25. C₈×C₁₀₉	D₄	semidirect
33. C₂×D₁₀₉	D₈	semidirect
43. C₁₀₉	#466	semidirect
45. C₂×D₈	D₁₀₉	semidirect
70. C₈	C₂²⋊D₁₀₉	semidirect