Subgroups — #20295303

Order 6976.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄₃₆	C₂	1	1	yes	maximal, characteristic
2	#16812632	C₂	1	2	yes	maximal, complemented
3	C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812619	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₀₉	—	2	1	no	—
8	C₂×#11724784	C₂²	1	2	yes	—
9	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
10	#13182006	—	2	1	no	—
11	C₂×Q₈×C₁₀₉	—	2	1	no	—
12	C₄⋊C₄×C₁₀₉	—	2	1	no	—
13	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₀₉	—	2	1	no	—
16	Q₈×C₁₀₉	—	4	2	no	—
17	C₂×C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	1	no	—
22	D₄₃₆	—	2	1	no	—
23	D₄₃₆	Q₈	1	2	yes	complemented
24	C₄×D₁₀₉	—	4	2	no	—
25	#11724784	—	2	2	no	—
26	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₀₉	—	2	2	no	—
28	C₂×D₁₀₉	—	4	1	no	—
29	C₂×D₁₀₉	—	4	1	no	—
30	C₄×C₁₀₉	—	2	1	no	—
31	C₄×C₁₀₉	—	4	2	no	—
32	C₄×C₁₀₉	D₈	1	2	yes	—
33	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₀₉	#41	1	1	yes	characteristic
35	Dic₁₀₉	—	4	1	no	—
36	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
38	C₂×C₁₀₉	#137	1	1	yes	characteristic
39	D₁₀₉	—	4	2	no	—
40	C₁₀₉	#474	1	1	yes	characteristic, complemented, Sylow
41	#474	—	109	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
43	C₄×D₄	—	109	1	no	—
44	#107	—	109	2	no	—
45	C₄⋊C₈	—	109	1	no	—
46	#102	—	109	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	109	1	no	—
49	C₂²×C₄	—	218	1	no	—
50	C₂×C₈	—	109	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	109	1	no	—
53	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
54	C₂²⋊C₄	—	218	1	no	—
55	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
56	C₂³	—	218	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	109	2	no	—
59	D₄	—	218	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	436	2	no	—
62	C₂×C₄	—	218	1	no	—
63	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
66	C₈	—	218	2	no	—
67	C₂²	—	436	1	no	—
68	C₂²	—	436	1	no	—
69	C₂²	—	218	2	no	—
70	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	436	1	no	—
74	C₄	#13182014	1	2	yes	—
75	C₄	Q₈×D₁₀₉	1	1	yes	characteristic
76	C₄	#13182030	1	1	yes	characteristic
77	C₂	—	436	2	no	—
78	C₂	#16812675	1	1	yes	characteristic
79	C₂	C₂×#13182014	1	1	yes	characteristic
80	C₂	#16812730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₄⋊Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#474
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16812632	C₂	semidirect
3. C₄⋊#11724784	C₂	semidirect
5. C₄⋊Q₈×C₁₀₉	C₂	semidirect
23. D₄₃₆	Q₈	semidirect
40. C₁₀₉	#474	semidirect
42. C₄⋊Q₈	D₁₀₉	semidirect