Subgroups — #14271835

Order 1856.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#90809	C₂	1	2	yes	maximal, complemented
2	C₄×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11745257	C₂	1	6	yes	maximal, complemented
4	#11745236	C₂	1	3	yes	maximal, complemented
5	C₄×D₁₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×C₂₉	C₂²	1	1	yes	characteristic
7	#90809	—	2	8	no	—
8	C₂×D₁₁₆	C₂²	1	6	yes	—
9	C₂×D₁₁₆	C₂²	1	3	yes	—
10	C₂×D₁₁₆	C₂²	1	6	yes	—
11	C₂×C₄×D₂₉	C₂²	1	6	yes	—
12	C₄⋊C₄×C₂₉	C₂²	1	3	yes	—
13	C₄²×C₂₉	C₂²	1	3	yes	—
14	#90781	C₂²	1	6	yes	—
15	C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
16	C₂²×D₂₉	C₂³	1	6	yes	—
17	Q₈×C₂₉	D₄	1	4	yes	complemented
18	C₂×C₄×C₂₉	C₂³	1	3	yes	—
19	C₂×C₄×C₂₉	C₂³	1	3	yes	—
20	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₂₉	C₂³	1	2	yes	—
22	D₁₁₆	—	2	12	no	—
23	D₁₁₆	—	2	6	no	—
24	D₁₁₆	—	4	6	no	—
25	C₄×D₂₉	—	2	12	no	—
26	C₂²×C₂₉	C₂⁴	1	1	yes	characteristic
27	C₂×D₂₉	—	2	6	no	—
28	C₂×D₂₉	—	2	6	no	—
29	C₂×D₂₉	—	2	6	no	—
30	C₄×C₂₉	—	2	3	no	—
31	C₄×C₂₉	C₂×D₄	1	6	yes	—
32	C₄×C₂₉	#41	1	2	yes	complemented
33	Dic₂₉	—	2	2	no	—
34	#550	—	29	1	no	maximal, Sylow
35	C₂×C₂₉	C₂²×D₄	1	1	yes	characteristic
36	C₂×C₂₉	C₂×#41	1	1	yes	characteristic
37	C₂×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
38	D₂₉	—	4	6	no	—
39	C₂×#41	—	29	2	no	—
40	#127	—	29	3	no	—
41	#121	—	29	6	no	—
42	C₄×Q₈	D₂₉	1	1	yes	characteristic, complemented
43	C₄×D₄	—	29	3	no	—
44	C₂₉	#550	1	1	yes	characteristic, complemented, Sylow
45	#41	—	58	8	no	—
46	C₂×Q₈	C₂×D₂₉	1	1	yes	characteristic
47	C₂×D₄	—	29	3	no	—
48	C₂×D₄	—	29	6	no	—
49	C₂×D₄	—	29	6	no	—
50	C₂²×C₄	—	29	6	no	—
51	C₄⋊C₄	—	29	1	no	—
52	C₄⋊C₄	C₂×D₂₉	1	3	yes	—
53	C₂²⋊C₄	—	29	6	no	—
54	C₄²	C₂×D₂₉	1	3	yes	—
55	C₂³	—	29	6	no	—
56	Q₈	D₁₁₆	1	4	yes	complemented
57	D₄	—	58	12	no	—
58	D₄	—	58	6	no	—
59	D₄	—	116	6	no	—
60	C₂×C₄	—	29	2	no	—
61	C₂×C₄	—	58	12	no	—
62	C₂×C₄	C₂²×D₂₉	1	3	yes	—
63	C₂×C₄	C₂²×D₂₉	1	3	yes	—
64	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
65	C₂²	—	58	6	no	—
66	C₂²	—	58	6	no	—
67	C₂²	—	58	6	no	—
68	C₂²	C₂³×D₂₉	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	58	2	no	—
71	C₄	C₂×D₁₁₆	1	6	yes	—
72	C₄	#90809	1	2	yes	complemented
73	C₂	—	116	6	no	—
74	C₂	C₂²×D₁₁₆	1	1	yes	characteristic
75	C₂	C₂×#90809	1	1	yes	characteristic
76	C₂	#11745364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₉
commutator, socle, cosocle	C₂²×C₂₉	C₂⁴
Fitting	C₄×Q₈×C₂₉	C₂
hypercenter	C₄×Q₈	D₂₉
hypocenter	C₂₉	#550
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#90809	C₂	semidirect
2. C₄×Q₈×C₂₉	C₂	semidirect
3. #11745257	C₂	semidirect
4. #11745236	C₂	semidirect
5. C₄×D₁₁₆	C₂	semidirect
17. Q₈×C₂₉	D₄	semidirect
32. C₄×C₂₉	#41	semidirect
42. C₄×Q₈	D₂₉	semidirect
44. C₂₉	#550	semidirect
56. Q₈	D₁₁₆	semidirect
72. C₄	#90809	semidirect

Maximal subgroups

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

subgroup	index
C₂×#90809	2
C₄×Q₈×C₂₉	2
#11745257	2
#11745236	2
C₄×D₁₁₆	2
#550	29