Subgroups — #29505508

Order 15296.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439947	C₂	1	4	yes	maximal, complemented
2	#21439941	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₄₇₈	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₂₃₉	C₂	1	4	yes	maximal, complemented
5	#127×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₈	C₂²	1	2	yes	—
7	C₂²×Dic₂₃₉	C₂²	1	4	yes	—
8	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₃₉	C₂²	1	8	yes	—
11	C₂²⋊Dic₂₃₉	C₂²	1	4	yes	—
12	C₄×Dic₂₃₉	C₂²	1	4	yes	—
13	C₂×D₄×C₂₃₉	C₂²	1	4	yes	—
14	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
15	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
16	Dic₄₇₈	—	2	4	no	—
17	C₂×Dic₂₃₉	—	4	4	no	—
18	C₂×Dic₂₃₉	C₂³	1	4	yes	—
19	C₂×Dic₂₃₉	C₂³	1	4	yes	—
20	C₂³×C₂₃₉	C₂³	1	4	yes	—
21	D₄×C₂₃₉	—	2	8	no	—
22	D₄×C₂₃₉	—	4	2	no	—
23	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₃₉	C₂³	1	2	yes	—
25	C₂²×C₂₃₉	—	2	8	no	—
26	C₂²×C₂₃₉	—	2	4	no	—
27	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₃₉	—	2	1	no	—
29	C₄×C₂₃₉	#41	1	4	yes	—
30	Dic₂₃₉	—	2	4	no	—
31	Dic₂₃₉	—	2	4	no	—
32	C₂×C₂₃₉	—	4	4	no	—
33	C₂×C₂₃₉	C₂×#41	1	2	yes	—
34	C₂×C₂₃₉	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂₃₉	#559	1	1	yes	characteristic, complemented, Sylow
36	#559	—	239	1	no	maximal, Sylow
37	#127	D₂₃₉	1	1	yes	characteristic
38	#124	—	239	4	no	—
39	#123	—	239	4	no	—
40	C₄×Q₈	—	239	2	no	—
41	C₄×D₄	—	239	4	no	—
42	C₂×Q₈	—	239	2	no	—
43	C₂×D₄	C₂×D₂₃₉	1	4	yes	—
44	C₂×D₄	C₂×D₂₃₉	1	2	yes	—
45	C₂²×C₄	—	239	4	no	—
46	C₄⋊C₄	—	239	2	no	—
47	C₄⋊C₄	—	239	4	no	—
48	C₂²⋊C₄	—	239	8	no	—
49	C₂²⋊C₄	—	239	4	no	—
50	C₄²	—	239	4	no	—
51	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
52	C₂³	C₂²×D₂₃₉	1	4	yes	—
53	Q₈	—	478	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	956	4	no	—
57	C₂×C₄	—	239	4	no	—
58	C₂×C₄	—	239	4	no	—
59	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₃₉	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₂₃₉	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	478	4	no	—
66	C₄	—	478	4	no	—
67	C₄	#16838547	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#16838547	1	2	yes	—
70	C₂	#21440005	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₉
commutator, socle, cosocle	C₂²×C₂₃₉	C₂⁴
Fitting, prime radical	#127×C₂₃₉	C₂
hypercenter	#127	D₂₃₉
hypocenter	C₂₃₉	#559
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #21439947	C₂	semidirect
2. #21439941	C₂	semidirect
3. C₄×Dic₄₇₈	C₂	semidirect
4. D₄×Dic₂₃₉	C₂	semidirect
35. C₂₃₉	#559	semidirect

Maximal subgroups

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

subgroup	index
#21439947	2
#21439941	2
C₄×Dic₄₇₈	2
D₄×Dic₂₃₉	2
#127×C₂₃₉	2
#559	239