Subgroups — #29504710

Order 15296.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₃₉⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂₃₉⋊#95	C₂	1	2	yes	maximal
3	C₂₃₉⋊#95	C₂	1	2	yes	maximal
4	C₂₃₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₉	C₂²	1	2	yes	—
7	C₂²×Dic₂₃₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₃₉	C₂²	1	1	yes	characteristic
9	Dic₂₃₉⋊C₄	—	2	2	no	—
10	C₂²×C₄×C₂₃₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃₉	—	2	2	no	—
13	C₂×Dic₂₃₉	—	2	1	no	—
14	C₂×Dic₂₃₉	—	2	1	no	—
15	C₂×Dic₂₃₉	—	2	1	no	—
16	C₂×Dic₂₃₉	—	2	2	no	—
17	C₂×Dic₂₃₉	—	2	2	no	—
18	C₂×Dic₂₃₉	—	2	2	no	—
19	C₂×Dic₂₃₉	—	2	2	no	—
20	C₂×Dic₂₃₉	Q₈	1	2	yes	—
21	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₃₉	—	2	2	no	—
23	C₂×C₄×C₂₃₉	—	2	2	no	—
24	C₂×C₄×C₂₃₉	—	2	4	no	—
25	C₂×C₄×C₂₃₉	D₄	1	2	yes	—
26	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₂₃₉	#41	1	2	yes	—
29	C₂²×C₂₃₉	#41	1	2	yes	—
30	C₂²×C₂₃₉	#41	1	1	yes	characteristic
31	C₄×C₂₃₉	—	2	2	no	—
32	C₄×C₂₃₉	—	4	2	no	—
33	Dic₂₃₉	—	2	2	no	—
34	Dic₂₃₉	—	4	1	no	—
35	Dic₂₃₉	—	4	2	no	—
36	C₂×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₂₃₉	#123	1	1	yes	characteristic
38	C₂×C₂₃₉	#124	1	1	yes	characteristic
39	C₂×C₂₃₉	#125	1	2	yes	—
40	C₂×C₂₃₉	#126	1	2	yes	—
41	C₂₃₉	#400	1	1	yes	characteristic, complemented, Sylow
42	#400	—	239	1	no	maximal, Sylow
43	C₂×C₄⋊C₄	—	239	1	no	—
44	C₂×C₄⋊C₄	D₂₃₉	1	1	yes	characteristic
45	#95	—	239	1	no	—
46	#95	—	239	2	no	—
47	#95	—	239	2	no	—
48	C₂²×C₄	—	239	2	no	—
49	C₂²×C₄	—	239	1	no	—
50	C₂²×C₄	—	239	1	no	—
51	C₂²×C₄	C₂×D₂₃₉	1	2	yes	—
52	C₂²×C₄	C₂×D₂₃₉	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	478	2	no	—
55	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	478	1	no	—
60	C₂×C₄	—	478	2	no	—
61	C₂×C₄	—	478	2	no	—
62	C₂×C₄	—	478	2	no	—
63	C₂×C₄	—	478	2	no	—
64	C₂×C₄	—	478	1	no	—
65	C₂×C₄	—	478	1	no	—
66	C₂×C₄	—	239	2	no	—
67	C₂×C₄	C₂²⋊D₂₃₉	1	2	yes	—
68	C₂²	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
69	C₂²	Q₈×D₂₃₉	1	1	yes	characteristic
70	C₂²	#16838547	1	2	yes	—
71	C₂²	#16838548	1	2	yes	—
72	C₂²	#16838549	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	4	2	no	—
75	C₄	—	956	1	no	—
76	C₄	—	956	2	no	—
77	C₄	—	478	2	no	—
78	C₂	#21439936	1	1	yes	characteristic
79	C₂	#21439943	1	1	yes	characteristic
80	C₂	#21439947	1	1	yes	characteristic
81	C₂	#21439950	1	2	yes	—
82	C₂	#21439955	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃₉
commutator, socle, prime radical, cosocle	C₂³×C₂₃₉	C₂³
Fitting	C₂×C₄⋊C₄×C₂₃₉	C₂
hypercenter	C₂×C₄⋊C₄	D₂₃₉
hypocenter	C₂₃₉	#400
LCS	G ▷ C₂³×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
41. C₂₃₉	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₃₉⋊C₄	2
C₂₃₉⋊#95	2
C₂×C₄⋊C₄×C₂₃₉	2
#400	239