Subgroups — #44482720

Order 30592.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505140	C₂	1	1	yes	maximal, characteristic
2	#29504571	C₂	1	1	yes	maximal, characteristic
3	#356×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#21439952	C₄	1	1	yes	characteristic
5	#21439884	—	2	1	no	—
6	#21439884	—	2	1	no	—
7	#21439879	—	2	1	no	—
8	#21439871	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	#101×C₂₃₉	—	2	1	no	—
11	#16838532	—	4	1	no	—
12	C₂×C₂₃₉⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₃₉	—	2	1	no	—
14	Dic₂₃₉⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₂₃₉	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₃₉	D₄	1	1	yes	characteristic
17	M₁₆×C₂₃₉	—	4	1	no	—
18	C₄⋊C₄×C₂₃₉	—	4	1	no	—
19	C₄²×C₂₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₃₉	—	4	1	no	—
21	C₂₃₉⋊C₈	—	4	1	no	—
22	C₂₃₉⋊C₈	—	8	1	no	—
23	Q₈×C₂₃₉	—	4	1	no	—
24	Q₈×C₂₃₉	—	4	1	no	—
25	C₂×C₄×C₂₃₉	—	2	1	no	—
26	C₂×C₄×C₂₃₉	—	2	1	no	—
27	C₂×C₄×C₂₃₉	—	2	1	no	—
28	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₃₉	—	8	1	no	—
30	C₂²×C₂₃₉	#99	1	1	yes	characteristic
31	C₄×C₂₃₉	—	2	1	no	—
32	C₄×C₂₃₉	—	4	1	no	—
33	C₄×C₂₃₉	—	4	1	no	—
34	C₄×C₂₃₉	—	4	1	no	—
35	Dic₂₃₉	—	8	1	no	—
36	C₂×C₂₃₉	—	2	1	no	—
37	C₂×C₂₃₉	#354	1	1	yes	characteristic
38	C₂₃₉	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	239	1	no	maximal, Sylow
40	#491	—	239	1	no	—
41	#356	—	239	1	no	—
42	#356	D₂₃₉	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
44	#125	—	239	1	no	—
45	#103	—	478	1	no	—
46	#103	—	478	1	no	—
47	#101	—	2	1	no	—
48	#101	—	478	1	no	—
49	#97	—	239	1	no	—
50	C₂×Q₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
51	C₂×Q₈	D₉₅₆	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	956	1	no	—
54	C₂×C₈	—	478	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	956	1	no	—
57	C₄⋊C₄	—	478	1	no	—
58	C₄²	C₄×D₂₃₉	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	956	1	no	—
65	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	1912	1	no	—
68	C₈	—	956	1	no	—
69	C₂²	#21439874	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1912	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#29504566	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#29504566
commutator	C₄²×C₂₃₉	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₃₉
Fitting	#356×C₂₃₉	C₂
socle	C₂×C₂₃₉	#354
prime radical	C₂²×C₂₃₉	#99
cosocle	C₄⋊Q₈×C₂₃₉	C₂²
hypercenter	#356	D₂₃₉
hypocenter	C₂₃₉	#1413
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂×C₄×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₂₃₉	#1413	semidirect

Maximal subgroups

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

subgroup	index
#29505140	2
#29504571	2
#356×C₂₃₉	2
#1413	239