Subgroups — #44482722

Order 30592.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505126	C₂	1	1	yes	maximal, characteristic
2	#29504569	C₂	1	1	yes	maximal, characteristic
3	#354×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#21439952	C₄	1	1	yes	characteristic, complemented
5	#21439884	—	2	1	no	—
6	#21439881	—	2	1	no	—
7	#21439879	—	2	1	no	—
8	C₄×C₂₃₉⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#124×C₂₃₉	C₂²	1	1	yes	characteristic
10	#99×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×D₄×C₂₃₉	D₄	1	1	yes	characteristic
17	C₂²⋊C₄×C₂₃₉	—	4	1	no	—
18	C₂²⋊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	C₂³×C₂₃₉	—	2	1	no	—
24	Q₈×C₂₃₉	—	4	1	no	—
25	D₄×C₂₃₉	—	4	1	no	—
26	C₂×C₄×C₂₃₉	—	2	1	no	—
27	C₂×C₄×C₂₃₉	—	2	1	no	—
28	C₂×C₄×C₂₃₉	—	8	1	no	—
29	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₃₉	—	4	1	no	—
31	C₂²×C₂₃₉	—	8	1	no	—
32	C₂²×C₂₃₉	#99	1	1	yes	characteristic
33	C₄×C₂₃₉	—	2	1	no	—
34	C₄×C₂₃₉	—	4	1	no	—
35	C₄×C₂₃₉	—	4	1	no	—
36	C₄×C₂₃₉	—	16	1	no	—
37	Dic₂₃₉	—	8	1	no	—
38	C₂×C₂₃₉	—	2	1	no	—
39	C₂×C₂₃₉	—	8	1	no	—
40	C₂×C₂₃₉	#354	1	1	yes	characteristic
41	C₂₃₉	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	239	1	no	maximal, Sylow
43	#488	—	239	1	no	—
44	#355	—	239	1	no	—
45	#354	D₂₃₉	1	1	yes	characteristic
46	#125	—	239	1	no	—
47	#124	C₂×D₂₃₉	1	1	yes	characteristic
48	#103	—	478	1	no	—
49	#102	—	478	1	no	—
50	#101	—	478	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	239	1	no	—
53	C₂×Q₈	D₉₅₆	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
55	M₁₆	—	956	1	no	—
56	C₂×C₈	—	478	1	no	—
57	C₄⋊C₄	—	956	1	no	—
58	C₄⋊C₄	—	478	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₂₃₉	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	956	1	no	—
69	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
70	C₈	—	1912	1	no	—
71	C₈	—	956	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#21439874	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	1912	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#29504566	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#29504566
commutator	C₄²×C₂₃₉	C₂×C₄
Frattini	#124	C₂×D₂₃₉
Fitting	#354×C₂₃₉	C₂
socle	C₂×C₂₃₉	#354
prime radical	C₂×D₄×C₂₃₉	D₄
cosocle	#124×C₂₃₉	C₂²
hypercenter	#354	D₂₃₉
hypocenter	C₂₃₉	#1414
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂×C₄×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #21439952	C₄	semidirect
8. C₄×C₂₃₉⋊C₈	C₄	semidirect
41. C₂₃₉	#1414	semidirect

Maximal subgroups

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

subgroup	index
#29505126	2
#29504569	2
#354×C₂₃₉	2
#1414	239