Subgroups — #29504683

Order 15296.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₂₃₉)⋊C₄	C₂	1	3	yes	maximal
2	C₂₃₉⋊#95	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₃₉	C₂²	1	1	yes	characteristic
5	C₂²×Dic₂₃₉	C₂²	1	3	yes	—
6	(C₂×D₂₃₉)⋊C₄	—	2	6	no	—
7	C₂×Q₈×C₂₃₉	—	2	6	no	—
8	C₂²×C₄×C₂₃₉	C₂²	1	3	yes	—
9	C₂²×D₂₃₉	—	2	3	no	—
10	C₂²×D₂₃₉	—	2	3	no	—
11	C₂²×D₂₃₉	Q₈	1	2	yes	complemented
12	C₂×Dic₂₃₉	—	2	3	no	—
13	C₂×Dic₂₃₉	—	2	6	no	—
14	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
15	Q₈×C₂₃₉	—	2	8	no	—
16	C₂×C₄×C₂₃₉	—	2	6	no	—
17	C₂×C₄×C₂₃₉	D₄	1	6	yes	—
18	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
19	C₂²×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₂₃₉	#41	1	3	yes	—
21	C₂×D₂₃₉	—	2	6	no	—
22	C₂×D₂₃₉	—	4	1	no	—
23	C₂×D₂₃₉	—	4	3	no	—
24	C₄×C₂₃₉	—	2	6	no	—
25	Dic₂₃₉	—	4	3	no	—
26	C₂×C₂₃₉	#120	1	1	yes	characteristic
27	C₂×C₂₃₉	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₂₃₉	#124	1	3	yes	—
29	D₂₃₉	—	4	2	no	—
30	C₂₃₉	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	239	1	no	maximal, Sylow
32	C₂²×Q₈	D₂₃₉	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	239	3	no	—
34	#95	—	239	3	no	—
35	C₂⁴	—	239	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	239	3	no	—
38	C₂²×C₄	C₂×D₂₃₉	1	3	yes	—
39	C₂²⋊C₄	—	478	6	no	—
40	C₂³	—	478	3	no	—
41	C₂³	—	478	3	no	—
42	C₂³	—	239	2	no	—
43	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	478	3	no	—
47	C₂×C₄	—	478	6	no	—
48	C₂×C₄	C₂²⋊D₂₃₉	1	6	yes	—
49	C₂²	—	956	1	no	—
50	C₂²	—	956	3	no	—
51	C₂²	—	478	6	no	—
52	C₂²	C₂×C₂²⋊D₂₃₉	1	3	yes	—
53	C₂²	Q₈×D₂₃₉	1	1	yes	characteristic
54	C₂²	#16838549	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	956	3	no	—
57	C₂	—	956	2	no	—
58	C₂	#21439929	1	1	yes	characteristic
59	C₂	#21439936	1	3	yes	—
60	C₂	#21439949	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃₉
commutator, socle, cosocle	C₂³×C₂₃₉	C₂³
Fitting	C₂²×Q₈×C₂₃₉	C₂
prime radical	C₂³×D₂₃₉	C₂²
hypercenter	C₂²×Q₈	D₂₃₉
hypocenter	C₂₃₉	#393
LCS	G ▷ C₂³×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂₃₉⋊#95	C₂	semidirect
3. C₂²×Q₈×C₂₃₉	C₂	semidirect
11. C₂²×D₂₃₉	Q₈	semidirect
30. C₂₃₉	#393	semidirect
32. C₂²×Q₈	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₂₃₉)⋊C₄	2
C₂₃₉⋊#95	2
C₂²×Q₈×C₂₃₉	2
#393	239