Subgroups — C₂²×#16838538 #29505548

Order 15296.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16838538	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₇₈	—	2	3	no	—
6	C₂×Dic₄₇₈	C₂²	1	6	yes	—
7	C₂²×Dic₂₃₉	—	2	1	no	—
8	#16838538	C₂²	1	16	yes	direct product factor
9	C₂×C₂₃₉⋊C₈	C₂²	1	6	yes	—
10	C₂×Q₈×C₂₃₉	—	2	3	no	—
11	C₂×Q₈×C₂₃₉	C₂²	1	6	yes	—
12	C₂²×C₄×C₂₃₉	—	2	1	no	—
13	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
14	Dic₄₇₈	—	2	6	no	—
15	Dic₄₇₈	C₂³	1	4	yes	—
16	C₂×Dic₂₃₉	—	2	6	no	—
17	C₂₃₉⋊C₈	C₂³	1	4	yes	—
18	C₂³×C₂₃₉	D₄	1	1	yes	characteristic
19	Q₈×C₂₃₉	—	2	6	no	—
20	Q₈×C₂₃₉	C₂³	1	4	yes	—
21	C₂×C₄×C₂₃₉	—	2	6	no	—
22	C₂×C₄×C₂₃₉	D₄	1	3	yes	—
23	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
24	C₂²×C₂₃₉	Q₁₆	1	4	yes	complemented
25	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
26	C₄×C₂₃₉	—	2	4	no	—
27	C₄×C₂₃₉	C₂×D₄	1	3	yes	—
28	C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
29	Dic₂₃₉	—	2	4	no	—
30	C₂×C₂₃₉	C₂×Q₁₆	1	6	yes	complemented
31	C₂×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
32	C₂₃₉	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
33	C₂²×Q₁₆	—	239	1	no	maximal, Sylow
34	C₂²×Q₈	—	239	1	no	—
35	C₂²×Q₈	D₂₃₉	1	1	yes	characteristic
36	C₂×Q₁₆	—	239	12	no	—
37	C₂²×C₈	—	239	1	no	—
38	C₂×Q₈	—	2	3	no	—
39	C₂×Q₈	—	478	3	no	—
40	C₂×Q₈	—	239	6	no	—
41	C₂×Q₈	C₂×D₂₃₉	1	6	yes	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	478	1	no	—
44	C₂²×C₄	C₂×D₂₃₉	1	1	yes	characteristic
45	Q₁₆	—	239	16	no	—
46	C₂×C₈	—	239	6	no	—
47	C₂³	C₂²⋊D₂₃₉	1	1	yes	characteristic
48	Q₈	—	2	6	no	—
49	Q₈	—	478	6	no	—
50	Q₈	—	239	4	no	—
51	Q₈	C₂²×D₂₃₉	1	4	yes	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	478	6	no	—
54	C₂×C₄	C₂²⋊D₂₃₉	1	3	yes	—
55	C₂×C₄	C₂²×D₂₃₉	1	3	yes	—
56	C₈	—	239	4	no	—
57	C₂²	#16838538	1	4	yes	direct product factor
58	C₂²	C₂×C₂²⋊D₂₃₉	1	3	yes	—
59	C₄	—	2	4	no	—
60	C₄	—	478	4	no	—
61	C₄	C₂×C₂²⋊D₂₃₉	1	3	yes	—
62	C₄	C₂³×D₂₃₉	1	1	yes	characteristic
63	C₂	C₂×#16838538	1	6	yes	direct product factor
64	C₂	C₂²×C₂²⋊D₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₃₉
commutator, cosocle	C₄×C₂₃₉	C₂⁴
Frattini	C₄	C₂³×D₂₃₉
Fitting	C₂²×Q₈×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	D₄
hypercenter	C₂²×Q₈	D₂₃₉
hypocenter	C₂₃₉	C₂²×Q₁₆
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16838538	C₂	direct (N × H)
8. #16838538	C₂²	direct (N × H)
24. C₂²×C₂₃₉	Q₁₆	semidirect
30. C₂×C₂₃₉	C₂×Q₁₆	semidirect
32. C₂₃₉	C₂²×Q₁₆	semidirect
57. C₂²	#16838538	direct (N × H)
63. C₂	C₂×#16838538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₇₈	2
C₂×#16838538	2
C₂²×C₂₃₉⋊C₈	2
C₂²×Q₈×C₂₃₉	2
C₂²×Q₁₆	239

Subgroups — C22×#16838538 #29505548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16838538 #29505548