Subgroups — #29505325

Order 15296.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439959	C₂	1	2	yes	maximal, complemented
2	#21439951	C₂	1	2	yes	maximal, complemented
3	#21439937	C₂	1	4	yes	maximal
4	C₄×Dic₄₇₈	C₂	1	4	yes	maximal, complemented
5	#21439919	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
7	#117×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₈	C₂²	1	4	yes	—
9	C₂²×Dic₂₃₉	C₂²	1	2	yes	—
10	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
11	C₄⋊Dic₂₃₉	C₂²	1	4	yes	—
12	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
13	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
14	C₂²⋊Dic₂₃₉	C₂²	1	2	yes	—
15	C₄×Dic₂₃₉	C₂²	1	2	yes	—
16	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₃₉	C₂²	1	2	yes	—
19	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
22	C₄²×C₂₃₉	C₂²	1	2	yes	—
23	Dic₄₇₈	—	2	8	no	—
24	C₂×Dic₂₃₉	—	2	4	no	—
25	C₂×Dic₂₃₉	C₂³	1	2	yes	—
26	C₂×Dic₂₃₉	C₂³	1	4	yes	—
27	C₂×Dic₂₃₉	C₂³	1	2	yes	—
28	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₃₉	Q₈	1	4	yes	complemented
30	C₂×C₄×C₂₃₉	C₂³	1	4	yes	—
31	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
33	C₂²×C₂₃₉	—	2	2	no	—
34	C₂²×C₂₃₉	C₂×Q₈	1	2	yes	—
35	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
36	C₄×C₂₃₉	—	2	4	no	—
37	C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
38	C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
39	Dic₂₃₉	—	2	2	no	—
40	Dic₂₃₉	—	2	4	no	—
41	Dic₂₃₉	#41	1	4	yes	—
42	C₂×C₂₃₉	—	2	2	no	—
43	C₂×C₂₃₉	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₂₃₉	C₂×#41	1	2	yes	—
45	C₂₃₉	#533	1	1	yes	characteristic, complemented, Sylow
46	#533	—	239	1	no	maximal, Sylow
47	C₄⋊Q₈	—	239	2	no	—
48	#125	—	239	2	no	—
49	C₂²⋊Q₈	—	239	4	no	—
50	C₄×Q₈	—	239	4	no	—
51	#117	—	239	1	no	—
52	#117	D₂₃₉	1	1	yes	characteristic
53	C₂×C₄²	—	239	1	no	—
54	C₂×Q₈	—	239	4	no	—
55	C₂²×C₄	—	239	2	no	—
56	C₂²×C₄	C₂×D₂₃₉	1	1	yes	characteristic
57	C₄⋊C₄	—	239	2	no	—
58	C₄⋊C₄	—	239	4	no	—
59	C₄⋊C₄	—	239	4	no	—
60	C₄⋊C₄	—	239	4	no	—
61	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
62	C₂²⋊C₄	—	239	2	no	—
63	C₂²⋊C₄	C₂×D₂₃₉	1	2	yes	—
64	C₄²	—	239	1	no	—
65	C₄²	—	239	1	no	—
66	C₄²	—	239	2	no	—
67	C₄²	—	239	2	no	—
68	C₄²	C₂×D₂₃₉	1	2	yes	—
69	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
70	Q₈	—	478	8	no	—
71	C₂×C₄	—	478	4	no	—
72	C₂×C₄	—	239	2	no	—
73	C₂×C₄	—	239	4	no	—
74	C₂×C₄	—	239	2	no	—
75	C₂×C₄	Dic₄₇₈	1	4	yes	complemented
76	C₂×C₄	C₂²×D₂₃₉	1	4	yes	—
77	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	C₂×Dic₄₇₈	1	2	yes	—
81	C₂²	C₂³×D₂₃₉	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	478	2	no	—
84	C₄	—	478	4	no	—
85	C₄	—	239	4	no	—
86	C₄	C₂×Dic₄₇₈	1	2	yes	—
87	C₄	C₂×Dic₄₇₈	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₄₇₈	1	1	yes	characteristic
90	C₂	#41×D₂₃₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₉
commutator, socle, cosocle	C₂²×C₂₃₉	C₂⁴
Frattini	C₂²	C₂³×D₂₃₉
Fitting	#117×C₂₃₉	C₂
prime radical	C₂³×C₂₃₉	C₂³
hypercenter	#117	D₂₃₉
hypocenter	C₂₃₉	#533
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #21439959	C₂	semidirect
2. #21439951	C₂	semidirect
4. C₄×Dic₄₇₈	C₂	semidirect
29. C₂×C₄×C₂₃₉	Q₈	semidirect
45. C₂₃₉	#533	semidirect
75. C₂×C₄	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
#21439959	2
#21439951	2
#21439937	2
C₄×Dic₄₇₈	2
#21439919	2
C₂×C₄×Dic₂₃₉	2
#117×C₂₃₉	2
#533	239