Subgroups — #448×C₂₃₉ #29504352

Order 15296.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₁₆×C₂₃₉	C₂	1	2	yes	maximal, complemented
3	#121×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#103×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₃₉	—	2	4	no	—
7	C₂×Q₈×C₂₃₉	—	2	1	no	—
8	C₂×Q₈×C₂₃₉	C₂²	1	2	yes	—
9	C₂×D₄×C₂₃₉	—	2	1	no	—
10	C₂×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₃₉	—	2	1	no	—
12	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
13	SD₁₆×C₂₃₉	—	2	4	no	—
14	C₂×C₈×C₂₃₉	C₂²	1	2	yes	—
15	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
17	C₂³×C₂₃₉	—	2	1	no	—
18	C₂³×C₂₃₉	D₄	1	1	yes	characteristic
19	Q₈×C₂₃₉	—	2	4	no	—
20	Q₈×C₂₃₉	—	2	2	no	—
21	Q₈×C₂₃₉	D₄	1	4	yes	complemented
22	D₄×C₂₃₉	—	2	2	no	—
23	D₄×C₂₃₉	—	4	2	no	—
24	C₂×C₄×C₂₃₉	—	2	4	no	—
25	C₂×C₄×C₂₃₉	—	2	2	no	—
26	C₂×C₄×C₂₃₉	—	2	2	no	—
27	C₂×C₄×C₂₃₉	—	2	1	no	—
28	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₈×C₂₃₉	—	2	2	no	—
31	C₂²×C₂₃₉	—	2	1	no	—
32	C₂²×C₂₃₉	—	2	1	no	—
33	C₂²×C₂₃₉	—	4	1	no	—
34	C₂²×C₂₃₉	—	4	1	no	—
35	C₂²×C₂₃₉	—	4	1	no	—
36	C₂²×C₂₃₉	SD₁₆	1	2	yes	complemented
37	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₃₉	—	2	4	no	—
39	C₄×C₂₃₉	—	2	1	no	—
40	C₄×C₂₃₉	—	4	1	no	—
41	C₄×C₂₃₉	C₂×D₄	1	2	yes	—
42	C₂×C₂₃₉	—	2	2	no	—
43	C₂×C₂₃₉	—	8	1	no	—
44	C₂×C₂₃₉	#120	1	1	yes	characteristic
45	C₂×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₂₃₉	#137	1	1	yes	characteristic
47	C₂₃₉	#448	1	1	yes	characteristic, direct product factor, Sylow
48	#448	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×Q₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
50	C₂×SD₁₆	C₂×C₂₃₉	1	2	yes	complemented
51	#121	C₂×C₂₃₉	1	1	yes	characteristic
52	#103	C₂×C₂₃₉	1	2	yes	complemented
53	C₂²⋊C₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
54	C₂×Q₈	—	2	4	no	—
55	C₂×Q₈	—	2	1	no	—
56	C₂×Q₈	C₂²×C₂₃₉	1	2	yes	—
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
61	SD₁₆	—	2	4	no	—
62	C₂×C₈	C₂²×C₂₃₉	1	2	yes	—
63	C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂³	—	2	1	no	—
66	C₂³	D₄×C₂₃₉	1	1	yes	characteristic
67	Q₈	—	2	4	no	—
68	Q₈	—	2	2	no	—
69	Q₈	D₄×C₂₃₉	1	4	yes	complemented
70	D₄	—	2	2	no	—
71	D₄	—	4	2	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
77	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	SD₁₆×C₂₃₉	1	2	yes	complemented
85	C₂²	C₂×D₄×C₂₃₉	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	C₂×D₄×C₂₃₉	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	—	8	1	no	—
92	C₂	#120×C₂₃₉	1	1	yes	characteristic
93	C₂	C₂×SD₁₆×C₂₃₉	1	1	yes	characteristic
94	C₂	#137×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₃₉
prime radical	#121×C₂₃₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₂²×C₄×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₂₃₉	C₂	semidirect
2. C₂×SD₁₆×C₂₃₉	C₂	semidirect
4. #103×C₂₃₉	C₂	semidirect
5. C₂²⋊C₈×C₂₃₉	C₂	semidirect
21. Q₈×C₂₃₉	D₄	semidirect
36. C₂²×C₂₃₉	SD₁₆	semidirect
47. C₂₃₉	#448	direct (N × H)
48. #448	C₂₃₉	direct (N × H)
49. C₂²×Q₈	C₂×C₂₃₉	semidirect
50. C₂×SD₁₆	C₂×C₂₃₉	semidirect
52. #103	C₂×C₂₃₉	semidirect
53. C₂²⋊C₈	C₂×C₂₃₉	semidirect
69. Q₈	D₄×C₂₃₉	semidirect
84. C₂²	SD₁₆×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₃₉	2
C₂×SD₁₆×C₂₃₉	2
#121×C₂₃₉	2
#103×C₂₃₉	2
C₂²⋊C₈×C₂₃₉	2
#448	239

Subgroups — #448×C239 #29504352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #448×C₂₃₉ #29504352