Subgroups — #31804382

Order 17792.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₇₈	C₄	1	1	yes	characteristic
5	C₂×C₁₃₉⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₁₃₉⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₁₃₉⋊C₈	—	2	1	no	—
8	C₈×Dic₁₃₉	—	2	1	no	—
9	C₂×C₁₆×C₁₃₉	—	2	1	no	—
10	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₇₈	C₈	1	1	yes	characteristic
12	C₂×C₁₃₉⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₃₉	C₈	1	1	yes	characteristic
14	Dic₁₃₉⋊C₄	—	2	1	no	—
15	C₄×Dic₁₃₉	—	2	1	no	—
16	C₁₃₉⋊C₁₆	—	2	2	no	—
17	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
19	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₃₉	—	4	1	no	—
21	Dic₂₇₈	—	2	1	no	—
22	Dic₂₇₈	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₃₉	—	2	1	no	—
24	C₂×Dic₁₃₉	—	2	1	no	—
25	C₁₃₉⋊C₈	—	4	1	no	—
26	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₉	M₁₆	1	1	yes	characteristic
29	C₈×C₁₃₉	—	2	1	no	—
30	C₈×C₁₃₉	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₃₉	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₃₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₃₉	—	2	1	no	—
34	C₄×C₁₃₉	#103	1	1	yes	characteristic
35	C₄×C₁₃₉	#104	1	1	yes	characteristic
36	C₄×C₁₃₉	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₃₉	M₃₂	1	1	yes	characteristic
38	Dic₁₃₉	—	2	2	no	—
39	Dic₁₃₉	—	4	1	no	—
40	C₂×C₁₃₉	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₃₉	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₃₉	#350	1	1	yes	characteristic
43	C₁₃₉	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	139	1	no	maximal, Sylow
45	C₈×Q₈	—	139	1	no	—
46	C₄⋊C₁₆	D₁₃₉	1	1	yes	characteristic
47	C₄×C₁₆	—	139	1	no	—
48	C₄×Q₈	—	139	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	278	1	no	—
51	C₄⋊C₈	—	278	1	no	—
52	C₄⋊C₈	—	139	1	no	—
53	C₄×C₈	—	278	1	no	—
54	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
55	C₂×Q₈	—	139	1	no	—
56	C₂×C₈	—	278	1	no	—
57	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
58	C₂×C₈	D₅₅₆	1	1	yes	characteristic
59	C₄⋊C₄	—	278	1	no	—
60	C₄⋊C₄	—	139	1	no	—
61	C₄²	—	278	1	no	—
62	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
63	C₁₆	—	4	1	no	—
64	C₁₆	—	278	2	no	—
65	Q₈	—	278	1	no	—
66	Q₈	—	139	2	no	—
67	C₂×C₄	—	278	1	no	—
68	C₂×C₄	—	278	1	no	—
69	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₃₉	1	1	yes	characteristic
71	C₂×C₄	#14555768	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	556	1	no	—
74	C₈	#14555773	1	1	yes	characteristic
75	C₈	#14555775	1	1	yes	characteristic
76	C₂²	(C₂×D₁₃₉)⋊C₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	278	2	no	—
79	C₄	—	556	1	no	—
80	C₄	#17984765	1	1	yes	characteristic
81	C₄	#17984768	1	1	yes	characteristic
82	C₄	C₁₆×D₁₃₉	1	1	yes	characteristic
83	C₄	#17984781	1	1	yes	characteristic
84	C₂	#22580683	1	1	yes	characteristic
85	C₂	(C₂×D₁₃₉)⋊C₁₆	1	1	yes	characteristic
86	C₂	#22580738	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₃₉
commutator	C₄×C₁₃₉	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₃₉
Fitting	C₄⋊C₁₆×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	C₂²⋊C₈
cosocle	C₄×C₈×C₁₃₉	C₂²
hypercenter	C₄⋊C₁₆	D₁₃₉
hypocenter	C₁₃₉	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
22. Dic₂₇₈	C₁₆	semidirect
43. C₁₃₉	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₇₈	2
C₄×C₁₃₉⋊C₁₆	2
C₄⋊C₁₆×C₁₃₉	2
Q₈⋊C₁₆	139