Subgroups — #31804576

Order 17792.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581321	C₂	1	1	yes	maximal, characteristic
2	#22580752	C₂	1	1	yes	maximal, characteristic
3	#356×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#17984834	C₄	1	1	yes	characteristic
5	#17984766	—	2	1	no	—
6	#17984766	—	2	1	no	—
7	#17984761	—	2	1	no	—
8	#17984753	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
10	#101×C₁₃₉	—	2	1	no	—
11	#14555769	—	4	1	no	—
12	C₂×C₁₃₉⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₃₉	—	2	1	no	—
14	Dic₁₃₉⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
17	M₁₆×C₁₃₉	—	4	1	no	—
18	C₄⋊C₄×C₁₃₉	—	4	1	no	—
19	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₃₉	—	4	1	no	—
21	C₁₃₉⋊C₈	—	4	1	no	—
22	C₁₃₉⋊C₈	—	8	1	no	—
23	Q₈×C₁₃₉	—	4	1	no	—
24	Q₈×C₁₃₉	—	4	1	no	—
25	C₂×C₄×C₁₃₉	—	2	1	no	—
26	C₂×C₄×C₁₃₉	—	2	1	no	—
27	C₂×C₄×C₁₃₉	—	2	1	no	—
28	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₁₃₉	—	8	1	no	—
30	C₂²×C₁₃₉	#99	1	1	yes	characteristic
31	C₄×C₁₃₉	—	2	1	no	—
32	C₄×C₁₃₉	—	4	1	no	—
33	C₄×C₁₃₉	—	4	1	no	—
34	C₄×C₁₃₉	—	4	1	no	—
35	Dic₁₃₉	—	8	1	no	—
36	C₂×C₁₃₉	—	2	1	no	—
37	C₂×C₁₃₉	#354	1	1	yes	characteristic
38	C₁₃₉	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	139	1	no	maximal, Sylow
40	#491	—	139	1	no	—
41	#356	—	139	1	no	—
42	#356	D₁₃₉	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
44	#125	—	139	1	no	—
45	#103	—	278	1	no	—
46	#103	—	278	1	no	—
47	#101	—	2	1	no	—
48	#101	—	278	1	no	—
49	#97	—	139	1	no	—
50	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
51	C₂×Q₈	D₅₅₆	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	556	1	no	—
54	C₂×C₈	—	278	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	278	1	no	—
57	C₄⋊C₄	—	556	1	no	—
58	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	556	1	no	—
65	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	556	1	no	—
68	C₈	—	1112	1	no	—
69	C₂²	#17984756	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1112	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#22580747	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22580747
commutator	C₄²×C₁₃₉	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₃₉
Fitting	#356×C₁₃₉	C₂
socle	C₂×C₁₃₉	#354
prime radical	C₂²×C₁₃₉	#99
cosocle	C₄⋊Q₈×C₁₃₉	C₂²
hypercenter	#356	D₁₃₉
hypocenter	C₁₃₉	#1413
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₁₃₉	#1413	semidirect

Maximal subgroups

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

subgroup	index
#22581321	2
#22580752	2
#356×C₁₃₉	2
#1413	139