Subgroups — #31808281

Order 17792.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₂₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₁₁₂	C₂	1	2	yes	maximal, complemented
3	#22580781	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580755	C₂	1	2	yes	maximal, complemented
5	#346×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₁₂	C₂²	1	2	yes	—
7	C₄×D₅₅₆	—	2	2	no	—
8	D₂₂₂₄	C₄	1	4	yes	complemented
9	#17984776	C₂²	1	2	yes	—
10	#17984764	—	2	2	no	—
11	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×D₅₅₆	—	2	2	no	—
15	C₂×C₄×D₁₃₉	—	4	2	no	—
16	D₁₁₁₂	—	2	2	no	—
17	D₁₁₁₂	C₂×C₄	1	4	yes	complemented
18	C₄⋊Dic₁₃₉	—	2	2	no	—
19	(C₂×D₁₃₉)⋊C₄	—	4	2	no	—
20	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
22	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₃₉	—	2	1	no	—
24	C₁₆×C₁₃₉	C₂×C₄	1	2	yes	complemented
25	C₂²×D₁₃₉	—	4	2	no	—
26	D₅₅₆	—	2	4	no	—
27	D₅₅₆	—	4	2	no	—
28	C₂×Dic₁₃₉	—	4	2	no	—
29	C₄×D₁₃₉	—	8	4	no	—
30	C₂×C₄×C₁₃₉	D₈	1	2	yes	—
31	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₃₉	—	2	1	no	—
33	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₃₉	#41	1	1	yes	characteristic
35	C₂²×C₁₃₉	C₂×D₈	1	1	yes	characteristic
36	C₂×D₁₃₉	—	4	4	no	—
37	C₂×D₁₃₉	—	8	2	no	—
38	C₂×D₁₃₉	—	8	2	no	—
39	C₄×C₁₃₉	—	2	2	no	—
40	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
41	C₄×C₁₃₉	#135	1	1	yes	characteristic
42	Dic₁₃₉	—	8	2	no	—
43	C₂×C₁₃₉	C₄×D₈	1	1	yes	characteristic
44	C₂×C₁₃₉	#509	1	2	yes	—
45	D₁₃₉	—	8	4	no	—
46	C₁₃₉	#2177	1	1	yes	characteristic, complemented, Sylow
47	#2177	—	139	1	no	maximal, Sylow
48	C₂×D₁₆	—	139	1	no	—
49	C₄×D₈	—	139	2	no	—
50	#367	—	139	1	no	—
51	#357	—	139	2	no	—
52	#346	D₁₃₉	1	1	yes	characteristic, complemented
53	C₂×D₈	—	139	2	no	—
54	C₄×D₄	—	278	2	no	—
55	D₁₆	—	139	4	no	—
56	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
57	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
58	#107	—	139	2	no	—
59	#102	—	278	2	no	—
60	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
61	C₂×D₄	—	278	2	no	—
62	C₂²×C₄	—	556	2	no	—
63	D₈	—	278	2	no	—
64	D₈	—	139	4	no	—
65	C₂×C₈	D₅₅₆	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
67	C₄⋊C₄	—	278	2	no	—
68	C₂²⋊C₄	—	556	2	no	—
69	C₄²	D₅₅₆	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₁₃₉	1	2	yes	complemented
72	C₂³	—	556	2	no	—
73	D₄	—	278	4	no	—
74	D₄	—	556	2	no	—
75	C₂×C₄	—	556	2	no	—
76	C₂×C₄	—	1112	4	no	—
77	C₂×C₄	D₁₁₁₂	1	2	yes	—
78	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
81	C₈	#14555785	1	1	yes	characteristic
82	C₂²	—	556	4	no	—
83	C₂²	—	1112	2	no	—
84	C₂²	—	1112	2	no	—
85	C₂²	C₂×D₁₁₁₂	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	1112	2	no	—
88	C₄	C₄×D₅₅₆	1	1	yes	characteristic
89	C₄	#17984864	1	1	yes	characteristic
90	C₂	—	1112	4	no	—
91	C₂	C₄×D₁₁₁₂	1	1	yes	characteristic
92	C₂	#22581391	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₁₁₂
commutator	C₈×C₁₃₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#346×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₈
prime radical	C₂×D₂₂₂₄	C₂
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	#346	D₁₃₉
hypocenter	C₁₃₉	#2177
LCS	G ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₁₁₂	C₂	semidirect
3. #22580781	C₂	semidirect
4. #22580755	C₂	semidirect
5. #346×C₁₃₉	C₂	semidirect
8. D₂₂₂₄	C₄	semidirect
17. D₁₁₁₂	C₂×C₄	semidirect
24. C₁₆×C₁₃₉	C₂×C₄	semidirect
46. C₁₃₉	#2177	semidirect
52. #346	D₁₃₉	semidirect
71. C₁₆	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₂₂₄	2
C₄×D₁₁₁₂	2
#22580781	2
#22580755	2
#346×C₁₃₉	2
#2177	139