Subgroups — #37510402

Order 22912.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26038487	C₂	1	3	yes	maximal
2	C₂×#19153031	C₂	1	3	yes	maximal
3	#338×C₁₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₅₈	C₂²	1	1	yes	characteristic
5	C₂×#15668243	C₂²	1	3	yes	—
6	#19153048	—	2	6	no	—
7	#19153031	—	2	6	no	—
8	C₂×M₁₆×C₁₇₉	C₂²	1	3	yes	—
9	C₂×Dic₃₅₈	—	2	3	no	—
10	C₂×Dic₃₅₈	—	2	3	no	—
11	C₂²×Dic₁₇₉	Q₈	1	2	yes	—
12	#15668243	—	2	3	no	—
13	#15668243	D₄	1	6	yes	—
14	C₂×C₁₇₉⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₇₉	—	2	3	no	—
17	M₁₆×C₁₇₉	—	2	3	no	—
18	C₂×C₈×C₁₇₉	—	2	3	no	—
19	Dic₃₅₈	—	4	3	no	—
20	Dic₃₅₈	—	4	1	no	—
21	C₂×Dic₁₇₉	—	2	6	no	—
22	C₁₇₉⋊C₈	—	2	6	no	—
23	C₂³×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₁₇₉	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₇₉	#41	1	3	yes	—
26	C₈×C₁₇₉	—	4	3	no	—
27	C₂²×C₁₇₉	—	4	1	no	—
28	C₂²×C₁₇₉	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₁₇₉	#121	1	3	yes	—
30	C₄×C₁₇₉	#126	1	1	yes	characteristic
31	Dic₁₇₉	—	4	2	no	—
32	C₂×C₁₇₉	—	2	3	no	—
33	C₂×C₁₇₉	#396	1	1	yes	characteristic
34	C₁₇₉	#2070	1	1	yes	characteristic, complemented, Sylow
35	#2070	—	179	1	no	maximal, Sylow
36	#430	—	179	3	no	—
37	C₂×#101	—	179	3	no	—
38	#338	D₁₇₉	1	1	yes	characteristic
39	C₂²×Q₈	—	179	1	no	—
40	C₂×M₁₆	—	179	3	no	—
41	C₂×M₁₆	C₂×D₁₇₉	1	3	yes	—
42	#108	—	358	6	no	—
43	#101	—	358	6	no	—
44	C₂×Q₈	—	358	3	no	—
45	C₂×Q₈	—	358	3	no	—
46	C₂²×C₄	—	179	2	no	—
47	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	358	3	no	—
51	M₁₆	—	179	6	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	358	3	no	—
54	C₂³	Q₈×D₁₇₉	1	1	yes	characteristic
55	Q₈	—	716	3	no	—
56	Q₈	—	716	1	no	—
57	C₂×C₄	—	358	6	no	—
58	C₂×C₄	D₄×D₁₇₉	1	3	yes	—
59	C₂×C₄	#15668259	1	3	yes	—
60	C₈	—	4	3	no	—
61	C₈	—	358	6	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#19153091	1	3	yes	—
64	C₄	—	716	2	no	—
65	C₄	#19153086	1	3	yes	—
66	C₄	#19153107	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#26038363	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26038363
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#338×C₁₇₉	C₂
socle	C₂×C₁₇₉	#396
prime radical	C₂³×C₁₇₉	C₂×Q₈
hypercenter	#338	D₁₇₉
hypocenter	C₁₇₉	#2070
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
34. C₁₇₉	#2070	semidirect

Maximal subgroups

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

subgroup	index
#26038487	2
C₂×#19153031	2
#338×C₁₇₉	2
#2070	179