Subgroups — #34070215

Order 19328.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23753012	C₂	1	3	yes	maximal, complemented
2	#23752180	C₂	1	3	yes	maximal, complemented
3	#338×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14559004	C₂²	1	3	yes	—
5	#18012772	—	2	6	no	—
6	#18012767	—	2	6	no	—
7	#18012761	—	2	6	no	—
8	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
9	(C₂×D₁₅₁)⋊C₈	—	2	6	no	—
10	C₂×M₁₆×C₁₅₁	C₂²	1	3	yes	—
11	#14559004	—	2	6	no	—
12	#14559004	—	2	6	no	—
13	C₂×Dic₃₀₂	D₄	1	3	yes	—
14	C₂×D₆₀₄	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₅₁	—	2	3	no	—
16	C₂×C₄×D₁₅₁	—	2	3	no	—
17	Dic₆₀₄	—	2	6	no	—
18	#14558993	—	2	12	no	—
19	D₁₂₀₈	—	2	6	no	—
20	C₂×C₁₅₁⋊C₈	—	2	3	no	—
21	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₅₁	—	2	6	no	—
23	C₂×C₈×C₁₅₁	D₄	1	6	yes	—
24	C₂²×D₁₅₁	—	2	3	no	—
25	Dic₃₀₂	—	2	6	no	—
26	D₆₀₄	—	2	6	no	—
27	C₂²⋊D₁₅₁	—	4	6	no	—
28	C₂×Dic₁₅₁	—	2	3	no	—
29	C₄×D₁₅₁	—	4	3	no	—
30	C₄×D₁₅₁	—	4	3	no	—
31	C₁₅₁⋊C₈	—	4	1	no	—
32	C₂³×C₁₅₁	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
35	C₈×C₁₅₁	—	2	6	no	—
36	C₂²×C₁₅₁	—	4	1	no	—
37	C₂²×C₁₅₁	#121	1	3	yes	—
38	C₂×D₁₅₁	—	4	6	no	—
39	C₂×D₁₅₁	—	4	3	no	—
40	C₄×C₁₅₁	#120	1	3	yes	—
41	C₄×C₁₅₁	#127	1	1	yes	characteristic
42	Dic₁₅₁	—	4	3	no	—
43	C₂×C₁₅₁	—	2	3	no	—
44	C₂×C₁₅₁	#392	1	1	yes	characteristic
45	D₁₅₁	—	8	3	no	—
46	C₁₅₁	#2017	1	1	yes	characteristic, complemented, Sylow
47	#2017	—	151	1	no	maximal, Sylow
48	#575	—	151	3	no	—
49	#408	—	151	3	no	—
50	#338	D₁₅₁	1	1	yes	characteristic, complemented
51	C₂×#41	—	151	3	no	—
52	#137	—	302	6	no	—
53	AGL(1,ℤ/8)	—	302	6	no	—
54	#135	—	302	6	no	—
55	C₂×M₁₆	C₂×D₁₅₁	1	3	yes	—
56	C₂²×C₈	—	151	1	no	—
57	C₂²⋊C₈	—	302	6	no	—
58	#41	—	302	6	no	—
59	#41	—	302	6	no	—
60	C₂×Q₈	—	151	3	no	—
61	C₂×D₄	—	302	3	no	—
62	C₂×D₄	—	151	3	no	—
63	C₂²×C₄	—	302	3	no	—
64	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
65	Q₁₆	—	302	6	no	—
66	SD₁₆	—	302	12	no	—
67	D₈	—	302	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	302	3	no	—
70	C₂×C₈	D₆₀₄	1	6	yes	—
71	C₂³	—	302	3	no	—
72	C₂³	#14559005	1	1	yes	characteristic
73	Q₈	—	302	6	no	—
74	D₄	—	302	6	no	—
75	D₄	—	604	6	no	—
76	C₂×C₄	—	302	3	no	—
77	C₂×C₄	—	604	3	no	—
78	C₂×C₄	—	604	3	no	—
79	C₂×C₄	D₄×D₁₅₁	1	3	yes	—
80	C₂×C₄	C₂×D₆₀₄	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	604	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	604	6	no	—
85	C₂²	—	604	3	no	—
86	C₂²	#18012714	1	3	yes	—
87	C₄	—	604	3	no	—
88	C₄	#18012709	1	3	yes	—
89	C₄	#18012736	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	1208	3	no	—
92	C₂	#23752122	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#18012736
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#338×C₁₅₁	C₂
socle	C₂×C₁₅₁	#392
hypercenter	#338	D₁₅₁
hypocenter	C₁₅₁	#2017
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #23753012	C₂	semidirect
2. #23752180	C₂	semidirect
3. #338×C₁₅₁	C₂	semidirect
46. C₁₅₁	#2017	semidirect
50. #338	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23753012	2
#23752180	2
#338×C₁₅₁	2
#2017	151