Subgroups — #34071339

Order 19328.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752592	C₂	1	1	yes	maximal, characteristic
2	#23752015	C₂	1	4	yes	maximal
3	C₄×C₁₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₀₄	—	2	1	no	—
6	C₂×Dic₆₀₄	C₂²	1	2	yes	—
7	#18012739	—	2	1	no	—
8	C₂×C₁₅₁⋊C₁₆	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
10	#107×C₁₅₁	—	2	1	no	—
11	#107×C₁₅₁	C₂²	1	2	yes	—
12	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₀₂	—	2	2	no	—
14	Dic₆₀₄	—	2	4	no	—
15	Dic₆₀₄	—	4	2	no	—
16	C₄⋊Dic₁₅₁	—	4	1	no	—
17	C₄⋊Dic₁₅₁	—	4	1	no	—
18	C₁₅₁⋊C₁₆	—	2	2	no	—
19	C₂×Q₈×C₁₅₁	—	4	1	no	—
20	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₅₁	—	2	2	no	—
23	C₄⋊C₄×C₁₅₁	—	4	1	no	—
24	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
25	Dic₃₀₂	—	4	2	no	—
26	Dic₃₀₂	—	4	2	no	—
27	C₂×Dic₁₅₁	—	4	2	no	—
28	Q₈×C₁₅₁	—	8	2	no	—
29	C₂×C₄×C₁₅₁	—	4	2	no	—
30	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₅₁	SD₁₆	1	2	yes	—
34	C₈×C₁₅₁	#41	1	2	yes	—
35	C₂²×C₁₅₁	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₅₁	—	8	2	no	—
37	C₄×C₁₅₁	SD₃₂	1	2	yes	—
38	C₄×C₁₅₁	Q₃₂	1	2	yes	—
39	C₄×C₁₅₁	#124	1	1	yes	characteristic
40	C₄×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₁₅₁	—	8	2	no	—
42	C₂×C₁₅₁	#486	1	1	yes	characteristic
43	C₂×C₁₅₁	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₁₅₁	C₂×Q₃₂	1	1	yes	characteristic
45	C₁₅₁	#2241	1	1	yes	characteristic, complemented, Sylow
46	#2241	—	151	1	no	maximal, Sylow
47	#500	D₁₅₁	1	1	yes	characteristic
48	#494	—	151	1	no	—
49	#358	—	151	4	no	—
50	C₄×C₁₆	—	151	1	no	—
51	C₂×Q₁₆	—	302	1	no	—
52	C₂×Q₁₆	—	151	2	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	302	1	no	—
55	C₂×C₁₆	—	151	2	no	—
56	#107	—	2	1	no	—
57	#107	C₂×D₁₅₁	1	2	yes	—
58	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	302	2	no	—
61	Q₁₆	—	302	4	no	—
62	Q₁₆	—	604	2	no	—
63	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	604	1	no	—
68	C₄⋊C₄	—	604	1	no	—
69	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
70	C₁₆	—	302	2	no	—
71	Q₈	—	8	2	no	—
72	Q₈	—	604	2	no	—
73	Q₈	—	604	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	604	2	no	—
76	C₂×C₄	#14558989	1	1	yes	characteristic
77	C₂×C₄	#14558989	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
79	C₈	#14558991	1	2	yes	—
80	C₈	#14559005	1	2	yes	—
81	C₂²	C₂×#14558989	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1208	2	no	—
84	C₄	#18012683	1	2	yes	—
85	C₄	#18012685	1	2	yes	—
86	C₄	#18012728	1	1	yes	characteristic
87	C₄	C₂×#14558991	1	1	yes	characteristic
88	C₂	#23752560	1	1	yes	characteristic
89	C₂	C₂×#18012683	1	1	yes	characteristic
90	C₂	C₂×#18012685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14558989
commutator, cosocle	C₂×C₈×C₁₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₁
Fitting	#500×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	C₂×D₈
hypercenter	#500	D₁₅₁
hypocenter	C₁₅₁	#2241
LCS	G ▷ C₂×C₈×C₁₅₁ ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
45. C₁₅₁	#2241	semidirect

Maximal subgroups

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

subgroup	index
#23752592	2
#23752015	2
C₄×C₁₅₁⋊C₁₆	2
#500×C₁₅₁	2
#2241	151