Subgroups — #34070995

Order 19328.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752644	C₂	1	1	yes	maximal, characteristic
2	#23752328	C₂	1	2	yes	maximal, complemented
3	#23752018	C₂	1	2	yes	maximal, complemented
4	#23751987	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#18012761	C₂²	1	2	yes	—
7	#18012747	—	2	2	no	—
8	Dic₁₂₀₈	C₄	1	1	yes	characteristic, complemented
9	#18012684	C₄	1	2	yes	complemented
10	D₂₄₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#18012665	—	2	2	no	—
13	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
15	#108×C₁₅₁	C₂²	1	2	yes	—
16	#14559004	—	2	2	no	—
17	Dic₆₀₄	C₂×C₄	1	2	yes	—
18	#14558993	—	2	2	no	—
19	D₁₂₀₈	C₂×C₄	1	2	yes	—
20	#14558987	—	4	2	no	—
21	C₂×C₁₅₁⋊C₈	D₄	1	1	yes	characteristic
22	C₈×D₁₅₁	—	4	2	no	—
23	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
24	C₁₅₁⋊C₁₆	—	2	1	no	—
25	M₁₆×C₁₅₁	—	2	2	no	—
26	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
29	Dic₃₀₂	—	2	2	no	—
30	D₆₀₄	—	2	2	no	—
31	C₂²⋊D₁₅₁	—	4	2	no	—
32	C₂×Dic₁₅₁	—	2	1	no	—
33	C₄×D₁₅₁	—	4	2	no	—
34	C₁₅₁⋊C₈	D₈	1	2	yes	—
35	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₅₁	—	4	2	no	—
37	C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₅₁	#41	1	1	yes	characteristic
39	C₂²×C₁₅₁	#135	1	1	yes	characteristic
40	C₂×D₁₅₁	—	4	2	no	—
41	C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
43	Dic₁₅₁	—	4	1	no	—
44	Dic₁₅₁	—	4	2	no	—
45	C₂×C₁₅₁	—	2	1	no	—
46	C₂×C₁₅₁	C₄×D₈	1	1	yes	characteristic
47	D₁₅₁	—	8	2	no	—
48	C₁₅₁	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	151	1	no	maximal, Sylow
50	#508	—	151	1	no	—
51	#443	—	151	2	no	—
52	#368	D₁₅₁	1	1	yes	characteristic, complemented
53	#359	—	151	2	no	—
54	#346	—	151	1	no	—
55	#135	—	151	2	no	—
56	#131	—	302	2	no	—
57	Q₃₂	—	151	1	no	—
58	SD₃₂	—	151	2	no	—
59	D₁₆	—	151	1	no	—
60	C₂×C₁₆	—	151	1	no	—
61	C₂×C₁₆	C₂×D₁₅₁	1	1	yes	characteristic
62	#108	C₂×D₁₅₁	1	2	yes	—
63	#104	—	302	2	no	—
64	C₄×C₈	—	151	1	no	—
65	#41	—	302	2	no	—
66	Q₁₆	—	151	2	no	—
67	SD₁₆	—	302	2	no	—
68	D₈	—	151	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	604	2	no	—
71	C₂×C₈	—	604	2	no	—
72	C₂×C₈	—	151	1	no	—
73	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
74	C₄²	—	151	1	no	—
75	C₁₆	—	302	1	no	—
76	C₁₆	C₄×D₁₅₁	1	1	yes	characteristic
77	C₁₆	C₄×D₁₅₁	1	1	yes	characteristic
78	Q₈	—	302	2	no	—
79	D₄	—	302	2	no	—
80	D₄	—	604	2	no	—
81	C₂×C₄	—	302	1	no	—
82	C₂×C₄	—	604	2	no	—
83	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	—	151	2	no	—
86	C₈	C₂×C₄×D₁₅₁	1	1	yes	characteristic
87	C₈	#14559005	1	1	yes	characteristic
88	C₂²	—	604	2	no	—
89	C₂²	#18012762	1	1	yes	characteristic
90	C₄	—	604	1	no	—
91	C₄	—	604	2	no	—
92	C₄	#18012701	1	1	yes	characteristic
93	C₄	D₈×D₁₅₁	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	1208	2	no	—
96	C₂	#23752294	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₅₁
commutator	C₈×C₁₅₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅₁
Fitting	#368×C₁₅₁	C₂
socle	C₂×C₁₅₁	C₄×D₈
prime radical	#23752644	C₂
cosocle	C₂×C₈×C₁₅₁	C₂³
hypercenter	#368	D₁₅₁
hypocenter	C₁₅₁	#2179
LCS	G ▷ C₈×C₁₅₁ ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #23752328	C₂	semidirect
3. #23752018	C₂	semidirect
4. #23751987	C₂	semidirect
5. #368×C₁₅₁	C₂	semidirect
8. Dic₁₂₀₈	C₄	semidirect
9. #18012684	C₄	semidirect
10. D₂₄₁₆	C₄	semidirect
48. C₁₅₁	#2179	semidirect
52. #368	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752644	2
#23752328	2
#23752018	2
#23751987	2
#368×C₁₅₁	2
#2179	151