Subgroups — #34042879

Order 19072.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750057	C₂	1	1	yes	maximal, characteristic
2	#23749453	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#18011933	—	2	1	no	—
5	Dic₁₁₉₂	C₄	1	1	yes	characteristic, complemented
6	#18011856	—	2	1	no	—
7	D₂₃₈₄	C₄	1	1	yes	characteristic, complemented
8	#18011845	—	2	1	no	—
9	C₂×C₁₆×C₁₄₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₄₉	—	2	1	no	—
11	#14558736	—	4	1	no	—
12	Dic₅₉₆	—	2	1	no	—
13	#14558725	—	4	1	no	—
14	D₁₁₉₂	—	2	1	no	—
15	C₄⋊Dic₁₄₉	—	4	1	no	—
16	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
17	C₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₄₉	D₄	1	1	yes	characteristic
19	Dic₂₉₈	—	4	1	no	—
20	D₅₉₆	—	4	1	no	—
21	C₂²⋊D₁₄₉	—	8	1	no	—
22	C₂×Dic₁₄₉	—	8	1	no	—
23	C₄×D₁₄₉	—	8	1	no	—
24	C₂×C₄×C₁₄₉	D₈	1	1	yes	characteristic
25	C₈×C₁₄₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₄₉	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₄₉	D₁₆	1	1	yes	characteristic
28	C₂×D₁₄₉	—	8	1	no	—
29	C₄×C₁₄₉	#102	1	1	yes	characteristic
30	C₄×C₁₄₉	SD₃₂	1	1	yes	characteristic
31	Dic₁₄₉	—	8	1	no	—
32	Dic₁₄₉	—	16	1	no	—
33	C₂×C₁₄₉	—	2	1	no	—
34	C₂×C₁₄₉	#357	1	1	yes	characteristic
35	D₁₄₉	—	16	1	no	—
36	C₁₄₉	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	149	1	no	maximal, Sylow
38	#508	—	149	1	no	—
39	M₆₄	D₁₄₉	1	1	yes	characteristic, complemented
40	#367	—	149	1	no	—
41	#135	—	298	1	no	—
42	Q₃₂	—	149	1	no	—
43	SD₃₂	—	298	1	no	—
44	D₁₆	—	149	1	no	—
45	C₂×C₁₆	C₂×D₁₄₉	1	1	yes	characteristic
46	#107	—	298	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	596	1	no	—
49	Q₁₆	—	298	1	no	—
50	SD₁₆	—	596	1	no	—
51	D₈	—	298	1	no	—
52	C₂×C₈	D₅₉₆	1	1	yes	characteristic
53	C₄⋊C₄	—	596	1	no	—
54	C₁₆	C₄×D₁₄₉	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₄₉	1	1	yes	characteristic
56	Q₈	—	596	1	no	—
57	D₄	—	596	1	no	—
58	D₄	—	1192	1	no	—
59	C₂×C₄	—	1192	1	no	—
60	C₂×C₄	—	1192	1	no	—
61	C₂×C₄	D₁₁₉₂	1	1	yes	characteristic
62	C₈	#14558713	1	1	yes	characteristic
63	C₈	#14558725	1	1	yes	characteristic
64	C₂²	—	1192	1	no	—
65	C₂²	D₂₃₈₄	1	1	yes	characteristic
66	C₄	—	2384	1	no	—
67	C₄	—	1192	1	no	—
68	C₄	#18011833	1	1	yes	characteristic
69	C₄	#18011856	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2384	1	no	—
72	C₂	#23749427	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23749427
commutator	C₁₆×C₁₄₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₄₉
Fitting	M₆₄×C₁₄₉	C₂
socle	C₂×C₁₄₉	#357
prime radical	#23750057	C₂
cosocle	C₂×C₁₆×C₁₄₉	C₂²
hypercenter	M₆₄	D₁₄₉
hypocenter	C₁₄₉	#1418
LCS	G ▷ C₁₆×C₁₄₉ ▷ C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #23749453	C₂	semidirect
3. M₆₄×C₁₄₉	C₂	semidirect
5. Dic₁₁₉₂	C₄	semidirect
7. D₂₃₈₄	C₄	semidirect
36. C₁₄₉	#1418	semidirect
39. M₆₄	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750057	2
#23749453	2
M₆₄×C₁₄₉	2
#1418	149