Subgroups — #34044289

Order 19072.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750021	C₂	1	2	yes	maximal
2	#23750005	C₂	1	1	yes	maximal, characteristic
3	#23749975	C₂	1	2	yes	maximal
4	C₈×#11924746	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₉₆	—	2	2	no	—
7	#18011911	C₂²	1	2	yes	—
8	#18011842	—	2	4	no	—
9	#18011835	—	2	4	no	—
10	C₄×#11924746	C₂²	1	1	yes	characteristic
11	C₄×#11924746	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
13	#107×C₁₄₉	—	2	2	no	—
14	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₉₈	—	2	2	no	—
16	Dic₅₉₆	—	4	4	no	—
17	C₂×#11924746	D₄	1	2	yes	—
18	C₂×#11924746	D₄	1	2	yes	—
19	C₄⋊Dic₁₄₉	—	2	2	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₂×Q₈×C₁₄₉	—	2	2	no	—
22	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₄₉	—	2	2	no	—
24	C₄⋊C₄×C₁₄₉	—	2	2	no	—
25	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
26	Dic₂₉₈	—	4	4	no	—
27	C₂×Dic₁₄₉	—	4	2	no	—
28	#11924746	—	2	2	no	—
29	#11924746	Q₁₆	1	4	yes	complemented
30	Q₈×C₁₄₉	—	4	4	no	—
31	C₂×C₄×C₁₄₉	—	4	2	no	—
32	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₄₉	SD₁₆	1	4	yes	—
36	C₂²×C₁₄₉	#127	1	1	yes	characteristic
37	C₄×C₁₄₉	—	8	2	no	—
38	C₄×C₁₄₉	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₄₉	C₂×Q₁₆	1	2	yes	—
40	C₄×C₁₄₉	#135	1	2	yes	—
41	Dic₁₄₉	—	8	2	no	—
42	C₂×C₁₄₉	#492	1	1	yes	characteristic
43	C₂×C₁₄₉	#494	1	1	yes	characteristic
44	C₂×C₁₄₉	#495	1	1	yes	characteristic
45	C₁₄₉	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	149	1	no	maximal, Sylow
47	#500	D₁₄₉	1	1	yes	characteristic
48	#498	—	149	2	no	—
49	#494	—	149	1	no	—
50	#487	—	149	2	no	—
51	C₈²	—	149	1	no	—
52	C₂×Q₁₆	—	298	2	no	—
53	C₄⋊Q₈	—	149	2	no	—
54	C₄⋊Q₈	C₂×D₁₄₉	1	2	yes	—
55	#107	—	2	2	no	—
56	#106	—	298	4	no	—
57	#103	—	298	4	no	—
58	C₄×C₈	—	149	1	no	—
59	C₄×C₈	—	149	1	no	—
60	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	298	2	no	—
63	Q₁₆	—	596	4	no	—
64	C₂×C₈	—	149	2	no	—
65	C₂×C₈	—	149	2	no	—
66	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	298	2	no	—
70	C₄⋊C₄	—	298	2	no	—
71	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	596	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	596	2	no	—
76	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
79	C₈	—	298	2	no	—
80	C₈	—	149	4	no	—
81	C₈	#14558723	1	4	yes	—
82	C₂²	#18011907	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	1192	2	no	—
85	C₄	C₂×#14558723	1	2	yes	—
86	C₄	Q₁₆×D₁₄₉	1	2	yes	—
87	C₄	#18011930	1	2	yes	—
88	C₂	#23749999	1	1	yes	characteristic
89	C₂	#23750003	1	1	yes	characteristic
90	C₂	#23750008	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#18011907
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	#500×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	#127
hypercenter	#500	D₁₄₉
hypocenter	C₁₄₉	#1707
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
29. #11924746	Q₁₆	semidirect
45. C₁₄₉	#1707	semidirect

Maximal subgroups

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

subgroup	index
#23750021	2
#23750005	2
#23749975	2
C₈×#11924746	2
#500×C₁₄₉	2
#1707	149