Subgroups — #34044303

Order 19072.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23749976	C₂	1	3	yes	maximal, complemented
2	#23749972	C₂	1	3	yes	maximal
3	C₈²×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#18011911	C₂²	1	3	yes	—
5	#18011908	C₂²	1	1	yes	characteristic
6	#18011836	—	2	6	no	—
7	#18011833	—	2	6	no	—
8	C₄×C₈×C₁₄₉	C₂²	1	3	yes	—
9	C₂×Dic₂₉₈	—	2	3	no	—
10	C₂×D₅₉₆	—	2	3	no	—
11	C₄⋊Dic₁₄₉	—	2	3	no	—
12	C₄⋊Dic₁₄₉	—	2	3	no	—
13	C₂×C₈×C₁₄₉	D₄	1	6	yes	—
14	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
15	C₂²×D₁₄₉	—	4	1	no	—
16	Dic₂₉₈	—	4	6	no	—
17	D₅₉₆	—	4	6	no	—
18	C₂×Dic₁₄₉	—	4	3	no	—
19	C₂×C₄×C₁₄₉	C₂×D₄	1	3	yes	—
20	C₈×C₁₄₉	—	2	6	no	—
21	C₂²×C₁₄₉	#127	1	1	yes	characteristic
22	C₂×D₁₄₉	—	8	3	no	—
23	C₄×C₁₄₉	#135	1	6	yes	—
24	Dic₁₄₉	—	8	3	no	—
25	C₂×C₁₄₉	#495	1	3	yes	—
26	D₁₄₉	—	16	1	no	—
27	C₁₄₉	#1711	1	1	yes	characteristic, complemented, Sylow
28	#1711	—	149	1	no	maximal, Sylow
29	#487	—	149	3	no	—
30	#486	—	149	3	no	—
31	C₈²	D₁₄₉	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	—	149	3	no	—
33	#127	—	149	1	no	—
34	#103	—	298	6	no	—
35	#102	—	298	6	no	—
36	C₄×C₈	C₂×D₁₄₉	1	3	yes	—
37	C₂×Q₈	—	298	3	no	—
38	C₂×D₄	—	298	3	no	—
39	C₂×C₈	D₅₉₆	1	6	yes	—
40	C₄⋊C₄	—	298	3	no	—
41	C₄⋊C₄	—	298	3	no	—
42	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
43	C₂³	—	596	1	no	—
44	Q₈	—	596	6	no	—
45	D₄	—	596	6	no	—
46	C₂×C₄	—	596	3	no	—
47	C₂×C₄	C₂×D₅₉₆	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	1192	3	no	—
50	C₂²	#18011908	1	1	yes	characteristic
51	C₄	—	1192	3	no	—
52	C₄	#18011933	1	6	yes	—
53	C₂	—	2384	1	no	—
54	C₂	#23750009	1	3	yes	—

	subgroup	G/H
center	C₂²	#18011908
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	C₈²×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#127
prime radical	#18011908	C₂²
hypercenter	C₈²	D₁₄₉
hypocenter	C₁₄₉	#1711
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #23749976	C₂	semidirect
3. C₈²×C₁₄₉	C₂	semidirect
27. C₁₄₉	#1711	semidirect
31. C₈²	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23749976	2
#23749972	2
C₈²×C₁₄₉	2
#1711	149