Subgroups — #34059824

Order 19072.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750530	C₂	1	2	yes	maximal
2	#23749999	C₂	1	1	yes	maximal, characteristic
3	#18011970	—	2	2	no	—
4	C₂×#14558723	—	2	2	no	—
5	#18011908	C₂²	1	1	yes	characteristic
6	C₄×#11924746	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈×C₁₄₉	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10597010	—	4	2	no	—
9	C₂×D₅₉₆	—	4	1	no	—
10	C₂×D₅₉₆	D₄	1	2	yes	—
11	#14558723	—	4	4	no	—
12	C₂×#11924746	—	2	1	no	—
13	C₂×Q₈×C₁₄₉	—	2	1	no	—
14	C₄⋊C₄×C₁₄₉	—	4	1	no	—
15	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
16	C₂×#10597010	—	8	2	no	—
17	C₂²×D₁₄₉	—	2	2	no	—
18	D₅₉₆	—	4	2	no	—
19	D₅₉₆	—	8	2	no	—
20	#11924746	—	2	2	no	—
21	Q₈×C₁₄₉	—	4	2	no	—
22	C₂×C₄×C₁₄₉	—	2	1	no	—
23	C₂×C₄×C₁₄₉	—	4	1	no	—
24	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₄₉	#99	1	1	yes	characteristic
26	C₂×D₁₄₉	—	4	2	no	—
27	C₂×D₁₄₉	—	8	2	no	—
28	#10597010	—	16	2	no	—
29	C₄×C₁₄₉	—	2	2	no	—
30	C₄×C₁₄₉	—	2	1	no	—
31	C₄×C₁₄₉	—	8	1	no	—
32	C₂×C₁₄₉	—	2	1	no	—
33	C₂×C₁₄₉	#353	1	1	yes	characteristic
34	D₁₄₉	—	8	2	no	—
35	C₁₄₉	#1409	1	1	yes	characteristic, complemented, Sylow
36	#1409	—	149	1	no	maximal, Sylow
37	#492	—	149	1	no	—
38	#353	—	149	2	no	—
39	C₂×SD₁₆	—	298	2	no	—
40	C₄⋊Q₈	#10597010	1	1	yes	characteristic, complemented
41	#127	—	149	1	no	—
42	#99	—	298	2	no	—
43	C₄×C₈	—	149	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	596	1	no	—
46	C₂×D₄	—	149	2	no	—
47	SD₁₆	—	596	4	no	—
48	C₂×C₈	—	298	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	596	2	no	—
51	C₄²	C₂×#10597010	1	1	yes	characteristic
52	C₂³	—	298	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	596	2	no	—
55	D₄	—	1192	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	1192	2	no	—
59	C₂×C₄	C₂²⋊#10597010	1	1	yes	characteristic
60	C₈	—	298	2	no	—
61	C₂²	—	596	2	no	—
62	C₂²	—	1192	2	no	—
63	C₂²	#18011969	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	2384	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	1192	2	no	—
70	C₂	#23750529	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23750529
commutator	C₄²×C₁₄₉	C₂×C₄
Frattini	C₄²	C₂×#10597010
Fitting	C₄⋊Q₈×C₁₄₉	C₄
socle	C₂×C₁₄₉	#353
prime radical, cosocle	#18011908	C₂²
hypercenter	C₄⋊Q₈	#10597010
hypocenter	C₁₄₉	#1409
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂×C₄×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×#11924746	C₄	semidirect
7. C₄⋊Q₈×C₁₄₉	C₄	semidirect
35. C₁₄₉	#1409	semidirect
40. C₄⋊Q₈	#10597010	semidirect

Maximal subgroups

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

subgroup	index
#23750530	2
#23749999	2
#1409	149