Subgroups — #34061105

Order 19072.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750546	C₂	1	2	yes	maximal
2	C₄⋊#14558739	C₂	1	1	yes	maximal, characteristic
3	#23750511	C₂	1	2	yes	maximal
4	C₄×#14558739	C₂	1	1	yes	maximal, characteristic
5	#23749749	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558739	C₂²	1	1	yes	characteristic
7	C₂×#14558739	C₂²	1	2	yes	—
8	C₂×#14558739	C₂²	1	1	yes	characteristic
9	C₄×Dic₂₉₈	C₄	1	1	yes	characteristic
10	#18011840	C₄	1	2	yes	—
11	C₄×#11924746	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₄₉	C₂²	1	2	yes	—
13	C₄⋊C₈×C₁₄₉	C₄	1	1	yes	characteristic
14	#14558739	—	2	1	no	—
15	#14558739	—	2	2	no	—
16	#14558739	Q₈	1	2	yes	complemented
17	C₂×Dic₂₉₈	C₈	1	1	yes	characteristic
18	C₂×#11924746	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924746	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₁₄₉	C₈	1	1	yes	characteristic
21	#14558715	C₈	1	2	yes	—
22	C₄×Dic₁₄₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₄₉	C₂×C₄	1	2	yes	—
24	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
25	Dic₂₉₈	C₁₆	1	4	yes	complemented
26	C₂×Dic₁₄₉	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₄₉	C₂×C₈	1	2	yes	—
28	#11924746	—	2	1	no	—
29	#11924746	C₂×Q₈	1	1	yes	characteristic
30	#11924746	#41	1	1	yes	characteristic
31	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₄₉	—	2	2	no	—
35	C₂²×C₁₄₉	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₄₉	—	2	1	no	—
37	C₄×C₁₄₉	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₄₉	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₄₉	#131	1	1	yes	characteristic
40	Dic₁₄₉	—	2	2	no	—
41	Dic₁₄₉	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₄₉	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₄₉	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₄₉	#504	1	1	yes	characteristic
45	C₁₄₉	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	Q₈×C₁₆	—	149	1	no	maximal, Sylow
47	C₈×Q₈	—	149	1	no	—
48	C₄⋊C₁₆	—	149	1	no	—
49	C₄⋊C₁₆	—	149	2	no	—
50	C₄×C₁₆	—	149	1	no	—
51	C₄×C₁₆	—	149	2	no	—
52	C₄×Q₈	—	149	1	no	—
53	C₂×C₁₆	—	149	1	no	—
54	C₂×C₁₆	—	149	2	no	—
55	C₂×C₁₆	—	149	1	no	—
56	C₄⋊C₈	—	149	2	no	—
57	C₄⋊C₈	#10597010	1	1	yes	characteristic
58	C₄×C₈	—	149	2	no	—
59	C₄×C₈	—	149	1	no	—
60	C₂×Q₈	—	149	1	no	—
61	C₂×C₈	—	149	1	no	—
62	C₂×C₈	—	149	1	no	—
63	C₂×C₈	C₂×#10597010	1	2	yes	—
64	C₄⋊C₄	—	149	2	no	—
65	C₄⋊C₄	—	149	1	no	—
66	C₄²	—	149	2	no	—
67	C₄²	C₂×#10597010	1	1	yes	characteristic
68	C₁₆	—	298	1	no	—
69	C₁₆	—	298	2	no	—
70	C₁₆	—	149	2	no	—
71	Q₈	—	149	4	no	—
72	C₂×C₄	—	149	2	no	—
73	C₂×C₄	—	149	2	no	—
74	C₂×C₄	D₁₄₉⋊C₈	1	1	yes	characteristic
75	C₂×C₄	D₁₄₉⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	298	1	no	—
79	C₈	—	149	1	no	—
80	C₈	—	149	1	no	—
81	C₂²	C₂×D₁₄₉⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	298	2	no	—
84	C₄	—	149	4	no	—
85	C₄	D₁₄₉⋊C₁₆	1	2	yes	—
86	C₄	Q₈×#10597010	1	1	yes	characteristic
87	C₄	#18011998	1	1	yes	characteristic
88	C₂	#23750674	1	1	yes	characteristic
89	C₂	C₂×D₁₄₉⋊C₁₆	1	1	yes	characteristic
90	C₂	#23750684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10597010
commutator	C₂×C₁₄₉	C₂²×C₁₆
Fitting	C₄⋊C₈×C₁₄₉	C₄
socle, prime radical	C₂²×C₁₄₉	C₂²×C₈
cosocle	C₂×#11924746	C₂³
hypercenter	C₄⋊C₈	#10597010
hypocenter	C₁₄₉	Q₈×C₁₆
LCS	G ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #14558739	Q₈	semidirect
25. Dic₂₉₈	C₁₆	semidirect
45. C₁₄₉	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23750546	2
C₄⋊#14558739	2
#23750511	2
C₄×#14558739	2
#23749749	2
Q₈×C₁₆	149