Subgroups — #34059658

Order 19072.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14558739	C₂	1	1	yes	maximal, characteristic
2	C₄×#14558739	C₂	1	1	yes	maximal, characteristic
3	#23749749	C₂	1	1	yes	maximal, characteristic
4	C₂×#14558739	—	2	1	no	—
5	C₂×#14558739	—	2	1	no	—
6	C₄×Dic₂₉₈	C₄	1	1	yes	characteristic
7	#18011840	—	2	1	no	—
8	C₄×#11924746	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₄₉	—	2	1	no	—
10	C₄⋊C₈×C₁₄₉	C₄	1	1	yes	characteristic
11	#14558739	—	2	2	no	—
12	#14558739	—	4	1	no	—
13	C₂×Dic₂₉₈	C₈	1	1	yes	characteristic
14	C₂×#11924746	D₄	1	1	yes	characteristic
15	C₂×#11924746	D₄	1	1	yes	characteristic
16	C₄⋊Dic₁₄₉	C₈	1	1	yes	characteristic
17	#14558715	—	2	1	no	—
18	C₄×Dic₁₄₉	—	2	1	no	—
19	C₂×C₈×C₁₄₉	—	2	1	no	—
20	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
21	Dic₂₉₈	—	2	1	no	—
22	Dic₂₉₈	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₄₉	—	2	1	no	—
24	C₂×Dic₁₄₉	—	2	1	no	—
25	#11924746	—	2	1	no	—
26	#11924746	SD₁₆	1	1	yes	characteristic
27	#11924746	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₄₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₄₉	M₁₆	1	1	yes	characteristic
31	C₈×C₁₄₉	—	4	1	no	—
32	C₂²×C₁₄₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₄₉	—	2	1	no	—
34	C₄×C₁₄₉	#103	1	1	yes	characteristic
35	C₄×C₁₄₉	#104	1	1	yes	characteristic
36	C₄×C₁₄₉	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₄₉	M₃₂	1	1	yes	characteristic
38	Dic₁₄₉	—	2	2	no	—
39	Dic₁₄₉	—	4	1	no	—
40	C₂×C₁₄₉	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₄₉	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₄₉	#350	1	1	yes	characteristic
43	C₁₄₉	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	149	1	no	maximal, Sylow
45	C₈×Q₈	—	149	1	no	—
46	C₄⋊C₁₆	—	149	1	no	—
47	C₄×C₁₆	—	149	1	no	—
48	C₄×Q₈	—	149	1	no	—
49	C₂×C₁₆	—	298	1	no	—
50	C₂×C₁₆	—	298	1	no	—
51	C₄⋊C₈	—	298	1	no	—
52	C₄⋊C₈	#10597010	1	1	yes	characteristic
53	C₄×C₈	—	298	1	no	—
54	C₄×C₈	—	149	1	no	—
55	C₂×Q₈	—	149	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	149	1	no	—
58	C₂×C₈	—	149	1	no	—
59	C₄⋊C₄	—	298	1	no	—
60	C₄⋊C₄	—	149	1	no	—
61	C₄²	—	298	1	no	—
62	C₄²	C₂×#10597010	1	1	yes	characteristic
63	C₁₆	—	298	2	no	—
64	C₁₆	—	596	1	no	—
65	Q₈	—	298	1	no	—
66	Q₈	—	149	2	no	—
67	C₂×C₄	—	298	1	no	—
68	C₂×C₄	—	298	1	no	—
69	C₂×C₄	C₂²⋊#10597010	1	1	yes	characteristic
70	C₂×C₄	D₁₄₉⋊C₈	1	1	yes	characteristic
71	C₂×C₄	#14558746	1	1	yes	characteristic
72	C₈	—	4	1	no	—
73	C₈	—	298	1	no	—
74	C₈	—	149	1	no	—
75	C₈	—	149	1	no	—
76	C₂²	#18011968	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	298	2	no	—
79	C₄	—	596	1	no	—
80	C₄	#18011974	1	1	yes	characteristic
81	C₄	#18011975	1	1	yes	characteristic
82	C₄	D₁₄₉⋊C₁₆	1	1	yes	characteristic
83	C₄	#18011986	1	1	yes	characteristic
84	C₂	#23750475	1	1	yes	characteristic
85	C₂	#23750520	1	1	yes	characteristic
86	C₂	#23750524	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10597010
commutator	C₄×C₁₄₉	C₂×C₁₆
Frattini	C₄²	C₂×#10597010
Fitting	C₄⋊C₈×C₁₄₉	C₄
socle, prime radical	C₂²×C₁₄₉	C₂²⋊C₈
cosocle	C₄×#11924746	C₂²
hypercenter	C₄⋊C₈	#10597010
hypocenter	C₁₄₉	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
22. Dic₂₉₈	C₁₆	semidirect
43. 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
C₄⋊#14558739	2
C₄×#14558739	2
#23749749	2
Q₈⋊C₁₆	149