Subgroups — #30657761

Order 16448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551659	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#14523346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14523346	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊#14523346	C₂	1	1	yes	maximal, characteristic
5	#22551647	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14523346	C₂	1	1	yes	maximal, characteristic, complemented
7	#22551548	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523346	C₂²	1	1	yes	characteristic
9	C₂×#14523346	C₂²	1	1	yes	characteristic
10	C₂×#14523346	C₂²	1	1	yes	characteristic
11	C₂×#14523346	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₅₇	C₂²	1	1	yes	characteristic
13	#16861597	C₄	1	1	yes	characteristic
14	#16861597	C₄	1	1	yes	characteristic
15	C₂²⋊Dic₂₅₇	C₄	1	1	yes	characteristic
16	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₂₅₇	C₄	1	1	yes	characteristic
19	#14523346	—	2	1	no	—
20	#14523346	—	2	1	no	—
21	#14523346	—	2	1	no	—
22	#14523346	—	2	1	no	—
23	C₂×Dic₂₅₇	—	2	1	no	—
24	C₂×Dic₂₅₇	—	2	1	no	—
25	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
29	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₂₅₇	—	2	1	no	—
33	C₂²×C₂₅₇	—	2	1	no	—
34	C₂²×C₂₅₇	—	2	1	no	—
35	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₂₅₇	—	2	1	no	—
37	C₄×C₂₅₇	—	2	1	no	—
38	Dic₂₅₇	—	2	1	no	—
39	Dic₂₅₇	—	2	1	no	—
40	Dic₂₅₇	—	2	1	no	—
41	Dic₂₅₇	#41	1	1	yes	characteristic
42	Dic₂₅₇	#41	1	1	yes	characteristic
43	C₂×C₂₅₇	—	4	1	no	—
44	C₂×C₂₅₇	#117	1	1	yes	characteristic
45	C₂×C₂₅₇	#131	1	1	yes	characteristic
46	C₂×C₂₅₇	#131	1	1	yes	characteristic
47	C₂₅₇	#433	1	1	yes	characteristic, complemented, Sylow
48	#433	—	257	1	no	maximal, Sylow
49	#117	—	257	1	no	—
50	C₄⋊C₈	—	257	1	no	—
51	C₄⋊C₈	—	257	1	no	—
52	C₂²⋊C₈	—	257	1	no	—
53	C₂²⋊C₈	—	257	1	no	—
54	#97	—	257	1	no	—
55	C₄×C₈	—	257	1	no	—
56	C₂²×C₄	—	257	1	no	—
57	C₂×C₈	—	257	1	no	—
58	C₂×C₈	—	257	1	no	—
59	C₂×C₈	—	257	1	no	—
60	C₂×C₈	—	257	1	no	—
61	C₄⋊C₄	—	257	1	no	—
62	C₄⋊C₄	—	257	1	no	—
63	C₂²⋊C₄	—	257	1	no	—
64	C₂²⋊C₄	#11759967	1	1	yes	characteristic
65	C₄²	—	257	1	no	—
66	C₄²	—	257	1	no	—
67	C₂³	C₂×#11759967	1	1	yes	characteristic
68	C₂×C₄	—	257	1	no	—
69	C₂×C₄	—	257	1	no	—
70	C₂×C₄	—	257	1	no	—
71	C₂×C₄	—	257	1	no	—
72	C₂×C₄	—	514	1	no	—
73	C₂×C₄	—	514	1	no	—
74	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
75	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
76	C₈	—	514	1	no	—
77	C₈	—	514	1	no	—
78	C₈	—	514	1	no	—
79	C₈	—	514	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂²×#11759967	1	1	yes	characteristic
84	C₄	—	257	1	no	—
85	C₄	—	257	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	514	1	no	—
89	C₄	—	514	1	no	—
90	C₄	—	514	1	no	—
91	C₂	—	4	1	no	—
92	C₂	#22551671	1	1	yes	characteristic
93	C₂	#22551679	1	1	yes	characteristic
94	C₂	#22551679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11759967
commutator, socle	C₂²×C₂₅₇	C₂²×C₄
Fitting	C₂²⋊C₄×C₂₅₇	C₄
prime radical	C₂³×C₂₅₇	C₂×C₄
cosocle	C₂×Dic₂₅₇	C₂³
hypercenter	C₂²⋊C₄	#11759967
hypocenter	C₂₅₇	#433
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #22551659	C₂	semidirect
2. C₄⋊#14523346	C₂	semidirect
5. #22551647	C₂	semidirect
6. C₄×#14523346	C₂	semidirect
47. C₂₅₇	#433	semidirect

Maximal subgroups

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

subgroup	index
#22551659	2
C₄⋊#14523346	2
C₂²⋊#14523346	2
#22551647	2
C₄×#14523346	2
#22551548	2
#433	257