Subgroups — #31800966

Order 17728.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984572	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555667	C₂	1	1	yes	maximal, characteristic
3	#22579935	C₂	1	2	yes	maximal, complemented
4	C₄×#14555667	C₂	1	2	yes	maximal, complemented
5	#22579836	C₂	1	1	yes	maximal, characteristic
6	#17984572	C₄	1	4	yes	complemented
7	C₂×#14555667	C₄	1	4	yes	complemented
8	C₂×#14555667	C₂²	1	1	yes	characteristic
9	C₂×#14555667	C₂²	1	2	yes	—
10	C₂×#14555667	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₇₇	C₂²	1	1	yes	characteristic
12	#17984542	C₄	1	2	yes	—
13	C₂²⋊Dic₂₇₇	C₄	1	1	yes	characteristic
14	C₄×Dic₂₇₇	C₂²	1	2	yes	—
15	C₂²⋊C₄×C₂₇₇	C₄	1	1	yes	characteristic
16	#14555667	C₂×C₄	1	2	yes	—
17	#14555667	C₂×C₄	1	2	yes	—
18	#14555667	C₂×C₄	1	4	yes	—
19	C₂×Dic₂₇₇	C₂×C₄	1	2	yes	—
20	C₂×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₇₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₇₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₂₇₇	C₂³	1	1	yes	characteristic
24	C₂³×C₂₇₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
26	C₂²×C₂₇₇	—	2	2	no	—
27	C₂²×C₂₇₇	C₄²	1	2	yes	—
28	C₂²×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₇₇	—	2	2	no	—
30	Dic₂₇₇	—	2	2	no	—
31	Dic₂₇₇	C₄²	1	2	yes	—
32	Dic₂₇₇	C₂²×C₄	1	1	yes	characteristic
33	Dic₂₇₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₂₇₇	—	2	2	no	—
35	C₂×C₂₇₇	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₂₇₇	#131	1	2	yes	—
37	C₂₇₇	#405	1	1	yes	characteristic, complemented, Sylow
38	#405	—	277	1	no	maximal, Sylow
39	C₂×M₁₆	—	277	1	no	—
40	C₂²×C₈	—	277	1	no	—
41	#117	—	277	1	no	—
42	#97	—	277	2	no	—
43	C₄×C₈	—	277	2	no	—
44	C₂²×C₄	—	277	1	no	—
45	M₁₆	—	277	4	no	—
46	C₂×C₈	—	277	4	no	—
47	C₂×C₈	—	277	1	no	—
48	C₂×C₈	—	277	2	no	—
49	C₂×C₈	—	277	1	no	—
50	C₄⋊C₄	—	277	2	no	—
51	C₂²⋊C₄	—	277	1	no	—
52	C₂²⋊C₄	#11764308	1	1	yes	characteristic
53	C₄²	—	277	2	no	—
54	C₂³	C₂×#11764308	1	1	yes	characteristic
55	C₂×C₄	—	277	2	no	—
56	C₂×C₄	—	277	2	no	—
57	C₂×C₄	—	277	1	no	—
58	C₂×C₄	—	277	1	no	—
59	C₂×C₄	—	277	2	no	—
60	C₂×C₄	C₂×#11764308	1	2	yes	—
61	C₈	—	277	2	no	—
62	C₈	—	277	2	no	—
63	C₈	—	277	4	no	—
64	C₂²	—	2	2	no	—
65	C₂²	C₄×#11764308	1	2	yes	—
66	C₂²	C₂²×#11764308	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	277	1	no	—
69	C₄	—	277	1	no	—
70	C₄	—	277	2	no	—
71	C₄	—	554	2	no	—
72	C₂	—	2	2	no	—
73	C₂	C₂×C₄×#11764308	1	1	yes	characteristic
74	C₂	#22579967	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764308
commutator	C₂×C₂₇₇	C₂×C₄²
Fitting	C₂²⋊C₄×C₂₇₇	C₄
socle	C₂²×C₂₇₇	C₂²×C₄
prime radical	C₂³×C₂₇₇	C₂×C₄
cosocle	C₂×Dic₂₇₇	C₂³
hypercenter	C₂²⋊C₄	#11764308
hypocenter	C₂₇₇	#405
LCS	G ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22579935	C₂	semidirect
4. C₄×#14555667	C₂	semidirect
6. #17984572	C₄	semidirect
7. C₂×#14555667	C₄	semidirect
37. C₂₇₇	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984572	2
C₂²×#14555667	2
#22579935	2
C₄×#14555667	2
#22579836	2
#405	277