Subgroups — C₂×#22579815 #31800047

Order 17728.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984547	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555660	C₂	1	1	yes	maximal, characteristic
3	#22579815	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	#17984547	—	2	1	no	—
6	#17984547	C₂²	1	2	yes	—
7	C₂×#14555660	—	2	1	no	—
8	C₂×#14555660	C₄	1	2	yes	—
9	C₂×#14555660	C₄	1	2	yes	—
10	C₂×#14555660	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
12	M₁₆×C₂₇₇	—	2	1	no	—
13	M₁₆×C₂₇₇	C₂²	1	2	yes	—
14	C₂×C₈×C₂₇₇	—	2	1	no	—
15	#14555660	—	2	2	no	—
16	#14555660	C₂×C₄	1	4	yes	—
17	C₂³×C₂₇₇	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₇₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₇₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₇₇	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
24	C₈×C₂₇₇	—	2	2	no	—
25	C₂²×C₂₇₇	—	2	2	no	—
26	C₂²×C₂₇₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₇₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₇₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₂₇₇	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₂₇₇	—	2	1	no	—
34	C₂×C₂₇₇	—	2	1	no	—
35	C₂×C₂₇₇	#108	1	2	yes	complemented
36	C₂×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂₇₇	C₂×#108	1	1	yes	characteristic, complemented, Sylow
38	C₂×#108	—	277	1	no	maximal, Sylow
39	C₂×M₁₆	—	277	1	no	—
40	C₂×M₁₆	D₂₇₇	1	1	yes	characteristic
41	C₂²×C₈	—	277	1	no	—
42	#108	—	277	4	no	—
43	C₂²×C₄	C₂×D₂₇₇	1	1	yes	characteristic
44	M₁₆	—	2	1	no	—
45	M₁₆	—	277	2	no	—
46	M₁₆	—	554	1	no	—
47	M₁₆	C₂×D₂₇₇	1	2	yes	—
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	277	2	no	—
50	C₂×C₈	—	277	2	no	—
51	C₂×C₈	—	277	2	no	—
52	C₂×C₈	—	554	1	no	—
53	C₂³	Dic₅₅₄	1	1	yes	characteristic
54	C₂×C₄	C₄×D₂₇₇	1	1	yes	characteristic
55	C₂×C₄	C₄×D₂₇₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₇₇	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₂₇₇	1	1	yes	characteristic
58	C₂×C₄	Dic₅₅₄	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	277	4	no	—
62	C₈	—	554	2	no	—
63	C₂²	—	2	2	no	—
64	C₂²	#17984542	1	1	yes	characteristic
65	C₂²	#17984542	1	1	yes	characteristic
66	C₂²	C₂×Dic₅₅₄	1	1	yes	characteristic
67	C₄	#17984542	1	1	yes	characteristic
68	C₄	#17984542	1	1	yes	characteristic
69	C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	#22579815	1	2	yes	direct product factor
74	C₂	C₂×#17984542	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₇₇
commutator	C₄×C₂₇₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₇₇
Fitting	C₂×M₁₆×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₄⋊C₄
prime radical	C₂³×C₂₇₇	Q₈
cosocle	C₂×C₄×C₂₇₇	C₂³
hypercenter	C₂×M₁₆	D₂₇₇
hypocenter	C₂₇₇	C₂×#108
LCS	G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22579815	C₂	direct (N × H)
35. C₂×C₂₇₇	#108	semidirect
37. C₂₇₇	C₂×#108	semidirect
73. C₂	#22579815	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984547	2
C₂²×#14555660	2
#22579815	2
C₂×M₁₆×C₂₇₇	2
C₂×#108	277

Subgroups — C2×#22579815 #31800047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22579815 #31800047