Subgroups — #41011385

Order 27008.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213404	C₂	1	1	yes	maximal, characteristic
2	#27212928	C₂	1	1	yes	maximal, characteristic
3	H(3,ℤ/4)×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₄₄	C₄	1	1	yes	characteristic, complemented
5	#20269956	—	2	1	no	—
6	#20269917	—	2	1	no	—
7	#20269909	C₄	1	1	yes	characteristic, complemented
8	#20269902	—	2	1	no	—
9	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
10	#117×C₂₁₁	—	2	1	no	—
11	C₂×Dic₄₂₂	—	2	1	no	—
12	Dic₈₄₄	—	4	1	no	—
13	C₄⋊Dic₂₁₁	—	2	1	no	—
14	Dic₂₁₁⋊C₄	—	4	1	no	—
15	C₂²⋊Dic₂₁₁	—	4	1	no	—
16	C₂₁₁⋊C₁₆	—	4	1	no	—
17	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
18	M₁₆×C₂₁₁	—	4	1	no	—
19	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₂₁₁	—	4	1	no	—
22	C₂²⋊C₄×C₂₁₁	—	4	1	no	—
23	C₄²×C₂₁₁	—	2	2	no	—
24	Dic₄₂₂	—	4	1	no	—
25	Dic₄₂₂	—	4	1	no	—
26	C₂×Dic₂₁₁	—	4	1	no	—
27	C₂×Dic₂₁₁	—	4	1	no	—
28	C₂³×C₂₁₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	—	2	1	no	—
30	C₂×C₄×C₂₁₁	—	2	1	no	—
31	C₂×C₄×C₂₁₁	—	4	2	no	—
32	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
34	C₈×C₂₁₁	—	2	1	no	—
35	C₈×C₂₁₁	—	2	1	no	—
36	C₂²×C₂₁₁	—	2	1	no	—
37	C₂²×C₂₁₁	—	4	1	no	—
38	C₂²×C₂₁₁	#102	1	1	yes	characteristic
39	C₄×C₂₁₁	—	2	1	no	—
40	C₄×C₂₁₁	—	8	2	no	—
41	C₄×C₂₁₁	#99	1	1	yes	characteristic
42	C₄×C₂₁₁	#104	1	1	yes	characteristic
43	Dic₂₁₁	—	8	1	no	—
44	Dic₂₁₁	—	8	1	no	—
45	C₂×C₂₁₁	—	2	1	no	—
46	C₂×C₂₁₁	—	4	1	no	—
47	C₂×C₂₁₁	#327	1	1	yes	characteristic
48	C₂₁₁	#1340	1	1	yes	characteristic, complemented, Sylow
49	#1340	—	211	1	no	maximal, Sylow
50	#470	—	211	1	no	—
51	#349	—	211	1	no	—
52	H(3,ℤ/4)	D₂₁₁	1	1	yes	characteristic
53	C₂×Q₁₆	—	211	1	no	—
54	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂²⋊Q₈	—	422	1	no	—
56	#117	—	2	1	no	—
57	M₃₂	—	422	1	no	—
58	#106	—	211	1	no	—
59	#103	—	422	1	no	—
60	C₂×Q₈	—	422	1	no	—
61	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
62	Q₁₆	—	844	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₄×D₂₁₁	1	1	yes	characteristic
65	C₂×C₈	D₈₄₄	1	1	yes	characteristic
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	844	1	no	—
68	C₄⋊C₄	—	422	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	844	1	no	—
71	C₄²	—	2	2	no	—
72	C₁₆	—	844	1	no	—
73	C₂³	#16809947	1	1	yes	characteristic
74	Q₈	—	844	1	no	—
75	Q₈	—	844	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	844	1	no	—
80	C₂×C₄	—	844	1	no	—
81	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
82	C₂×C₄	#16809949	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#20269897	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	2	no	—
90	C₄	—	1688	1	no	—
91	C₄	—	1688	1	no	—
92	C₄	#20269891	1	1	yes	characteristic
93	C₄	#20269905	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#27212879	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212879
commutator	C₂×C₈×C₂₁₁	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₂₁₁
Fitting	H(3,ℤ/4)×C₂₁₁	C₂
socle	C₂×C₂₁₁	#327
prime radical	C₂³×C₂₁₁	D₈
cosocle	C₂×M₁₆×C₂₁₁	C₂²
hypercenter	H(3,ℤ/4)	D₂₁₁
hypocenter	C₂₁₁	#1340
LCS	G ▷ C₂×C₈×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₈₄₄	C₄	semidirect
7. #20269909	C₄	semidirect
48. C₂₁₁	#1340	semidirect

Maximal subgroups

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

subgroup	index
#27213404	2
#27212928	2
H(3,ℤ/4)×C₂₁₁	2
#1340	211