Finite Group Databasebeta

Subgroups — #46838231

Order 34688.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22576103	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#31773979	C₂	1	1	yes	maximal, characteristic
4	#31773722	C₂	1	1	yes	maximal, characteristic, complemented
5	#31773699	C₂	1	1	yes	maximal, characteristic
6	#31773672	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₁₆₈	C₂²	1	1	yes	characteristic
9	Q₈×Dic₂₇₁	—	2	1	no	—
10	#22576122	—	2	1	no	—
11	#22576103	C₄	1	4	yes	complemented
12	C₂×C₂₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#22576094	C₂²	1	1	yes	characteristic
14	#22576084	—	2	1	no	—
15	#22576080	—	2	1	no	—
16	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
19	#107×C₂₇₁	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₈₄	—	2	1	no	—
21	C₂×C₄×D₂₇₁	—	4	1	no	—
22	D₂₁₆₈	—	2	1	no	—
23	D₂₁₆₈	C₂×C₄	1	2	yes	—
24	C₂×C₂₇₁⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₂₇₁	—	2	1	no	—
26	C₄⋊Dic₂₇₁	—	4	1	no	—
27	(C₂×D₂₇₁)⋊C₄	—	4	1	no	—
28	C₄×Dic₂₇₁	—	4	1	no	—
29	C₄×Dic₂₇₁	D₄	1	1	yes	characteristic
30	C₂₇₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₂₇₁	—	2	1	no	—
32	Q₁₆×C₂₇₁	—	2	1	no	—
33	Q₁₆×C₂₇₁	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₇₁	—	2	1	no	—
36	C₁₆×C₂₇₁	—	2	1	no	—
37	C₂²×D₂₇₁	—	4	1	no	—
38	D₁₀₈₄	—	2	2	no	—
39	D₁₀₈₄	—	4	1	no	—
40	C₂×Dic₂₇₁	—	4	1	no	—
41	C₂×Dic₂₇₁	D₈	1	1	yes	characteristic
42	C₂×Dic₂₇₁	D₈	1	1	yes	characteristic
43	C₄×D₂₇₁	—	8	2	no	—
44	C₂₇₁⋊C₈	—	2	1	no	—
45	Q₈×C₂₇₁	—	2	2	no	—
46	Q₈×C₂₇₁	—	4	1	no	—
47	C₂×C₄×C₂₇₁	—	4	1	no	—
48	C₂×C₄×C₂₇₁	—	4	1	no	—
49	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
50	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₂₇₁	#41	1	1	yes	characteristic
52	C₂²×C₂₇₁	C₂×D₈	1	1	yes	characteristic
53	C₂×D₂₇₁	—	4	2	no	—
54	C₂×D₂₇₁	—	8	1	no	—
55	C₂×D₂₇₁	—	8	1	no	—
56	C₄×C₂₇₁	—	4	2	no	—
57	C₄×C₂₇₁	—	8	1	no	—
58	C₄×C₂₇₁	C₄×D₄	1	1	yes	characteristic
59	C₄×C₂₇₁	#135	1	1	yes	characteristic
60	Dic₂₇₁	—	2	1	no	—
61	Dic₂₇₁	—	2	1	no	—
62	Dic₂₇₁	—	8	1	no	—
63	C₂×C₂₇₁	C₄×D₈	1	1	yes	characteristic
64	C₂×C₂₇₁	#509	1	1	yes	characteristic
65	C₂×C₂₇₁	#510	1	1	yes	characteristic
66	D₂₇₁	—	8	2	no	—
67	C₂₇₁	#2175	1	1	yes	characteristic, complemented, Sylow
68	#2175	—	271	1	no	maximal, Sylow
69	C₂×SD₃₂	—	271	1	no	—
70	C₄×Q₁₆	—	271	1	no	—
71	C₄×D₈	—	271	1	no	—
72	#366	—	271	1	no	—
73	#358	D₂₇₁	1	1	yes	characteristic, complemented
74	#357	—	271	1	no	—
75	#346	—	271	1	no	—
76	C₂×Q₁₆	C₂×D₂₇₁	1	1	yes	characteristic
77	C₂×D₈	—	271	1	no	—
78	C₄×Q₈	—	542	1	no	—
79	C₄×D₄	—	542	1	no	—
80	SD₃₂	—	271	4	no	—
81	C₂×C₁₆	—	271	1	no	—
82	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
83	#107	—	271	1	no	—
84	#107	C₂×D₂₇₁	1	1	yes	characteristic
85	#103	—	542	1	no	—
86	#102	—	542	1	no	—
87	C₄×C₈	—	271	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	542	1	no	—
90	C₂²×C₄	—	1084	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₂₇₁	1	2	yes	complemented
93	D₈	—	271	2	no	—
94	D₈	—	542	1	no	—
95	C₂×C₈	—	271	1	no	—
96	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	542	1	no	—
99	C₄⋊C₄	—	1084	1	no	—
100	C₂²⋊C₄	—	1084	1	no	—
101	C₄²	—	271	1	no	—
102	C₄²	—	1084	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	271	2	no	—
105	C₂³	—	1084	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	542	2	no	—
109	D₄	—	1084	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	271	1	no	—
113	C₂×C₄	—	271	1	no	—
114	C₂×C₄	—	1084	1	no	—
115	C₂×C₄	—	2168	2	no	—
116	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
117	C₈	—	542	1	no	—
118	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
119	C₈	#16864653	1	1	yes	characteristic
120	C₂²	—	1084	2	no	—
121	C₂²	—	2168	1	no	—
122	C₂²	—	2168	1	no	—
123	C₂²	D₈×D₂₇₁	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	542	1	no	—
127	C₄	—	542	1	no	—
128	C₄	—	2168	1	no	—
129	C₄	(C₂²⋊D₂₇₁)⋊C₄	1	1	yes	characteristic
130	C₄	#22576181	1	1	yes	characteristic
131	C₂	—	2168	2	no	—
132	C₂	#31773977	1	1	yes	characteristic
133	C₂	#31774334	1	1	yes	characteristic
134	C₂	#31774339	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₇₁
commutator	C₈×C₂₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	#358×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×D₈
prime radical	C₂×D₂₁₆₈	C₂²
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	#358	D₂₇₁
hypocenter	C₂₇₁	#2175
LCS	G ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₂₇₁	C₂	semidirect
4. #31773722	C₂	semidirect
6. #31773672	C₂	semidirect
7. #358×C₂₇₁	C₂	semidirect
11. #22576103	C₄	semidirect
30. C₂₇₁⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₂₇₁	C₂×C₄	semidirect
67. C₂₇₁	#2175	semidirect
73. #358	D₂₇₁	semidirect
92. Q₁₆	C₄×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22576103	2
Q₁₆×Dic₂₇₁	2
#31773979	2
#31773722	2
#31773699	2
#31773672	2
#358×C₂₇₁	2
#2175	271