Finite Group Databasebeta

Subgroups — #12013605

Order 1280.

218 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 65 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	#10602183	C₂	1	1	yes	maximal, characteristic
2	#10602155	C₂	1	1	yes	maximal, characteristic
3	#10600089	C₂	1	1	yes	maximal, characteristic
4	#10599350	C₂	1	1	yes	maximal, characteristic
5	C₅×#1365	C₂	1	1	yes	maximal, characteristic
6	#10598046	C₂	1	1	yes	maximal, characteristic
7	#10597967	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	#65381	—	2	1	no	—
10	#65367	—	2	1	no	—
11	#65119	—	2	1	no	—
12	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
13	#64971	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#335	C₂²	1	1	yes	characteristic
16	#64724	—	2	1	no	—
17	#64723	—	2	1	no	—
18	#64712	—	2	1	no	—
19	#64709	—	2	1	no	—
20	#64704	—	2	1	no	—
21	C₄⋊#672	C₂²	1	1	yes	characteristic
22	C₈×#168	C₂²	1	1	yes	characteristic
23	#12448	—	5	1	no	maximal, Sylow
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₂×#689	—	4	2	no	—
26	C₂×Dic₂₀	D₄	1	1	yes	characteristic
27	#4015	—	2	1	no	—
28	C₄×Dic₁₀	—	2	1	no	—
29	C₂×C₅×C₁₆	—	2	1	no	—
30	C₅×#107	—	2	1	no	—
31	C₅×#107	—	2	1	no	—
32	C₅×C₄⋊C₈	—	2	1	no	—
33	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
34	#3950	—	2	1	no	—
35	#3950	—	2	1	no	—
36	#3950	D₄	1	1	yes	characteristic
37	#3948	—	2	1	no	—
38	#3946	—	4	1	no	—
39	C₈×Dic₅	—	4	1	no	—
40	C₂×#672	—	2	1	no	—
41	#3939	—	4	1	no	—
42	#3939	—	4	1	no	—
43	C₄×#168	D₄	1	1	yes	characteristic
44	C₄×#168	D₄	1	1	yes	characteristic
45	#2227	—	5	1	no	—
46	#2227	—	5	1	no	—
47	#1713	—	5	1	no	—
48	C₈×Q₁₆	—	5	1	no	—
49	#1370	—	5	1	no	—
50	#1365	—	5	1	no	—
51	#1365	D₅	1	1	yes	characteristic
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×Dic₁₀	—	2	1	no	—
54	C₂×Dic₁₀	—	4	1	no	—
55	C₂×C₅×C₈	—	4	1	no	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	4	1	no	—
59	C₅×C₄⋊C₄	—	4	1	no	—
60	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
61	#689	—	8	4	no	—
62	C₄⋊Dic₅	—	2	1	no	—
63	C₄⋊Dic₅	—	4	1	no	—
64	C₄⋊Dic₅	—	4	1	no	—
65	#683	—	4	1	no	—
66	C₄×Dic₅	—	4	1	no	—
67	C₂×#168	—	2	1	no	—
68	C₂×#168	D₈	1	1	yes	characteristic
69	C₂×#168	D₈	1	1	yes	characteristic
70	Dic₂₀	—	2	1	no	—
71	Dic₂₀	Q₁₆	1	2	yes	complemented
72	C₅×C₁₆	—	4	1	no	—
73	#672	—	4	1	no	—
74	#500	—	5	1	no	—
75	#500	—	10	1	no	—
76	#500	C₂×D₅	1	1	yes	characteristic
77	#494	—	10	1	no	—
78	C₈×Q₈	—	10	1	no	—
79	C₄×Q₁₆	—	5	1	no	—
80	#366	—	10	1	no	—
81	#366	—	10	1	no	—
82	C₄⋊C₁₆	—	5	1	no	—
83	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
84	#358	—	10	1	no	—
85	#358	—	10	1	no	—
86	#335	C₂×D₅	1	1	yes	characteristic
87	Q₈⋊C₈	—	10	1	no	—
88	C₈²	—	5	1	no	—
89	C₅×Q₈	—	8	2	no	—
90	C₂×C₄×C₅	—	8	1	no	—
91	C₂×C₄×C₅	#121	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₅	—	4	1	no	—
95	C₂×Dic₅	—	4	1	no	—
96	C₂×Dic₅	—	8	1	no	—
97	Dic₁₀	—	2	2	no	—
98	Dic₁₀	—	4	1	no	—
99	Dic₁₀	—	8	2	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	8	1	no	—
102	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
103	C₅×C₈	#135	1	1	yes	characteristic
104	#168	—	2	2	no	—
105	#168	—	2	1	no	—
106	#168	Q₃₂	1	2	yes	—
107	C₂×Q₁₆	—	5	1	no	—
108	C₂×Q₁₆	—	20	2	no	—
109	C₄⋊Q₈	—	2	1	no	—
110	C₄⋊Q₈	—	10	1	no	—
111	C₄×Q₈	—	10	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	10	1	no	—
114	#107	—	2	1	no	—
115	#107	—	2	1	no	—
116	#107	—	5	1	no	—
117	#107	—	10	1	no	—
118	#107	—	10	1	no	—
119	#107	—	20	1	no	—
120	#107	—	20	1	no	—
121	C₄⋊C₈	—	2	1	no	—
122	C₄⋊C₈	—	20	1	no	—
123	#103	—	10	1	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	20	1	no	—
127	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₂²×C₅	#459	1	1	yes	characteristic
129	C₄×C₅	—	2	1	no	—
130	C₄×C₅	—	16	1	no	—
131	C₄×C₅	#467	1	1	yes	characteristic
132	C₄×C₅	#473	1	1	yes	characteristic
133	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
134	C₄×C₅	#510	1	1	yes	characteristic
135	Dic₅	—	4	2	no	—
136	Dic₅	—	8	1	no	—
137	Dic₅	—	16	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂×Q₈	—	10	1	no	—
140	C₂×Q₈	—	20	1	no	—
141	Q₁₆	—	5	2	no	—
142	Q₁₆	—	10	1	no	—
143	Q₁₆	—	40	4	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	5	1	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	10	1	no	—
148	C₂×C₈	#709	1	1	yes	characteristic
149	C₂×C₈	D₄×D₅	1	1	yes	characteristic
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	10	1	no	—
153	C₄⋊C₄	—	20	1	no	—
154	C₄⋊C₄	—	20	1	no	—
155	C₄⋊C₄	—	20	1	no	—
156	C₄²	—	20	1	no	—
157	C₄²	D₄×D₅	1	1	yes	characteristic
158	C₁₆	—	4	1	no	—
159	C₁₆	—	20	1	no	—
160	C₂×C₅	#1669	1	1	yes	characteristic
161	C₂×C₅	#2209	1	1	yes	characteristic
162	C₂×C₅	#2213	1	1	yes	characteristic
163	Q₈	—	8	2	no	—
164	Q₈	—	10	2	no	—
165	Q₈	—	20	1	no	—
166	Q₈	—	40	2	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	—	20	1	no	—
169	C₂×C₄	—	20	1	no	—
170	C₂×C₄	—	40	1	no	—
171	C₂×C₄	#4030	1	1	yes	characteristic
172	C₂×C₄	D₅×D₈	1	1	yes	characteristic
173	C₂×C₄	#4060	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	5	2	no	—
176	C₈	—	8	1	no	—
177	C₈	—	10	2	no	—
178	C₈	—	10	1	no	—
179	C₈	#4062	1	1	yes	characteristic
180	C₈	D₅×Q₁₆	1	1	yes	characteristic
181	C₅	#12448	1	1	yes	characteristic, complemented, Sylow
182	C₂²	#65075	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	16	1	no	—
185	C₄	—	20	2	no	—
186	C₄	—	40	1	no	—
187	C₄	—	80	1	no	—
188	C₄	#65044	1	1	yes	characteristic
189	C₄	#65102	1	1	yes	characteristic
190	C₄	#65204	1	1	yes	characteristic
191	C₄	D₅×Q₃₂	1	1	yes	characteristic
192	C₂	#10599234	1	1	yes	characteristic
193	C₂	#10600029	1	1	yes	characteristic
194	C₂	#10600056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65075
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1365	C₂
socle, prime radical	C₂²×C₅	#459
hypercenter	#1365	D₅
hypocenter	C₅	#12448
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1365

Complements

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

N	H ≅ G/N	type
71. Dic₂₀	Q₁₆	semidirect
181. C₅	#12448	semidirect

Maximal subgroups

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

subgroup	index
#10602183	2
#10602155	2
#10600089	2
#10599350	2
C₅×#1365	2
#10598046	2
#10597967	2
#12448	5