Finite Group Databasebeta

Subgroups — #12013674

Order 1280.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₅×#1674	C₂	1	1	yes	maximal, characteristic
2	#10602177	C₂	1	1	yes	maximal, characteristic
3	#10601742	C₂	1	1	yes	maximal, characteristic
4	#10601737	C₂	1	1	yes	maximal, characteristic
5	#10598292	C₂	1	1	yes	maximal, characteristic
6	#10597966	C₂	1	1	yes	maximal, characteristic
7	#10597961	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#462	—	2	1	no	—
10	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
11	#65320	—	2	1	no	—
12	#64971	C₂²	1	1	yes	characteristic
13	C₅×#335	C₂²	1	1	yes	characteristic
14	C₅×#332	—	2	1	no	—
15	#64709	—	2	1	no	—
16	#64709	—	2	1	no	—
17	#64706	—	2	1	no	—
18	C₄⋊#672	—	2	1	no	—
19	C₄⋊#672	C₂²	1	1	yes	characteristic
20	#64681	—	2	1	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	#64678	C₂²	1	1	yes	characteristic
23	#12460	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₁₆	—	4	1	no	—
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
27	C₂×#689	—	4	1	no	—
28	#4015	—	2	1	no	—
29	C₅×#107	—	2	1	no	—
30	C₅×#107	—	2	1	no	—
31	C₅×C₄⋊C₈	—	2	1	no	—
32	C₅×C₄⋊C₈	—	2	1	no	—
33	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
34	C₅×#103	—	4	1	no	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
37	#3950	—	2	1	no	—
38	#3950	—	2	1	no	—
39	C₂×#672	—	2	1	no	—
40	C₂×#672	D₄	1	1	yes	characteristic
41	C₂×#672	D₄	1	1	yes	characteristic
42	#3942	—	4	1	no	—
43	C₄⋊#168	—	2	1	no	—
44	#2252	—	5	1	no	—
45	#2183	—	5	1	no	—
46	#1674	—	5	1	no	—
47	#1674	D₅	1	1	yes	characteristic
48	#1372	—	5	1	no	—
49	#1364	—	5	1	no	—
50	#1364	—	5	1	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
53	C₂×Dic₁₀	—	4	1	no	—
54	C₅×Q₁₆	—	8	2	no	—
55	C₂×C₅×C₈	—	2	1	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	1	no	—
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	—	4	1	no	—
62	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
63	C₄²×C₅	—	2	1	no	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	#689	—	8	2	no	—
66	C₄⋊Dic₅	—	4	1	no	—
67	C₄⋊Dic₅	—	4	1	no	—
68	C₂×#168	—	4	1	no	—
69	#672	—	2	1	no	—
70	#672	—	4	1	no	—
71	#672	Q₁₆	1	2	yes	complemented
72	#500	—	5	1	no	—
73	#500	C₂×D₅	1	1	yes	characteristic
74	#462	—	2	1	no	—
75	#462	—	10	1	no	—
76	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
77	#366	—	10	1	no	—
78	#366	—	10	1	no	—
79	C₄⋊C₁₆	—	5	1	no	—
80	C₄⋊C₁₆	—	10	1	no	—
81	#346	—	10	1	no	—
82	C₄×C₁₆	—	5	1	no	—
83	#335	—	5	1	no	—
84	#335	C₂×D₅	1	1	yes	characteristic
85	#332	—	2	1	no	—
86	#332	—	10	1	no	—
87	C₅×Q₈	—	2	1	no	—
88	C₅×Q₈	—	2	1	no	—
89	C₅×Q₈	—	8	1	no	—
90	C₅×Q₈	—	8	1	no	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	8	1	no	—
94	C₂×C₄×C₅	#121	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅	#135	1	1	yes	characteristic
97	C₂×Dic₅	—	8	1	no	—
98	Dic₁₀	—	8	1	no	—
99	Dic₁₀	—	8	1	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	4	1	no	—
102	C₅×C₈	—	8	1	no	—
103	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
104	C₅×C₈	#137	1	1	yes	characteristic
105	#168	—	8	1	no	—
106	C₂×Q₁₆	—	4	1	no	—
107	C₂×Q₁₆	—	20	1	no	—
108	C₄⋊Q₈	—	2	1	no	—
109	C₄⋊Q₈	—	10	1	no	—
110	C₄×Q₈	C₂²⋊D₅	1	1	yes	characteristic
111	C₂×C₁₆	—	5	1	no	—
112	C₂×C₁₆	—	5	1	no	—
113	C₂×C₁₆	—	10	1	no	—
114	#107	—	2	1	no	—
115	#107	—	2	1	no	—
116	#107	—	10	1	no	—
117	#107	—	10	1	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	—	10	1	no	—
121	C₄⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
122	#103	—	4	1	no	—
123	#103	—	20	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
126	C₂²×C₅	#466	1	1	yes	characteristic
127	C₄×C₅	—	2	1	no	—
128	C₄×C₅	—	4	1	no	—
129	C₄×C₅	—	4	1	no	—
130	C₄×C₅	—	16	1	no	—
131	C₄×C₅	#462	1	1	yes	characteristic
132	C₄×C₅	#473	1	1	yes	characteristic
133	C₄×C₅	#509	1	1	yes	characteristic
134	C₄×C₅	#509	1	1	yes	characteristic
135	Dic₅	—	16	1	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	20	1	no	—
138	C₂×Q₈	#686	1	1	yes	characteristic
139	Q₁₆	—	8	2	no	—
140	Q₁₆	—	40	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	20	1	no	—
144	C₂×C₈	D₄×D₅	1	1	yes	characteristic
145	C₂×C₈	#711	1	1	yes	characteristic
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	20	1	no	—
150	C₄⋊C₄	—	20	1	no	—
151	C₄⋊C₄	#686	1	1	yes	characteristic
152	C₄²	—	2	1	no	—
153	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
154	C₁₆	—	5	2	no	—
155	C₁₆	—	10	1	no	—
156	C₁₆	—	20	1	no	—
157	C₂×C₅	#1674	1	1	yes	characteristic
158	C₂×C₅	#2218	1	1	yes	characteristic
159	C₂×C₅	#2222	1	1	yes	characteristic
160	Q₈	—	2	1	no	—
161	Q₈	—	2	1	no	—
162	Q₈	—	8	1	no	—
163	Q₈	—	8	1	no	—
164	Q₈	—	40	1	no	—
165	Q₈	—	40	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	40	1	no	—
170	C₂×C₄	C₂×#686	1	1	yes	characteristic
171	C₂×C₄	#4085	1	1	yes	characteristic
172	C₂×C₄	#4096	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	8	1	no	—
176	C₈	—	40	1	no	—
177	C₈	D₅×Q₁₆	1	1	yes	characteristic
178	C₈	#4064	1	1	yes	characteristic
179	C₅	#12460	1	1	yes	characteristic, complemented, Sylow
180	C₂²	#65327	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	16	1	no	—
185	C₄	—	80	1	no	—
186	C₄	#65436	1	1	yes	characteristic
187	C₄	#65471	1	1	yes	characteristic
188	C₄	#65482	1	1	yes	characteristic
189	C₄	#65493	1	1	yes	characteristic
190	C₂	#10601846	1	1	yes	characteristic
191	C₂	#10601866	1	1	yes	characteristic
192	C₂	#10601889	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65327
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1674	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#1674	D₅
hypocenter	C₅	#12460
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #1674

Complements

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

N	H ≅ G/N	type
71. #672	Q₁₆	semidirect
179. C₅	#12460	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1674	2
#10602177	2
#10601742	2
#10601737	2
#10598292	2
#10597966	2
#10597961	2
#12460	5