Finite Group Databasebeta

Subgroups — #12014006

Order 1280.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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×#2241	C₂	1	1	yes	maximal, characteristic
2	#10602175	C₂	1	1	yes	maximal, characteristic
3	#10602154	C₂	1	1	yes	maximal, characteristic
4	#10602103	C₂	1	1	yes	maximal, characteristic
5	#10602091	C₂	1	1	yes	maximal, characteristic
6	#10599858	C₂	1	1	yes	maximal, characteristic
7	#10597989	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#494	C₂²	1	1	yes	characteristic
10	#65382	—	2	1	no	—
11	#65370	—	2	1	no	—
12	#65367	—	2	1	no	—
13	#65367	—	2	1	no	—
14	#64988	C₂²	1	1	yes	characteristic
15	#64971	C₂²	1	1	yes	characteristic
16	C₅×#358	—	2	2	no	—
17	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	#64784	—	2	2	no	—
19	#64723	—	2	2	no	—
20	#64712	—	2	2	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	C₈×#168	C₂²	1	1	yes	characteristic
23	#12514	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₁₆	—	2	1	no	—
25	C₂×C₅×Q₁₆	—	2	1	no	—
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₂×Dic₂₀	—	2	1	no	—
29	C₂×Dic₂₀	—	2	1	no	—
30	#4015	—	2	1	no	—
31	#4015	—	2	1	no	—
32	C₂×C₅×C₁₆	D₄	1	2	yes	—
33	C₅×#107	—	2	1	no	—
34	C₅×#107	—	2	1	no	—
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	#3967	—	4	2	no	—
37	#3950	—	2	1	no	—
38	#3950	—	2	1	no	—
39	C₂×#672	D₄	1	2	yes	—
40	#3942	—	4	2	no	—
41	#3939	—	4	2	no	—
42	#3939	—	4	2	no	—
43	C₄×#168	D₄	1	1	yes	characteristic
44	C₄×#168	D₄	1	1	yes	characteristic
45	#2241	—	5	1	no	—
46	#2241	—	5	1	no	—
47	#2241	—	5	1	no	—
48	#2241	D₅	1	1	yes	characteristic
49	#1713	—	5	1	no	—
50	#1710	—	5	1	no	—
51	#1311	—	5	1	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×Dic₁₀	—	4	1	no	—
55	C₂×Dic₁₀	—	4	1	no	—
56	C₅×Q₁₆	—	4	2	no	—
57	C₅×Q₁₆	—	4	2	no	—
58	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	—	4	1	no	—
62	C₅×C₄⋊C₄	—	4	1	no	—
63	C₅×C₄⋊C₄	—	4	1	no	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	C₄⋊Dic₅	—	4	1	no	—
66	C₄⋊Dic₅	—	4	1	no	—
67	C₄⋊Dic₅	—	4	1	no	—
68	C₄⋊Dic₅	—	4	1	no	—
69	C₂×#168	D₈	1	2	yes	—
70	C₂×#168	D₈	1	2	yes	—
71	Dic₂₀	—	4	2	no	—
72	Dic₂₀	—	4	2	no	—
73	C₅×C₁₆	—	2	2	no	—
74	#672	—	2	2	no	—
75	#500	—	5	1	no	—
76	#500	—	10	1	no	—
77	#500	—	10	1	no	—
78	#500	C₂×D₅	1	1	yes	characteristic
79	#494	—	5	1	no	—
80	#494	C₂×D₅	1	1	yes	characteristic
81	#487	—	10	1	no	—
82	#487	—	10	1	no	—
83	#358	—	2	2	no	—
84	#358	—	10	2	no	—
85	#358	—	10	2	no	—
86	#358	—	10	2	no	—
87	C₄×C₁₆	—	5	1	no	—
88	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
89	C₈²	—	5	1	no	—
90	C₅×Q₈	—	8	1	no	—
91	C₅×Q₈	—	8	1	no	—
92	C₅×Q₈	—	8	2	no	—
93	C₂×C₄×C₅	—	8	1	no	—
94	C₂×C₄×C₅	—	8	1	no	—
95	C₂×C₄×C₅	#127	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
98	C₂×Dic₅	—	8	1	no	—
99	C₂×Dic₅	—	8	1	no	—
100	Dic₁₀	—	8	1	no	—
101	Dic₁₀	—	8	1	no	—
102	Dic₁₀	—	8	2	no	—
103	C₅×C₈	#135	1	2	yes	—
104	C₅×C₈	#135	1	2	yes	—
105	#168	—	2	2	no	—
106	#168	—	2	2	no	—
107	C₂×Q₁₆	—	2	1	no	—
108	C₂×Q₁₆	—	2	1	no	—
109	C₂×Q₁₆	—	10	1	no	—
110	C₂×Q₁₆	—	10	1	no	—
111	C₄⋊Q₈	—	2	1	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	10	1	no	—
114	C₄⋊Q₈	—	10	1	no	—
115	C₂×C₁₆	—	5	2	no	—
116	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
117	#107	—	2	1	no	—
118	#107	—	2	1	no	—
119	#107	—	10	1	no	—
120	#107	—	10	1	no	—
121	#107	—	20	2	no	—
122	#107	—	20	2	no	—
123	#103	—	20	2	no	—
124	#103	—	20	2	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₂²×C₅	#493	1	1	yes	characteristic
129	C₄×C₅	—	16	1	no	—
130	C₄×C₅	—	16	1	no	—
131	C₄×C₅	#495	1	1	yes	characteristic
132	C₄×C₅	#495	1	1	yes	characteristic
133	C₄×C₅	#510	1	2	yes	—
134	C₄×C₅	#510	1	2	yes	—
135	Dic₅	—	16	1	no	—
136	Dic₅	—	16	1	no	—
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂×Q₈	—	20	1	no	—
140	C₂×Q₈	—	20	1	no	—
141	Q₁₆	—	4	2	no	—
142	Q₁₆	—	4	2	no	—
143	Q₁₆	—	20	2	no	—
144	Q₁₆	—	20	2	no	—
145	C₂×C₈	—	5	2	no	—
146	C₂×C₈	—	5	2	no	—
147	C₂×C₈	D₄×D₅	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	20	1	no	—
154	C₄⋊C₄	—	20	1	no	—
155	C₄⋊C₄	—	20	1	no	—
156	C₄⋊C₄	—	20	1	no	—
157	C₄²	D₄×D₅	1	1	yes	characteristic
158	C₁₆	—	2	2	no	—
159	C₁₆	—	10	2	no	—
160	C₂×C₅	#1709	1	1	yes	characteristic
161	C₂×C₅	#2251	1	1	yes	characteristic
162	C₂×C₅	#2251	1	1	yes	characteristic
163	Q₈	—	8	1	no	—
164	Q₈	—	8	1	no	—
165	Q₈	—	8	2	no	—
166	Q₈	—	40	1	no	—
167	Q₈	—	40	1	no	—
168	Q₈	—	40	2	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	—	40	1	no	—
172	C₂×C₄	—	40	1	no	—
173	C₂×C₄	D₅×D₈	1	1	yes	characteristic
174	C₂×C₄	D₅×D₈	1	1	yes	characteristic
175	C₂×C₄	#4086	1	1	yes	characteristic
176	C₈	—	10	2	no	—
177	C₈	—	10	2	no	—
178	C₈	#4062	1	2	yes	—
179	C₈	#4096	1	2	yes	—
180	C₅	#12514	1	1	yes	characteristic, complemented, Sylow
181	C₂²	#65440	1	1	yes	characteristic
182	C₄	—	16	1	no	—
183	C₄	—	16	1	no	—
184	C₄	—	80	1	no	—
185	C₄	—	80	1	no	—
186	C₄	#65204	1	2	yes	—
187	C₄	#65207	1	2	yes	—
188	C₄	#65348	1	1	yes	characteristic
189	C₄	#65457	1	1	yes	characteristic
190	C₂	#10601975	1	1	yes	characteristic
191	C₂	#10602275	1	1	yes	characteristic
192	C₂	#10602294	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2241	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#2241	D₅
hypocenter	C₅	#12514
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
180. C₅	#12514	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2241	2
#10602175	2
#10602154	2
#10602103	2
#10602091	2
#10599858	2
#10597989	2
#12514	5