Finite Group Databasebeta

Subgroups — #12014046

Order 1280.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#10602178	C₂	1	1	yes	maximal, characteristic
3	#10602177	C₂	1	1	yes	maximal, characteristic
4	#10602155	C₂	1	1	yes	maximal, characteristic
5	#10602154	C₂	1	1	yes	maximal, characteristic
6	#10599839	C₂	1	1	yes	maximal, characteristic
7	#10597989	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#500	C₂²	1	1	yes	characteristic
10	#65382	—	2	1	no	—
11	#65381	—	2	1	no	—
12	#65370	—	2	1	no	—
13	#65367	—	2	1	no	—
14	#64971	C₂²	1	1	yes	characteristic
15	#64971	C₂²	1	1	yes	characteristic
16	C₅×#366	—	2	2	no	—
17	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	#64725	—	2	2	no	—
19	#64710	—	2	2	no	—
20	#64709	—	2	2	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	C₈×#168	C₂²	1	1	yes	characteristic
23	#12524	—	5	1	no	maximal, Sylow
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₂×#689	—	4	2	no	—
27	#4015	—	2	1	no	—
28	#4015	—	2	1	no	—
29	C₂×C₅×C₁₆	D₄	1	2	yes	—
30	C₅×#107	—	2	1	no	—
31	C₅×#107	—	2	1	no	—
32	C₅×#107	—	2	1	no	—
33	C₅×#107	—	2	1	no	—
34	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
35	#3967	—	4	2	no	—
36	#3950	—	2	1	no	—
37	#3950	—	2	1	no	—
38	#3950	—	2	1	no	—
39	#3950	—	2	1	no	—
40	C₂×#672	D₄	1	2	yes	—
41	#3942	—	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	#2254	—	5	1	no	—
46	#2254	—	5	1	no	—
47	#2252	—	5	1	no	—
48	#2252	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	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	4	1	no	—
59	C₅×C₄⋊C₄	—	4	1	no	—
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	—	4	1	no	—
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	#689	—	8	4	no	—
64	C₄⋊Dic₅	—	4	1	no	—
65	C₄⋊Dic₅	—	4	1	no	—
66	C₄⋊Dic₅	—	4	1	no	—
67	C₄⋊Dic₅	—	4	1	no	—
68	C₂×#168	D₈	1	2	yes	—
69	C₂×#168	D₈	1	2	yes	—
70	C₅×C₁₆	Q₁₆	1	4	yes	complemented
71	#672	Q₁₆	1	4	yes	complemented
72	#500	—	5	1	no	—
73	#500	—	5	1	no	—
74	#500	—	10	1	no	—
75	#500	C₂×D₅	1	1	yes	characteristic
76	#500	C₂×D₅	1	1	yes	characteristic
77	#494	—	10	1	no	—
78	#487	—	10	1	no	—
79	#487	—	10	1	no	—
80	#367	—	10	2	no	—
81	#367	—	10	2	no	—
82	#366	—	2	2	no	—
83	#366	—	10	2	no	—
84	C₄×C₁₆	—	5	1	no	—
85	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
86	C₈²	—	5	1	no	—
87	C₅×Q₈	—	8	2	no	—
88	C₅×Q₈	—	8	2	no	—
89	C₂×C₄×C₅	—	8	1	no	—
90	C₂×C₄×C₅	—	8	1	no	—
91	C₂×C₄×C₅	#127	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
94	C₂×Dic₅	—	8	1	no	—
95	C₂×Dic₅	—	8	1	no	—
96	Dic₁₀	—	8	2	no	—
97	Dic₁₀	—	8	2	no	—
98	C₅×C₈	C₂×Q₁₆	1	2	yes	—
99	C₅×C₈	C₂×Q₁₆	1	2	yes	—
100	#168	—	2	2	no	—
101	#168	—	2	2	no	—
102	C₂×Q₁₆	—	20	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	2	1	no	—
105	C₄⋊Q₈	—	10	1	no	—
106	C₄⋊Q₈	—	10	1	no	—
107	C₂×C₁₆	—	5	2	no	—
108	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
109	#107	—	2	1	no	—
110	#107	—	2	1	no	—
111	#107	—	2	1	no	—
112	#107	—	2	1	no	—
113	#107	—	10	1	no	—
114	#107	—	10	1	no	—
115	#107	—	10	1	no	—
116	#107	—	10	1	no	—
117	#107	—	20	2	no	—
118	#103	—	20	2	no	—
119	#103	—	20	2	no	—
120	C₄×C₈	—	5	1	no	—
121	C₄×C₈	—	5	1	no	—
122	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
123	C₂²×C₅	#493	1	1	yes	characteristic
124	C₄×C₅	—	16	1	no	—
125	C₄×C₅	—	16	1	no	—
126	C₄×C₅	#494	1	1	yes	characteristic
127	C₄×C₅	#494	1	1	yes	characteristic
128	C₄×C₅	#509	1	2	yes	—
129	C₄×C₅	#510	1	2	yes	—
130	Dic₅	—	16	1	no	—
131	Dic₅	—	16	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	20	1	no	—
135	C₂×Q₈	—	20	1	no	—
136	Q₁₆	—	40	4	no	—
137	C₂×C₈	—	5	2	no	—
138	C₂×C₈	—	5	2	no	—
139	C₂×C₈	D₄×D₅	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	20	1	no	—
146	C₄⋊C₄	—	20	1	no	—
147	C₄⋊C₄	—	20	1	no	—
148	C₄⋊C₄	—	20	1	no	—
149	C₄²	D₄×D₅	1	1	yes	characteristic
150	C₁₆	—	5	4	no	—
151	C₁₆	#689	1	4	yes	complemented
152	C₂×C₅	#1713	1	1	yes	characteristic
153	C₂×C₅	#2250	1	1	yes	characteristic
154	C₂×C₅	#2251	1	1	yes	characteristic
155	Q₈	—	8	2	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	40	2	no	—
158	Q₈	—	40	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	40	1	no	—
162	C₂×C₄	—	40	1	no	—
163	C₂×C₄	D₅×D₈	1	1	yes	characteristic
164	C₂×C₄	D₅×D₈	1	1	yes	characteristic
165	C₂×C₄	#4086	1	1	yes	characteristic
166	C₈	—	10	2	no	—
167	C₈	—	10	2	no	—
168	C₈	D₅×Q₁₆	1	2	yes	—
169	C₈	C₂×#689	1	2	yes	—
170	C₅	#12524	1	1	yes	characteristic, complemented, Sylow
171	C₂²	#65440	1	1	yes	characteristic
172	C₄	—	16	1	no	—
173	C₄	—	16	1	no	—
174	C₄	—	80	1	no	—
175	C₄	—	80	1	no	—
176	C₄	#65200	1	2	yes	—
177	C₄	#65207	1	2	yes	—
178	C₄	#65381	1	1	yes	characteristic
179	C₄	#65470	1	1	yes	characteristic
180	C₂	#10602155	1	1	yes	characteristic
181	C₂	#10602259	1	1	yes	characteristic
182	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×#2252	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#2252	D₅
hypocenter	C₅	#12524
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
70. C₅×C₁₆	Q₁₆	semidirect
71. #672	Q₁₆	semidirect
151. C₁₆	#689	semidirect
170. C₅	#12524	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2252	2
#10602178	2
#10602177	2
#10602155	2
#10602154	2
#10599839	2
#10597989	2
#12524	5