Finite Group Databasebeta

Subgroups — #12002444

Order 1280.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#10605129	C₂	1	1	yes	maximal, characteristic
2	#10601585	C₂	1	2	yes	maximal
3	#10600234	C₂	1	1	yes	maximal, characteristic
4	C₅×#1274	C₂	1	1	yes	maximal, characteristic
5	#10598237	C₂	1	2	yes	maximal
6	#65804	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	2	yes	—
8	C₅×#376	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₄×#683	—	2	2	no	—
11	#65213	C₂²	1	1	yes	characteristic
12	#64949	—	2	2	no	—
13	#64946	—	2	2	no	—
14	#64743	—	2	2	no	—
15	#64687	—	2	2	no	—
16	#64687	—	2	2	no	—
17	#10373	—	5	1	no	maximal, Sylow
18	C₂×C₄²×C₅	D₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	#4072	—	2	1	no	—
21	C₂×C₄⋊Dic₅	—	2	2	no	—
22	#4070	—	2	2	no	—
23	C₂×#683	—	2	2	no	—
24	C₂×#683	—	2	2	no	—
25	C₂×C₄×Dic₅	—	2	2	no	—
26	C₂×#681	—	2	1	no	—
27	#4016	—	2	1	no	—
28	#4015	Q₈	1	2	yes	—
29	C₄×Dic₁₀	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₂²⋊C₈	—	2	2	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₅×#95	—	2	2	no	—
34	#3963	—	2	2	no	—
35	#3963	—	2	2	no	—
36	#3963	D₄	1	2	yes	—
37	#3963	D₄	1	2	yes	—
38	C₄⋊#168	—	2	2	no	—
39	#2402	—	5	1	no	—
40	#1995	—	5	1	no	—
41	#1503	—	5	2	no	—
42	#1303	—	5	2	no	—
43	#1274	D₅	1	1	yes	characteristic
44	C₂²×C₄×C₅	—	2	2	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
47	C₂²×Dic₅	—	2	2	no	—
48	C₂²×Dic₅	—	2	2	no	—
49	C₂×Dic₁₀	—	2	2	no	—
50	C₂×C₅×C₈	—	4	2	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅	#41	1	2	yes	—
55	C₄²×C₅	#41	1	1	yes	characteristic
56	C₂²⋊Dic₅	—	4	1	no	—
57	C₄⋊Dic₅	—	2	2	no	—
58	C₄⋊Dic₅	—	2	2	no	—
59	C₄⋊Dic₅	—	4	4	no	—
60	#683	—	4	4	no	—
61	#683	—	4	4	no	—
62	#683	—	4	2	no	—
63	C₄×Dic₅	—	4	1	no	—
64	C₄×Dic₅	—	4	2	no	—
65	C₄×Dic₅	—	4	2	no	—
66	#681	—	4	2	no	—
67	C₂×#168	—	4	1	no	—
68	#533	—	5	1	no	—
69	#431	C₂×D₅	1	2	yes	—
70	#423	—	5	1	no	—
71	#400	—	10	2	no	—
72	#398	—	10	2	no	—
73	C₄×C₄⋊C₄	—	5	2	no	—
74	C₄×C₄⋊C₄	—	10	2	no	—
75	#376	C₂×D₅	1	1	yes	characteristic
76	#339	—	10	2	no	—
77	#328	—	10	2	no	—
78	#328	—	10	2	no	—
79	C₂³×C₅	#120	1	1	yes	characteristic
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅	#124	1	1	yes	characteristic
87	C₂×C₄×C₅	#124	1	2	yes	—
88	C₂×C₄×C₅	#135	1	2	yes	—
89	C₂×C₄×C₅	#137	1	2	yes	—
90	C₂×Dic₅	—	2	4	no	—
91	C₂×Dic₅	—	4	2	no	—
92	C₂×Dic₅	—	4	2	no	—
93	C₂×Dic₅	—	4	2	no	—
94	C₂×Dic₅	—	4	2	no	—
95	C₂×Dic₅	—	4	2	no	—
96	C₂×Dic₅	—	4	2	no	—
97	Dic₁₀	—	8	2	no	—
98	C₅×C₈	—	8	2	no	—
99	#168	—	8	1	no	—
100	C₂×M₁₆	—	10	1	no	—
101	C₄⋊Q₈	—	5	2	no	—
102	#125	—	10	1	no	—
103	C₂²⋊Q₈	—	10	2	no	—
104	C₄×Q₈	—	10	2	no	—
105	#117	—	10	1	no	—
106	C₂×C₄⋊C₄	—	10	2	no	—
107	C₂×C₄⋊C₄	—	10	2	no	—
108	C₂×C₄⋊C₄	—	10	2	no	—
109	C₂×C₄²	—	10	2	no	—
110	C₂×C₄²	D₂₀	1	2	yes	—
111	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	10	2	no	—
114	C₂²⋊C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	2	2	no	—
117	#95	—	5	2	no	—
118	#95	—	5	2	no	—
119	#95	—	10	2	no	—
120	#95	—	10	2	no	—
121	C₂²×C₅	—	2	2	no	—
122	C₂²×C₅	#393	1	1	yes	characteristic
123	C₂²×C₅	#452	1	2	yes	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	8	2	no	—
127	C₄×C₅	#479	1	2	yes	—
128	C₄×C₅	#488	1	1	yes	characteristic
129	C₄×C₅	#491	1	1	yes	characteristic
130	Dic₅	—	4	4	no	—
131	Dic₅	—	8	2	no	—
132	Dic₅	—	8	2	no	—
133	C₂×Q₈	—	10	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	10	2	no	—
136	C₂²×C₄	—	10	2	no	—
137	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
138	C₂²×C₄	D₄×D₅	1	2	yes	—
139	M₁₆	—	20	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	20	1	no	—
142	C₄⋊C₄	—	10	2	no	—
143	C₄⋊C₄	—	10	2	no	—
144	C₄⋊C₄	—	20	4	no	—
145	C₄⋊C₄	—	20	4	no	—
146	C₄⋊C₄	—	20	4	no	—
147	C₄⋊C₄	—	20	2	no	—
148	C₂²⋊C₄	—	20	1	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	20	1	no	—
152	C₄²	—	20	2	no	—
153	C₄²	—	20	2	no	—
154	C₄²	#709	1	2	yes	—
155	C₄²	#711	1	1	yes	characteristic
156	C₄²	Q₈×D₅	1	1	yes	characteristic
157	C₂×C₅	—	2	2	no	—
158	C₂×C₅	#2026	1	1	yes	characteristic
159	C₂×C₅	#2031	1	2	yes	—
160	C₂³	#4028	1	1	yes	characteristic
161	Q₈	—	40	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	10	4	no	—
167	C₂×C₄	—	20	2	no	—
168	C₂×C₄	—	20	2	no	—
169	C₂×C₄	—	20	2	no	—
170	C₂×C₄	—	20	2	no	—
171	C₂×C₄	—	20	2	no	—
172	C₂×C₄	—	20	2	no	—
173	C₂×C₄	#4021	1	1	yes	characteristic
174	C₂×C₄	#4031	1	2	yes	—
175	C₂×C₄	#4042	1	2	yes	—
176	C₂×C₄	#4043	1	1	yes	characteristic
177	C₂×C₄	#4050	1	2	yes	—
178	C₂×C₄	#4055	1	2	yes	—
179	C₈	—	8	2	no	—
180	C₈	—	40	1	no	—
181	C₅	#10373	1	1	yes	characteristic, complemented, Sylow
182	C₂²	—	2	2	no	—
183	C₂²	#64962	1	1	yes	characteristic
184	C₂²	#65022	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	20	4	no	—
189	C₄	—	40	2	no	—
190	C₄	—	40	2	no	—
191	C₄	#64986	1	1	yes	characteristic
192	C₄	#64992	1	1	yes	characteristic
193	C₄	#65118	1	2	yes	—
194	C₂	—	2	2	no	—
195	C₂	#10599483	1	1	yes	characteristic
196	C₂	#10599507	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#4021
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1274	C₂
socle	C₂²×C₅	#393
prime radical	C₂³×C₅	#120
hypercenter	#1274	D₅
hypocenter	C₅	#10373
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

Complements

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

N	H ≅ G/N	type
181. C₅	#10373	semidirect

Maximal subgroups

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

subgroup	index
#10605129	2
#10601585	2
#10600234	2
C₅×#1274	2
#10598237	2
#10373	5