Finite Group Databasebeta

Subgroups — #12088626

Order 1280.

736 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 151 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	#10613327	C₂	1	3	yes	maximal, complemented
2	#10613295	C₂	1	3	yes	maximal
3	C₅×#1711	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602018	C₂	1	3	yes	maximal, complemented
5	#10601985	C₂	1	3	yes	maximal, complemented
6	#10598854	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598843	C₂	1	1	yes	maximal, characteristic, complemented
8	#66062	C₂²	1	3	yes	—
9	D₅×#127	C₂²	1	1	yes	characteristic
10	C₅×#487	C₂²	1	3	yes	complemented
11	C₅×#486	C₂²	1	3	yes	—
12	#65381	C₂²	1	3	yes	complemented
13	#65377	C₂²	1	3	yes	—
14	#65368	C₂²	1	3	yes	complemented
15	#65359	C₂²	1	3	yes	—
16	#65105	—	2	6	no	—
17	#65078	—	2	6	no	—
18	#65059	—	2	6	no	—
19	#65053	—	2	6	no	—
20	#64983	C₂²	1	3	yes	—
21	C₄×#676	C₂²	1	3	yes	—
22	#64970	C₂²	1	3	yes	complemented
23	C₅×C₈²	C₂²	1	1	yes	characteristic, complemented
24	#64675	C₂²	1	3	yes	complemented
25	#19753	—	5	1	no	maximal, Sylow
26	C₂×D₄×D₅	—	2	3	no	—
27	C₅×C₄⋊Q₈	C₂³	1	3	yes	—
28	C₅×#127	C₂³	1	1	yes	characteristic
29	#4092	—	2	3	no	—
30	Q₈×Dic₅	—	2	3	no	—
31	C₂×#689	—	2	6	no	—
32	#4086	—	2	3	no	—
33	C₂×#676	D₄	1	6	yes	—
34	#4043	—	2	3	no	—
35	#4038	—	2	3	no	—
36	#4036	—	2	3	no	—
37	#4033	—	2	3	no	—
38	#4020	C₂³	1	1	yes	characteristic
39	C₄²×D₅	C₂³	1	1	yes	characteristic
40	#4015	C₂³	1	3	yes	—
41	C₅×#103	—	2	6	no	—
42	C₅×#102	—	2	6	no	—
43	C₄×C₅×C₈	C₂³	1	3	yes	—
44	#3964	—	2	6	no	—
45	#3953	—	2	6	no	—
46	#3948	—	2	6	no	—
47	#3947	D₄	1	6	yes	—
48	#3941	—	2	6	no	—
49	#3940	—	2	6	no	—
50	C₄×#168	C₂³	1	3	yes	—
51	#3185	—	5	3	no	—
52	#3138	—	5	3	no	—
53	#1720	—	5	3	no	—
54	#1720	—	5	3	no	—
55	#1711	—	5	1	no	—
56	#1711	D₅	1	1	yes	characteristic, complemented
57	#1455	—	5	1	no	—
58	C₂³×D₅	—	4	1	no	—
59	C₂×C₅×Q₈	—	2	3	no	—
60	C₂×C₅×D₄	—	2	3	no	—
61	C₂×C₂²⋊D₅	—	4	3	no	—
62	D₄×D₅	—	4	12	no	—
63	C₂×D₂₀	—	2	3	no	—
64	C₂×C₄×D₅	C₂×D₄	1	3	yes	—
65	C₂×Dic₁₀	—	2	3	no	—
66	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
67	C₅×C₄⋊C₄	—	2	3	no	—
68	C₅×C₄⋊C₄	—	2	3	no	—
69	C₄²×C₅	C₂⁴	1	1	yes	characteristic
70	#689	—	4	12	no	—
71	#685	—	4	3	no	—
72	C₄⋊Dic₅	—	2	3	no	—
73	C₄⋊Dic₅	—	2	3	no	—
74	C₄⋊Dic₅	—	4	3	no	—
75	#683	—	4	3	no	—
76	C₄×Dic₅	—	4	3	no	—
77	C₄×Dic₅	C₂×D₄	1	3	yes	—
78	C₂×#168	C₂×D₄	1	6	yes	—
79	#676	—	4	6	no	—
80	#533	—	5	3	no	—
81	C₂×#127	—	5	1	no	—
82	#498	—	5	3	no	—
83	#494	—	5	3	no	—
84	#490	—	10	6	no	—
85	#487	—	5	3	no	—
86	#487	C₂×D₅	1	3	yes	complemented
87	#486	—	5	3	no	—
88	#486	—	5	3	no	—
89	#486	—	5	3	no	—
90	#486	C₂×D₅	1	3	yes	—
91	#470	—	10	6	no	—
92	#441	—	10	6	no	—
93	#418	—	10	6	no	—
94	C₄×M₁₆	—	5	3	no	—
95	#322	—	5	3	no	—
96	C₈²	C₂×D₅	1	1	yes	characteristic, complemented
97	C₂³×C₅	—	4	1	no	—
98	C₂²×D₅	—	4	1	no	—
99	C₂²×D₅	—	8	6	no	—
100	C₂²×D₅	#127	1	1	yes	characteristic
101	C₅×Q₈	—	4	6	no	—
102	C₅×D₄	—	4	6	no	—
103	C₂×C₄×C₅	—	4	3	no	—
104	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
105	C₂²⋊D₅	—	8	6	no	—
106	C₂×Dic₅	—	4	3	no	—
107	C₂×Dic₅	#127	1	3	yes	—
108	D₂₀	—	4	6	no	—
109	C₄×D₅	—	2	6	no	—
110	Dic₁₀	—	4	6	no	—
111	C₅×C₈	—	2	6	no	—
112	#168	—	2	6	no	—
113	C₂²×D₄	—	10	3	no	—
114	C₂×Q₁₆	—	10	6	no	—
115	C₂×M₁₆	—	5	6	no	—
116	C₄⋊Q₈	—	5	3	no	—
117	C₄⋊Q₈	C₂²×D₅	1	3	yes	—
118	#127	—	5	1	no	—
119	#127	—	10	3	no	—
120	#127	C₂²×D₅	1	1	yes	characteristic
121	#125	—	10	3	no	—
122	C₂²⋊Q₈	—	10	3	no	—
123	C₂²⋊Q₈	—	10	3	no	—
124	C₄×Q₈	—	10	3	no	—
125	C₄×Q₈	—	10	3	no	—
126	#117	—	10	3	no	—
127	C₂×C₄²	—	5	1	no	—
128	#106	—	10	6	no	—
129	#103	—	2	6	no	—
130	#103	—	10	6	no	—
131	#102	—	2	6	no	—
132	#102	—	10	6	no	—
133	#102	—	10	6	no	—
134	#102	—	10	6	no	—
135	#97	—	5	6	no	—
136	C₄×C₈	—	5	3	no	—
137	C₄×C₈	C₂²×D₅	1	3	yes	—
138	C₂²×C₅	—	8	3	no	—
139	C₂²×C₅	C₂×#127	1	1	yes	characteristic
140	C₂×D₅	—	2	3	no	—
141	C₂×D₅	—	8	3	no	—
142	C₂×D₅	—	16	4	no	—
143	C₄×C₅	—	8	3	no	—
144	C₄×C₅	#575	1	6	yes	—
145	Dic₅	—	2	3	no	—
146	Dic₅	—	8	3	no	—
147	C₂⁴	—	20	1	no	—
148	C₂×Q₈	—	2	3	no	—
149	C₂×Q₈	—	10	3	no	—
150	C₂×D₄	—	2	3	no	—
151	C₂×D₄	—	10	3	no	—
152	C₂×D₄	—	20	3	no	—
153	C₂×D₄	—	20	12	no	—
154	C₂²×C₄	—	5	3	no	—
155	Q₁₆	—	20	12	no	—
156	M₁₆	—	20	6	no	—
157	C₂×C₈	—	5	6	no	—
158	C₂×C₈	D₄×D₅	1	6	yes	—
159	C₄⋊C₄	—	2	3	no	—
160	C₄⋊C₄	—	2	3	no	—
161	C₄⋊C₄	—	10	3	no	—
162	C₄⋊C₄	—	10	3	no	—
163	C₄⋊C₄	—	20	3	no	—
164	C₄⋊C₄	—	20	3	no	—
165	C₂²⋊C₄	—	20	3	no	—
166	C₄²	—	5	3	no	—
167	C₄²	—	20	3	no	—
168	C₄²	C₂³×D₅	1	1	yes	characteristic
169	C₂×C₅	—	16	1	no	—
170	C₂×C₅	#3153	1	3	yes	—
171	D₅	—	4	1	no	—
172	D₅	—	16	1	no	—
173	C₂³	—	4	1	no	—
174	C₂³	—	5	1	no	—
175	C₂³	—	20	1	no	—
176	C₂³	—	40	6	no	—
177	Q₈	—	4	6	no	—
178	Q₈	—	20	6	no	—
179	D₄	—	4	6	no	—
180	D₄	—	20	6	no	—
181	D₄	—	40	6	no	—
182	C₂×C₄	—	4	3	no	—
183	C₂×C₄	—	5	3	no	—
184	C₂×C₄	—	10	6	no	—
185	C₂×C₄	—	20	3	no	—
186	C₂×C₄	C₂×D₄×D₅	1	3	yes	—
187	C₈	—	2	6	no	—
188	C₈	—	10	6	no	—
189	C₅	#19753	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	8	3	no	—
191	C₂²	—	10	3	no	—
192	C₂²	—	40	3	no	—
193	C₂²	—	80	4	no	—
194	C₂²	D₅×#127	1	1	yes	characteristic
195	C₄	—	8	3	no	—
196	C₄	—	10	3	no	—
197	C₄	—	40	3	no	—
198	C₄	#66102	1	6	yes	—
199	C₂	—	16	1	no	—
200	C₂	—	20	1	no	—
201	C₂	—	80	1	no	—
202	C₂	#10613552	1	3	yes	—

	subgroup	G/H
center	C₂²	D₅×#127
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1711	C₂
socle	C₂²×C₅	C₂×#127
prime radical	D₅×#127	C₂²
hypercenter	#1711	D₅
hypocenter	C₅	#19753
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1711

Complements

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

N	H ≅ G/N	type
1. #10613327	C₂	semidirect
3. C₅×#1711	C₂	semidirect
4. #10602018	C₂	semidirect
5. #10601985	C₂	semidirect
6. #10598854	C₂	semidirect
7. #10598843	C₂	semidirect
10. C₅×#487	C₂²	semidirect
12. #65381	C₂²	semidirect
14. #65368	C₂²	semidirect
22. #64970	C₂²	semidirect
23. C₅×C₈²	C₂²	semidirect
24. #64675	C₂²	semidirect
56. #1711	D₅	semidirect
86. #487	C₂×D₅	semidirect
96. C₈²	C₂×D₅	semidirect
189. C₅	#19753	semidirect

Maximal subgroups

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

subgroup	index
#10613327	2
#10613295	2
C₅×#1711	2
#10602018	2
#10601985	2
#10598854	2
#10598843	2
#19753	5