Finite Group Databasebeta

Subgroups — #11993035

Order 1280.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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	#10600642	C₂	1	1	yes	maximal, characteristic
2	C₅×#1291	C₂	1	1	yes	maximal, characteristic
3	#10598245	C₂	1	1	yes	maximal, characteristic
4	#10598244	C₂	1	1	yes	maximal, characteristic
5	#10598243	C₂	1	1	yes	maximal, characteristic
6	#10598243	C₂	1	1	yes	maximal, characteristic
7	#10598157	C₂	1	1	yes	maximal, characteristic
8	C₅×#432	C₂²	1	1	yes	characteristic
9	C₅×#432	C₂²	1	1	yes	characteristic
10	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	#65226	C₂²	1	1	yes	characteristic
12	#65215	C₂²	1	1	yes	characteristic
13	C₄×#681	C₂²	1	1	yes	characteristic
14	C₄×#681	C₂²	1	1	yes	characteristic
15	#64675	C₄	1	1	yes	characteristic, complemented
16	#64674	C₄	1	1	yes	characteristic, complemented
17	#64674	C₄	1	1	yes	characteristic, complemented
18	C₈×#168	C₄	1	1	yes	characteristic, complemented
19	#8804	—	5	1	no	maximal, Sylow
20	C₂×C₅×M₁₆	—	2	1	no	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
22	C₂×#681	—	2	1	no	—
23	C₂×#681	—	2	1	no	—
24	C₅×C₄⋊C₈	—	2	1	no	—
25	C₅×C₄⋊C₈	—	2	1	no	—
26	C₅×C₂²⋊C₈	—	2	1	no	—
27	C₅×C₂²⋊C₈	—	2	1	no	—
28	C₅×#97	—	2	1	no	—
29	C₅×#97	—	2	1	no	—
30	C₅×#97	—	2	1	no	—
31	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	#3963	—	2	1	no	—
34	#3963	—	2	1	no	—
35	#3963	—	2	1	no	—
36	C₂²⋊#168	—	2	1	no	—
37	C₄⋊#168	—	2	1	no	—
38	#3935	—	2	1	no	—
39	#3935	—	2	1	no	—
40	#3935	—	2	1	no	—
41	C₄×#168	C₂×C₄	1	1	yes	characteristic
42	C₄×#168	C₂×C₄	1	1	yes	characteristic
43	C₄×#168	C₂×C₄	1	1	yes	characteristic
44	C₄×#168	C₂×C₄	1	1	yes	characteristic
45	#1455	—	5	1	no	—
46	#1291	—	5	1	no	—
47	#1291	—	5	1	no	—
48	#1291	D₅	1	1	yes	characteristic
49	#1288	—	5	1	no	—
50	#1288	—	5	1	no	—
51	#1288	—	5	1	no	—
52	C₂²×C₄×C₅	#41	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	1	yes	characteristic
54	C₂²×C₄×C₅	#41	1	1	yes	characteristic
55	C₂²×Dic₅	—	4	1	no	—
56	C₅×M₁₆	—	4	2	no	—
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	1	no	—
61	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅	#41	1	1	yes	characteristic
63	C₄²×C₅	#41	1	1	yes	characteristic
64	C₄²×C₅	#41	1	1	yes	characteristic
65	#681	—	4	2	no	—
66	#681	—	4	2	no	—
67	C₂×#168	—	2	1	no	—
68	C₂×#168	—	2	1	no	—
69	C₂×#168	—	2	1	no	—
70	C₂×#168	—	2	1	no	—
71	C₂×#168	—	4	1	no	—
72	#432	—	5	1	no	—
73	#432	C₂×D₅	1	1	yes	characteristic
74	#432	C₂×D₅	1	1	yes	characteristic
75	C₄×M₁₆	—	5	1	no	—
76	C₄×M₁₆	—	5	1	no	—
77	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
78	#383	—	5	1	no	—
79	#322	—	5	1	no	—
80	#322	—	5	1	no	—
81	#322	—	5	1	no	—
82	C₈²	—	5	1	no	—
83	C₂³×C₅	#126	1	1	yes	characteristic
84	C₂×C₄×C₅	—	4	1	no	—
85	C₂×C₄×C₅	—	4	1	no	—
86	C₂×C₄×C₅	—	4	1	no	—
87	C₂×C₄×C₅	#117	1	1	yes	characteristic
88	C₂×C₄×C₅	#117	1	1	yes	characteristic
89	C₂×C₄×C₅	#117	1	1	yes	characteristic
90	C₂×C₄×C₅	#126	1	1	yes	characteristic
91	C₂×C₄×C₅	#126	1	1	yes	characteristic
92	C₂×C₄×C₅	#126	1	1	yes	characteristic
93	C₂×Dic₅	—	8	1	no	—
94	C₂×Dic₅	—	8	1	no	—
95	C₂×Dic₅	—	8	1	no	—
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	—	4	1	no	—
98	C₅×C₈	—	8	1	no	—
99	C₅×C₈	—	8	1	no	—
100	#168	—	2	2	no	—
101	#168	—	2	2	no	—
102	#168	—	4	1	no	—
103	#168	—	4	1	no	—
104	#168	—	8	1	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂×M₁₆	—	10	1	no	—
107	C₂×M₁₆	—	10	1	no	—
108	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	—	10	1	no	—
112	C₂²⋊C₈	—	2	1	no	—
113	C₂²⋊C₈	—	2	1	no	—
114	C₂²⋊C₈	—	10	1	no	—
115	#97	—	2	1	no	—
116	#97	—	2	1	no	—
117	#97	—	2	1	no	—
118	#97	—	10	1	no	—
119	#97	—	10	1	no	—
120	#97	—	10	1	no	—
121	C₄×C₈	—	5	1	no	—
122	C₄×C₈	—	5	1	no	—
123	C₄×C₈	—	5	1	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	C₄×D₅	1	1	yes	characteristic
126	C₄×C₈	C₄×D₅	1	1	yes	characteristic
127	#95	—	10	1	no	—
128	#95	—	10	1	no	—
129	#95	—	10	1	no	—
130	C₂²×C₅	—	2	1	no	—
131	C₂²×C₅	—	2	1	no	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	#383	1	1	yes	characteristic
134	C₄×C₅	—	2	1	no	—
135	C₄×C₅	—	2	1	no	—
136	C₄×C₅	—	2	1	no	—
137	C₄×C₅	—	2	1	no	—
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	—	2	1	no	—
140	Dic₅	—	16	1	no	—
141	C₂²×C₄	—	20	1	no	—
142	C₂²×C₄	#709	1	1	yes	characteristic
143	C₂²×C₄	#711	1	1	yes	characteristic
144	C₂²×C₄	#713	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	20	2	no	—
147	M₁₆	—	20	2	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	10	1	no	—
153	C₂×C₈	—	10	1	no	—
154	C₂×C₈	—	10	1	no	—
155	C₂×C₈	—	10	1	no	—
156	C₂×C₈	—	20	1	no	—
157	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
158	C₄²	#709	1	1	yes	characteristic
159	C₄²	#711	1	1	yes	characteristic
160	C₄²	#711	1	1	yes	characteristic
161	C₂×C₅	—	4	1	no	—
162	C₂×C₅	#1841	1	1	yes	characteristic
163	C₂×C₅	#1841	1	1	yes	characteristic
164	C₂×C₅	#1841	1	1	yes	characteristic
165	C₂³	#4044	1	1	yes	characteristic
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	40	1	no	—
170	C₂×C₄	—	40	1	no	—
171	C₂×C₄	—	40	1	no	—
172	C₂×C₄	#4018	1	1	yes	characteristic
173	C₂×C₄	#4023	1	1	yes	characteristic
174	C₂×C₄	#4025	1	1	yes	characteristic
175	C₂×C₄	#4025	1	1	yes	characteristic
176	C₂×C₄	#4038	1	1	yes	characteristic
177	C₂×C₄	#4044	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	4	1	no	—
180	C₈	—	8	1	no	—
181	C₈	—	8	1	no	—
182	C₈	—	10	2	no	—
183	C₈	—	10	2	no	—
184	C₈	—	20	1	no	—
185	C₈	—	20	1	no	—
186	C₈	—	40	1	no	—
187	C₅	#8804	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	#64940	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	80	1	no	—
199	C₂	—	4	1	no	—
200	C₂	#10599509	1	1	yes	characteristic
201	C₂	#10599542	1	1	yes	characteristic
202	C₂	#10599810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64940
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1291	C₂
socle	C₂²×C₅	#383
prime radical	C₂³×C₅	#126
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1291	D₅
hypocenter	C₅	#8804
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1291

Complements

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

N	H ≅ G/N	type
15. #64675	C₄	semidirect
16. #64674	C₄	semidirect
17. #64674	C₄	semidirect
18. C₈×#168	C₄	semidirect
187. C₅	#8804	semidirect

Maximal subgroups

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

subgroup	index
#10600642	2
C₅×#1291	2
#10598245	2
#10598244	2
#10598243	2
#10598157	2
#8804	5