Finite Group Databasebeta

Subgroups — #11993021

Order 1280.

306 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₄×#3935	C₂	1	1	yes	maximal, characteristic
2	C₅×#1276	C₂	1	1	yes	maximal, characteristic
3	#10598232	C₂	1	2	yes	maximal
4	#10598231	C₂	1	2	yes	maximal
5	#10598102	C₂	1	1	yes	maximal, characteristic
6	C₅×#431	C₂²	1	2	yes	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	#65214	C₂²	1	2	yes	—
9	C₂×#3935	C₄	1	1	yes	characteristic
10	C₂×#3935	C₄	1	1	yes	characteristic
11	C₂×#3935	C₂²	1	1	yes	characteristic
12	C₂×#3935	C₂²	1	1	yes	characteristic
13	C₂×C₄×#168	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	#8801	—	5	1	no	maximal, Sylow
16	C₂×C₄²×C₅	—	2	2	no	—
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂²×#168	C₂×C₄	1	1	yes	characteristic
21	C₂²×#168	C₂×C₄	1	1	yes	characteristic
22	C₂²×#168	C₂×C₄	1	1	yes	characteristic
23	C₂²×#168	C₂×C₄	1	1	yes	characteristic
24	C₅×C₄⋊C₈	—	2	2	no	—
25	C₅×C₂²⋊C₈	—	2	2	no	—
26	C₄×C₅×C₈	—	2	2	no	—
27	C₂²⋊#168	—	2	2	no	—
28	C₄⋊#168	—	2	2	no	—
29	#3935	—	2	2	no	—
30	#3935	—	2	1	no	—
31	#3935	—	2	1	no	—
32	#3935	—	2	2	no	—
33	#3935	C₈	1	4	yes	complemented
34	C₄×#168	—	2	2	no	—
35	C₄×#168	—	2	2	no	—
36	C₄×#168	—	2	2	no	—
37	C₄×#97	—	5	1	no	—
38	#1281	—	5	2	no	—
39	#1276	—	5	1	no	—
40	#1276	D₅	1	1	yes	characteristic
41	#1275	—	5	2	no	—
42	C₂²×C₄×C₅	—	2	1	no	—
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	2	yes	—
46	C₂×C₅×C₈	—	4	2	no	—
47	C₄²×C₅	—	2	1	no	—
48	C₄²×C₅	—	2	1	no	—
49	C₄²×C₅	—	4	4	no	—
50	C₄²×C₅	C₂×C₈	1	2	yes	—
51	C₄²×C₅	M₁₆	1	2	yes	—
52	C₄²×C₅	#41	1	2	yes	—
53	C₄²×C₅	#41	1	1	yes	characteristic
54	C₄²×C₅	#41	1	1	yes	characteristic
55	C₂×#168	—	2	1	no	—
56	C₂×#168	—	2	1	no	—
57	C₂×#168	—	2	1	no	—
58	C₂×#168	—	2	1	no	—
59	C₂×#168	—	2	1	no	—
60	C₂×#168	—	2	1	no	—
61	C₂×#168	—	2	1	no	—
62	C₂×#168	—	2	1	no	—
63	C₂×#168	—	4	2	no	—
64	C₂×#168	C₂×C₈	1	2	yes	—
65	C₂×#168	C₂×C₈	1	2	yes	—
66	C₂×#168	M₁₆	1	2	yes	—
67	C₂×#168	M₁₆	1	2	yes	—
68	#431	—	5	2	no	—
69	#431	C₂×D₅	1	2	yes	—
70	C₂×#97	—	5	1	no	—
71	C₂×#97	—	5	1	no	—
72	C₂×#97	—	5	1	no	—
73	C₂×#97	—	5	1	no	—
74	C₂×C₄×C₈	—	5	1	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	C₄³	C₂×D₅	1	1	yes	characteristic
77	C₂³×C₅	#117	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	#117	1	1	yes	characteristic
84	C₂×C₄×C₅	#117	1	1	yes	characteristic
85	C₂×C₄×C₅	#126	1	2	yes	—
86	C₂×C₄×C₅	#126	1	2	yes	—
87	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₅×C₈	—	8	2	no	—
92	#168	—	2	2	no	—
93	#168	—	2	2	no	—
94	#168	—	2	2	no	—
95	#168	—	2	2	no	—
96	#168	—	8	2	no	—
97	C₂²×C₈	—	5	1	no	—
98	C₂²×C₈	—	5	1	no	—
99	C₂²×C₈	—	5	1	no	—
100	C₂²×C₈	—	5	1	no	—
101	C₂×C₄²	—	2	2	no	—
102	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
104	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	10	2	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	—	10	2	no	—
109	#97	—	5	4	no	—
110	#97	—	10	2	no	—
111	#97	—	10	2	no	—
112	#97	—	10	1	no	—
113	#97	—	10	1	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	10	2	no	—
116	C₄×C₈	—	10	2	no	—
117	C₄×C₈	—	10	2	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	#383	1	1	yes	characteristic
120	C₂²×C₅	#431	1	1	yes	characteristic
121	C₂²×C₅	#432	1	1	yes	characteristic
122	C₄×C₅	—	2	4	no	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	4	2	no	—
126	C₄×C₅	#431	1	1	yes	characteristic
127	C₄×C₅	#431	1	1	yes	characteristic
128	C₄×C₅	#432	1	1	yes	characteristic
129	C₄×C₅	#432	1	1	yes	characteristic
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
133	C₂²×C₄	#711	1	2	yes	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	5	2	no	—
136	C₂×C₈	—	5	2	no	—
137	C₂×C₈	—	5	2	no	—
138	C₂×C₈	—	5	2	no	—
139	C₂×C₈	—	10	1	no	—
140	C₂×C₈	—	10	1	no	—
141	C₂×C₈	—	10	1	no	—
142	C₂×C₈	—	10	1	no	—
143	C₂×C₈	—	10	1	no	—
144	C₂×C₈	—	10	1	no	—
145	C₂×C₈	—	10	1	no	—
146	C₂×C₈	—	10	1	no	—
147	C₂×C₈	—	20	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	4	4	no	—
151	C₄²	C₈×D₅	1	2	yes	—
152	C₄²	#676	1	2	yes	—
153	C₄²	#709	1	2	yes	—
154	C₄²	#711	1	1	yes	characteristic
155	C₄²	#713	1	1	yes	characteristic
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	#1839	1	1	yes	characteristic
159	C₂×C₅	#1841	1	2	yes	—
160	C₂³	#4023	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	#4018	1	1	yes	characteristic
167	C₂×C₄	#4025	1	2	yes	—
168	C₂×C₄	#4038	1	1	yes	characteristic
169	C₂×C₄	#4044	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
171	C₂×C₄	C₂×#676	1	1	yes	characteristic
172	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
173	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
174	C₈	—	8	2	no	—
175	C₈	—	10	2	no	—
176	C₈	—	10	2	no	—
177	C₈	—	10	2	no	—
178	C₈	—	10	2	no	—
179	C₈	—	40	2	no	—
180	C₅	#8801	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	#64940	1	1	yes	characteristic
183	C₂²	#65007	1	1	yes	characteristic
184	C₂²	#65008	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	4	2	no	—
189	C₄	#64974	1	1	yes	characteristic
190	C₄	#64994	1	1	yes	characteristic
191	C₄	#65122	1	1	yes	characteristic
192	C₄	#65124	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#10599400	1	1	yes	characteristic
196	C₂	#10599542	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#4038
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1276	C₂
socle	C₂²×C₅	#383
prime radical	C₂³×C₅	#117
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1276	D₅
hypocenter	C₅	#8801
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
33. #3935	C₈	semidirect
180. C₅	#8801	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3935	2
C₅×#1276	2
#10598232	2
#10598231	2
#10598102	2
#8801	5