Subgroups — #11997845

Order 1280.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
2	#10601261	C₂	1	2	yes	maximal
3	C₄×#3950	C₂	1	1	yes	maximal, characteristic
4	C₂×#64675	C₂	1	1	yes	maximal, characteristic
5	#10598195	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3950	C₄	1	1	yes	characteristic
9	C₂×#3950	C₄	1	1	yes	characteristic
10	C₂×#3946	—	2	2	no	—
11	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	C₅×#334	—	2	2	no	—
15	#64771	C₄	1	2	yes	—
16	#64770	—	2	2	no	—
17	#64675	—	2	2	no	—
18	#9595	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅	—	2	2	no	—
26	C₂²×#168	—	2	1	no	—
27	C₂²×#168	D₄	1	2	yes	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₄×C₅×C₈	D₄	1	2	yes	—
31	C₄×C₅×C₈	Q₈	1	2	yes	—
32	#3963	—	2	2	no	—
33	#3950	—	2	2	no	—
34	#3950	C₈	1	4	yes	complemented
35	#3946	—	4	4	no	—
36	C₄×#168	—	2	1	no	—
37	C₄×#168	—	2	1	no	—
38	C₄×#168	—	2	1	no	—
39	C₄×#168	Q₈	1	2	yes	—
40	#1939	—	5	2	no	—
41	C₄×#107	—	5	1	no	—
42	C₂×#334	D₅	1	1	yes	characteristic
43	C₂×#322	—	5	1	no	—
44	#1279	—	5	2	no	—
45	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	#41	1	1	yes	characteristic
48	C₂²×Dic₅	—	2	2	no	—
49	C₂²×Dic₅	—	2	2	no	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	—	4	2	no	—
54	C₂×C₅×C₈	—	4	2	no	—
55	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
58	C₂×C₅×C₈	M₁₆	1	2	yes	—
59	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₅	#41	1	1	yes	characteristic
63	C₄⋊Dic₅	—	2	2	no	—
64	C₄⋊Dic₅	—	2	2	no	—
65	C₄⋊Dic₅	—	2	2	no	—
66	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
67	C₄×Dic₅	—	4	2	no	—
68	C₄×Dic₅	—	4	2	no	—
69	C₂×#168	—	2	2	no	—
70	C₂×#168	—	2	2	no	—
71	C₂×#168	—	2	2	no	—
72	C₂×#168	—	2	2	no	—
73	C₂×#168	—	4	1	no	—
74	C₂×#168	—	4	1	no	—
75	C₂×#107	—	5	1	no	—
76	C₂×#107	—	5	1	no	—
77	C₂×C₄⋊C₈	—	10	2	no	—
78	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
79	C₂×C₄×C₈	—	5	1	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	C₄×C₄⋊C₄	—	5	2	no	—
83	#340	—	5	2	no	—
84	#336	—	10	2	no	—
85	#334	—	2	2	no	—
86	#322	—	10	2	no	—
87	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅	#121	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅	#125	1	1	yes	characteristic
95	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	#131	1	1	yes	characteristic
99	C₂×C₄×C₅	#131	1	1	yes	characteristic
100	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₅	#137	1	2	yes	—
102	C₂×Dic₅	—	2	4	no	—
103	C₂×Dic₅	—	4	2	no	—
104	C₂×Dic₅	—	4	2	no	—
105	C₂×Dic₅	—	4	2	no	—
106	C₂×Dic₅	—	4	2	no	—
107	C₂×Dic₅	—	4	2	no	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	—	8	2	no	—
110	C₅×C₈	C₄⋊C₈	1	4	yes	complemented
111	#168	—	4	2	no	—
112	#168	—	4	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	5	2	no	—
115	C₂²×C₈	—	10	1	no	—
116	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
117	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	5	2	no	—
119	C₂×C₄⋊C₄	—	5	2	no	—
120	C₂×C₄²	—	10	2	no	—
121	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
122	#107	—	5	4	no	—
123	#107	—	10	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	20	4	no	—
127	C₄×C₈	—	5	2	no	—
128	C₄×C₈	—	10	1	no	—
129	C₄×C₈	—	10	1	no	—
130	C₄×C₈	—	10	1	no	—
131	C₄×C₈	Dic₁₀	1	2	yes	—
132	C₄×C₈	C₂²⋊D₅	1	2	yes	—
133	#95	—	10	2	no	—
134	C₂²×C₅	#389	1	1	yes	characteristic
135	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
136	C₂²×C₅	#436	1	1	yes	characteristic
137	C₂²×C₅	#441	1	1	yes	characteristic
138	C₂²×C₅	#442	1	1	yes	characteristic
139	C₂²×C₅	#444	1	1	yes	characteristic
140	C₂²×C₅	#444	1	1	yes	characteristic
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₄×C₅	#424	1	1	yes	characteristic
145	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
146	C₄×C₅	#446	1	1	yes	characteristic
147	C₄×C₅	#468	1	2	yes	—
148	C₄×C₅	#501	1	2	yes	—
149	Dic₅	—	4	4	no	—
150	Dic₅	—	8	2	no	—
151	C₂²×C₄	—	10	2	no	—
152	C₂²×C₄	—	10	2	no	—
153	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
155	C₂²×C₄	#713	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	10	2	no	—
162	C₂×C₈	—	10	2	no	—
163	C₂×C₈	—	10	2	no	—
164	C₂×C₈	—	10	2	no	—
165	C₂×C₈	—	20	1	no	—
166	C₂×C₈	—	20	1	no	—
167	C₂×C₈	C₈×D₅	1	2	yes	—
168	C₂×C₈	#676	1	2	yes	—
169	C₂×C₈	#683	1	2	yes	—
170	C₂×C₈	#683	1	2	yes	—
171	C₄⋊C₄	—	5	4	no	—
172	C₄⋊C₄	—	10	2	no	—
173	C₄⋊C₄	—	10	2	no	—
174	C₄⋊C₄	—	10	2	no	—
175	C₄²	—	20	2	no	—
176	C₄²	—	20	2	no	—
177	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
178	C₄²	#711	1	1	yes	characteristic
179	C₄²	Q₈×D₅	1	1	yes	characteristic
180	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
181	C₂×C₅	#1579	1	1	yes	characteristic
182	C₂×C₅	#1580	1	1	yes	characteristic
183	C₂×C₅	#1593	1	1	yes	characteristic
184	C₂×C₅	#1594	1	1	yes	characteristic
185	C₂×C₅	#1939	1	1	yes	characteristic
186	C₂×C₅	#1943	1	1	yes	characteristic
187	C₂×C₅	#1946	1	1	yes	characteristic
188	C₂³	#4039	1	1	yes	characteristic
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	10	4	no	—
192	C₂×C₄	—	20	2	no	—
193	C₂×C₄	—	20	2	no	—
194	C₂×C₄	—	20	2	no	—
195	C₂×C₄	—	20	2	no	—
196	C₂×C₄	—	20	2	no	—
197	C₂×C₄	#4033	1	1	yes	characteristic
198	C₂×C₄	#4034	1	1	yes	characteristic
199	C₂×C₄	#4036	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
201	C₂×C₄	C₂×#676	1	1	yes	characteristic
202	C₂×C₄	#4053	1	1	yes	characteristic
203	C₂×C₄	#4053	1	1	yes	characteristic
204	C₂×C₄	#4060	1	2	yes	—
205	C₂×C₄	#4061	1	2	yes	—
206	C₂×C₄	C₂×#683	1	1	yes	characteristic
207	C₂×C₄	#4084	1	1	yes	characteristic
208	C₂×C₄	#4092	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	8	2	no	—
211	C₈	—	20	2	no	—
212	C₈	—	20	2	no	—
213	C₈	#3946	1	4	yes	complemented
214	C₅	#9595	1	1	yes	characteristic, complemented, Sylow
215	C₂²	#65123	1	1	yes	characteristic
216	C₂²	#65127	1	1	yes	characteristic
217	C₂²	#65142	1	1	yes	characteristic
218	C₂²	#65158	1	1	yes	characteristic
219	C₂²	#65183	1	1	yes	characteristic
220	C₂²	#65183	1	1	yes	characteristic
221	C₂²	#65262	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	20	4	no	—
225	C₄	—	40	2	no	—
226	C₄	#65119	1	1	yes	characteristic
227	C₄	#65120	1	1	yes	characteristic
228	C₄	#65145	1	2	yes	—
229	C₄	C₂×#3946	1	1	yes	characteristic
230	C₄	#65408	1	1	yes	characteristic
231	C₄	#65463	1	2	yes	—
232	C₂	#10599310	1	1	yes	characteristic
233	C₂	#10599311	1	1	yes	characteristic
234	C₂	#10599317	1	1	yes	characteristic
235	C₂	#10599318	1	1	yes	characteristic
236	C₂	#10601261	1	1	yes	characteristic
237	C₂	#10601354	1	1	yes	characteristic
238	C₂	#10601498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#334	C₂
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#334	D₅
hypocenter	C₅	#9595
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
34. #3950	C₈	semidirect
110. C₅×C₈	C₄⋊C₈	semidirect
213. C₈	#3946	semidirect
214. C₅	#9595	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#334	2
#10601261	2
C₄×#3950	2
C₂×#64675	2
#10598195	2
#9595	5