Subgroups — #12008636

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	C₂	1	1	yes	maximal, characteristic
3	#10600537	C₂	1	1	yes	maximal, characteristic
4	#10598015	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₁₀	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3946	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅	C₄	1	2	yes	—
9	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
10	C₈×Dic₁₀	—	2	1	no	—
11	C₈×Dic₁₀	—	2	1	no	—
12	C₈×Dic₁₀	—	2	1	no	—
13	C₈×Dic₁₀	—	2	1	no	—
14	C₅×C₄⋊C₁₆	—	2	1	no	—
15	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₂²⋊#672	C₂²	1	2	yes	—
18	C₄⋊#672	—	2	1	no	—
19	C₄×#672	—	2	1	no	—
20	#11493	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	C₈	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
26	C₂×#683	C₈	1	2	yes	—
27	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
28	C₂²×#168	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₀	—	2	1	no	—
30	C₄×Dic₁₀	—	2	1	no	—
31	C₄×Dic₁₀	—	2	1	no	—
32	C₄×Dic₁₀	—	2	1	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₄×C₅×C₈	D₄	1	1	yes	characteristic
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	C₄×C₅×C₈	D₄	1	1	yes	characteristic
37	C₄×C₅×C₈	D₄	1	1	yes	characteristic
38	#3946	—	2	2	no	—
39	#3946	—	2	2	no	—
40	C₈×Dic₅	—	2	2	no	—
41	C₈×Dic₅	—	2	2	no	—
42	C₂×#672	—	2	2	no	—
43	C₄⋊#168	—	2	1	no	—
44	C₄⋊#168	—	2	1	no	—
45	C₂×C₈×Q₈	—	5	1	no	—
46	#2162	—	5	1	no	—
47	#2162	D₅	1	1	yes	characteristic
48	#1322	—	5	4	no	—
49	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₅	C₂×C₈	1	2	yes	—
53	C₂²×Dic₅	C₂×C₈	1	2	yes	—
54	C₂×Dic₁₀	—	2	1	no	—
55	C₂×Dic₁₀	—	2	1	no	—
56	C₂×Dic₁₀	—	2	2	no	—
57	C₂×Dic₁₀	C₁₆	1	4	yes	complemented
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅	M₁₆	1	1	yes	characteristic
67	C₄²×C₅	M₁₆	1	1	yes	characteristic
68	C₄⋊Dic₅	—	2	1	no	—
69	C₄⋊Dic₅	—	2	1	no	—
70	#683	—	2	2	no	—
71	#683	—	2	2	no	—
72	C₄×Dic₅	—	2	2	no	—
73	C₄×Dic₅	—	2	2	no	—
74	C₂×#168	—	2	2	no	—
75	C₂×#168	—	4	2	no	—
76	C₅×C₁₆	—	4	2	no	—
77	#672	—	4	2	no	—
78	C₂×C₄×Q₈	—	5	1	no	—
79	C₈×Q₈	—	10	1	no	—
80	C₈×Q₈	—	10	1	no	—
81	C₈×Q₈	—	10	1	no	—
82	C₈×Q₈	—	10	1	no	—
83	C₂×C₄⋊C₈	—	5	2	no	—
84	C₂×C₄⋊C₈	—	5	1	no	—
85	C₂×C₄×C₈	—	5	2	no	—
86	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	1	no	—
88	C₄⋊C₁₆	—	10	1	no	—
89	C₂²⋊C₁₆	—	5	2	no	—
90	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	—	10	1	no	—
93	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	4	1	no	—
97	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
102	C₂×C₄×C₅	M₃₂	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
105	C₂×Dic₅	—	2	2	no	—
106	C₂×Dic₅	—	2	2	no	—
107	C₂×Dic₅	—	2	2	no	—
108	C₂×Dic₅	—	4	2	no	—
109	C₂×Dic₅	C₂×C₁₆	1	4	yes	—
110	Dic₁₀	—	2	4	no	—
111	Dic₁₀	—	2	4	no	—
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	#101	1	2	yes	—
116	#168	—	4	2	no	—
117	C₂²×Q₈	—	5	1	no	—
118	C₂²×C₈	—	5	2	no	—
119	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
121	C₄×Q₈	—	10	1	no	—
122	C₄×Q₈	—	10	1	no	—
123	C₄×Q₈	—	10	1	no	—
124	C₄×Q₈	—	10	1	no	—
125	C₂×C₄⋊C₄	—	5	2	no	—
126	C₂×C₄⋊C₄	—	5	1	no	—
127	C₂×C₄²	—	5	2	no	—
128	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	10	2	no	—
131	C₄⋊C₈	—	10	2	no	—
132	C₄⋊C₈	—	10	2	no	—
133	C₄⋊C₈	—	10	1	no	—
134	C₄⋊C₈	—	10	1	no	—
135	C₄×C₈	—	10	2	no	—
136	C₄×C₈	—	10	2	no	—
137	C₄×C₈	D₂₀	1	1	yes	characteristic
138	C₄×C₈	D₂₀	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
145	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
150	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
151	C₄×C₅	#410	1	1	yes	characteristic
152	C₄×C₅	C₂×#101	1	1	yes	characteristic
153	Dic₅	—	2	4	no	—
154	Dic₅	—	4	2	no	—
155	C₂×Q₈	—	5	4	no	—
156	C₂×Q₈	—	10	2	no	—
157	C₂×Q₈	—	10	1	no	—
158	C₂×Q₈	—	10	1	no	—
159	C₂²×C₄	—	5	2	no	—
160	C₂²×C₄	—	5	2	no	—
161	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
162	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	10	2	no	—
167	C₂×C₈	—	20	2	no	—
168	C₂×C₈	#685	1	1	yes	characteristic
169	C₂×C₈	#685	1	1	yes	characteristic
170	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
172	C₄⋊C₄	—	10	2	no	—
173	C₄⋊C₄	—	10	2	no	—
174	C₄⋊C₄	—	10	1	no	—
175	C₄⋊C₄	—	10	1	no	—
176	C₄²	—	10	2	no	—
177	C₄²	—	10	2	no	—
178	C₄²	#676	1	1	yes	characteristic
179	C₄²	#676	1	1	yes	characteristic
180	C₄²	#685	1	1	yes	characteristic
181	C₄²	#685	1	1	yes	characteristic
182	C₁₆	—	4	2	no	—
183	C₁₆	—	20	2	no	—
184	C₂×C₅	—	2	1	no	—
185	C₂×C₅	—	2	1	no	—
186	C₂×C₅	#1461	1	1	yes	characteristic
187	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
188	C₂×C₅	#2115	1	1	yes	characteristic
189	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
190	Q₈	—	10	4	no	—
191	Q₈	—	10	4	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	5	4	no	—
196	C₂×C₄	—	10	2	no	—
197	C₂×C₄	—	10	2	no	—
198	C₂×C₄	—	10	2	no	—
199	C₂×C₄	—	20	2	no	—
200	C₂×C₄	D₅×C₁₆	1	2	yes	—
201	C₂×C₄	#3930	1	2	yes	—
202	C₂×C₄	#3952	1	1	yes	characteristic
203	C₂×C₄	#3952	1	1	yes	characteristic
204	C₂×C₄	#3952	1	1	yes	characteristic
205	C₂×C₄	#3952	1	1	yes	characteristic
206	C₂×C₄	C₂×#676	1	1	yes	characteristic
207	C₂×C₄	C₂×#685	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	20	2	no	—
212	C₈	#3956	1	2	yes	—
213	C₅	#11493	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂×D₅×C₁₆	1	1	yes	characteristic
217	C₂²	C₂×#3930	1	1	yes	characteristic
218	C₂²	C₂×#3952	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	10	4	no	—
223	C₄	—	20	2	no	—
224	C₄	#64727	1	2	yes	—
225	C₄	#64727	1	2	yes	—
226	C₄	#65295	1	1	yes	characteristic
227	C₄	C₂×#3956	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#10601579	1	1	yes	characteristic
231	C₂	C₂×#64727	1	1	yes	characteristic
232	C₂	#10602228	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂²×C₅	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2162	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2162	D₅
hypocenter	C₅	#11493
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
57. C₂×Dic₁₀	C₁₆	semidirect
213. C₅	#11493	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀	2
C₅×#2162	2
#10600537	2
#10598015	2
#11493	5