Subgroups — #12014162

Order 1280.

586 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×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3944	C₂	1	2	yes	maximal
3	#10598155	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₅×#117	C₄	1	1	yes	characteristic
7	C₅×#405	—	2	2	no	—
8	C₅×#405	—	2	2	no	—
9	C₂×C₅×#97	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₂×#3944	—	2	4	no	—
12	C₂×#3944	—	2	2	no	—
13	C₂×#3944	C₄	1	8	yes	complemented
14	C₂×#3944	C₂²	1	4	yes	—
15	C₂²×#672	—	2	2	no	—
16	#12551	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×M₁₆	—	2	2	no	—
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	C₈	1	4	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	2	yes	—
27	C₂²×C₅×C₈	Q₈	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₅×#117	—	2	2	no	—
30	C₅×#117	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
32	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₅×#97	—	2	4	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄×C₅×C₈	—	2	2	no	—
37	#3944	—	2	8	no	—
38	#3944	—	2	4	no	—
39	#3944	C₂×C₄	1	8	yes	—
40	C₂×#672	—	2	8	no	—
41	C₂×#672	—	2	4	no	—
42	C₂²×M₃₂	—	5	2	no	—
43	C₂×#405	D₅	1	1	yes	characteristic
44	#1377	—	5	4	no	—
45	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₄²	1	2	yes	—
47	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅	M₁₆	1	2	yes	—
53	C₂²×C₄×C₅	M₁₆	1	2	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₅×M₁₆	—	2	8	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	C₄²	1	2	yes	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
64	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
65	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₅×C₂²⋊C₄	—	2	4	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	#672	—	2	8	no	—
76	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
77	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
78	C₂×#117	Dic₅	1	1	yes	characteristic
79	C₂×M₃₂	—	5	8	no	—
80	C₂×M₃₂	—	5	4	no	—
81	C₂×M₃₂	—	10	4	no	—
82	C₂×M₃₂	—	10	2	no	—
83	C₂²×C₁₆	—	10	2	no	—
84	#405	—	2	2	no	—
85	#405	—	2	2	no	—
86	C₂×#97	C₂×D₅	1	2	yes	—
87	C₂×C₄×C₈	Dic₅	1	2	yes	—
88	C₂³×C₅	—	4	2	no	—
89	C₂³×C₅	C₄×C₈	1	2	yes	—
90	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
91	C₂³×C₅	C₄⋊C₈	1	2	yes	—
92	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	4	4	no	—
95	C₂×C₄×C₅	#95	1	4	yes	—
96	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₅	#97	1	4	yes	—
98	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
102	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
108	C₅×C₈	—	2	4	no	—
109	C₅×C₈	#95	1	2	yes	—
110	C₅×C₈	#95	1	2	yes	—
111	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
112	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	#168	1	4	yes	—
117	C₂²×C₈	Dic₁₀	1	2	yes	—
118	C₂²×C₈	C₄×D₅	1	4	yes	—
119	C₂²×C₈	D₂₀	1	2	yes	—
120	C₂²×C₈	C₂×Dic₅	1	2	yes	—
121	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
123	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
124	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
125	#117	—	2	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	#168	1	2	yes	—
128	C₂×C₂²⋊C₄	#168	1	2	yes	—
129	C₂×C₄²	C₂×Dic₅	1	2	yes	—
130	M₃₂	—	5	8	no	—
131	M₃₂	—	10	8	no	—
132	M₃₂	—	10	4	no	—
133	C₂×C₁₆	—	10	8	no	—
134	C₂×C₁₆	—	10	4	no	—
135	#97	—	2	4	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	4	2	no	—
141	C₂²×C₅	—	4	2	no	—
142	C₂²×C₅	#336	1	4	yes	—
143	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₄×C₅	—	4	4	no	—
147	C₄×C₅	#336	1	4	yes	—
148	C₄×C₅	C₂×#95	1	1	yes	characteristic
149	C₄×C₅	C₂×#97	1	1	yes	characteristic
150	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂⁴	C₂×#168	1	1	yes	characteristic
153	C₂²×C₄	C₂×#168	1	4	yes	—
154	C₂²×C₄	C₂×#168	1	1	yes	characteristic
155	C₂²×C₄	C₂×#168	1	2	yes	—
156	C₂²×C₄	#681	1	2	yes	—
157	C₂²×C₄	#681	1	2	yes	—
158	C₂²×C₄	C₄×Dic₅	1	2	yes	—
159	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
162	M₁₆	—	2	8	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	C₂×#168	1	4	yes	—
166	C₂×C₈	C₄×Dic₅	1	2	yes	—
167	C₂×C₈	#683	1	4	yes	—
168	C₂×C₈	#683	1	4	yes	—
169	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
170	C₂×C₈	#685	1	4	yes	—
171	C₂×C₈	#685	1	4	yes	—
172	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
174	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
176	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₁₆	—	10	8	no	—
183	C₂×C₅	—	2	4	no	—
184	C₂×C₅	—	2	1	no	—
185	C₂×C₅	—	2	1	no	—
186	C₂×C₅	C₂×#336	1	1	yes	characteristic
187	C₂×C₅	#2115	1	2	yes	—
188	C₂³	—	4	2	no	—
189	C₂³	C₄×#168	1	2	yes	—
190	C₂³	C₄⋊#168	1	2	yes	—
191	C₂³	C₂²⋊#168	1	2	yes	—
192	C₂³	C₂²×#168	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₄×#168	1	2	yes	—
196	C₂×C₄	#3935	1	4	yes	—
197	C₂×C₄	C₄⋊#168	1	4	yes	—
198	C₂×C₄	C₄⋊#168	1	2	yes	—
199	C₂×C₄	C₂²⋊#168	1	4	yes	—
200	C₂×C₄	C₂²⋊#168	1	2	yes	—
201	C₂×C₄	#3963	1	4	yes	—
202	C₂×C₄	C₂²×#168	1	1	yes	characteristic
203	C₂×C₄	C₂×#681	1	1	yes	characteristic
204	C₂×C₄	C₂×#681	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₈	#3963	1	2	yes	—
210	C₈	#3963	1	2	yes	—
211	C₈	C₂×#683	1	2	yes	—
212	C₈	C₂×#685	1	2	yes	—
213	C₅	#12551	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	#64744	1	4	yes	—
219	C₂²	C₂×C₄×#168	1	1	yes	characteristic
220	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
221	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	#64744	1	4	yes	—
224	C₄	C₂×#3935	1	1	yes	characteristic
225	C₄	C₂×C₄⋊#168	1	1	yes	characteristic
226	C₄	C₂×C₂²⋊#168	1	1	yes	characteristic
227	C₄	C₂×#3963	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	4	no	—
231	C₂	C₂×#64744	1	1	yes	characteristic
232	C₂	#10602800	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#405	D₅
hypocenter	C₅	#12551
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #10598155	C₂	semidirect
13. C₂×#3944	C₄	semidirect
213. C₅	#12551	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂²×#3944	2
#10598155	2
#12551	5