Subgroups — #12015225

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 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	C₂×C₂²⋊#672	C₂	1	2	yes	maximal
3	C₂×C₄⋊#672	C₂	1	2	yes	maximal, complemented
4	#10598148	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	C₄	1	1	yes	characteristic
8	C₅×#405	—	2	4	no	—
9	C₂×C₅×#97	C₄	1	1	yes	characteristic
10	C₂×C₅×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	4	yes	—
14	C₂²⋊#672	—	2	4	no	—
15	C₄⋊#672	—	2	4	no	—
16	#12784	—	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	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₅×#97	—	2	2	no	—
36	C₅×#97	—	2	2	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	C₄×C₅×C₈	D₄	1	4	yes	—
39	C₂×#672	—	2	4	no	—
40	C₂×#672	—	2	4	no	—
41	C₂×#672	—	2	4	no	—
42	C₂×#405	D₅	1	1	yes	characteristic
43	C₂×C₄⋊C₁₆	—	5	2	no	—
44	C₂×C₂²⋊C₁₆	—	5	2	no	—
45	#1374	—	5	2	no	—
46	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	1	no	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	M₁₆	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₅×M₁₆	—	2	8	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	M₁₆	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₅×C₈	#41	1	2	yes	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₂²⋊C₄	—	2	4	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₄²×C₅	—	2	1	no	—
77	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₅	M₁₆	1	2	yes	—
79	#672	—	4	4	no	—
80	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
81	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×#117	Dic₅	1	1	yes	characteristic
83	C₂²×C₁₆	—	5	4	no	—
84	#405	—	2	4	no	—
85	C₂×#97	Dic₅	1	1	yes	characteristic
86	C₂×#97	Dic₅	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
89	C₄⋊C₁₆	—	10	4	no	—
90	C₂²⋊C₁₆	—	10	4	no	—
91	C₂³×C₅	—	2	1	no	—
92	C₂³×C₅	—	2	1	no	—
93	C₂³×C₅	—	2	4	no	—
94	C₂³×C₅	C₄⋊C₈	1	2	yes	—
95	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
106	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	#117	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	—	2	2	no	—
116	C₅×C₈	#121	1	2	yes	—
117	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	Dic₁₀	1	2	yes	—
125	C₂²×C₈	D₂₀	1	2	yes	—
126	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
127	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
128	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
130	#117	—	2	2	no	—
131	#117	—	2	2	no	—
132	C₂×C₄⋊C₄	#168	1	2	yes	—
133	C₂×C₂²⋊C₄	#168	1	2	yes	—
134	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
135	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
136	C₂×C₁₆	—	10	4	no	—
137	C₂×C₁₆	—	10	4	no	—
138	C₂×C₁₆	—	10	4	no	—
139	#97	—	2	2	no	—
140	#97	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	C₂²⋊D₅	1	4	yes	—
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	4	no	—
145	C₂²×C₅	—	4	2	no	—
146	C₂²×C₅	—	4	2	no	—
147	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₂²×C₅	#431	1	1	yes	characteristic
150	C₂²×C₅	#504	1	4	yes	—
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	4	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	#380	1	1	yes	characteristic
155	C₄×C₅	#407	1	1	yes	characteristic
156	C₄×C₅	#423	1	1	yes	characteristic
157	C₄×C₅	#432	1	1	yes	characteristic
158	C₂⁴	C₂×#168	1	1	yes	characteristic
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	C₂×#168	1	2	yes	—
164	C₂²×C₄	C₂×#168	1	1	yes	characteristic
165	C₂²×C₄	C₂×#168	1	2	yes	—
166	C₂²×C₄	#681	1	2	yes	—
167	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
168	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
169	M₁₆	—	2	8	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	#681	1	2	yes	—
179	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
180	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
181	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
182	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
183	C₂×C₈	#709	1	2	yes	—
184	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	#681	1	2	yes	—
190	C₄²	C₂²⋊Dic₅	1	2	yes	—
191	C₁₆	—	20	4	no	—
192	C₂×C₅	—	4	2	no	—
193	C₂×C₅	#1797	1	1	yes	characteristic
194	C₂×C₅	#2113	1	2	yes	—
195	C₂×C₅	#2151	1	2	yes	—
196	C₂×C₅	#2164	1	2	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	C₄⋊#168	1	2	yes	—
201	C₂³	C₂²×#168	1	1	yes	characteristic
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₄⋊#168	1	2	yes	—
212	C₂×C₄	C₂²⋊#168	1	4	yes	—
213	C₂×C₄	C₂×#681	1	1	yes	characteristic
214	C₂×C₄	C₂×#681	1	1	yes	characteristic
215	C₂×C₄	C₂×#681	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
217	C₂×C₄	#4072	1	1	yes	characteristic
218	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	#4070	1	2	yes	—
223	C₈	#4076	1	2	yes	—
224	C₅	#12784	1	1	yes	characteristic, complemented, Sylow
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
230	C₂²	#65214	1	1	yes	characteristic
231	C₂²	#65429	1	4	yes	—
232	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	#65213	1	1	yes	characteristic
237	C₄	#65215	1	1	yes	characteristic
238	C₄	#65496	1	1	yes	characteristic
239	C₄	#65501	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	#10600675	1	2	yes	—
242	C₂	#10600676	1	2	yes	—
243	C₂	#10602323	1	1	yes	characteristic
244	C₂	#10602797	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle	C₂³×C₅	C₂²×C₈
Fitting	C₂×C₅×#405	C₂
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#405	D₅
hypocenter	C₅	#12784
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊#672	C₂	semidirect
4. #10598148	C₂	semidirect
224. C₅	#12784	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
C₂×C₂²⋊#672	2
C₂×C₄⋊#672	2
#10598148	2
#12784	5