Subgroups — #12014338

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64978	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64727	C₂	1	2	yes	maximal
4	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598295	C₂	1	2	yes	maximal, complemented
6	C₂²×#676	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×#4018	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	2	yes	—
14	C₂×#3935	C₄	1	1	yes	characteristic
15	#64978	—	2	2	no	—
16	#64978	—	2	2	no	—
17	C₅×#346	—	2	2	no	—
18	#64727	C₄	1	8	yes	complemented
19	C₄×#672	—	2	2	no	—
20	#12600	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	C₂×C₄	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×#685	C₈	1	2	yes	—
26	C₂×#683	C₈	1	2	yes	—
27	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×#676	—	2	2	no	—
31	C₂×#676	—	2	2	no	—
32	C₂×#676	C₈	1	4	yes	—
33	C₂×C₈×D₅	—	2	1	no	—
34	C₂×C₈×D₅	—	2	1	no	—
35	C₂×C₈×D₅	—	2	2	no	—
36	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
37	#4018	—	2	2	no	—
38	#4018	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	C₄×C₅×C₈	D₄	1	2	yes	—
44	#3947	—	2	2	no	—
45	C₈×Dic₅	—	2	1	no	—
46	C₈×Dic₅	—	2	1	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	C₂×C₄	1	4	yes	—
50	#3935	—	2	2	no	—
51	C₂×#405	—	5	1	no	—
52	C₂×C₂²⋊C₁₆	—	5	2	no	—
53	C₂×#346	D₅	1	1	yes	characteristic, complemented
54	C₂×C₄×C₁₆	—	5	1	no	—
55	#1374	—	5	2	no	—
56	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₅	C₂×C₈	1	2	yes	—
60	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅	—	2	1	no	—
62	C₂×C₄×D₅	—	2	1	no	—
63	C₂×C₄×D₅	—	2	1	no	—
64	C₂×C₄×D₅	—	2	1	no	—
65	C₂×C₄×D₅	C₄²	1	2	yes	—
66	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
69	C₂×C₅×C₈	M₁₆	1	2	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₅×C₈	#41	1	2	yes	—
74	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₅	M₁₆	1	2	yes	—
76	#685	—	2	4	no	—
77	#683	—	2	4	no	—
78	C₄×Dic₅	—	2	1	no	—
79	C₄×Dic₅	—	2	1	no	—
80	C₂×#168	—	2	1	no	—
81	C₂×#168	—	2	1	no	—
82	C₂×#168	—	2	1	no	—
83	C₂×#168	—	2	1	no	—
84	C₂×#168	C₄²	1	2	yes	—
85	C₂×#168	C₂×C₈	1	2	yes	—
86	#676	—	2	8	no	—
87	C₈×D₅	—	2	4	no	—
88	C₈×D₅	—	2	4	no	—
89	C₅×C₁₆	—	2	4	no	—
90	#672	—	2	4	no	—
91	C₂²×M₁₆	—	5	1	no	—
92	C₂³×C₈	—	5	1	no	—
93	C₂×#117	—	5	1	no	—
94	C₂²×C₁₆	—	5	2	no	—
95	C₂²×C₁₆	C₂×D₅	1	2	yes	—
96	#405	—	10	2	no	—
97	#405	—	10	2	no	—
98	C₂×#97	—	5	1	no	—
99	C₂×#97	—	5	1	no	—
100	C₂×C₄×C₈	—	5	1	no	—
101	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
102	C₂²⋊C₁₆	—	5	8	no	—
103	#346	—	2	2	no	—
104	C₄×C₁₆	—	10	2	no	—
105	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
106	C₂²×D₅	—	2	1	no	—
107	C₂²×D₅	—	2	1	no	—
108	C₂²×D₅	—	2	4	no	—
109	C₂²×D₅	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
112	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	#117	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₂×Dic₅	—	2	1	no	—
119	C₂×Dic₅	—	2	1	no	—
120	C₂×Dic₅	—	2	2	no	—
121	C₂×Dic₅	—	2	4	no	—
122	C₂×Dic₅	C₄×C₈	1	2	yes	—
123	C₄×D₅	—	2	2	no	—
124	C₄×D₅	—	2	2	no	—
125	C₄×D₅	—	2	2	no	—
126	C₄×D₅	—	2	2	no	—
127	C₅×C₈	C₂²⋊C₈	1	4	yes	—
128	C₅×C₈	C₄×D₄	1	2	yes	—
129	C₅×C₈	C₄×D₄	1	2	yes	—
130	#168	—	2	2	no	—
131	#168	—	2	2	no	—
132	C₂³×C₄	—	5	1	no	—
133	C₂×M₁₆	—	5	4	no	—
134	C₂×M₁₆	—	10	2	no	—
135	C₂×M₁₆	—	10	2	no	—
136	C₂²×C₈	—	5	1	no	—
137	C₂²×C₈	—	5	1	no	—
138	C₂²×C₈	—	5	4	no	—
139	C₂²×C₈	—	10	1	no	—
140	C₂²×C₈	—	10	2	no	—
141	C₂²×C₈	—	10	1	no	—
142	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
144	#117	—	10	2	no	—
145	#117	—	10	2	no	—
146	C₂×C₄⋊C₄	—	5	2	no	—
147	C₂×C₂²⋊C₄	—	5	2	no	—
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	5	4	no	—
153	C₂×C₁₆	—	10	2	no	—
154	C₂×C₁₆	—	10	2	no	—
155	C₂×C₁₆	C₄×D₅	1	4	yes	—
156	#97	—	10	2	no	—
157	#97	—	10	2	no	—
158	C₄×C₈	—	10	1	no	—
159	C₄×C₈	—	10	1	no	—
160	C₄×C₈	D₂₀	1	2	yes	—
161	C₄×C₈	C₂²⋊D₅	1	2	yes	—
162	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
163	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₅	#431	1	1	yes	characteristic
165	C₂²×C₅	#504	1	4	yes	—
166	C₂×D₅	—	2	4	no	—
167	C₂×D₅	—	2	2	no	—
168	C₂×D₅	—	4	2	no	—
169	C₂×D₅	—	4	2	no	—
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
173	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
174	C₄×C₅	#431	1	1	yes	characteristic
175	Dic₅	—	2	2	no	—
176	Dic₅	—	4	2	no	—
177	C₂⁴	—	5	1	no	—
178	C₂²×C₄	—	5	1	no	—
179	C₂²×C₄	—	5	2	no	—
180	C₂²×C₄	—	5	2	no	—
181	C₂²×C₄	—	5	2	no	—
182	C₂²×C₄	—	10	1	no	—
183	C₂²×C₄	—	10	1	no	—
184	C₂²×C₄	—	10	1	no	—
185	C₂²×C₄	—	10	1	no	—
186	C₂²×C₄	C₈×D₅	1	2	yes	—
187	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
188	M₁₆	—	10	8	no	—
189	C₂×C₈	—	5	2	no	—
190	C₂×C₈	—	5	2	no	—
191	C₂×C₈	—	10	1	no	—
192	C₂×C₈	—	10	1	no	—
193	C₂×C₈	—	10	1	no	—
194	C₂×C₈	—	10	1	no	—
195	C₂×C₈	—	10	4	no	—
196	C₂×C₈	—	10	4	no	—
197	C₂×C₈	C₈×D₅	1	2	yes	—
198	C₂×C₈	#676	1	2	yes	—
199	C₂×C₈	#685	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
201	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
202	C₂×C₈	#709	1	2	yes	—
203	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₄⋊C₄	—	10	4	no	—
205	C₂²⋊C₄	—	10	4	no	—
206	C₄²	—	10	1	no	—
207	C₄²	—	10	1	no	—
208	C₄²	#676	1	2	yes	—
209	C₄²	#685	1	2	yes	—
210	C₁₆	—	2	4	no	—
211	C₁₆	—	10	4	no	—
212	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
213	C₂×C₅	#2108	1	2	yes	—
214	C₂×C₅	#2113	1	2	yes	—
215	C₂×C₅	#2164	1	2	yes	—
216	D₅	—	4	2	no	—
217	C₂³	—	5	2	no	—
218	C₂³	—	10	1	no	—
219	C₂³	—	10	4	no	—
220	C₂³	—	10	1	no	—
221	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	10	1	no	—
225	C₂×C₄	—	10	1	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	10	2	no	—
229	C₂×C₄	—	10	2	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	4	no	—
232	C₂×C₄	#3952	1	4	yes	—
233	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
234	C₂×C₄	#4018	1	1	yes	characteristic
235	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂×#676	1	1	yes	characteristic
237	C₂×C₄	C₂×#676	1	1	yes	characteristic
238	C₂×C₄	C₂×#685	1	1	yes	characteristic
239	C₈	—	10	2	no	—
240	C₈	—	10	2	no	—
241	C₈	#3952	1	4	yes	—
242	C₈	C₄×D₂₀	1	2	yes	—
243	C₈	C₄×C₂²⋊D₅	1	2	yes	—
244	C₅	#12600	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	10	2	no	—
246	C₂²	—	10	4	no	—
247	C₂²	—	20	2	no	—
248	C₂²	—	20	2	no	—
249	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
250	C₂²	#64974	1	1	yes	characteristic
251	C₂²	#65196	1	4	yes	—
252	C₂²	C₂×#3952	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	10	2	no	—
255	C₄	—	20	2	no	—
256	C₄	#64974	1	1	yes	characteristic
257	C₄	C₄×#676	1	1	yes	characteristic
258	C₄	C₄×#685	1	1	yes	characteristic
259	C₄	C₂×#3952	1	1	yes	characteristic
260	C₂	—	20	2	no	—
261	C₂	#10599865	1	2	yes	—
262	C₂	#10599867	1	2	yes	—
263	C₂	C₄×#3952	1	1	yes	characteristic
264	C₂	#10602225	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂×C₅×#346	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12600
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#346	C₂	semidirect
4. C₂×C₄×#672	C₂	semidirect
5. #10598295	C₂	semidirect
18. #64727	C₄	semidirect
53. C₂×#346	D₅	semidirect
244. C₅	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64978	2
C₂×C₅×#346	2
C₂×#64727	2
C₂×C₄×#672	2
#10598295	2
#12600	5