Subgroups — #12129282

Order 1280.

648 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
2	#10615481	C₂	1	1	yes	maximal, characteristic
3	C₂×#64780	C₂	1	1	yes	maximal, characteristic
4	#10602787	C₂	1	2	yes	maximal
5	C₂×#64776	C₂	1	2	yes	maximal
6	#10601519	C₂	1	8	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	1	yes	characteristic
10	C₂²×#681	C₂²	1	2	yes	—
11	C₂²×#681	C₂²	1	1	yes	characteristic
12	C₅×#430	—	2	2	no	—
13	C₅×#430	C₂²	1	4	yes	—
14	C₂×C₅×#108	—	2	2	no	—
15	#65408	C₂²	1	4	yes	—
16	#65407	—	2	2	no	—
17	#65391	—	2	2	no	—
18	#64780	C₂²	1	4	yes	—
19	#64778	C₂²	1	8	yes	—
20	#64776	C₂²	1	8	yes	—
21	#23753	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	D₄	1	2	yes	—
31	C₂×C₅×M₁₆	C₂³	1	4	yes	—
32	C₂×C₅×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₂²⋊Dic₅	Q₈	1	2	yes	—
36	#4072	—	2	1	no	—
37	#4072	—	2	1	no	—
38	#4072	Q₈	1	2	yes	—
39	#4072	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₅	—	2	1	no	—
41	C₂×C₄×Dic₅	—	2	1	no	—
42	C₂×#681	—	2	1	no	—
43	C₂×#681	—	2	1	no	—
44	C₂×#681	—	2	2	no	—
45	C₂×#681	—	2	2	no	—
46	C₂×#681	—	2	2	no	—
47	C₂×#681	—	2	2	no	—
48	C₂×#681	—	2	2	no	—
49	C₂×#681	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	C₂³	1	4	yes	—
52	C₂²×#168	—	2	2	no	—
53	C₂²×#168	C₂³	1	2	yes	—
54	C₅×#108	—	2	8	no	—
55	#3946	—	2	8	no	—
56	C₂×#430	D₅	1	1	yes	characteristic
57	#2905	—	5	1	no	—
58	#2103	—	5	8	no	—
59	C₂×#344	—	5	2	no	—
60	#1738	—	5	2	no	—
61	C₂×#338	—	5	1	no	—
62	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	#41	1	2	yes	—
67	C₂²×C₄×C₅	#41	1	2	yes	—
68	C₂²×C₄×C₅	#41	1	2	yes	—
69	C₂²×C₄×C₅	#41	1	2	yes	—
70	C₂²×C₄×C₅	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅	—	2	2	no	—
74	C₅×M₁₆	—	2	4	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
86	C₂²⋊Dic₅	—	2	2	no	—
87	C₂²⋊Dic₅	C₂×Q₈	1	4	yes	—
88	C₄⋊Dic₅	—	2	2	no	—
89	C₄⋊Dic₅	—	2	2	no	—
90	C₄×Dic₅	—	2	2	no	—
91	C₄×Dic₅	—	2	2	no	—
92	#681	—	2	4	no	—
93	#681	—	2	4	no	—
94	#681	—	4	1	no	—
95	#681	—	4	1	no	—
96	#681	—	4	2	no	—
97	#681	—	4	2	no	—
98	#681	—	4	2	no	—
99	C₂×#168	—	2	2	no	—
100	C₂×#168	—	2	4	no	—
101	C₂×#168	—	2	4	no	—
102	C₂×#168	—	4	2	no	—
103	C₂×#168	—	4	2	no	—
104	C₂²×M₁₆	—	5	1	no	—
105	C₂²×M₁₆	—	5	2	no	—
106	C₂²×M₁₆	C₂×D₅	1	2	yes	—
107	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
108	C₂×#117	—	5	1	no	—
109	#430	—	2	2	no	—
110	#430	C₂×D₅	1	4	yes	—
111	C₂×#108	—	2	2	no	—
112	#424	—	5	4	no	—
113	#424	—	10	2	no	—
114	#423	—	10	2	no	—
115	#344	—	5	8	no	—
116	#341	—	5	8	no	—
117	#338	—	5	4	no	—
118	C₂³×C₅	—	8	1	no	—
119	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂³×C₅	#125	1	2	yes	—
122	C₂³×C₅	#125	1	2	yes	—
123	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅	#123	1	1	yes	characteristic
127	C₂×C₄×C₅	#123	1	1	yes	characteristic
128	C₂×C₄×C₅	#123	1	2	yes	—
129	C₂×C₄×C₅	#124	1	1	yes	characteristic
130	C₂×C₄×C₅	#124	1	2	yes	—
131	C₂×C₄×C₅	#124	1	1	yes	characteristic
132	C₂×C₄×C₅	#125	1	2	yes	—
133	C₂×C₄×C₅	#125	1	2	yes	—
134	C₂×C₄×C₅	#126	1	2	yes	—
135	C₂×C₄×C₅	#126	1	2	yes	—
136	C₂×C₄×C₅	#126	1	2	yes	—
137	C₂×C₄×C₅	#126	1	2	yes	—
138	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×#41	1	2	yes	—
140	C₂×C₄×C₅	C₂×#41	1	2	yes	—
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₅	—	2	4	no	—
143	C₂×Dic₅	—	8	2	no	—
144	C₅×C₈	—	2	4	no	—
145	C₅×C₈	—	4	4	no	—
146	#168	—	4	2	no	—
147	#168	—	4	4	no	—
148	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	5	4	no	—
157	C₂×M₁₆	—	10	2	no	—
158	C₂×M₁₆	—	10	2	no	—
159	C₂×M₁₆	—	10	2	no	—
160	C₂×M₁₆	—	10	1	no	—
161	C₂×M₁₆	—	10	1	no	—
162	C₂×M₁₆	—	10	2	no	—
163	C₂×M₁₆	—	10	2	no	—
164	C₂×M₁₆	—	10	2	no	—
165	C₂×M₁₆	—	10	2	no	—
166	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
167	C₂×M₁₆	C₂²×D₅	1	4	yes	—
168	C₂×M₁₆	C₂²×D₅	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	5	2	no	—
171	C₂²×C₈	—	10	2	no	—
172	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
173	#117	—	5	2	no	—
174	#117	—	5	2	no	—
175	#117	—	10	1	no	—
176	#117	—	10	1	no	—
177	C₂×C₄⋊C₄	—	10	1	no	—
178	C₂×C₂²⋊C₄	—	5	2	no	—
179	C₂×C₄²	—	10	1	no	—
180	#108	—	2	8	no	—
181	C₄⋊C₈	—	10	8	no	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	—	4	1	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	4	1	no	—
187	C₂²×C₅	#400	1	1	yes	characteristic
188	C₂²×C₅	#400	1	1	yes	characteristic
189	C₂²×C₅	#400	1	2	yes	—
190	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
191	C₂²×C₅	C₂×#125	1	2	yes	—
192	C₄×C₅	#400	1	2	yes	—
193	C₄×C₅	#400	1	1	yes	characteristic
194	C₄×C₅	#400	1	1	yes	characteristic
195	C₄×C₅	C₂×#123	1	1	yes	characteristic
196	C₄×C₅	C₂×#124	1	1	yes	characteristic
197	C₄×C₅	C₂×#126	1	2	yes	—
198	Dic₅	—	4	4	no	—
199	C₂⁴	Q₈×D₅	1	1	yes	characteristic
200	C₂²×C₄	—	10	2	no	—
201	C₂²×C₄	#709	1	2	yes	—
202	C₂²×C₄	#709	1	2	yes	—
203	C₂²×C₄	#711	1	2	yes	—
204	C₂²×C₄	#711	1	2	yes	—
205	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
206	C₂²×C₄	#713	1	1	yes	characteristic
207	C₂²×C₄	#713	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	10	4	no	—
218	M₁₆	—	10	4	no	—
219	M₁₆	—	20	2	no	—
220	M₁₆	—	20	2	no	—
221	M₁₆	—	20	1	no	—
222	M₁₆	—	20	1	no	—
223	M₁₆	—	20	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	10	4	no	—
230	C₂×C₈	—	10	4	no	—
231	C₂×C₈	—	10	2	no	—
232	C₂×C₈	—	20	2	no	—
233	C₂×C₈	—	20	2	no	—
234	C₂×C₈	C₂×C₂²⋊D₅	1	4	yes	—
235	C₄⋊C₄	—	10	2	no	—
236	C₄⋊C₄	—	10	2	no	—
237	C₂²⋊C₄	—	5	4	no	—
238	C₂²⋊C₄	—	10	2	no	—
239	C₄²	—	10	2	no	—
240	C₄²	—	10	2	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	C₂×#400	1	1	yes	characteristic
247	C₂³	—	8	1	no	—
248	C₂³	#4035	1	2	yes	—
249	C₂³	#4035	1	2	yes	—
250	C₂³	#4092	1	1	yes	characteristic
251	C₂³	#4092	1	1	yes	characteristic
252	C₂³	C₂×Q₈×D₅	1	1	yes	characteristic
253	C₂×C₄	—	10	4	no	—
254	C₂×C₄	—	40	2	no	—
255	C₂×C₄	#4035	1	2	yes	—
256	C₂×C₄	#4035	1	2	yes	—
257	C₂×C₄	#4044	1	2	yes	—
258	C₂×C₄	#4044	1	2	yes	—
259	C₂×C₄	#4044	1	2	yes	—
260	C₂×C₄	#4044	1	2	yes	—
261	C₂×C₄	#4075	1	1	yes	characteristic
262	C₂×C₄	#4075	1	1	yes	characteristic
263	C₂×C₄	#4075	1	2	yes	—
264	C₂×C₄	#4082	1	1	yes	characteristic
265	C₂×C₄	#4082	1	2	yes	—
266	C₂×C₄	#4082	1	1	yes	characteristic
267	C₂×C₄	#4092	1	1	yes	characteristic
268	C₂×C₄	#4092	1	1	yes	characteristic
269	C₂×C₄	C₂×#709	1	2	yes	—
270	C₂×C₄	C₂×#711	1	2	yes	—
271	C₂×C₄	C₂×#713	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	4	no	—
275	C₈	—	20	4	no	—
276	C₈	—	20	2	no	—
277	C₅	#23753	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#65271	1	1	yes	characteristic
284	C₂²	#65271	1	2	yes	—
285	C₂²	#65271	1	1	yes	characteristic
286	C₂²	C₂×#4035	1	2	yes	—
287	C₂²	C₂×#4092	1	1	yes	characteristic
288	C₄	—	20	4	no	—
289	C₄	#65271	1	1	yes	characteristic
290	C₄	#65271	1	2	yes	—
291	C₄	#65271	1	1	yes	characteristic
292	C₄	C₂×#4044	1	2	yes	—
293	C₄	C₂×#4075	1	1	yes	characteristic
294	C₄	C₂×#4082	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂×#65271	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4082
commutator, cosocle	C₂²×C₄×C₅	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#430	C₂
socle	C₂×C₅	C₂×#400
prime radical	C₂⁴×C₅	C₂×Q₈
hypercenter	C₂×#430	D₅
hypocenter	C₅	#23753
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
6. #10601519	C₂	semidirect
277. C₅	#23753	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#430	2
#10615481	2
C₂×#64780	2
#10602787	2
C₂×#64776	2
#10601519	2
#23753	5