Subgroups — #12015226

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#676	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64734	C₂	1	2	yes	maximal
4	#10600536	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598303	C₂	1	2	yes	maximal, complemented
6	C₂²×#676	C₄	1	1	yes	characteristic
7	C₂²×#676	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×#3947	C₂²	1	1	yes	characteristic
13	C₂×#3944	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	C₄×#676	—	2	2	no	—
16	C₄×#676	—	2	2	no	—
17	C₅×C₄⋊C₁₆	—	2	2	no	—
18	#64734	—	2	4	no	—
19	C₄⋊#672	—	2	2	no	—
20	#12785	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	C₈	1	2	yes	—
22	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅	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	—	2	1	no	—
33	C₂×#676	—	2	1	no	—
34	C₂×#676	—	2	1	no	—
35	C₂×#676	—	2	1	no	—
36	C₂×#676	—	2	2	no	—
37	C₂×#676	D₄	1	2	yes	—
38	C₂×#676	Q₈	1	2	yes	—
39	C₄²×D₅	—	2	2	no	—
40	C₄²×D₅	—	2	1	no	—
41	C₄²×D₅	—	2	1	no	—
42	C₅×M₃₂	—	2	2	no	—
43	C₅×M₃₂	—	2	2	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₄×C₅×C₈	D₄	1	2	yes	—
46	C₄×C₅×C₈	D₄	1	2	yes	—
47	#3947	—	2	2	no	—
48	#3947	—	2	1	no	—
49	#3947	—	2	1	no	—
50	#3944	—	2	2	no	—
51	#3944	—	2	2	no	—
52	C₂×#672	—	2	2	no	—
53	C₄×#168	—	2	2	no	—
54	C₂×C₄×M₁₆	—	5	1	no	—
55	#2150	—	5	1	no	—
56	#2150	D₅	1	1	yes	characteristic, complemented
57	C₂×#349	—	5	2	no	—
58	#1377	—	5	2	no	—
59	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₅	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₅	—	2	1	no	—
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×C₄×D₅	—	2	1	no	—
68	C₂×C₄×D₅	—	2	2	no	—
69	C₂×C₄×D₅	—	2	2	no	—
70	C₂×C₄×D₅	—	4	4	no	—
71	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₅	M₁₆	1	2	yes	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₅×C₈	#41	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	1	yes	characteristic
82	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅	M₁₆	1	1	yes	characteristic
84	C₄²×C₅	M₁₆	1	1	yes	characteristic
85	C₄×Dic₅	—	2	2	no	—
86	C₄×Dic₅	—	2	2	no	—
87	C₄×Dic₅	—	2	1	no	—
88	C₄×Dic₅	—	2	1	no	—
89	C₂×#168	—	2	1	no	—
90	C₂×#168	—	2	1	no	—
91	C₂×#168	—	2	1	no	—
92	C₂×#168	—	2	1	no	—
93	C₂×#168	C₄⋊C₄	1	2	yes	—
94	C₂×#168	M₁₆	1	2	yes	—
95	#676	—	2	8	no	—
96	#676	—	2	2	no	—
97	#676	—	2	2	no	—
98	#676	—	2	2	no	—
99	#676	—	2	2	no	—
100	C₅×C₁₆	—	4	2	no	—
101	#672	—	4	2	no	—
102	C₂²×M₁₆	—	5	1	no	—
103	C₂²×M₁₆	—	5	1	no	—
104	C₂²×C₄²	—	5	1	no	—
105	C₂×M₃₂	—	5	2	no	—
106	C₂×M₃₂	C₂×D₅	1	2	yes	—
107	C₄×M₁₆	—	10	2	no	—
108	C₄×M₁₆	—	10	2	no	—
109	C₂×#97	—	5	1	no	—
110	C₂×#97	—	5	1	no	—
111	C₂×C₄×C₈	—	5	1	no	—
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	C₄⋊C₁₆	—	2	2	no	—
114	C₄⋊C₁₆	—	10	2	no	—
115	#349	—	10	4	no	—
116	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₅	—	2	1	no	—
118	C₂²×D₅	—	2	1	no	—
119	C₂²×D₅	—	4	2	no	—
120	C₂²×D₅	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
123	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	#117	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
130	C₂×Dic₅	—	2	1	no	—
131	C₂×Dic₅	—	2	1	no	—
132	C₂×Dic₅	—	2	2	no	—
133	C₂×Dic₅	—	4	2	no	—
134	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
135	C₄×D₅	—	2	2	no	—
136	C₄×D₅	—	2	2	no	—
137	C₄×D₅	—	2	2	no	—
138	C₄×D₅	—	2	2	no	—
139	C₄×D₅	—	4	4	no	—
140	C₄×D₅	—	4	4	no	—
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	#121	1	1	yes	characteristic
143	C₅×C₈	#121	1	1	yes	characteristic
144	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
145	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
146	#168	—	2	2	no	—
147	#168	—	2	2	no	—
148	C₂³×C₄	—	5	1	no	—
149	C₂³×C₄	—	5	2	no	—
150	C₂×M₁₆	—	5	2	no	—
151	C₂×M₁₆	—	5	2	no	—
152	C₂×M₁₆	—	10	2	no	—
153	C₂×M₁₆	—	10	2	no	—
154	C₂×M₁₆	—	10	2	no	—
155	C₂×M₁₆	—	10	1	no	—
156	C₂×M₁₆	—	10	1	no	—
157	C₂×M₁₆	—	10	1	no	—
158	C₂×M₁₆	—	10	1	no	—
159	C₂²×C₈	—	5	1	no	—
160	C₂²×C₈	—	5	1	no	—
161	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄²	—	5	2	no	—
164	C₂×C₄²	—	5	1	no	—
165	C₂×C₄²	—	10	1	no	—
166	C₂×C₄²	—	10	1	no	—
167	C₂×C₄²	—	10	2	no	—
168	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	10	2	no	—
172	M₃₂	—	10	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	10	2	no	—
175	#97	—	10	2	no	—
176	#97	—	10	1	no	—
177	#97	—	10	1	no	—
178	C₄×C₈	—	10	2	no	—
179	C₄×C₈	D₂₀	1	2	yes	—
180	C₄×C₈	C₂²⋊D₅	1	2	yes	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
184	C₂²×C₅	#431	1	1	yes	characteristic
185	C₂×D₅	—	2	2	no	—
186	C₂×D₅	—	4	2	no	—
187	C₂×D₅	—	4	2	no	—
188	C₂×D₅	—	4	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	#349	1	4	yes	complemented
191	C₄×C₅	#380	1	1	yes	characteristic
192	C₄×C₅	#407	1	1	yes	characteristic
193	C₄×C₅	#423	1	1	yes	characteristic
194	C₄×C₅	#432	1	1	yes	characteristic
195	Dic₅	—	2	2	no	—
196	Dic₅	—	4	2	no	—
197	C₂⁴	—	5	1	no	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	2	no	—
200	C₂²×C₄	—	5	2	no	—
201	C₂²×C₄	—	5	2	no	—
202	C₂²×C₄	—	10	1	no	—
203	C₂²×C₄	—	10	1	no	—
204	C₂²×C₄	—	10	1	no	—
205	C₂²×C₄	—	10	1	no	—
206	C₂²×C₄	—	10	2	no	—
207	C₂²×C₄	—	10	2	no	—
208	C₂²×C₄	—	20	4	no	—
209	C₂²×C₄	C₈×D₅	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
211	M₁₆	—	10	8	no	—
212	M₁₆	—	10	2	no	—
213	M₁₆	—	10	2	no	—
214	M₁₆	—	10	2	no	—
215	M₁₆	—	10	2	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	5	2	no	—
220	C₂×C₈	—	10	1	no	—
221	C₂×C₈	—	10	1	no	—
222	C₂×C₈	—	10	1	no	—
223	C₂×C₈	—	10	1	no	—
224	C₂×C₈	#685	1	2	yes	—
225	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
226	C₂×C₈	D₄×D₅	1	1	yes	characteristic
227	C₂×C₈	#711	1	1	yes	characteristic
228	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
229	C₂×C₈	#713	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₄²	—	10	2	no	—
232	C₄²	—	10	2	no	—
233	C₄²	—	10	1	no	—
234	C₄²	—	10	1	no	—
235	C₄²	#676	1	1	yes	characteristic
236	C₄²	#676	1	1	yes	characteristic
237	C₄²	#685	1	2	yes	—
238	C₁₆	—	4	2	no	—
239	C₁₆	—	20	2	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	#1797	1	1	yes	characteristic
243	C₂×C₅	C₂×#349	1	2	yes	—
244	D₅	—	4	2	no	—
245	C₂³	—	5	2	no	—
246	C₂³	—	10	1	no	—
247	C₂³	—	10	1	no	—
248	C₂³	—	20	2	no	—
249	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	10	1	no	—
253	C₂×C₄	—	10	1	no	—
254	C₂×C₄	—	10	2	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	—	20	4	no	—
260	C₂×C₄	—	20	4	no	—
261	C₂×C₄	—	20	2	no	—
262	C₂×C₄	#3952	1	2	yes	—
263	C₂×C₄	#3952	1	2	yes	—
264	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₄	#4038	1	1	yes	characteristic
266	C₂×C₄	C₂×#676	1	1	yes	characteristic
267	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
268	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×#685	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	10	2	no	—
272	C₈	—	10	2	no	—
273	C₈	#4041	1	1	yes	characteristic
274	C₈	#4043	1	1	yes	characteristic
275	C₈	#4084	1	1	yes	characteristic
276	C₈	#4092	1	1	yes	characteristic
277	C₅	#12785	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	2	2	no	—
279	C₂²	—	10	2	no	—
280	C₂²	—	20	2	no	—
281	C₂²	—	20	2	no	—
282	C₂²	—	20	2	no	—
283	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
284	C₂²	#65122	1	1	yes	characteristic
285	C₂²	C₂×#3952	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	10	2	no	—
288	C₄	—	20	2	no	—
289	C₄	#64734	1	4	yes	complemented
290	C₄	#65124	1	1	yes	characteristic
291	C₄	#65126	1	1	yes	characteristic
292	C₄	#65276	1	1	yes	characteristic
293	C₄	#65415	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	20	2	no	—
297	C₂	#10601268	1	1	yes	characteristic
298	C₂	C₂×#64734	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂³×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2150	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2150	D₅
hypocenter	C₅	#12785
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. C₅×#2150	C₂	semidirect
4. #10600536	C₂	semidirect
5. #10598303	C₂	semidirect
56. #2150	D₅	semidirect
190. C₄×C₅	#349	semidirect
277. C₅	#12785	semidirect
289. C₄	#64734	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#676	2
C₅×#2150	2
C₂×#64734	2
#10600536	2
#10598303	2
#12785	5