Subgroups — C₂²×C₅×#361 #11956109

Order 1280.

1390 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#361	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×D₄	—	2	1	no	—
6	C₂²×C₅×D₈	—	2	3	no	—
7	C₂²×C₅×D₈	—	2	1	no	—
8	C₂²×C₅×D₈	—	2	3	no	—
9	C₂²×C₅×D₈	C₄	1	6	yes	—
10	C₂²×C₅×D₈	C₄	1	2	yes	—
11	C₂²×C₅×D₈	C₂²	1	6	yes	—
12	C₂²×C₅×M₁₆	—	2	1	no	—
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₃₂	—	2	3	no	—
15	C₂×C₅×M₃₂	C₂²	1	6	yes	complemented
16	C₂²×C₅×C₁₆	—	2	1	no	—
17	C₂×C₅×#108	—	2	3	no	—
18	C₂×C₅×#108	C₂²	1	6	yes	complemented
19	C₅×#361	C₂²	1	16	yes	direct product factor
20	C₂²×#361	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂⁵×C₅	—	4	1	no	—
22	C₂²×C₅×D₄	—	2	6	no	—
23	C₂²×C₅×D₄	—	2	2	no	—
24	C₂²×C₅×D₄	—	2	6	no	—
25	C₂²×C₅×D₄	—	4	3	no	—
26	C₂²×C₅×D₄	—	4	1	no	—
27	C₂²×C₅×D₄	—	4	3	no	—
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×D₈	—	2	6	no	—
30	C₂×C₅×D₈	—	2	12	no	—
31	C₂×C₅×D₈	—	2	6	no	—
32	C₂×C₅×D₈	—	2	6	no	—
33	C₂×C₅×D₈	—	2	6	no	—
34	C₂×C₅×D₈	—	2	6	no	—
35	C₂×C₅×D₈	C₂×C₄	1	12	yes	—
36	C₂×C₅×D₈	C₂×C₄	1	12	yes	—
37	C₂×C₅×D₈	C₂³	1	4	yes	—
38	C₂×C₅×M₁₆	—	2	6	no	—
39	C₂×C₅×M₁₆	—	4	3	no	—
40	C₂²×C₅×C₈	—	4	1	no	—
41	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	D₄	1	3	yes	—
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	D₄	1	3	yes	—
46	C₂²×C₅×C₈	C₂³	1	3	yes	—
47	C₅×M₃₂	—	2	6	no	—
48	C₅×M₃₂	C₂³	1	4	yes	complemented
49	C₂×C₅×C₁₆	—	2	6	no	—
50	C₅×#108	—	2	6	no	—
51	C₅×#108	C₂³	1	4	yes	complemented
52	C₂³×D₈	C₂×C₅	1	1	yes	characteristic
53	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂²×#108	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#361	C₂×C₅	1	12	yes	direct product factor
56	C₂⁴×C₅	—	4	6	no	—
57	C₂⁴×C₅	—	4	2	no	—
58	C₂⁴×C₅	—	4	6	no	—
59	C₂⁴×C₅	—	8	8	no	—
60	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
61	C₂×C₅×D₄	—	2	12	no	—
62	C₂×C₅×D₄	—	2	12	no	—
63	C₂×C₅×D₄	—	2	4	no	—
64	C₂×C₅×D₄	—	4	12	no	—
65	C₂×C₅×D₄	—	4	6	no	—
66	C₂×C₅×D₄	—	4	6	no	—
67	C₂×C₅×D₄	—	4	6	no	—
68	C₂×C₅×D₄	—	4	6	no	—
69	C₂×C₅×D₄	—	4	6	no	—
70	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
71	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	D₈	1	3	yes	—
73	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
76	C₅×D₈	—	2	12	no	—
77	C₅×D₈	—	2	12	no	—
78	C₅×D₈	—	2	4	no	—
79	C₅×D₈	C₂²×C₄	1	8	yes	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	4	6	no	—
82	C₂×C₅×C₈	—	4	6	no	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
91	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
92	C₅×C₁₆	—	2	4	no	—
93	C₂³×D₄	—	2	1	no	—
94	C₂²×D₈	—	2	3	no	—
95	C₂²×D₈	—	2	1	no	—
96	C₂²×D₈	—	2	3	no	—
97	C₂²×D₈	C₄×C₅	1	6	yes	—
98	C₂²×D₈	C₄×C₅	1	2	yes	—
99	C₂²×D₈	C₂²×C₅	1	6	yes	—
100	C₂²×M₁₆	—	2	1	no	—
101	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂×M₃₂	—	2	3	no	—
103	C₂×M₃₂	C₂²×C₅	1	6	yes	complemented
104	C₂²×C₁₆	—	2	1	no	—
105	C₂×#108	—	2	3	no	—
106	C₂×#108	C₂²×C₅	1	6	yes	complemented
107	#361	C₂²×C₅	1	16	yes	direct product factor
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	—	4	4	no	—
110	C₂³×C₅	—	4	12	no	—
111	C₂³×C₅	—	4	12	no	—
112	C₂³×C₅	—	8	24	no	—
113	C₂³×C₅	—	8	24	no	—
114	C₂³×C₅	—	8	8	no	—
115	C₂³×C₅	#102	1	3	yes	—
116	C₂³×C₅	#102	1	1	yes	characteristic
117	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
118	C₅×D₄	—	2	8	no	—
119	C₅×D₄	—	4	4	no	—
120	C₅×D₄	—	4	12	no	—
121	C₅×D₄	—	4	12	no	—
122	C₂×C₄×C₅	#102	1	6	yes	—
123	C₂×C₄×C₅	#102	1	3	yes	—
124	C₂×C₄×C₅	#102	1	3	yes	—
125	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
126	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
127	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
128	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
129	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
130	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₅×C₈	—	4	4	no	—
132	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
133	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
134	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
135	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₂⁵	—	4	1	no	—
137	C₂²×D₄	—	2	6	no	—
138	C₂²×D₄	—	2	2	no	—
139	C₂²×D₄	—	2	6	no	—
140	C₂²×D₄	—	4	1	no	—
141	C₂²×D₄	—	4	3	no	—
142	C₂²×D₄	—	4	3	no	—
143	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂×D₈	—	2	6	no	—
145	C₂×D₈	—	2	12	no	—
146	C₂×D₈	—	2	6	no	—
147	C₂×D₈	—	2	6	no	—
148	C₂×D₈	—	2	6	no	—
149	C₂×D₈	—	2	6	no	—
150	C₂×D₈	C₂×C₄×C₅	1	12	yes	—
151	C₂×D₈	C₂×C₄×C₅	1	12	yes	—
152	C₂×D₈	C₂³×C₅	1	4	yes	—
153	C₂×M₁₆	—	2	6	no	—
154	C₂×M₁₆	—	4	3	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
157	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	3	yes	—
159	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₅×D₄	1	3	yes	—
161	C₂²×C₈	C₂³×C₅	1	3	yes	—
162	M₃₂	—	2	6	no	—
163	M₃₂	C₂³×C₅	1	4	yes	complemented
164	C₂×C₁₆	—	2	6	no	—
165	#108	—	2	6	no	—
166	#108	C₂³×C₅	1	4	yes	complemented
167	C₂²×C₅	—	2	6	no	—
168	C₂²×C₅	—	2	6	no	—
169	C₂²×C₅	—	4	8	no	—
170	C₂²×C₅	—	8	8	no	—
171	C₂²×C₅	—	8	24	no	—
172	C₂²×C₅	—	8	24	no	—
173	C₂²×C₅	#361	1	4	yes	direct product factor
174	C₂²×C₅	C₂×#102	1	3	yes	—
175	C₂²×C₅	C₂×#102	1	3	yes	—
176	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
177	C₄×C₅	C₂×#102	1	3	yes	—
178	C₄×C₅	C₂×#102	1	3	yes	—
179	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
180	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
181	C₂⁴	—	4	6	no	—
182	C₂⁴	—	4	6	no	—
183	C₂⁴	—	4	2	no	—
184	C₂⁴	—	8	8	no	—
185	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
186	C₂×D₄	—	2	12	no	—
187	C₂×D₄	—	2	12	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	4	6	no	—
190	C₂×D₄	—	4	6	no	—
191	C₂×D₄	—	4	6	no	—
192	C₂×D₄	—	4	12	no	—
193	C₂×D₄	—	4	6	no	—
194	C₂×D₄	—	4	6	no	—
195	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
196	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₅×D₈	1	3	yes	—
199	C₂²×C₄	C₅×SD₁₆	1	3	yes	—
200	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
201	D₈	—	2	12	no	—
202	D₈	—	2	12	no	—
203	D₈	—	2	4	no	—
204	D₈	C₂²×C₄×C₅	1	8	yes	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	6	no	—
207	C₂×C₈	—	4	6	no	—
208	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
209	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
210	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
211	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
212	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
213	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
214	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
215	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
216	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
217	C₁₆	—	2	4	no	—
218	C₂×C₅	—	2	1	no	—
219	C₂×C₅	—	2	3	no	—
220	C₂×C₅	—	8	8	no	—
221	C₂×C₅	C₂×#361	1	6	yes	direct product factor
222	C₂×C₅	C₂²×#102	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	—	4	12	no	—
225	C₂³	—	4	12	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	8	24	no	—
228	C₂³	—	8	24	no	—
229	C₂³	—	8	8	no	—
230	C₂³	C₅×#102	1	3	yes	—
231	C₂³	C₅×#102	1	1	yes	characteristic
232	C₂³	C₂×C₅×SD₁₆	1	3	yes	—
233	D₄	—	2	8	no	—
234	D₄	—	4	12	no	—
235	D₄	—	4	12	no	—
236	D₄	—	4	4	no	—
237	C₂×C₄	C₅×#102	1	6	yes	—
238	C₂×C₄	C₅×#102	1	3	yes	—
239	C₂×C₄	C₅×#102	1	3	yes	—
240	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
241	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
242	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
243	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
244	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
245	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₈	—	4	4	no	—
247	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
248	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
249	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
250	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
251	C₅	C₂²×#361	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	6	no	—
253	C₂²	—	2	6	no	—
254	C₂²	—	4	8	no	—
255	C₂²	—	8	24	no	—
256	C₂²	—	8	8	no	—
257	C₂²	—	8	24	no	—
258	C₂²	C₅×#361	1	4	yes	direct product factor
259	C₂²	C₂×C₅×#102	1	3	yes	—
260	C₂²	C₂×C₅×#102	1	3	yes	—
261	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
262	C₄	C₂×C₅×#102	1	3	yes	—
263	C₄	C₂×C₅×#102	1	3	yes	—
264	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
266	C₂	—	2	3	no	—
267	C₂	—	2	1	no	—
268	C₂	—	8	8	no	—
269	C₂	C₂×C₅×#361	1	6	yes	direct product factor
270	C₂	C₂²×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#102
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂³×C₅×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×M₃₂	C₂	semidirect
3. C₂²×C₅×#108	C₂	semidirect
4. C₂×C₅×#361	C₂	direct (N × H)
15. C₂×C₅×M₃₂	C₂²	semidirect
18. C₂×C₅×#108	C₂²	semidirect
19. C₅×#361	C₂²	direct (N × H)
20. C₂²×#361	C₅	direct (N × H)
48. C₅×M₃₂	C₂³	semidirect
51. C₅×#108	C₂³	semidirect
53. C₂²×M₃₂	C₂×C₅	semidirect
54. C₂²×#108	C₂×C₅	semidirect
55. C₂×#361	C₂×C₅	direct (N × H)
103. C₂×M₃₂	C₂²×C₅	semidirect
106. C₂×#108	C₂²×C₅	semidirect
107. #361	C₂²×C₅	direct (N × H)
163. M₃₂	C₂³×C₅	semidirect
166. #108	C₂³×C₅	semidirect
173. C₂²×C₅	#361	direct (N × H)
221. C₂×C₅	C₂×#361	direct (N × H)
251. C₅	C₂²×#361	direct (N × H)
258. C₂²	C₅×#361	direct (N × H)
269. C₂	C₂×C₅×#361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×D₈	2
C₂²×C₅×M₃₂	2
C₂²×C₅×#108	2
C₂×C₅×#361	2
C₂²×#361	5

Subgroups — C22×C5×#361 #11956109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#361 #11956109