Subgroups — C₂²×C₅×#359 #11956103

Order 1280.

1174 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#359	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₅×#41	—	2	1	no	—
6	C₂×C₅×#135	—	2	3	no	—
7	C₂×C₅×#135	—	2	3	no	—
8	C₂×C₅×#135	C₄	1	6	yes	—
9	C₂×C₅×#135	C₂²	1	6	yes	—
10	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
11	C₂²×C₅×SD₁₆	—	2	1	no	—
12	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
13	C₂²×C₅×M₁₆	—	2	1	no	—
14	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	—	2	1	no	—
16	C₂²×C₅×C₁₆	—	2	3	no	—
17	C₂²×C₅×C₁₆	C₂²	1	6	yes	complemented
18	C₂×C₅×#108	—	2	3	no	—
19	C₂×C₅×#108	C₂²	1	6	yes	complemented
20	C₅×#359	C₂²	1	16	yes	direct product factor
21	C₂²×#359	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	6	no	—
23	C₂×C₅×#41	—	2	6	no	—
24	C₂×C₅×#41	—	4	3	no	—
25	C₂×C₅×#41	—	4	3	no	—
26	C₂²×C₅×Q₈	—	2	1	no	—
27	C₂²×C₅×D₄	—	2	1	no	—
28	C₂²×C₅×D₄	—	4	1	no	—
29	C₂³×C₄×C₅	—	4	1	no	—
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₅×#135	—	2	12	no	—
32	C₅×#135	—	2	6	no	—
33	C₅×#135	—	2	6	no	—
34	C₅×#135	C₂×C₄	1	12	yes	—
35	C₅×#135	C₂³	1	4	yes	—
36	C₂×C₅×Q₁₆	—	2	3	no	—
37	C₂×C₅×Q₁₆	C₂×C₄	1	6	yes	—
38	C₂×C₅×SD₁₆	—	2	6	no	—
39	C₂×C₅×SD₁₆	—	2	6	no	—
40	C₂×C₅×D₈	—	2	3	no	—
41	C₂×C₅×D₈	C₂×C₄	1	6	yes	—
42	C₂×C₅×M₁₆	—	2	6	no	—
43	C₂×C₅×M₁₆	—	4	3	no	—
44	C₂²×C₅×C₈	—	4	1	no	—
45	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
47	C₂²×C₅×C₈	D₄	1	3	yes	—
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	D₄	1	3	yes	—
50	C₂²×C₅×C₈	C₂³	1	3	yes	—
51	C₂×C₅×C₁₆	—	2	6	no	—
52	C₂×C₅×C₁₆	—	2	6	no	—
53	C₂×C₅×C₁₆	C₂³	1	4	yes	complemented
54	C₅×#108	—	2	6	no	—
55	C₅×#108	C₂³	1	4	yes	complemented
56	C₂²×#135	C₂×C₅	1	1	yes	characteristic
57	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×#108	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#359	C₂×C₅	1	12	yes	direct product factor
60	C₂⁴×C₅	—	4	1	no	—
61	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
62	C₅×#41	—	2	12	no	—
63	C₅×#41	—	2	4	no	—
64	C₅×#41	—	4	12	no	—
65	C₅×#41	—	4	6	no	—
66	C₅×#41	—	4	6	no	—
67	C₂×C₅×Q₈	—	2	6	no	—
68	C₂×C₅×Q₈	—	4	3	no	—
69	C₂×C₅×D₄	—	2	6	no	—
70	C₂×C₅×D₄	—	4	6	no	—
71	C₂×C₅×D₄	—	4	6	no	—
72	C₂×C₅×D₄	—	4	3	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	—	4	6	no	—
75	C₂²×C₄×C₅	—	4	6	no	—
76	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
78	C₂²×C₄×C₅	D₈	1	3	yes	—
79	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
81	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
82	C₅×Q₁₆	—	2	6	no	—
83	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
84	C₅×SD₁₆	—	2	12	no	—
85	C₅×SD₁₆	—	2	4	no	—
86	C₅×D₈	—	2	6	no	—
87	C₅×D₈	C₂²×C₄	1	4	yes	—
88	C₅×M₁₆	—	2	4	no	—
89	C₅×M₁₆	—	4	6	no	—
90	C₂×C₅×C₈	—	4	6	no	—
91	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
93	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₅×C₁₆	—	2	4	no	—
101	C₂²×#41	—	2	1	no	—
102	C₂×#135	—	2	3	no	—
103	C₂×#135	—	2	3	no	—
104	C₂×#135	C₄×C₅	1	6	yes	—
105	C₂×#135	C₂²×C₅	1	6	yes	—
106	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic
107	C₂²×SD₁₆	—	2	1	no	—
108	C₂²×D₈	C₄×C₅	1	1	yes	characteristic
109	C₂²×M₁₆	—	2	1	no	—
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂²×C₁₆	—	2	1	no	—
112	C₂²×C₁₆	—	2	3	no	—
113	C₂²×C₁₆	C₂²×C₅	1	6	yes	complemented
114	C₂×#108	—	2	3	no	—
115	C₂×#108	C₂²×C₅	1	6	yes	complemented
116	#359	C₂²×C₅	1	16	yes	direct product factor
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	4	6	no	—
119	C₂³×C₅	—	8	4	no	—
120	C₂³×C₅	#102	1	3	yes	—
121	C₂³×C₅	#102	1	1	yes	characteristic
122	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
123	C₅×Q₈	—	2	4	no	—
124	C₅×Q₈	—	4	6	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	4	4	no	—
127	C₅×D₄	—	4	12	no	—
128	C₅×D₄	—	4	6	no	—
129	C₂×C₄×C₅	—	4	6	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	12	no	—
132	C₂×C₄×C₅	#102	1	3	yes	—
133	C₂×C₄×C₅	#102	1	3	yes	—
134	C₂×C₄×C₅	#102	1	6	yes	—
135	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
137	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
138	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
139	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
140	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
141	C₅×C₈	—	4	4	no	—
142	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
143	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
144	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
145	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₂×#41	—	2	6	no	—
147	C₂×#41	—	2	6	no	—
148	C₂×#41	—	4	3	no	—
149	C₂×#41	—	4	3	no	—
150	C₂²×Q₈	—	2	1	no	—
151	C₂²×D₄	—	2	1	no	—
152	C₂²×D₄	—	4	1	no	—
153	C₂³×C₄	—	4	1	no	—
154	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
155	#135	—	2	12	no	—
156	#135	—	2	6	no	—
157	#135	—	2	6	no	—
158	#135	C₂×C₄×C₅	1	12	yes	—
159	#135	C₂³×C₅	1	4	yes	—
160	C₂×Q₁₆	—	2	3	no	—
161	C₂×Q₁₆	C₂×C₄×C₅	1	6	yes	—
162	C₂×SD₁₆	—	2	6	no	—
163	C₂×SD₁₆	—	2	6	no	—
164	C₂×D₈	—	2	3	no	—
165	C₂×D₈	C₂×C₄×C₅	1	6	yes	—
166	C₂×M₁₆	—	2	6	no	—
167	C₂×M₁₆	—	4	3	no	—
168	C₂²×C₈	—	4	1	no	—
169	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
171	C₂²×C₈	C₅×D₄	1	3	yes	—
172	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
173	C₂²×C₈	C₅×D₄	1	3	yes	—
174	C₂²×C₈	C₂³×C₅	1	3	yes	—
175	C₂×C₁₆	—	2	6	no	—
176	C₂×C₁₆	—	2	6	no	—
177	C₂×C₁₆	C₂³×C₅	1	4	yes	complemented
178	#108	—	2	6	no	—
179	#108	C₂³×C₅	1	4	yes	complemented
180	C₂²×C₅	—	2	6	no	—
181	C₂²×C₅	—	2	6	no	—
182	C₂²×C₅	—	4	4	no	—
183	C₂²×C₅	—	8	12	no	—
184	C₂²×C₅	#359	1	4	yes	direct product factor
185	C₂²×C₅	C₂×#102	1	3	yes	—
186	C₂²×C₅	C₂×#102	1	3	yes	—
187	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
188	C₄×C₅	—	4	4	no	—
189	C₄×C₅	C₂×#102	1	3	yes	—
190	C₄×C₅	C₂×#102	1	3	yes	—
191	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
192	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
193	C₂⁴	—	4	1	no	—
194	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
195	#41	—	2	12	no	—
196	#41	—	2	4	no	—
197	#41	—	4	6	no	—
198	#41	—	4	6	no	—
199	#41	—	4	12	no	—
200	C₂×Q₈	—	2	6	no	—
201	C₂×Q₈	—	4	3	no	—
202	C₂×D₄	—	2	6	no	—
203	C₂×D₄	—	4	6	no	—
204	C₂×D₄	—	4	6	no	—
205	C₂×D₄	—	4	3	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	6	no	—
208	C₂²×C₄	—	4	6	no	—
209	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
211	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
212	C₂²×C₄	C₅×D₈	1	3	yes	—
213	C₂²×C₄	C₅×SD₁₆	1	3	yes	—
214	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
215	Q₁₆	—	2	6	no	—
216	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
217	SD₁₆	—	2	12	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	6	no	—
220	D₈	C₂²×C₄×C₅	1	4	yes	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	6	no	—
223	C₂×C₈	—	4	6	no	—
224	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
225	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
227	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
228	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
229	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
230	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
231	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
232	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
233	C₁₆	—	2	4	no	—
234	C₂×C₅	—	2	1	no	—
235	C₂×C₅	—	2	3	no	—
236	C₂×C₅	—	8	4	no	—
237	C₂×C₅	C₂×#359	1	6	yes	direct product factor
238	C₂×C₅	C₂²×#102	1	1	yes	characteristic
239	C₂³	—	2	4	no	—
240	C₂³	—	4	6	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₅×#102	1	3	yes	—
243	C₂³	C₅×#102	1	1	yes	characteristic
244	C₂³	C₂×C₅×SD₁₆	1	3	yes	—
245	Q₈	—	2	4	no	—
246	Q₈	—	4	6	no	—
247	D₄	—	2	4	no	—
248	D₄	—	4	12	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	6	no	—
251	C₂×C₄	—	4	6	no	—
252	C₂×C₄	—	4	12	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	C₅×#102	1	3	yes	—
255	C₂×C₄	C₅×#102	1	3	yes	—
256	C₂×C₄	C₅×#102	1	6	yes	—
257	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
258	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
259	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
260	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
261	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
262	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
265	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
266	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
267	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
268	C₅	C₂²×#359	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	12	no	—
273	C₂²	C₅×#359	1	4	yes	direct product factor
274	C₂²	C₂×C₅×#102	1	3	yes	—
275	C₂²	C₂×C₅×#102	1	3	yes	—
276	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
277	C₄	—	4	4	no	—
278	C₄	C₂×C₅×#102	1	3	yes	—
279	C₄	C₂×C₅×#102	1	3	yes	—
280	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
282	C₂	—	2	3	no	—
283	C₂	—	2	1	no	—
284	C₂	—	8	4	no	—
285	C₂	C₂×C₅×#359	1	6	yes	direct product factor
286	C₂	C₂²×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	#102
prime radical	C₂²×C₅×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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. C₂³×C₅×C₁₆	C₂	semidirect
3. C₂²×C₅×#108	C₂	semidirect
4. C₂×C₅×#359	C₂	direct (N × H)
17. C₂²×C₅×C₁₆	C₂²	semidirect
19. C₂×C₅×#108	C₂²	semidirect
20. C₅×#359	C₂²	direct (N × H)
21. C₂²×#359	C₅	direct (N × H)
53. C₂×C₅×C₁₆	C₂³	semidirect
55. C₅×#108	C₂³	semidirect
57. C₂³×C₁₆	C₂×C₅	semidirect
58. C₂²×#108	C₂×C₅	semidirect
59. C₂×#359	C₂×C₅	direct (N × H)
113. C₂²×C₁₆	C₂²×C₅	semidirect
115. C₂×#108	C₂²×C₅	semidirect
116. #359	C₂²×C₅	direct (N × H)
177. C₂×C₁₆	C₂³×C₅	semidirect
179. #108	C₂³×C₅	semidirect
184. C₂²×C₅	#359	direct (N × H)
237. C₂×C₅	C₂×#359	direct (N × H)
268. C₅	C₂²×#359	direct (N × H)
273. C₂²	C₅×#359	direct (N × H)
285. C₂	C₂×C₅×#359	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#135	2
C₂³×C₅×C₁₆	2
C₂²×C₅×#108	2
C₂×C₅×#359	2
C₂²×#359	5

Subgroups — C22×C5×#359 #11956103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#359 #11956103