Subgroups — C₂×C₅×#2222 #11956335

Order 1280.

638 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	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#473	C₂	1	2	yes	maximal
4	C₅×#2222	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#362	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#137	—	2	2	no	—
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₂×C₅×Q₃₂	—	2	1	no	—
11	C₂×C₅×Q₃₂	—	2	1	no	—
12	C₂×C₅×Q₃₂	—	2	2	no	—
13	C₂×C₅×Q₃₂	C₂²	1	4	yes	complemented
14	C₅×#504	—	2	2	no	—
15	C₅×#504	C₂²	1	4	yes	—
16	C₂×C₅×M₃₂	—	2	1	no	—
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	—	2	1	no	—
19	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	C₅×#473	—	2	4	no	—
21	C₅×#473	C₂²	1	8	yes	—
22	C₂×C₅×#108	C₂²	1	2	yes	—
23	C₂×C₅×#101	—	2	2	no	—
24	C₅×#368	C₂²	1	4	yes	complemented
25	C₅×#362	C₂²	1	8	yes	complemented
26	C₂×#2222	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	D₄	1	1	yes	characteristic
28	C₂²×C₅×Q₈	—	2	2	no	—
29	C₅×#137	—	2	8	no	—
30	C₅×#137	—	4	4	no	—
31	C₂×C₅×Q₁₆	—	2	2	no	—
32	C₂×C₅×Q₁₆	—	2	2	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	2	2	no	—
36	C₂×C₅×Q₁₆	—	4	2	no	—
37	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
38	C₂×C₅×SD₁₆	—	4	2	no	—
39	C₅×#131	—	2	1	no	—
40	C₅×#131	—	2	1	no	—
41	C₅×#131	D₄	1	2	yes	—
42	C₅×#131	C₂³	1	2	yes	—
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	—	2	1	no	—
47	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×Q₃₂	—	2	4	no	—
49	C₅×Q₃₂	—	2	4	no	—
50	C₅×M₃₂	—	2	1	no	—
51	C₅×M₃₂	—	2	4	no	—
52	C₅×M₃₂	C₂³	1	2	yes	—
53	C₂×C₅×C₁₆	—	2	1	no	—
54	C₂×C₅×C₁₆	—	2	4	no	—
55	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
56	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
57	C₂×C₅×C₁₆	D₄	1	2	yes	—
58	C₂×C₅×C₁₆	C₂³	1	2	yes	—
59	C₅×#108	—	2	2	no	—
60	C₅×#108	C₂³	1	4	yes	—
61	C₅×#101	—	2	8	no	—
62	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×#504	C₂×C₅	1	1	yes	characteristic
64	C₂×#473	C₂×C₅	1	2	yes	—
65	#2222	C₂×C₅	1	8	yes	direct product factor
66	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#362	C₂×C₅	1	2	yes	complemented
68	C₅×#41	—	2	1	no	—
69	C₅×#41	—	2	1	no	—
70	C₅×#41	D₈	1	2	yes	—
71	C₅×#41	C₂×D₄	1	2	yes	—
72	C₂×C₅×Q₈	—	2	4	no	—
73	C₂×C₅×Q₈	—	4	2	no	—
74	C₂×C₅×Q₈	—	4	2	no	—
75	C₂×C₅×Q₈	—	4	2	no	—
76	C₂×C₅×Q₈	—	4	2	no	—
77	C₂×C₅×Q₈	—	4	2	no	—
78	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
79	C₂×C₅×D₄	—	2	1	no	—
80	C₂×C₅×D₄	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	—	2	1	no	—
82	C₂²×C₄×C₅	—	4	2	no	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×Q₁₆	—	2	4	no	—
85	C₅×Q₁₆	—	2	4	no	—
86	C₅×Q₁₆	—	4	8	no	—
87	C₅×Q₁₆	—	4	2	no	—
88	C₅×Q₁₆	—	4	2	no	—
89	C₅×SD₁₆	—	4	8	no	—
90	C₅×M₁₆	—	2	1	no	—
91	C₅×M₁₆	—	2	2	no	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₅×M₁₆	—	4	2	no	—
95	C₅×M₁₆	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	#41	1	1	yes	characteristic
104	C₂×C₅×C₈	#41	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₁₆	—	2	2	no	—
107	C₅×C₁₆	C₂×D₄	1	2	yes	—
108	C₅×C₁₆	C₂×D₄	1	2	yes	—
109	C₂×#137	—	2	2	no	—
110	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
111	C₂×#131	C₂²×C₅	1	1	yes	characteristic
112	C₂×Q₃₂	—	2	1	no	—
113	C₂×Q₃₂	—	2	1	no	—
114	C₂×Q₃₂	—	2	2	no	—
115	C₂×Q₃₂	C₂²×C₅	1	4	yes	complemented
116	#504	—	2	2	no	—
117	#504	C₂²×C₅	1	4	yes	—
118	C₂×M₃₂	—	2	1	no	—
119	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
120	C₂²×C₁₆	—	2	1	no	—
121	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
122	#473	—	2	4	no	—
123	#473	C₂²×C₅	1	8	yes	—
124	C₂×#108	C₂²×C₅	1	2	yes	—
125	C₂×#101	—	2	2	no	—
126	#368	C₂²×C₅	1	4	yes	complemented
127	#362	C₂²×C₅	1	8	yes	complemented
128	C₂³×C₅	—	2	1	no	—
129	C₂³×C₅	#135	1	1	yes	characteristic
130	C₅×Q₈	—	2	1	no	—
131	C₅×Q₈	—	4	4	no	—
132	C₅×Q₈	—	4	4	no	—
133	C₅×Q₈	—	8	2	no	—
134	C₅×Q₈	—	8	2	no	—
135	C₅×Q₈	C₂×D₈	1	2	yes	—
136	C₅×D₄	—	2	2	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	2	1	no	—
139	C₅×D₄	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	8	2	no	—
145	C₂×C₄×C₅	—	8	2	no	—
146	C₂×C₄×C₅	#121	1	1	yes	characteristic
147	C₂×C₄×C₅	#121	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	#135	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	4	4	no	—
154	C₅×C₈	#121	1	1	yes	characteristic
155	C₅×C₈	#121	1	1	yes	characteristic
156	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
157	C₅×C₈	C₂×#41	1	1	yes	characteristic
158	C₂×#41	C₅×D₄	1	1	yes	characteristic
159	C₂²×Q₈	—	2	2	no	—
160	#137	—	2	8	no	—
161	#137	—	4	4	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	4	2	no	—
168	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
169	C₂×SD₁₆	—	4	2	no	—
170	#131	—	2	1	no	—
171	#131	—	2	1	no	—
172	#131	C₅×D₄	1	2	yes	—
173	#131	C₂³×C₅	1	2	yes	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
179	Q₃₂	—	2	4	no	—
180	Q₃₂	—	2	4	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	4	no	—
183	M₃₂	C₂³×C₅	1	2	yes	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
187	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
188	C₂×C₁₆	C₅×D₄	1	2	yes	—
189	C₂×C₁₆	C₂³×C₅	1	2	yes	—
190	#108	—	2	2	no	—
191	#108	C₂³×C₅	1	4	yes	—
192	#101	—	2	8	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	#466	1	1	yes	characteristic
197	C₂²×C₅	#466	1	1	yes	characteristic
198	C₂²×C₅	C₂×#135	1	1	yes	characteristic
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	8	4	no	—
201	C₄×C₅	#466	1	1	yes	characteristic
202	C₄×C₅	#466	1	1	yes	characteristic
203	C₄×C₅	C₂×#121	1	1	yes	characteristic
204	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
205	#41	—	2	1	no	—
206	#41	—	2	1	no	—
207	#41	C₅×D₈	1	2	yes	—
208	#41	C₂×C₅×D₄	1	2	yes	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₅×D₈	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	2	4	no	—
223	Q₁₆	—	4	8	no	—
224	Q₁₆	—	4	2	no	—
225	Q₁₆	—	4	2	no	—
226	SD₁₆	—	4	8	no	—
227	M₁₆	—	2	1	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	C₂×C₅×D₄	1	2	yes	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂×C₈	C₅×#41	1	1	yes	characteristic
241	C₂×C₈	C₅×#41	1	1	yes	characteristic
242	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
243	C₁₆	—	2	2	no	—
244	C₁₆	C₂×C₅×D₄	1	2	yes	—
245	C₁₆	C₂×C₅×D₄	1	2	yes	—
246	C₂×C₅	—	2	1	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	—	4	2	no	—
249	C₂×C₅	#2222	1	2	yes	direct product factor
250	C₂×C₅	C₂×#466	1	1	yes	characteristic
251	C₂³	—	2	1	no	—
252	C₂³	C₅×#135	1	1	yes	characteristic
253	Q₈	—	2	1	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	8	2	no	—
257	Q₈	—	8	2	no	—
258	Q₈	C₂×C₅×D₈	1	2	yes	—
259	D₄	—	2	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	2	1	no	—
262	D₄	C₂×C₅×D₈	1	2	yes	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	C₅×#121	1	1	yes	characteristic
270	C₂×C₄	C₅×#121	1	1	yes	characteristic
271	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
272	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
273	C₂×C₄	C₅×#135	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	4	no	—
277	C₈	C₅×#121	1	1	yes	characteristic
278	C₈	C₅×#121	1	1	yes	characteristic
279	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
280	C₈	C₂×C₅×#41	1	1	yes	characteristic
281	C₅	C₂×#2222	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₅×#466	1	1	yes	characteristic
286	C₂²	C₅×#466	1	1	yes	characteristic
287	C₂²	C₂×C₅×#135	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	8	4	no	—
290	C₄	C₅×#466	1	1	yes	characteristic
291	C₄	C₅×#466	1	1	yes	characteristic
292	C₄	C₂×C₅×#121	1	1	yes	characteristic
293	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	C₅×#2222	1	2	yes	direct product factor
298	C₂	C₂×C₅×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Q₃₂	C₂	semidirect
4. C₅×#2222	C₂	direct (N × H)
5. C₂×C₅×#368	C₂	semidirect
6. C₂×C₅×#362	C₂	semidirect
13. C₂×C₅×Q₃₂	C₂²	semidirect
24. C₅×#368	C₂²	semidirect
25. C₅×#362	C₂²	semidirect
26. C₂×#2222	C₅	direct (N × H)
62. C₂²×Q₃₂	C₂×C₅	semidirect
65. #2222	C₂×C₅	direct (N × H)
66. C₂×#368	C₂×C₅	semidirect
67. C₂×#362	C₂×C₅	semidirect
115. C₂×Q₃₂	C₂²×C₅	semidirect
126. #368	C₂²×C₅	semidirect
127. #362	C₂²×C₅	semidirect
249. C₂×C₅	#2222	direct (N × H)
281. C₅	C₂×#2222	direct (N × H)
297. C₂	C₅×#2222	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×Q₃₂	2
C₂×C₅×#504	2
C₂×C₅×#473	2
C₅×#2222	2
C₂×C₅×#368	2
C₂×C₅×#362	2
C₂×#2222	5

Subgroups — C2×C5×#2222 #11956335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2222 #11956335