Subgroups — C₅×D₄×Q₃₂ #11956572

Order 1280.

664 conjugacy classes of subgroups.

The subgroup lattice has 256 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	2	yes	maximal, complemented
2	C₅×D₄×Q₁₆	C₂	1	2	yes	maximal
3	C₅×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2216	C₂	1	2	yes	maximal, complemented
5	C₅×#2209	C₂	1	2	yes	maximal, complemented
6	C₅×#2188	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
10	C₅×D₄×Q₈	—	2	2	no	—
11	C₂×C₅×Q₃₂	—	2	8	no	—
12	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
13	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
17	C₅×#494	C₂²	1	2	yes	—
18	C₅×#467	C₂²	1	4	yes	—
19	C₅×#462	—	2	2	no	—
20	C₅×#451	—	2	4	no	—
21	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
22	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
23	C₅×#366	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₅×#358	C₂²	1	2	yes	—
26	C₅×#358	C₂²	1	4	yes	—
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
29	D₄×Q₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×Q₈	—	2	4	no	—
31	C₂×C₅×Q₁₆	—	2	8	no	—
32	C₂×C₅×Q₁₆	—	2	4	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	4	4	no	—
35	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
36	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×C₄⋊Q₈	—	4	2	no	—
40	C₅×C₂²⋊Q₈	—	2	4	no	—
41	C₅×C₂²⋊Q₈	—	4	4	no	—
42	C₄×C₅×Q₈	—	2	2	no	—
43	C₄×C₅×D₄	—	4	2	no	—
44	C₄×C₅×D₄	D₄	1	1	yes	characteristic
45	C₅×Q₃₂	—	2	8	no	—
46	C₅×Q₃₂	—	2	4	no	—
47	C₅×Q₃₂	—	4	2	no	—
48	C₅×Q₃₂	D₄	1	4	yes	direct product factor
49	C₂×C₅×C₁₆	—	2	4	no	—
50	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
53	C₅×#107	C₂³	1	2	yes	—
54	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
55	C₅×#103	—	2	2	no	—
56	C₅×#103	—	2	4	no	—
57	C₅×C₂²⋊C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂²×Q₃₂	C₂×C₅	1	2	yes	complemented
60	D₄×Q₁₆	C₂×C₅	1	2	yes	—
61	#2247	C₂×C₅	1	1	yes	characteristic, complemented
62	#2216	C₂×C₅	1	2	yes	complemented
63	#2209	C₂×C₅	1	2	yes	complemented
64	#2188	C₂×C₅	1	4	yes	complemented
65	C₄×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
66	D₄×C₁₆	C₂×C₅	1	1	yes	characteristic
67	C₂×C₅×Q₈	—	2	2	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	—	4	8	no	—
70	C₂×C₅×Q₈	—	4	4	no	—
71	C₂×C₅×Q₈	—	8	4	no	—
72	C₂×C₅×D₄	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	4	4	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₅×Q₁₆	—	2	4	no	—
76	C₅×Q₁₆	—	2	2	no	—
77	C₅×Q₁₆	—	2	4	no	—
78	C₅×Q₁₆	—	4	8	no	—
79	C₅×Q₁₆	—	8	2	no	—
80	C₅×Q₁₆	C₂×D₄	1	4	yes	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₅×C₂²⋊C₄	D₈	1	2	yes	—
94	C₄²×C₅	—	4	2	no	—
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₅×C₁₆	—	2	2	no	—
97	C₅×C₁₆	—	2	1	no	—
98	C₅×C₁₆	C₂×D₄	1	2	yes	—
99	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
100	D₄×Q₈	—	2	2	no	—
101	C₂×Q₃₂	—	2	8	no	—
102	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
103	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
104	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
105	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
107	#494	C₂²×C₅	1	2	yes	—
108	#467	C₂²×C₅	1	4	yes	—
109	#462	—	2	2	no	—
110	#451	—	2	4	no	—
111	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
112	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
113	#366	C₂²×C₅	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
115	#358	C₂²×C₅	1	2	yes	—
116	#358	C₂²×C₅	1	4	yes	—
117	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
118	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
119	C₂³×C₅	C₂×D₈	1	2	yes	—
120	C₅×Q₈	—	2	4	no	—
121	C₅×Q₈	—	4	2	no	—
122	C₅×Q₈	—	4	4	no	—
123	C₅×Q₈	—	4	4	no	—
124	C₅×Q₈	—	8	8	no	—
125	C₅×D₄	Q₃₂	1	4	yes	direct product factor
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	8	8	no	—
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
139	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂²×Q₈	—	2	4	no	—
141	C₂×Q₁₆	—	2	8	no	—
142	C₂×Q₁₆	—	2	4	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	4	4	no	—
145	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
146	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
147	C₂²×C₈	C₂³×C₅	1	2	yes	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₂²⋊Q₈	—	2	4	no	—
151	C₂²⋊Q₈	—	4	4	no	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×D₄	—	4	2	no	—
154	C₄×D₄	C₅×D₄	1	1	yes	characteristic
155	Q₃₂	—	2	8	no	—
156	Q₃₂	—	2	4	no	—
157	Q₃₂	—	4	2	no	—
158	Q₃₂	C₅×D₄	1	4	yes	direct product factor
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₅	1	2	yes	—
162	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
163	#107	C₂³×C₅	1	2	yes	—
164	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
165	#103	—	2	2	no	—
166	#103	—	2	4	no	—
167	C₂²⋊C₈	C₅×D₄	1	2	yes	—
168	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	C₂×Q₃₂	1	4	yes	complemented
172	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₅	—	2	1	no	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	4	4	no	—
176	C₄×C₅	—	8	2	no	—
177	C₄×C₅	—	8	4	no	—
178	C₄×C₅	C₂×Q₃₂	1	2	yes	complemented
179	C₄×C₅	D₄²	1	1	yes	characteristic
180	C₄×C₅	#576	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	8	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	8	4	no	—
186	C₂×D₄	C₅×D₈	1	1	yes	characteristic
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
189	Q₁₆	—	2	4	no	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	4	no	—
192	Q₁₆	—	4	8	no	—
193	Q₁₆	—	8	2	no	—
194	Q₁₆	C₂×C₅×D₄	1	4	yes	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
199	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
206	C₂²⋊C₄	—	4	4	no	—
207	C₂²⋊C₄	C₅×D₈	1	2	yes	—
208	C₄²	—	4	2	no	—
209	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
210	C₁₆	—	2	2	no	—
211	C₁₆	—	2	1	no	—
212	C₁₆	C₂×C₅×D₄	1	2	yes	—
213	C₂×C₅	—	2	4	no	—
214	C₂×C₅	D₄×D₈	1	1	yes	characteristic
215	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
216	C₂×C₅	#3417	1	1	yes	characteristic
217	C₂³	C₂×C₅×D₈	1	2	yes	—
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	8	8	no	—
223	D₄	C₅×Q₃₂	1	4	yes	direct product factor
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	8	no	—
230	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
232	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	1	no	—
236	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
237	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
238	C₅	D₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×C₅×Q₃₂	1	4	yes	complemented
242	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	4	no	—
246	C₄	—	8	4	no	—
247	C₄	—	8	2	no	—
248	C₄	C₂×C₅×Q₃₂	1	2	yes	complemented
249	C₄	C₅×D₄²	1	1	yes	characteristic
250	C₄	C₅×#576	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	C₅×D₄×D₈	1	1	yes	characteristic
253	C₂	C₂²×C₅×Q₃₂	1	1	yes	characteristic
254	C₂	C₅×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Q₃₂	C₂	semidirect
3. C₅×#2247	C₂	semidirect
4. C₅×#2216	C₂	semidirect
5. C₅×#2209	C₂	semidirect
6. C₅×#2188	C₂	semidirect
7. C₄×C₅×Q₃₂	C₂	semidirect
29. D₄×Q₃₂	C₅	direct (N × H)
48. C₅×Q₃₂	D₄	direct (N × H)
59. C₂²×Q₃₂	C₂×C₅	semidirect
61. #2247	C₂×C₅	semidirect
62. #2216	C₂×C₅	semidirect
63. #2209	C₂×C₅	semidirect
64. #2188	C₂×C₅	semidirect
65. C₄×Q₃₂	C₂×C₅	semidirect
125. C₅×D₄	Q₃₂	direct (N × H)
158. Q₃₂	C₅×D₄	direct (N × H)
171. C₂²×C₅	C₂×Q₃₂	semidirect
178. C₄×C₅	C₂×Q₃₂	semidirect
223. D₄	C₅×Q₃₂	direct (N × H)
238. C₅	D₄×Q₃₂	direct (N × H)
241. C₂²	C₂×C₅×Q₃₂	semidirect
248. C₄	C₂×C₅×Q₃₂	semidirect

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
C₅×D₄×Q₁₆	2
C₅×#2247	2
C₅×#2216	2
C₅×#2209	2
C₅×#2188	2
C₄×C₅×Q₃₂	2
C₅×D₄×C₁₆	2
D₄×Q₃₂	5

Subgroups — C5×D4×Q32 #11956572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₄×Q₃₂ #11956572