Subgroups — C₅×#33608 #11956328

Order 1280.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₅×#3059	C₂	1	2	yes	maximal
4	C₅×#2219	C₂	1	4	yes	maximal, complemented
5	C₅×#2216	C₂	1	4	yes	maximal, complemented
6	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2141	C₂	1	2	yes	maximal, complemented
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₅×#536	—	2	2	no	—
11	C₂×C₅×Q₃₂	—	2	4	no	—
12	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
14	C₅×#504	—	2	4	no	—
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×M₃₂	C₂²	1	2	yes	—
17	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₅×#470	—	2	4	no	—
20	C₅×#467	C₂²	1	8	yes	—
21	C₅×#427	C₂²	1	2	yes	—
22	C₅×#419	—	2	2	no	—
23	C₅×#367	C₂²	1	2	yes	—
24	C₅×#366	C₂²	1	2	yes	—
25	C₅×#358	C₂²	1	8	yes	—
26	#33608	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	C₂×C₅×Q₁₆	—	2	4	no	—
30	C₂×C₅×Q₁₆	—	2	2	no	—
31	C₂×C₅×Q₁₆	—	2	4	no	—
32	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
33	C₅×#131	—	2	2	no	—
34	C₅×#131	—	2	2	no	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₅×M₁₆	D₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊Q₈	—	4	2	no	—
40	C₅×#124	—	4	2	no	—
41	C₅×#123	—	4	4	no	—
42	C₅×C₂²⋊Q₈	—	2	8	no	—
43	C₅×#117	—	2	2	no	—
44	C₅×Q₃₂	—	2	4	no	—
45	C₅×Q₃₂	—	2	4	no	—
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	—	2	4	no	—
48	C₂×C₅×C₁₆	—	2	4	no	—
49	C₂×C₅×C₁₆	D₄	1	4	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
53	C₅×#107	C₂³	1	4	yes	—
54	C₅×#106	—	2	2	no	—
55	C₅×#103	—	2	8	no	—
56	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#504	C₂×C₅	1	1	yes	characteristic
58	#3059	C₂×C₅	1	2	yes	—
59	#2219	C₂×C₅	1	4	yes	complemented
60	#2216	C₂×C₅	1	4	yes	complemented
61	#2158	C₂×C₅	1	1	yes	characteristic, complemented
62	#2141	C₂×C₅	1	2	yes	complemented
63	C₅×#41	—	2	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₂×C₅×Q₈	—	2	4	no	—
66	C₂×C₅×Q₈	—	4	2	no	—
67	C₂×C₅×Q₈	—	4	4	no	—
68	C₂×C₅×Q₈	—	4	4	no	—
69	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
70	C₂×C₅×D₄	D₈	1	2	yes	—
71	C₂×C₅×D₄	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₅×Q₁₆	—	2	4	no	—
76	C₅×Q₁₆	—	2	4	no	—
77	C₅×Q₁₆	—	4	4	no	—
78	C₅×M₁₆	—	2	2	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	4	8	no	—
89	C₅×C₂²⋊C₄	—	4	2	no	—
90	C₅×C₂²⋊C₄	—	4	8	no	—
91	C₄²×C₅	—	4	2	no	—
92	C₅×C₁₆	—	2	2	no	—
93	C₅×C₁₆	C₂×D₄	1	2	yes	—
94	C₅×C₁₆	C₂×D₄	1	2	yes	—
95	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
96	C₂×#131	C₂²×C₅	1	1	yes	characteristic
97	#536	—	2	2	no	—
98	C₂×Q₃₂	—	2	4	no	—
99	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
100	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
101	#504	—	2	4	no	—
102	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
103	C₂×M₃₂	C₂²×C₅	1	2	yes	—
104	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
105	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
106	#470	—	2	4	no	—
107	#467	C₂²×C₅	1	8	yes	—
108	#427	C₂²×C₅	1	2	yes	—
109	#419	—	2	2	no	—
110	#367	C₂²×C₅	1	2	yes	—
111	#366	C₂²×C₅	1	2	yes	—
112	#358	C₂²×C₅	1	8	yes	—
113	C₂³×C₅	C₂×D₈	1	2	yes	—
114	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
115	C₅×Q₈	—	2	2	no	—
116	C₅×Q₈	—	4	4	no	—
117	C₅×Q₈	—	4	4	no	—
118	C₅×Q₈	—	8	4	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	8	4	no	—
126	C₂×C₄×C₅	#121	1	2	yes	—
127	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅	#135	1	2	yes	—
130	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	#121	1	2	yes	—
133	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
134	C₅×C₈	C₂×#41	1	1	yes	characteristic
135	C₂×#41	C₅×D₄	1	1	yes	characteristic
136	C₂²×Q₈	—	2	2	no	—
137	C₂×Q₁₆	—	2	4	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
141	#131	—	2	2	no	—
142	#131	—	2	2	no	—
143	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
144	C₂×M₁₆	C₅×D₄	1	2	yes	—
145	C₂²×C₈	C₂³×C₅	1	2	yes	—
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₄⋊Q₈	—	4	2	no	—
148	#124	—	4	2	no	—
149	#123	—	4	4	no	—
150	C₂²⋊Q₈	—	2	8	no	—
151	#117	—	2	2	no	—
152	Q₃₂	—	2	4	no	—
153	Q₃₂	—	2	4	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	4	no	—
156	C₂×C₁₆	—	2	4	no	—
157	C₂×C₁₆	C₅×D₄	1	4	yes	—
158	C₂×C₁₆	C₂³×C₅	1	2	yes	—
159	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
160	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
161	#107	C₂³×C₅	1	4	yes	—
162	#106	—	2	2	no	—
163	#103	—	2	8	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	#466	1	2	yes	—
168	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	8	4	no	—
171	C₄×C₅	—	8	4	no	—
172	C₄×C₅	#466	1	2	yes	—
173	C₄×C₅	C₂×#121	1	1	yes	characteristic
174	C₄×C₅	C₂×#135	1	1	yes	characteristic
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	4	4	no	—
181	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
182	C₂×D₄	C₅×D₈	1	2	yes	—
183	C₂×D₄	C₅×D₈	1	1	yes	characteristic
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
186	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	Q₁₆	—	2	4	no	—
188	Q₁₆	—	2	4	no	—
189	Q₁₆	—	4	4	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
195	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
197	C₂×C₈	C₅×#41	1	2	yes	—
198	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	8	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	8	no	—
203	C₄²	—	4	2	no	—
204	C₁₆	—	2	2	no	—
205	C₁₆	C₂×C₅×D₄	1	2	yes	—
206	C₁₆	C₂×C₅×D₄	1	2	yes	—
207	C₂×C₅	—	2	2	no	—
208	C₂×C₅	—	4	2	no	—
209	C₂×C₅	C₂×#466	1	1	yes	characteristic
210	C₂×C₅	#3417	1	2	yes	—
211	C₂³	C₂×C₅×D₈	1	2	yes	—
212	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	8	4	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	C₅×#121	1	2	yes	—
225	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
226	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
227	C₂×C₄	C₅×#135	1	2	yes	—
228	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	C₅×#121	1	2	yes	—
231	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
232	C₈	C₂×C₅×#41	1	1	yes	characteristic
233	C₅	#33608	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	C₅×#466	1	2	yes	—
238	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	8	4	no	—
241	C₄	—	8	4	no	—
242	C₄	C₅×#466	1	2	yes	—
243	C₄	C₂×C₅×#121	1	1	yes	characteristic
244	C₄	C₂×C₅×#135	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	C₂×C₅×#466	1	1	yes	characteristic
248	C₂	C₅×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ 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×#2219	C₂	semidirect
5. C₅×#2216	C₂	semidirect
6. C₅×#2158	C₂	semidirect
7. C₅×#2141	C₂	semidirect
26. #33608	C₅	direct (N × H)
56. C₂²×Q₃₂	C₂×C₅	semidirect
59. #2219	C₂×C₅	semidirect
60. #2216	C₂×C₅	semidirect
61. #2158	C₂×C₅	semidirect
62. #2141	C₂×C₅	semidirect
233. C₅	#33608	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
C₅×#3059	2
C₅×#2219	2
C₅×#2216	2
C₅×#2158	2
C₅×#2141	2
#33608	5

Subgroups — C5×#33608 #11956328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33608 #11956328