Subgroups — C₅×#33607 #11956327

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3058	C₂	1	2	yes	maximal, complemented
4	C₅×#2218	C₂	1	4	yes	maximal, complemented
5	C₅×#2215	C₂	1	4	yes	maximal, complemented
6	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2140	C₂	1	2	yes	maximal, complemented
8	C₂²×C₅×D₈	C₂²	1	2	yes	—
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₅×#535	—	2	2	no	—
11	C₂×C₅×D₁₆	—	2	4	no	—
12	C₂×C₅×D₁₆	C₂²	1	2	yes	—
13	C₂×C₅×D₁₆	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	complemented
17	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
19	C₅×#469	—	2	4	no	—
20	C₅×#466	C₂²	1	8	yes	complemented
21	C₅×#427	C₂²	1	2	yes	—
22	C₅×#418	—	2	2	no	—
23	C₅×#367	C₂²	1	2	yes	complemented
24	C₅×#366	C₂²	1	2	yes	complemented
25	C₅×#357	C₂²	1	8	yes	complemented
26	#33607	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	D₄	1	1	yes	characteristic
28	C₂²×C₅×D₄	—	2	2	no	—
29	C₂×C₅×D₈	—	2	4	no	—
30	C₂×C₅×D₈	—	2	2	no	—
31	C₂×C₅×D₈	—	2	4	no	—
32	C₂×C₅×D₈	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×#127	—	4	2	no	—
40	C₅×#124	—	4	2	no	—
41	C₅×#121	—	2	8	no	—
42	C₅×#120	—	4	4	no	—
43	C₅×#117	—	2	2	no	—
44	C₅×D₁₆	—	2	4	no	—
45	C₅×D₁₆	—	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	complemented
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×#102	—	2	8	no	—
56	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
58	#3058	C₂×C₅	1	2	yes	complemented
59	#2218	C₂×C₅	1	4	yes	complemented
60	#2215	C₂×C₅	1	4	yes	complemented
61	#2158	C₂×C₅	1	1	yes	characteristic, complemented
62	#2140	C₂×C₅	1	2	yes	complemented
63	C₂⁴×C₅	—	4	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₅×#41	—	2	2	no	—
66	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	4	no	—
68	C₂×C₅×D₄	—	4	8	no	—
69	C₂×C₅×D₄	—	4	4	no	—
70	C₂×C₅×D₄	—	4	2	no	—
71	C₂×C₅×D₄	—	4	4	no	—
72	C₂×C₅×D₄	D₈	1	2	yes	—
73	C₂×C₅×D₄	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×D₈	—	2	4	no	—
77	C₅×D₈	—	2	4	no	—
78	C₅×D₈	—	4	4	no	—
79	C₅×M₁₆	—	2	2	no	—
80	C₅×M₁₆	—	2	4	no	—
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	C₂×D₄	1	2	yes	—
86	C₂×C₅×C₈	#41	1	2	yes	—
87	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	4	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×D₈	C₂²×C₅	1	2	yes	—
96	C₂×#131	C₂²×C₅	1	1	yes	characteristic
97	#535	—	2	2	no	—
98	C₂×D₁₆	—	2	4	no	—
99	C₂×D₁₆	C₂²×C₅	1	2	yes	—
100	C₂×D₁₆	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	complemented
104	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
106	#469	—	2	4	no	—
107	#466	C₂²×C₅	1	8	yes	complemented
108	#427	C₂²×C₅	1	2	yes	—
109	#418	—	2	2	no	—
110	#367	C₂²×C₅	1	2	yes	complemented
111	#366	C₂²×C₅	1	2	yes	complemented
112	#357	C₂²×C₅	1	8	yes	complemented
113	C₂³×C₅	—	4	4	no	—
114	C₂³×C₅	—	8	4	no	—
115	C₂³×C₅	—	16	4	no	—
116	C₂³×C₅	C₂×D₈	1	2	yes	—
117	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
118	C₅×Q₈	—	2	2	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	2	2	no	—
121	C₅×D₄	—	4	4	no	—
122	C₅×D₄	—	4	4	no	—
123	C₅×D₄	—	8	4	no	—
124	C₅×D₄	—	16	8	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	#121	1	2	yes	—
129	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅	#135	1	2	yes	—
132	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	#121	1	2	yes	—
135	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₅×C₈	C₂×#41	1	1	yes	characteristic
137	C₂×#41	C₅×D₄	1	1	yes	characteristic
138	C₂²×D₄	—	2	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	4	no	—
142	C₂×D₈	C₂³×C₅	1	4	yes	—
143	#131	—	2	2	no	—
144	#131	—	2	2	no	—
145	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
146	C₂×M₁₆	C₅×D₄	1	2	yes	—
147	C₂²×C₈	C₂³×C₅	1	2	yes	—
148	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
149	#127	—	4	2	no	—
150	#124	—	4	2	no	—
151	#121	—	2	8	no	—
152	#120	—	4	4	no	—
153	#117	—	2	2	no	—
154	D₁₆	—	2	4	no	—
155	D₁₆	—	2	4	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	4	no	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
160	C₂×C₁₆	C₂³×C₅	1	2	yes	—
161	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
163	#107	C₂³×C₅	1	4	yes	—
164	#106	—	2	2	no	—
165	#102	—	2	8	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	8	8	no	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	—	16	8	no	—
172	C₂²×C₅	#466	1	2	yes	—
173	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	8	4	no	—
176	C₄×C₅	#466	1	2	yes	—
177	C₄×C₅	C₂×#121	1	1	yes	characteristic
178	C₄×C₅	C₂×#135	1	1	yes	characteristic
179	C₂⁴	—	4	2	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	4	8	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	C₅×D₈	1	2	yes	—
189	C₂×D₄	C₅×D₈	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	D₈	—	2	4	no	—
193	D₈	—	2	4	no	—
194	D₈	—	4	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
202	C₂×C₈	C₅×#41	1	2	yes	—
203	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	8	no	—
207	C₄²	—	4	2	no	—
208	C₁₆	—	2	2	no	—
209	C₁₆	C₂×C₅×D₄	1	2	yes	—
210	C₁₆	C₂×C₅×D₄	1	2	yes	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	4	2	no	—
213	C₂×C₅	—	16	4	no	—
214	C₂×C₅	C₂×#466	1	1	yes	characteristic
215	C₂×C₅	#3415	1	2	yes	—
216	C₂³	—	4	4	no	—
217	C₂³	—	8	4	no	—
218	C₂³	—	16	4	no	—
219	C₂³	C₂×C₅×D₈	1	2	yes	—
220	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	4	no	—
226	D₄	—	8	4	no	—
227	D₄	—	16	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₅×#121	1	2	yes	—
232	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
233	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
234	C₂×C₄	C₅×#135	1	2	yes	—
235	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	C₅×#121	1	2	yes	—
238	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
239	C₈	C₂×C₅×#41	1	1	yes	characteristic
240	C₅	#33607	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₅×#466	1	2	yes	—
248	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	C₅×#466	1	2	yes	—
252	C₄	C₂×C₅×#121	1	1	yes	characteristic
253	C₄	C₂×C₅×#135	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	16	4	no	—
257	C₂	C₂×C₅×#466	1	1	yes	characteristic
258	C₂	C₅×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
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×D₁₆	C₂	semidirect
2. C₂×C₅×#504	C₂	semidirect
3. C₅×#3058	C₂	semidirect
4. C₅×#2218	C₂	semidirect
5. C₅×#2215	C₂	semidirect
6. C₅×#2158	C₂	semidirect
7. C₅×#2140	C₂	semidirect
16. C₂×C₅×M₃₂	C₂²	semidirect
18. C₂²×C₅×C₁₆	C₂²	semidirect
20. C₅×#466	C₂²	semidirect
23. C₅×#367	C₂²	semidirect
24. C₅×#366	C₂²	semidirect
25. C₅×#357	C₂²	semidirect
26. #33607	C₅	direct (N × H)
49. C₂×C₅×C₁₆	D₄	semidirect
56. C₂²×D₁₆	C₂×C₅	semidirect
57. C₂×#504	C₂×C₅	semidirect
58. #3058	C₂×C₅	semidirect
59. #2218	C₂×C₅	semidirect
60. #2215	C₂×C₅	semidirect
61. #2158	C₂×C₅	semidirect
62. #2140	C₂×C₅	semidirect
103. C₂×M₃₂	C₂²×C₅	semidirect
105. C₂²×C₁₆	C₂²×C₅	semidirect
107. #466	C₂²×C₅	semidirect
110. #367	C₂²×C₅	semidirect
111. #366	C₂²×C₅	semidirect
112. #357	C₂²×C₅	semidirect
159. C₂×C₁₆	C₅×D₄	semidirect
240. C₅	#33607	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×D₁₆	2
C₂×C₅×#504	2
C₅×#3058	2
C₅×#2218	2
C₅×#2215	2
C₅×#2158	2
C₅×#2140	2
#33607	5

Subgroups — C5×#33607 #11956327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33607 #11956327