Subgroups — C₅×#33458 #11956178

Order 1280.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 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
2	C₅×#2950	C₂	1	2	yes	maximal
3	C₅×#2176	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×Q₃₂	C₂	1	4	yes	maximal, complemented
5	C₅×#2158	C₂	1	1	yes	maximal, characteristic
6	C₅×#2141	C₂	1	2	yes	maximal
7	C₅×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₅×#519	—	2	2	no	—
10	C₂×C₅×Q₃₂	C₄	1	8	yes	complemented
11	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#441	—	2	4	no	—
16	C₄×C₅×Q₁₆	C₂²	1	8	yes	—
17	C₅×#427	C₂²	1	2	yes	—
18	C₅×#419	—	2	2	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#367	C₂²	1	2	yes	—
21	C₅×#366	C₂²	1	2	yes	—
22	C₅×#358	C₂²	1	8	yes	—
23	C₅×#346	C₂²	1	2	yes	—
24	C₄×C₅×C₁₆	C₂²	1	2	yes	—
25	#33458	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	2	no	—
28	C₂×C₅×Q₁₆	—	2	4	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
31	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₅×Q₈	—	2	8	no	—
34	C₄×C₅×Q₈	—	4	4	no	—
35	C₅×#117	—	2	2	no	—
36	C₅×#117	—	4	4	no	—
37	C₅×#117	D₄	1	1	yes	characteristic
38	C₅×Q₃₂	C₂×C₄	1	8	yes	—
39	C₅×Q₃₂	C₂×C₄	1	8	yes	—
40	C₅×M₃₂	—	2	2	no	—
41	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
42	C₂×C₅×C₁₆	C₂³	1	2	yes	—
43	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
45	C₅×#107	C₂³	1	4	yes	—
46	C₅×#106	—	2	2	no	—
47	C₅×#103	—	2	8	no	—
48	C₅×#97	D₄	1	2	yes	—
49	C₄×C₅×C₈	C₂³	1	2	yes	—
50	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic
51	#2950	C₂×C₅	1	2	yes	—
52	#2176	C₂×C₅	1	4	yes	complemented
53	C₄×Q₃₂	C₂×C₅	1	4	yes	complemented
54	#2158	C₂×C₅	1	1	yes	characteristic
55	#2141	C₂×C₅	1	2	yes	—
56	#2108	C₂×C₅	1	1	yes	characteristic
57	C₂×C₅×Q₈	—	2	8	no	—
58	C₂×C₅×Q₈	—	2	4	no	—
59	C₂×C₅×Q₈	—	4	2	no	—
60	C₂×C₅×Q₈	—	4	4	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₅×Q₁₆	—	2	8	no	—
64	C₅×Q₁₆	—	2	4	no	—
65	C₅×Q₁₆	C₂²×C₄	1	8	yes	—
66	C₅×M₁₆	—	2	2	no	—
67	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	#41	1	2	yes	—
71	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	2	4	no	—
73	C₅×C₄⋊C₄	—	4	8	no	—
74	C₅×C₄⋊C₄	D₈	1	2	yes	—
75	C₅×C₂²⋊C₄	—	4	2	no	—
76	C₅×C₂²⋊C₄	D₈	1	2	yes	—
77	C₄²×C₅	—	4	2	no	—
78	C₄²×C₅	—	4	8	no	—
79	C₄²×C₅	C₂×D₄	1	2	yes	—
80	C₅×C₁₆	—	2	2	no	—
81	C₅×C₁₆	C₂²×C₄	1	2	yes	—
82	C₅×C₁₆	C₂²×C₄	1	2	yes	—
83	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
84	#519	—	2	2	no	—
85	C₂×Q₃₂	C₄×C₅	1	8	yes	complemented
86	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
87	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
88	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
90	#441	—	2	4	no	—
91	C₄×Q₁₆	C₂²×C₅	1	8	yes	—
92	#427	C₂²×C₅	1	2	yes	—
93	#419	—	2	2	no	—
94	#405	C₂²×C₅	1	1	yes	characteristic
95	#367	C₂²×C₅	1	2	yes	—
96	#366	C₂²×C₅	1	2	yes	—
97	#358	C₂²×C₅	1	8	yes	—
98	#346	C₂²×C₅	1	2	yes	—
99	C₄×C₁₆	C₂²×C₅	1	2	yes	—
100	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
101	C₅×Q₈	—	2	8	no	—
102	C₅×Q₈	—	4	4	no	—
103	C₅×Q₈	—	4	8	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	4	8	no	—
106	C₂×C₄×C₅	—	4	4	no	—
107	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅	#135	1	2	yes	—
111	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	C₄×D₄	1	2	yes	—
114	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
115	C₅×C₈	C₂×#41	1	1	yes	characteristic
116	C₂²×Q₈	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	4	no	—
119	C₂×Q₁₆	C₂×C₄×C₅	1	8	yes	—
120	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
121	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
123	C₄×Q₈	—	2	8	no	—
124	C₄×Q₈	—	4	4	no	—
125	#117	—	2	2	no	—
126	#117	—	4	4	no	—
127	#117	C₅×D₄	1	1	yes	characteristic
128	Q₃₂	C₂×C₄×C₅	1	8	yes	—
129	Q₃₂	C₂×C₄×C₅	1	8	yes	—
130	M₃₂	—	2	2	no	—
131	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
132	C₂×C₁₆	C₂³×C₅	1	2	yes	—
133	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
135	#107	C₂³×C₅	1	4	yes	—
136	#106	—	2	2	no	—
137	#103	—	2	8	no	—
138	#97	C₅×D₄	1	2	yes	—
139	C₄×C₈	C₂³×C₅	1	2	yes	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	C₄×D₈	1	2	yes	—
142	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₅	—	2	4	no	—
144	C₄×C₅	—	4	8	no	—
145	C₄×C₅	—	8	4	no	—
146	C₄×C₅	C₄×D₈	1	2	yes	—
147	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₅	C₂×#135	1	1	yes	characteristic
149	C₂×Q₈	—	2	8	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	Q₁₆	—	2	8	no	—
156	Q₁₆	—	2	4	no	—
157	Q₁₆	C₂²×C₄×C₅	1	8	yes	—
158	M₁₆	—	2	2	no	—
159	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
160	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
161	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₈	C₅×#41	1	2	yes	—
163	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	8	no	—
166	C₄⋊C₄	C₅×D₈	1	2	yes	—
167	C₂²⋊C₄	—	4	2	no	—
168	C₂²⋊C₄	C₅×D₈	1	2	yes	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	8	no	—
171	C₄²	C₂×C₅×D₄	1	2	yes	—
172	C₁₆	—	2	2	no	—
173	C₁₆	C₂²×C₄×C₅	1	2	yes	—
174	C₁₆	C₂²×C₄×C₅	1	2	yes	—
175	C₂×C₅	—	2	2	no	—
176	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
177	C₂×C₅	#3417	1	2	yes	—
178	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	4	8	no	—
181	Q₈	—	4	4	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
186	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
187	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
188	C₂×C₄	C₅×#135	1	2	yes	—
189	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	C₄×C₅×D₄	1	2	yes	—
192	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
193	C₈	C₂×C₅×#41	1	1	yes	characteristic
194	C₅	#33458	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	C₄×C₅×D₈	1	2	yes	—
197	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	4	8	no	—
200	C₄	—	8	4	no	—
201	C₄	C₄×C₅×D₈	1	2	yes	—
202	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
203	C₄	C₂×C₅×#135	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
206	C₂	C₅×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2176	C₂	semidirect
4. C₄×C₅×Q₃₂	C₂	semidirect
10. C₂×C₅×Q₃₂	C₄	semidirect
25. #33458	C₅	direct (N × H)
52. #2176	C₂×C₅	semidirect
53. C₄×Q₃₂	C₂×C₅	semidirect
85. C₂×Q₃₂	C₄×C₅	semidirect
194. C₅	#33458	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
C₅×#2950	2
C₅×#2176	2
C₄×C₅×Q₃₂	2
C₅×#2158	2
C₅×#2141	2
C₅×#2108	2
#33458	5

Subgroups — C5×#33458 #11956178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33458 #11956178