Subgroups — C₅×#7182 #11929902

Order 1280.

258 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic
2	C₅×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×M₃₂	—	2	1	no	—
5	C₂×C₅×#108	C₄	1	1	yes	characteristic
6	C₅×#427	C₄	1	1	yes	characteristic
7	C₅×#405	—	2	1	no	—
8	C₂×C₄×C₅×C₈	—	2	1	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#364	—	2	1	no	—
11	C₅×#364	C₄	1	2	yes	complemented
12	C₅×#339	—	2	1	no	—
13	C₅×#322	—	2	1	no	—
14	C₅×#322	C₄	1	2	yes	complemented
15	#7182	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₅×M₁₆	—	2	1	no	—
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
20	C₅×#117	—	2	1	no	—
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₅×M₃₂	—	2	2	no	—
23	C₅×M₃₂	—	4	1	no	—
24	C₂×C₅×C₁₆	—	4	1	no	—
25	C₅×#108	—	2	1	no	—
26	C₅×#108	C₈	1	2	yes	complemented
27	C₅×#107	C₈	1	2	yes	complemented
28	C₅×#106	—	2	1	no	—
29	C₅×#97	—	4	1	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	C₄×C₅×C₈	—	4	1	no	—
33	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
37	C₂×#364	C₂×C₅	1	1	yes	characteristic
38	#1772	C₂×C₅	1	1	yes	characteristic
39	C₂×#322	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
42	C₅×M₁₆	—	2	2	no	—
43	C₅×M₁₆	—	4	1	no	—
44	C₂×C₅×C₈	—	2	4	no	—
45	C₂×C₅×C₈	—	4	4	no	—
46	C₂×C₅×C₈	—	4	1	no	—
47	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
53	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
54	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
55	C₂×C₅×C₈	D₈	1	1	yes	characteristic
56	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
57	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
58	C₂×C₅×C₈	Q₁₆	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	2	no	—
60	C₅×C₂²⋊C₄	—	4	1	no	—
61	C₄²×C₅	—	2	1	no	—
62	C₄²×C₅	—	4	1	no	—
63	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₅×C₁₆	—	4	2	no	—
66	C₂×M₃₂	—	2	1	no	—
67	C₂×#108	C₄×C₅	1	1	yes	characteristic
68	#427	C₄×C₅	1	1	yes	characteristic
69	#405	—	2	1	no	—
70	C₂×C₄×C₈	—	2	1	no	—
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	#364	—	2	1	no	—
73	#364	C₄×C₅	1	2	yes	complemented
74	#339	—	2	1	no	—
75	#322	—	2	1	no	—
76	#322	C₄×C₅	1	2	yes	complemented
77	C₂³×C₅	#104	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	#95	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	#102	1	1	yes	characteristic
86	C₂×C₄×C₅	#103	1	1	yes	characteristic
87	C₂×C₄×C₅	#104	1	1	yes	characteristic
88	C₅×C₈	—	4	4	no	—
89	C₅×C₈	—	4	2	no	—
90	C₅×C₈	C₄×C₈	1	1	yes	characteristic
91	C₅×C₈	#97	1	1	yes	characteristic
92	C₅×C₈	#102	1	1	yes	characteristic
93	C₅×C₈	#103	1	1	yes	characteristic
94	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
95	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
96	C₅×C₈	#106	1	1	yes	characteristic
97	C₅×C₈	#107	1	1	yes	characteristic
98	C₂×M₁₆	—	2	1	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	4	1	no	—
106	C₂×C₁₆	—	4	1	no	—
107	#108	—	2	1	no	—
108	#108	C₅×C₈	1	2	yes	complemented
109	#107	C₅×C₈	1	2	yes	complemented
110	#106	—	2	1	no	—
111	#97	—	4	1	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
117	C₄×C₈	C₅×D₄	1	1	yes	characteristic
118	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	D₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₅	Q₈⋊C₈	1	1	yes	characteristic
122	C₂²×C₅	#339	1	1	yes	characteristic
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	8	2	no	—
126	C₄×C₅	D₄⋊C₈	1	1	yes	characteristic
127	C₄×C₅	Q₈⋊C₈	1	1	yes	characteristic
128	C₄×C₅	#336	1	1	yes	characteristic
129	C₄×C₅	#340	1	1	yes	characteristic
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
132	M₁₆	—	2	2	no	—
133	M₁₆	—	4	1	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
138	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
142	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
143	C₂×C₈	C₅×M₁₆	1	1	yes	characteristic
144	C₂×C₈	C₅×M₁₆	1	1	yes	characteristic
145	C₂×C₈	C₅×D₈	1	1	yes	characteristic
146	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
147	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
148	C₂×C₈	C₅×Q₁₆	1	1	yes	characteristic
149	C₄⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	4	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	4	1	no	—
153	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
154	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₁₆	—	4	2	no	—
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	#1279	1	1	yes	characteristic
159	C₂×C₅	#1336	1	2	yes	—
160	C₂³	C₅×#104	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	C₅×#95	1	1	yes	characteristic
166	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₄	C₅×#102	1	1	yes	characteristic
169	C₂×C₄	C₅×#103	1	1	yes	characteristic
170	C₂×C₄	C₅×#104	1	1	yes	characteristic
171	C₈	—	4	4	no	—
172	C₈	—	4	2	no	—
173	C₈	C₄×C₅×C₈	1	1	yes	characteristic
174	C₈	C₅×#97	1	1	yes	characteristic
175	C₈	C₅×#102	1	1	yes	characteristic
176	C₈	C₅×#103	1	1	yes	characteristic
177	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
178	C₈	C₅×C₄⋊C₈	1	1	yes	characteristic
179	C₈	C₅×#106	1	1	yes	characteristic
180	C₈	C₅×#107	1	1	yes	characteristic
181	C₅	#7182	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	—	2	2	no	—
183	C₂²	C₅×D₄⋊C₈	1	1	yes	characteristic
184	C₂²	C₅×Q₈⋊C₈	1	1	yes	characteristic
185	C₂²	C₅×#339	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	2	no	—
189	C₄	C₅×D₄⋊C₈	1	1	yes	characteristic
190	C₄	C₅×Q₈⋊C₈	1	1	yes	characteristic
191	C₄	C₅×#336	1	1	yes	characteristic
192	C₄	C₅×#340	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₅×#1279	1	1	yes	characteristic
196	C₂	C₅×#1336	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₈	C₄×C₅×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	D₄⋊C₈
prime radical	C₂³×C₅	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅×#364	C₄	semidirect
14. C₅×#322	C₄	semidirect
15. #7182	C₅	direct (N × H)
26. C₅×#108	C₈	semidirect
27. C₅×#107	C₈	semidirect
73. #364	C₄×C₅	semidirect
76. #322	C₄×C₅	semidirect
108. #108	C₅×C₈	semidirect
109. #107	C₅×C₈	semidirect
181. C₅	#7182	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#364	2
C₅×#1772	2
C₂×C₅×#322	2
#7182	5

Subgroups — C5×#7182 #11929902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7182 #11929902