Subgroups — C₅×#13172 #11935892

Order 1280.

274 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic
3	C₅×#2108	C₂	1	1	yes	maximal, characteristic
4	C₅×#1952	C₂	1	2	yes	maximal
5	C₅×#1387	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₅×#428	—	2	2	no	—
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₅×#405	C₂²	1	1	yes	characteristic
12	C₅×#368	C₄	1	4	yes	complemented
13	C₅×#367	—	2	1	no	—
14	C₅×#367	—	2	1	no	—
15	C₅×#346	—	2	1	no	—
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₅×#343	—	2	2	no	—
18	#13172	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₅×#117	D₄	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₅×M₃₂	—	2	1	no	—
25	C₅×M₃₂	Q₈	1	2	yes	—
26	C₂×C₅×C₁₆	—	2	1	no	—
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₅×C₁₆	Q₈	1	2	yes	—
31	C₅×#108	—	2	2	no	—
32	C₅×#108	C₂×C₄	1	4	yes	—
33	C₅×#107	—	2	2	no	—
34	C₅×#107	—	2	2	no	—
35	C₅×#107	—	4	2	no	—
36	C₅×#106	—	4	2	no	—
37	C₅×#97	—	2	1	no	—
38	C₄×C₅×C₈	—	2	1	no	—
39	C₂×#368	C₂×C₅	1	1	yes	characteristic
40	C₂×#367	C₂×C₅	1	1	yes	characteristic
41	#2108	C₂×C₅	1	1	yes	characteristic
42	#1952	C₂×C₅	1	2	yes	—
43	#1387	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₅	—	4	2	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₅×M₁₆	—	2	1	no	—
47	C₅×M₁₆	—	2	4	no	—
48	C₅×M₁₆	—	4	2	no	—
49	C₅×M₁₆	Q₁₆	1	2	yes	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₅×C₄⋊C₄	D₈	1	2	yes	—
61	C₅×C₂²⋊C₄	—	2	1	no	—
62	C₄²×C₅	—	2	1	no	—
63	C₅×C₁₆	—	2	2	no	—
64	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
65	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
66	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₂×#108	C₂²×C₅	1	2	yes	—
69	#428	—	2	2	no	—
70	C₂×#107	C₂²×C₅	1	2	yes	—
71	#405	C₂²×C₅	1	1	yes	characteristic
72	#368	C₄×C₅	1	4	yes	complemented
73	#367	—	2	1	no	—
74	#367	—	2	1	no	—
75	#346	—	2	1	no	—
76	C₄×C₁₆	—	2	1	no	—
77	#343	—	2	2	no	—
78	C₂³×C₅	#135	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	8	2	no	—
81	C₂×C₄×C₅	—	16	2	no	—
82	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅	#121	1	1	yes	characteristic
84	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅	#135	1	1	yes	characteristic
88	C₅×C₈	—	2	2	no	—
89	C₅×C₈	—	4	4	no	—
90	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
92	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
93	C₅×C₈	#125	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
97	#117	C₅×D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	2	no	—
99	M₃₂	—	2	1	no	—
100	M₃₂	C₅×Q₈	1	2	yes	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
103	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
104	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
105	C₂×C₁₆	C₅×Q₈	1	2	yes	—
106	#108	—	2	2	no	—
107	#108	C₂×C₄×C₅	1	4	yes	—
108	#107	—	2	2	no	—
109	#107	—	2	2	no	—
110	#107	—	4	2	no	—
111	#106	—	4	2	no	—
112	#97	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
116	C₂²×C₅	#466	1	1	yes	characteristic
117	C₂²×C₅	#499	1	1	yes	characteristic
118	C₄×C₅	—	4	2	no	—
119	C₄×C₅	—	16	2	no	—
120	C₄×C₅	#389	1	1	yes	characteristic
121	C₄×C₅	C₄×D₈	1	1	yes	characteristic
122	C₄×C₅	#467	1	1	yes	characteristic
123	C₄×C₅	#500	1	1	yes	characteristic
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
126	M₁₆	—	2	1	no	—
127	M₁₆	—	2	4	no	—
128	M₁₆	—	4	2	no	—
129	M₁₆	C₅×Q₁₆	1	2	yes	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
133	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
134	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
135	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
136	C₂×C₈	C₅×#41	1	1	yes	characteristic
137	C₂×C₈	C₅×#41	1	1	yes	characteristic
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	C₅×D₈	1	2	yes	—
141	C₂²⋊C₄	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₁₆	—	2	2	no	—
144	C₁₆	C₅×C₄⋊C₄	1	2	yes	—
145	C₁₆	C₅×C₄⋊C₄	1	2	yes	—
146	C₂×C₅	—	2	1	no	—
147	C₂×C₅	—	2	1	no	—
148	C₂×C₅	#1942	1	1	yes	characteristic
149	C₂×C₅	#2223	1	2	yes	—
150	C₂³	C₅×#135	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
155	C₂×C₄	C₅×#121	1	1	yes	characteristic
156	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
159	C₂×C₄	C₅×#135	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	4	4	no	—
162	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
163	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
164	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
165	C₈	C₅×#125	1	1	yes	characteristic
166	C₅	#13172	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²	—	2	2	no	—
168	C₂²	C₄×C₅×Q₁₆	1	1	yes	characteristic
169	C₂²	C₅×#466	1	1	yes	characteristic
170	C₂²	C₅×#499	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	2	no	—
173	C₄	C₅×#389	1	1	yes	characteristic
174	C₄	C₄×C₅×D₈	1	1	yes	characteristic
175	C₄	C₅×#467	1	1	yes	characteristic
176	C₄	C₅×#500	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₅×#1942	1	1	yes	characteristic
180	C₂	C₅×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×#368	C₄	semidirect
18. #13172	C₅	direct (N × H)
72. #368	C₄×C₅	semidirect
166. C₅	#13172	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×#368	2
C₂×C₅×#367	2
C₅×#2108	2
C₅×#1952	2
C₅×#1387	2
#13172	5

Subgroups — C5×#13172 #11935892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13172 #11935892