Subgroups — C₅×#10705 #11933425

Order 1280.

870 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	2	yes	maximal
2	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#341	C₂	1	4	yes	maximal
4	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×M₁₆	—	2	4	no	—
8	C₄×C₅×M₁₆	C₄	1	8	yes	—
9	C₂×C₅×#97	—	2	4	no	—
10	C₂×C₅×#97	—	2	2	no	—
11	C₂×C₅×#97	C₄	1	4	yes	—
12	C₂×C₄×C₅×C₈	—	2	2	no	—
13	C₅×#341	C₄	1	16	yes	complemented
14	#10705	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₅	Q₈	1	2	yes	—
16	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×C₅×M₁₆	—	2	4	no	—
18	C₂×C₅×M₁₆	—	2	8	no	—
19	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
20	C₂×C₅×M₁₆	C₂×C₄	1	16	yes	—
21	C₂²×C₅×C₈	—	2	4	no	—
22	C₂²×C₅×C₈	—	2	4	no	—
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅	D₄	1	4	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₅×#97	—	2	8	no	—
31	C₅×#97	—	2	8	no	—
32	C₄×C₅×C₈	—	2	8	no	—
33	C₂×C₄×M₁₆	C₂×C₅	1	2	yes	—
34	C₂²×#97	C₂×C₅	1	1	yes	characteristic
35	C₂×#341	C₂×C₅	1	4	yes	—
36	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₅	—	4	4	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	4	yes	—
46	C₅×M₁₆	—	2	8	no	—
47	C₅×M₁₆	—	2	16	no	—
48	C₅×M₁₆	C₄²	1	16	yes	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	2	8	no	—
51	C₂×C₅×C₈	—	2	4	no	—
52	C₂×C₅×C₈	—	2	16	no	—
53	C₂×C₅×C₈	—	2	16	no	—
54	C₂×C₅×C₈	C₄²	1	8	yes	—
55	C₄²×C₅	C₄²	1	4	yes	—
56	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
59	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
60	C₂³×C₈	C₂²×C₅	1	2	yes	—
61	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
62	C₄×M₁₆	—	2	4	no	—
63	C₄×M₁₆	C₄×C₅	1	8	yes	—
64	C₂×#97	—	2	4	no	—
65	C₂×#97	—	2	2	no	—
66	C₂×#97	C₄×C₅	1	4	yes	—
67	C₂×C₄×C₈	—	2	2	no	—
68	#341	C₄×C₅	1	16	yes	complemented
69	C₂³×C₅	—	2	4	no	—
70	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂³×C₅	C₄×Q₈	1	4	yes	—
73	C₂×C₄×C₅	—	2	8	no	—
74	C₂×C₄×C₅	—	4	8	no	—
75	C₂×C₄×C₅	#95	1	8	yes	—
76	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
78	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
81	C₂×C₄×C₅	#117	1	2	yes	—
82	C₂×C₄×C₅	#117	1	4	yes	—
83	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
85	C₅×C₈	—	2	8	no	—
86	C₅×C₈	—	2	16	no	—
87	C₂³×C₄	C₅×Q₈	1	2	yes	—
88	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
89	C₂×M₁₆	—	2	4	no	—
90	C₂×M₁₆	—	2	8	no	—
91	C₂×M₁₆	C₂×C₄×C₅	1	16	yes	—
92	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	—	2	4	no	—
95	C₂²×C₈	—	2	4	no	—
96	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
97	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
98	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
99	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
100	C₂×C₄²	C₅×D₄	1	2	yes	—
101	C₂×C₄²	C₅×D₄	1	4	yes	—
102	#97	—	2	8	no	—
103	#97	—	2	8	no	—
104	C₄×C₈	—	2	8	no	—
105	C₂²×C₅	—	2	4	no	—
106	C₂²×C₅	—	2	8	no	—
107	C₂²×C₅	C₂×#95	1	1	yes	characteristic
108	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
109	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
110	C₂²×C₅	#430	1	4	yes	—
111	C₄×C₅	—	2	8	no	—
112	C₄×C₅	C₄³	1	1	yes	characteristic
113	C₄×C₅	#376	1	1	yes	characteristic
114	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
115	C₄×C₅	C₄×C₂²⋊C₄	1	4	yes	—
116	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
119	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
120	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
122	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
123	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
125	C₂²×C₄	C₅×#41	1	4	yes	—
126	M₁₆	—	2	8	no	—
127	M₁₆	—	2	16	no	—
128	M₁₆	C₄²×C₅	1	16	yes	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	2	16	no	—
133	C₂×C₈	—	2	16	no	—
134	C₂×C₈	C₄²×C₅	1	8	yes	—
135	C₄²	C₄²×C₅	1	4	yes	—
136	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
137	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
138	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
139	C₂×C₅	—	2	4	no	—
140	C₂×C₅	C₄×#95	1	1	yes	characteristic
141	C₂×C₅	#1738	1	2	yes	—
142	C₂×C₅	#1778	1	4	yes	—
143	C₂³	—	2	4	no	—
144	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
145	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
146	C₂³	C₄×C₅×Q₈	1	4	yes	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	C₅×#95	1	8	yes	—
150	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
152	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
153	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₅×#117	1	2	yes	—
156	C₂×C₄	C₅×#117	1	4	yes	—
157	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
158	C₂×C₄	C₄×C₅×D₄	1	8	yes	—
159	C₈	—	2	8	no	—
160	C₈	—	2	16	no	—
161	C₅	#10705	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²	—	2	8	no	—
163	C₂²	—	2	4	no	—
164	C₂²	C₂×C₅×#95	1	1	yes	characteristic
165	C₂²	C₄×C₅×C₄⋊C₄	1	4	yes	—
166	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
167	C₂²	C₅×#430	1	4	yes	—
168	C₄	—	2	8	no	—
169	C₄	C₄³×C₅	1	1	yes	characteristic
170	C₄	C₅×#376	1	1	yes	characteristic
171	C₄	C₄×C₅×C₂²⋊C₄	1	2	yes	—
172	C₄	C₄×C₅×C₂²⋊C₄	1	4	yes	—
173	C₂	—	2	4	no	—
174	C₂	C₄×C₅×#95	1	1	yes	characteristic
175	C₂	C₅×#1738	1	2	yes	—
176	C₂	C₅×#1778	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₄	C₄³×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×#341	C₄	semidirect
14. #10705	C₅	direct (N × H)
68. #341	C₄×C₅	semidirect
161. C₅	#10705	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×C₄×C₅×M₁₆	2
C₂²×C₅×#97	2
C₂×C₅×#341	2
#10705	5

Subgroups — C5×#10705 #11933425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10705 #11933425