Subgroups — C₂×C₅×#2642 #11957348

Order 1280.

2286 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₅×#2642	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#400	C₂	1	8	yes	maximal
5	C₂×C₅×#391	C₂	1	4	yes	maximal
6	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₅×Q₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Q₈	—	2	8	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	8	yes	—
10	C₂×C₅×#125	—	2	8	no	—
11	C₂×C₅×#125	C₂²	1	8	yes	—
12	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂²×C₄²×C₅	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₅×#400	C₂²	1	64	yes	—
18	C₅×#391	C₂²	1	32	yes	—
19	C₅×#376	C₂²	1	8	yes	—
20	C₂×C₅×#95	C₂²	1	4	yes	—
21	C₂×C₅×#95	C₂²	1	16	yes	—
22	C₂×#2642	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	—	2	8	no	—
24	C₂²×C₅×Q₈	—	2	16	no	—
25	C₂²×C₅×Q₈	C₂³	1	8	yes	—
26	C₂³×C₄×C₅	C₂³	1	4	yes	—
27	C₂³×C₄×C₅	C₂³	1	4	yes	—
28	C₂³×C₄×C₅	C₂³	1	4	yes	—
29	C₂³×C₄×C₅	C₂³	1	2	yes	—
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₅×C₄⋊Q₈	—	2	32	no	—
32	C₅×#125	—	2	32	no	—
33	C₂×C₅×C₄⋊C₄	—	2	16	no	—
34	C₂×C₅×C₄⋊C₄	—	2	8	no	—
35	C₂×C₅×C₄⋊C₄	—	2	16	no	—
36	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
39	C₂×C₄²×C₅	—	2	8	no	—
40	C₂×C₄²×C₅	D₄	1	8	yes	—
41	C₂×C₄²×C₅	C₂³	1	8	yes	—
42	C₂×C₄²×C₅	C₂³	1	4	yes	—
43	C₅×#95	C₂³	1	16	yes	—
44	C₅×#95	C₂³	1	64	yes	—
45	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
46	C₂²×#125	C₂×C₅	1	1	yes	characteristic
47	#2642	C₂×C₅	1	16	yes	direct product factor
48	C₂×#400	C₂×C₅	1	8	yes	—
49	C₂×#391	C₂×C₅	1	4	yes	—
50	C₂×#376	C₂×C₅	1	1	yes	characteristic
51	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
52	C₂×C₅×Q₈	—	2	16	no	—
53	C₂×C₅×Q₈	—	2	32	no	—
54	C₂×C₅×Q₈	—	4	32	no	—
55	C₂²×C₄×C₅	—	2	8	no	—
56	C₂²×C₄×C₅	—	2	8	no	—
57	C₂²×C₄×C₅	—	2	8	no	—
58	C₂²×C₄×C₅	—	2	8	no	—
59	C₂²×C₄×C₅	—	2	16	no	—
60	C₂²×C₄×C₅	—	2	16	no	—
61	C₂²×C₄×C₅	—	2	16	no	—
62	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₅	#41	1	8	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	32	no	—
71	C₅×C₄⋊C₄	—	2	16	no	—
72	C₅×C₄⋊C₄	—	2	32	no	—
73	C₄²×C₅	—	2	16	no	—
74	C₄²×C₅	C₂×D₄	1	16	yes	—
75	C₂³×Q₈	C₂²×C₅	1	2	yes	—
76	C₂×C₄⋊Q₈	—	2	8	no	—
77	C₂×C₄⋊Q₈	C₂²×C₅	1	8	yes	—
78	C₂×#125	—	2	8	no	—
79	C₂×#125	C₂²×C₅	1	8	yes	—
80	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
81	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
82	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
83	C₂²×C₄²	C₂²×C₅	1	2	yes	—
84	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
85	#400	C₂²×C₅	1	64	yes	—
86	#391	C₂²×C₅	1	32	yes	—
87	#376	C₂²×C₅	1	8	yes	—
88	C₂×#95	C₂²×C₅	1	4	yes	—
89	C₂×#95	C₂²×C₅	1	16	yes	—
90	C₂³×C₅	C₂²×D₄	1	2	yes	—
91	C₂³×C₅	C₂×#41	1	4	yes	—
92	C₂³×C₅	2₋¹⁺⁴	1	8	yes	—
93	C₂³×C₅	C₂⁵	1	1	yes	characteristic
94	C₅×Q₈	—	4	32	no	—
95	C₂×C₄×C₅	—	2	16	no	—
96	C₂×C₄×C₅	—	2	8	no	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	—	2	16	no	—
102	C₂×C₄×C₅	—	2	16	no	—
103	C₂×C₄×C₅	—	4	16	no	—
104	C₂×C₄×C₅	—	4	16	no	—
105	C₂×C₄×C₅	—	4	16	no	—
106	C₂×C₄×C₅	#124	1	16	yes	—
107	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
108	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₅	C₂×#41	1	8	yes	—
110	C₂²×Q₈	—	2	8	no	—
111	C₂²×Q₈	—	2	16	no	—
112	C₂²×Q₈	C₂³×C₅	1	8	yes	—
113	C₂³×C₄	C₂³×C₅	1	4	yes	—
114	C₂³×C₄	C₂³×C₅	1	4	yes	—
115	C₂³×C₄	C₂³×C₅	1	4	yes	—
116	C₂³×C₄	C₂³×C₅	1	2	yes	—
117	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₄⋊Q₈	—	2	32	no	—
119	#125	—	2	32	no	—
120	C₂×C₄⋊C₄	—	2	16	no	—
121	C₂×C₄⋊C₄	—	2	8	no	—
122	C₂×C₄⋊C₄	—	2	16	no	—
123	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
124	C₂×C₄⋊C₄	C₂³×C₅	1	16	yes	—
125	C₂×C₄⋊C₄	C₂³×C₅	1	16	yes	—
126	C₂×C₄²	—	2	8	no	—
127	C₂×C₄²	C₅×D₄	1	8	yes	—
128	C₂×C₄²	C₂³×C₅	1	8	yes	—
129	C₂×C₄²	C₂³×C₅	1	4	yes	—
130	#95	C₂³×C₅	1	16	yes	—
131	#95	C₂³×C₅	1	64	yes	—
132	C₂²×C₅	C₂×#124	1	4	yes	—
133	C₂²×C₅	#536	1	8	yes	—
134	C₂²×C₅	#541	1	16	yes	—
135	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
136	C₂²×C₅	C₂²×#41	1	2	yes	—
137	C₂²×C₅	C₂×2₋¹⁺⁴	1	4	yes	—
138	C₄×C₅	—	2	8	no	—
139	C₄×C₅	—	4	8	no	—
140	C₄×C₅	—	4	8	no	—
141	C₄×C₅	—	4	8	no	—
142	C₄×C₅	C₂×#124	1	8	yes	—
143	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
144	C₂×Q₈	—	2	16	no	—
145	C₂×Q₈	—	2	32	no	—
146	C₂×Q₈	—	4	32	no	—
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	16	no	—
152	C₂²×C₄	—	2	16	no	—
153	C₂²×C₄	—	2	16	no	—
154	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
155	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
156	C₂²×C₄	C₅×#41	1	8	yes	—
157	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
158	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
159	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
160	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
161	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
162	C₄⋊C₄	—	2	32	no	—
163	C₄⋊C₄	—	2	16	no	—
164	C₄⋊C₄	—	2	32	no	—
165	C₄²	—	2	16	no	—
166	C₄²	C₂×C₅×D₄	1	16	yes	—
167	C₂×C₅	#2642	1	8	yes	direct product factor
168	C₂×C₅	C₂²×#124	1	1	yes	characteristic
169	C₂×C₅	C₂×#536	1	2	yes	—
170	C₂×C₅	C₂×#541	1	4	yes	—
171	C₂³	C₂²×C₅×D₄	1	2	yes	—
172	C₂³	C₂×C₅×#41	1	4	yes	—
173	C₂³	C₅×2₋¹⁺⁴	1	8	yes	—
174	C₂³	C₂⁵×C₅	1	1	yes	characteristic
175	Q₈	—	4	32	no	—
176	C₂×C₄	—	2	16	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	4	16	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	—	4	16	no	—
187	C₂×C₄	C₅×#124	1	16	yes	—
188	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
189	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
190	C₂×C₄	C₂×C₅×#41	1	8	yes	—
191	C₅	C₂×#2642	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	C₂×C₅×#124	1	4	yes	—
193	C₂²	C₅×#536	1	8	yes	—
194	C₂²	C₅×#541	1	16	yes	—
195	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
196	C₂²	C₂²×C₅×#41	1	2	yes	—
197	C₂²	C₂×C₅×2₋¹⁺⁴	1	4	yes	—
198	C₄	—	2	8	no	—
199	C₄	—	4	8	no	—
200	C₄	—	4	8	no	—
201	C₄	—	4	8	no	—
202	C₄	C₂×C₅×#124	1	8	yes	—
203	C₂	C₅×#2642	1	8	yes	direct product factor
204	C₂	C₂²×C₅×#124	1	1	yes	characteristic
205	C₂	C₂×C₅×#536	1	2	yes	—
206	C₂	C₂×C₅×#541	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2642	C₂	direct (N × H)
22. C₂×#2642	C₅	direct (N × H)
47. #2642	C₂×C₅	direct (N × H)
167. C₂×C₅	#2642	direct (N × H)
191. C₅	C₂×#2642	direct (N × H)
203. C₂	C₅×#2642	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×C₄⋊Q₈	2
C₂²×C₅×#125	2
C₅×#2642	2
C₂×C₅×#400	2
C₂×C₅×#391	2
C₂×C₅×#376	2
C₂×#2642	5

Subgroups — C2×C5×#2642 #11957348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2642 #11957348