Subgroups — C₂²×C₅×#491 #11983328

Order 1280.

2142 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	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#491	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×#103	C₂	1	4	yes	maximal
5	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₅×Q₈	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	—
10	C₂×C₅×#125	—	2	6	no	—
11	C₂×C₅×#125	C₂²	1	12	yes	—
12	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
13	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
14	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₅×#491	C₂²	1	64	yes	direct product factor
17	C₂×C₅×#103	C₂²	1	48	yes	—
18	C₂×C₅×#97	C₂²	1	12	yes	—
19	C₂²×#491	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×Q₈	—	2	6	no	—
21	C₂²×C₅×Q₈	—	2	16	no	—
22	C₂²×C₅×Q₈	C₂³	1	12	yes	—
23	C₂³×C₄×C₅	—	2	2	no	—
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	D₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	—	2	8	no	—
28	C₂²×C₅×C₈	C₂³	1	12	yes	—
29	C₅×C₄⋊Q₈	—	2	24	no	—
30	C₅×C₄⋊Q₈	C₂³	1	16	yes	—
31	C₅×#125	—	2	24	no	—
32	C₅×#125	C₂³	1	16	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	24	no	—
34	C₂×C₅×C₄⋊C₄	—	2	24	no	—
35	C₂×C₅×C₄⋊C₄	—	2	6	no	—
36	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
37	C₂×C₄²×C₅	D₄	1	6	yes	—
38	C₂×C₄²×C₅	C₂³	1	6	yes	—
39	C₅×#103	C₂³	1	64	yes	—
40	C₅×#97	C₂³	1	16	yes	—
41	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
42	C₂²×#125	C₂×C₅	1	1	yes	characteristic
43	C₂×#491	C₂×C₅	1	24	yes	direct product factor
44	C₂²×#103	C₂×C₅	1	4	yes	—
45	C₂²×#97	C₂×C₅	1	1	yes	characteristic
46	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₅×Q₈	—	2	12	no	—
48	C₂×C₅×Q₈	—	2	48	no	—
49	C₂×C₅×Q₈	—	4	24	no	—
50	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
51	C₂²×C₄×C₅	—	2	8	no	—
52	C₂²×C₄×C₅	—	2	12	no	—
53	C₂²×C₄×C₅	—	2	12	no	—
54	C₂²×C₄×C₅	—	4	8	no	—
55	C₂²×C₄×C₅	—	4	8	no	—
56	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₅	#41	1	8	yes	—
59	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
60	C₂×C₅×C₈	—	2	24	no	—
61	C₂×C₅×C₈	C₂⁴	1	8	yes	—
62	C₅×C₄⋊C₄	—	2	32	no	—
63	C₅×C₄⋊C₄	—	2	32	no	—
64	C₅×C₄⋊C₄	—	2	12	no	—
65	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
66	C₄²×C₅	C₂×D₄	1	12	yes	—
67	C₄²×C₅	C₂⁴	1	4	yes	—
68	C₂³×Q₈	C₂²×C₅	1	2	yes	—
69	C₂³×C₈	C₂²×C₅	1	2	yes	—
70	C₂×C₄⋊Q₈	—	2	6	no	—
71	C₂×C₄⋊Q₈	C₂²×C₅	1	12	yes	—
72	C₂×#125	—	2	6	no	—
73	C₂×#125	C₂²×C₅	1	12	yes	—
74	C₂²×C₄⋊C₄	—	2	2	no	—
75	C₂²×C₄⋊C₄	—	2	2	no	—
76	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
77	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
78	#491	C₂²×C₅	1	64	yes	direct product factor
79	C₂×#103	C₂²×C₅	1	48	yes	—
80	C₂×#97	C₂²×C₅	1	12	yes	—
81	C₂³×C₅	#124	1	4	yes	—
82	C₂³×C₅	#137	1	8	yes	—
83	C₂³×C₅	C₂²×D₄	1	3	yes	—
84	C₅×Q₈	—	2	16	no	—
85	C₅×Q₈	—	4	24	no	—
86	C₂×C₄×C₅	—	2	24	no	—
87	C₂×C₄×C₅	—	2	8	no	—
88	C₂×C₄×C₅	—	2	8	no	—
89	C₂×C₄×C₅	—	4	24	no	—
90	C₂×C₄×C₅	—	4	24	no	—
91	C₂×C₄×C₅	#124	1	12	yes	—
92	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
93	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
94	C₂×C₄×C₅	C₂×#41	1	12	yes	—
95	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
96	C₅×C₈	—	2	8	no	—
97	C₂²×Q₈	—	2	6	no	—
98	C₂²×Q₈	—	2	16	no	—
99	C₂²×Q₈	C₂³×C₅	1	12	yes	—
100	C₂³×C₄	—	2	2	no	—
101	C₂³×C₄	—	2	2	no	—
102	C₂³×C₄	C₅×D₄	1	2	yes	—
103	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
104	C₂²×C₈	—	2	8	no	—
105	C₂²×C₈	C₂³×C₅	1	12	yes	—
106	C₄⋊Q₈	—	2	24	no	—
107	C₄⋊Q₈	C₂³×C₅	1	16	yes	—
108	#125	—	2	24	no	—
109	#125	C₂³×C₅	1	16	yes	—
110	C₂×C₄⋊C₄	—	2	24	no	—
111	C₂×C₄⋊C₄	—	2	24	no	—
112	C₂×C₄⋊C₄	—	2	6	no	—
113	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
114	C₂×C₄²	C₅×D₄	1	6	yes	—
115	C₂×C₄²	C₂³×C₅	1	6	yes	—
116	#103	C₂³×C₅	1	64	yes	—
117	#97	C₂³×C₅	1	16	yes	—
118	C₂²×C₅	#491	1	16	yes	direct product factor
119	C₂²×C₅	C₂×#124	1	6	yes	—
120	C₂²×C₅	C₂×#137	1	12	yes	—
121	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
122	C₄×C₅	—	2	8	no	—
123	C₄×C₅	—	4	8	no	—
124	C₄×C₅	—	4	8	no	—
125	C₄×C₅	C₂×#124	1	6	yes	—
126	C₄×C₅	C₂²×#41	1	2	yes	—
127	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂×Q₈	—	2	12	no	—
129	C₂×Q₈	—	2	48	no	—
130	C₂×Q₈	—	4	24	no	—
131	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	—	2	12	no	—
134	C₂²×C₄	—	2	12	no	—
135	C₂²×C₄	—	4	8	no	—
136	C₂²×C₄	—	4	8	no	—
137	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
138	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
139	C₂²×C₄	C₅×#41	1	8	yes	—
140	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
141	C₂×C₈	—	2	24	no	—
142	C₂×C₈	C₂⁴×C₅	1	8	yes	—
143	C₄⋊C₄	—	2	32	no	—
144	C₄⋊C₄	—	2	32	no	—
145	C₄⋊C₄	—	2	12	no	—
146	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
147	C₄²	C₂×C₅×D₄	1	12	yes	—
148	C₄²	C₂⁴×C₅	1	4	yes	—
149	C₂×C₅	C₂×#491	1	12	yes	direct product factor
150	C₂×C₅	C₂²×#124	1	1	yes	characteristic
151	C₂×C₅	C₂²×#137	1	2	yes	—
152	C₂³	C₅×#124	1	4	yes	—
153	C₂³	C₅×#137	1	8	yes	—
154	C₂³	C₂²×C₅×D₄	1	3	yes	—
155	Q₈	—	2	16	no	—
156	Q₈	—	4	24	no	—
157	C₂×C₄	—	2	24	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	24	no	—
161	C₂×C₄	—	4	24	no	—
162	C₂×C₄	C₅×#124	1	12	yes	—
163	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
164	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
165	C₂×C₄	C₂×C₅×#41	1	12	yes	—
166	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
167	C₈	—	2	8	no	—
168	C₅	C₂²×#491	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²	C₅×#491	1	16	yes	direct product factor
170	C₂²	C₂×C₅×#124	1	6	yes	—
171	C₂²	C₂×C₅×#137	1	12	yes	—
172	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	4	8	no	—
175	C₄	—	4	8	no	—
176	C₄	C₂×C₅×#124	1	6	yes	—
177	C₄	C₂²×C₅×#41	1	2	yes	—
178	C₂	C₂×C₅×#491	1	12	yes	direct product factor
179	C₂	C₂²×C₅×#124	1	1	yes	characteristic
180	C₂	C₂²×C₅×#137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#491	C₂	direct (N × H)
16. C₅×#491	C₂²	direct (N × H)
19. C₂²×#491	C₅	direct (N × H)
43. C₂×#491	C₂×C₅	direct (N × H)
78. #491	C₂²×C₅	direct (N × H)
118. C₂²×C₅	#491	direct (N × H)
149. C₂×C₅	C₂×#491	direct (N × H)
168. C₅	C₂²×#491	direct (N × H)
169. C₂²	C₅×#491	direct (N × H)
178. C₂	C₂×C₅×#491	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
C₂×C₅×#491	2
C₂²×C₅×#103	2
C₂²×C₅×#97	2
C₂²×#491	5

Subgroups — C22×C5×#491 #11983328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#491 #11983328