Subgroups — C₂²×C₅×#492 #11983348

Order 1280.

3950 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#492	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₅×Q₈	C₂²	1	2	yes	—
7	C₂³×C₅×D₄	—	2	2	no	—
8	C₂³×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₅×SD₁₆	—	2	32	no	—
10	C₂²×C₅×SD₁₆	C₂²	1	48	yes	complemented
11	C₂³×C₅×C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	complemented
14	C₂×C₅×#127	—	2	6	no	—
15	C₂×C₅×#127	C₂²	1	12	yes	—
16	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#492	C₂²	1	64	yes	direct product factor
19	C₂×C₄×C₅×C₈	C₂²	1	12	yes	complemented
20	C₂²×#492	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂⁵×C₅	—	2	2	no	—
22	C₂²×C₅×Q₈	—	2	6	no	—
23	C₂²×C₅×Q₈	—	2	16	no	—
24	C₂²×C₅×Q₈	C₂³	1	12	yes	—
25	C₂²×C₅×D₄	—	2	16	no	—
26	C₂²×C₅×D₄	—	2	6	no	—
27	C₂²×C₅×D₄	—	2	24	no	—
28	C₂²×C₅×D₄	—	4	16	no	—
29	C₂²×C₅×D₄	C₂³	1	12	yes	—
30	C₂³×C₄×C₅	—	2	2	no	—
31	C₂³×C₄×C₅	D₄	1	2	yes	—
32	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×SD₁₆	—	2	192	no	—
34	C₂×C₅×SD₁₆	C₂³	1	64	yes	complemented
35	C₂²×C₅×C₈	D₄	1	16	yes	complemented
36	C₂²×C₅×C₈	C₂³	1	12	yes	—
37	C₅×C₄⋊Q₈	—	2	24	no	—
38	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
39	C₅×#127	—	2	24	no	—
40	C₅×#127	C₂³	1	16	yes	—
41	C₂×C₅×C₄⋊C₄	—	2	24	no	—
42	C₂×C₄²×C₅	D₄	1	6	yes	—
43	C₂×C₄²×C₅	C₂³	1	6	yes	—
44	C₄×C₅×C₈	C₂³	1	16	yes	complemented
45	C₂³×SD₁₆	C₂×C₅	1	4	yes	complemented
46	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂²×#127	C₂×C₅	1	1	yes	characteristic
48	C₂×#492	C₂×C₅	1	24	yes	direct product factor
49	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂⁴×C₅	—	2	12	no	—
51	C₂⁴×C₅	—	4	8	no	—
52	C₂⁴×C₅	—	4	16	no	—
53	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×Q₈	—	2	12	no	—
55	C₂×C₅×Q₈	—	2	48	no	—
56	C₂×C₅×Q₈	—	4	24	no	—
57	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
58	C₂×C₅×D₄	—	2	48	no	—
59	C₂×C₅×D₄	—	2	12	no	—
60	C₂×C₅×D₄	—	2	32	no	—
61	C₂×C₅×D₄	—	4	24	no	—
62	C₂×C₅×D₄	—	4	96	no	—
63	C₂×C₅×D₄	C₂⁴	1	8	yes	—
64	C₂²×C₄×C₅	—	2	12	no	—
65	C₂²×C₄×C₅	—	4	8	no	—
66	C₂²×C₄×C₅	SD₁₆	1	16	yes	complemented
67	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
70	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
71	C₅×SD₁₆	—	2	128	no	—
72	C₂×C₅×C₈	C₂×D₄	1	48	yes	complemented
73	C₂×C₅×C₈	C₂⁴	1	8	yes	—
74	C₅×C₄⋊C₄	—	2	32	no	—
75	C₄²×C₅	C₂×D₄	1	12	yes	—
76	C₄²×C₅	C₂⁴	1	4	yes	—
77	C₂³×Q₈	C₂²×C₅	1	2	yes	—
78	C₂³×D₄	—	2	2	no	—
79	C₂³×D₄	C₂²×C₅	1	2	yes	—
80	C₂²×SD₁₆	—	2	32	no	—
81	C₂²×SD₁₆	C₂²×C₅	1	48	yes	complemented
82	C₂³×C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₄⋊Q₈	—	2	6	no	—
84	C₂×C₄⋊Q₈	C₂²×C₅	1	12	yes	complemented
85	C₂×#127	—	2	6	no	—
86	C₂×#127	C₂²×C₅	1	12	yes	—
87	C₂²×C₄⋊C₄	—	2	2	no	—
88	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
89	#492	C₂²×C₅	1	64	yes	direct product factor
90	C₂×C₄×C₈	C₂²×C₅	1	12	yes	complemented
91	C₂³×C₅	—	2	8	no	—
92	C₂³×C₅	—	4	24	no	—
93	C₂³×C₅	—	4	48	no	—
94	C₂³×C₅	—	8	32	no	—
95	C₂³×C₅	#127	1	4	yes	—
96	C₂³×C₅	C₂×SD₁₆	1	8	yes	—
97	C₂³×C₅	C₂²×D₄	1	3	yes	—
98	C₅×Q₈	—	2	16	no	—
99	C₅×Q₈	—	4	24	no	—
100	C₅×D₄	—	2	16	no	—
101	C₅×D₄	—	4	24	no	—
102	C₅×D₄	—	4	64	no	—
103	C₂×C₄×C₅	—	2	8	no	—
104	C₂×C₄×C₅	—	4	24	no	—
105	C₂×C₄×C₅	#127	1	12	yes	—
106	C₂×C₄×C₅	C₂×SD₁₆	1	48	yes	complemented
107	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
109	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
110	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
111	C₅×C₈	C₂²×D₄	1	16	yes	complemented
112	C₂⁵	—	2	2	no	—
113	C₂²×Q₈	—	2	6	no	—
114	C₂²×Q₈	—	2	16	no	—
115	C₂²×Q₈	C₂³×C₅	1	12	yes	—
116	C₂²×D₄	—	2	16	no	—
117	C₂²×D₄	—	2	6	no	—
118	C₂²×D₄	—	2	24	no	—
119	C₂²×D₄	—	4	16	no	—
120	C₂²×D₄	C₂³×C₅	1	12	yes	—
121	C₂³×C₄	—	2	2	no	—
122	C₂³×C₄	C₅×D₄	1	2	yes	—
123	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂×SD₁₆	—	2	192	no	—
125	C₂×SD₁₆	C₂³×C₅	1	64	yes	complemented
126	C₂²×C₈	C₅×D₄	1	16	yes	complemented
127	C₂²×C₈	C₂³×C₅	1	12	yes	—
128	C₄⋊Q₈	—	2	24	no	—
129	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
130	#127	—	2	24	no	—
131	#127	C₂³×C₅	1	16	yes	—
132	C₂×C₄⋊C₄	—	2	24	no	—
133	C₂×C₄²	C₅×D₄	1	6	yes	—
134	C₂×C₄²	C₂³×C₅	1	6	yes	—
135	C₄×C₈	C₂³×C₅	1	16	yes	complemented
136	C₂²×C₅	—	4	8	no	—
137	C₂²×C₅	—	4	16	no	—
138	C₂²×C₅	—	8	48	no	—
139	C₂²×C₅	#492	1	16	yes	direct product factor
140	C₂²×C₅	C₂×#127	1	6	yes	—
141	C₂²×C₅	C₂²×SD₁₆	1	12	yes	—
142	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₅	—	4	8	no	—
144	C₄×C₅	C₂×#127	1	6	yes	—
145	C₄×C₅	C₂²×SD₁₆	1	16	yes	complemented
146	C₄×C₅	C₂³×D₄	1	2	yes	—
147	C₂⁴	—	2	12	no	—
148	C₂⁴	—	4	8	no	—
149	C₂⁴	—	4	16	no	—
150	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
151	C₂×Q₈	—	2	12	no	—
152	C₂×Q₈	—	2	48	no	—
153	C₂×Q₈	—	4	24	no	—
154	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
155	C₂×D₄	—	2	48	no	—
156	C₂×D₄	—	2	12	no	—
157	C₂×D₄	—	2	32	no	—
158	C₂×D₄	—	4	24	no	—
159	C₂×D₄	—	4	96	no	—
160	C₂×D₄	C₂⁴×C₅	1	8	yes	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	4	8	no	—
163	C₂²×C₄	C₅×SD₁₆	1	16	yes	complemented
164	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
165	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
166	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
167	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
168	SD₁₆	—	2	128	no	—
169	C₂×C₈	C₂×C₅×D₄	1	48	yes	complemented
170	C₂×C₈	C₂⁴×C₅	1	8	yes	—
171	C₄⋊C₄	—	2	32	no	—
172	C₄²	C₂×C₅×D₄	1	12	yes	—
173	C₄²	C₂⁴×C₅	1	4	yes	—
174	C₂×C₅	—	8	8	no	—
175	C₂×C₅	C₂×#492	1	12	yes	direct product factor
176	C₂×C₅	C₂²×#127	1	1	yes	characteristic
177	C₂×C₅	C₂³×SD₁₆	1	2	yes	—
178	C₂³	—	2	8	no	—
179	C₂³	—	4	24	no	—
180	C₂³	—	4	48	no	—
181	C₂³	—	8	32	no	—
182	C₂³	C₅×#127	1	4	yes	—
183	C₂³	C₂×C₅×SD₁₆	1	8	yes	—
184	C₂³	C₂²×C₅×D₄	1	3	yes	—
185	Q₈	—	2	16	no	—
186	Q₈	—	4	24	no	—
187	D₄	—	2	16	no	—
188	D₄	—	4	24	no	—
189	D₄	—	4	64	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	4	24	no	—
192	C₂×C₄	C₅×#127	1	12	yes	—
193	C₂×C₄	C₂×C₅×SD₁₆	1	48	yes	complemented
194	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
195	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
196	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
197	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
198	C₈	C₂²×C₅×D₄	1	16	yes	complemented
199	C₅	C₂²×#492	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	4	8	no	—
201	C₂²	—	4	16	no	—
202	C₂²	—	8	48	no	—
203	C₂²	C₅×#492	1	16	yes	direct product factor
204	C₂²	C₂×C₅×#127	1	6	yes	—
205	C₂²	C₂²×C₅×SD₁₆	1	12	yes	—
206	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
207	C₄	—	4	8	no	—
208	C₄	C₂×C₅×#127	1	6	yes	—
209	C₄	C₂²×C₅×SD₁₆	1	16	yes	complemented
210	C₄	C₂³×C₅×D₄	1	2	yes	—
211	C₂	—	8	8	no	—
212	C₂	C₂×C₅×#492	1	12	yes	direct product factor
213	C₂	C₂²×C₅×#127	1	1	yes	characteristic
214	C₂	C₂³×C₅×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂²×C₅×#127	C₂
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
1. C₂³×C₅×SD₁₆	C₂	semidirect
2. C₂²×C₅×C₄⋊Q₈	C₂	semidirect
4. C₂×C₅×#492	C₂	direct (N × H)
5. C₂²×C₄×C₅×C₈	C₂	semidirect
10. C₂²×C₅×SD₁₆	C₂²	semidirect
13. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
18. C₅×#492	C₂²	direct (N × H)
19. C₂×C₄×C₅×C₈	C₂²	semidirect
20. C₂²×#492	C₅	direct (N × H)
34. C₂×C₅×SD₁₆	C₂³	semidirect
35. C₂²×C₅×C₈	D₄	semidirect
38. C₅×C₄⋊Q₈	C₂³	semidirect
44. C₄×C₅×C₈	C₂³	semidirect
45. C₂³×SD₁₆	C₂×C₅	semidirect
46. C₂²×C₄⋊Q₈	C₂×C₅	semidirect
48. C₂×#492	C₂×C₅	direct (N × H)
49. C₂²×C₄×C₈	C₂×C₅	semidirect
66. C₂²×C₄×C₅	SD₁₆	semidirect
72. C₂×C₅×C₈	C₂×D₄	semidirect
81. C₂²×SD₁₆	C₂²×C₅	semidirect
84. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
89. #492	C₂²×C₅	direct (N × H)
90. C₂×C₄×C₈	C₂²×C₅	semidirect
106. C₂×C₄×C₅	C₂×SD₁₆	semidirect
111. C₅×C₈	C₂²×D₄	semidirect
125. C₂×SD₁₆	C₂³×C₅	semidirect
126. C₂²×C₈	C₅×D₄	semidirect
129. C₄⋊Q₈	C₂³×C₅	semidirect
135. C₄×C₈	C₂³×C₅	semidirect
139. C₂²×C₅	#492	direct (N × H)
145. C₄×C₅	C₂²×SD₁₆	semidirect
163. C₂²×C₄	C₅×SD₁₆	semidirect
169. C₂×C₈	C₂×C₅×D₄	semidirect
175. C₂×C₅	C₂×#492	direct (N × H)
193. C₂×C₄	C₂×C₅×SD₁₆	semidirect
198. C₈	C₂²×C₅×D₄	semidirect
199. C₅	C₂²×#492	direct (N × H)
203. C₂²	C₅×#492	direct (N × H)
209. C₄	C₂²×C₅×SD₁₆	semidirect
212. C₂	C₂×C₅×#492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×SD₁₆	2
C₂²×C₅×C₄⋊Q₈	2
C₂²×C₅×#127	2
C₂×C₅×#492	2
C₂²×C₄×C₅×C₈	2
C₂²×#492	5

Subgroups — C22×C5×#492 #11983348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#492 #11983348