Subgroups — C₅×#22738 #11945458

Order 1280.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₅×#1948	C₂	1	6	yes	maximal
4	C₅×#1834	C₂	1	3	yes	maximal
5	C₄×C₅×#106	C₂	1	3	yes	maximal
6	C₂×C₄×C₅×Q₈	—	2	2	no	—
7	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₅×C₈×Q₈	C₄	1	8	yes	complemented
9	C₂×C₅×#106	C₂²	1	6	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
12	C₅×#389	—	2	6	no	—
13	C₅×#384	C₂²	1	6	yes	—
14	C₄×C₅×C₄⋊C₄	C₂²	1	6	yes	—
15	C₅×#340	C₂²	1	1	yes	characteristic
16	C₅×#340	C₂²	1	3	yes	—
17	C₅×#340	C₂²	1	6	yes	—
18	#22738	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	3	yes	—
22	C₄×C₅×Q₈	—	4	8	no	—
23	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
24	C₄×C₅×Q₈	Q₈	1	4	yes	—
25	C₂×C₅×C₄⋊C₄	—	2	6	no	—
26	C₂×C₅×C₄⋊C₄	—	2	6	no	—
27	C₂×C₅×C₄⋊C₄	D₄	1	3	yes	—
28	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
29	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
30	C₂×C₄²×C₅	—	2	6	no	—
31	C₂×C₄²×C₅	C₂³	1	3	yes	—
32	C₅×#106	—	2	12	no	—
33	C₅×C₄⋊C₈	C₂×C₄	1	12	yes	—
34	C₄×C₅×C₈	C₂×C₄	1	12	yes	—
35	C₅×#95	—	2	6	no	—
36	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
37	Q₈×C₄⋊C₄	C₂×C₅	1	2	yes	—
38	#1948	C₂×C₅	1	6	yes	—
39	#1834	C₂×C₅	1	3	yes	—
40	C₄×#106	C₂×C₅	1	3	yes	—
41	C₂×C₅×Q₈	C₄⋊C₄	1	4	yes	—
42	C₂×C₅×Q₈	SD₁₆	1	8	yes	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	—	2	6	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
48	C₂×C₅×C₈	—	2	6	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
52	C₅×C₄⋊C₄	—	2	2	no	—
53	C₅×C₄⋊C₄	—	2	6	no	—
54	C₅×C₄⋊C₄	—	2	6	no	—
55	C₅×C₄⋊C₄	—	4	12	no	—
56	C₅×C₄⋊C₄	—	8	6	no	—
57	C₅×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
58	C₅×C₄⋊C₄	#41	1	4	yes	—
59	C₄²×C₅	—	4	12	no	—
60	C₄²×C₅	C₂²×C₄	1	6	yes	—
61	C₄²×C₅	C₂×Q₈	1	6	yes	—
62	C₂×C₄×Q₈	—	2	2	no	—
63	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
64	C₈×Q₈	C₄×C₅	1	8	yes	complemented
65	C₂×#106	C₂²×C₅	1	6	yes	—
66	C₂×C₄⋊C₈	C₂²×C₅	1	3	yes	—
67	C₂×C₄×C₈	C₂²×C₅	1	3	yes	—
68	#389	—	2	6	no	—
69	#384	C₂²×C₅	1	6	yes	—
70	C₄×C₄⋊C₄	C₂²×C₅	1	6	yes	—
71	#340	C₂²×C₅	1	3	yes	—
72	#340	C₂²×C₅	1	6	yes	—
73	#340	C₂²×C₅	1	1	yes	characteristic
74	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
75	C₅×Q₈	#106	1	16	yes	complemented
76	C₂×C₄×C₅	—	2	3	no	—
77	C₂×C₄×C₅	—	2	3	no	—
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	—	4	6	no	—
82	C₂×C₄×C₅	—	4	6	no	—
83	C₂×C₄×C₅	—	4	6	no	—
84	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
85	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
86	C₂×C₄×C₅	C₂×SD₁₆	1	12	yes	—
87	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×#41	1	2	yes	—
90	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
91	C₅×C₈	—	2	6	no	—
92	C₅×C₈	—	2	2	no	—
93	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
95	C₂²×C₈	C₂³×C₅	1	3	yes	—
96	C₄×Q₈	—	4	8	no	—
97	C₄×Q₈	C₂×C₄×C₅	1	4	yes	—
98	C₄×Q₈	C₅×Q₈	1	4	yes	—
99	C₂×C₄⋊C₄	—	2	6	no	—
100	C₂×C₄⋊C₄	—	2	6	no	—
101	C₂×C₄⋊C₄	C₅×D₄	1	3	yes	—
102	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
103	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
104	C₂×C₄²	—	2	6	no	—
105	C₂×C₄²	C₂³×C₅	1	3	yes	—
106	#106	—	2	12	no	—
107	C₄⋊C₈	C₂×C₄×C₅	1	12	yes	—
108	C₄×C₈	C₂×C₄×C₅	1	12	yes	—
109	#95	—	2	6	no	—
110	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₅	#548	1	2	yes	—
112	C₂²×C₅	C₂²×SD₁₆	1	2	yes	—
113	C₂²×C₅	#576	1	1	yes	characteristic
114	C₂²×C₅	#578	1	1	yes	characteristic
115	C₄×C₅	—	2	6	no	—
116	C₄×C₅	—	4	4	no	—
117	C₄×C₅	—	8	6	no	—
118	C₄×C₅	C₂×#106	1	12	yes	—
119	C₄×C₅	C₄×#41	1	1	yes	characteristic
120	C₄×C₅	#519	1	1	yes	characteristic
121	C₄×C₅	#557	1	2	yes	—
122	C₂×Q₈	C₅×C₄⋊C₄	1	4	yes	—
123	C₂×Q₈	C₅×SD₁₆	1	8	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	6	no	—
126	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
127	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
128	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
129	C₂×C₈	—	2	6	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
132	C₂×C₈	C₂²×C₄×C₅	1	6	yes	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	6	no	—
135	C₄⋊C₄	—	2	6	no	—
136	C₄⋊C₄	—	4	12	no	—
137	C₄⋊C₄	—	8	6	no	—
138	C₄⋊C₄	C₅×C₄⋊C₄	1	12	yes	—
139	C₄⋊C₄	C₅×#41	1	4	yes	—
140	C₄²	—	4	12	no	—
141	C₄²	C₂²×C₄×C₅	1	6	yes	—
142	C₄²	C₂×C₅×Q₈	1	6	yes	—
143	C₂×C₅	#2351	1	1	yes	characteristic
144	C₂×C₅	C₂²×#106	1	1	yes	characteristic
145	C₂×C₅	#2913	1	1	yes	characteristic
146	C₂×C₅	#3359	1	2	yes	—
147	C₂×C₅	#3362	1	2	yes	—
148	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
149	Q₈	C₅×#106	1	16	yes	complemented
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	6	no	—
156	C₂×C₄	—	4	6	no	—
157	C₂×C₄	—	4	6	no	—
158	C₂×C₄	C₂×C₅×C₄⋊C₄	1	6	yes	—
159	C₂×C₄	C₂×C₅×C₄⋊C₄	1	6	yes	—
160	C₂×C₄	C₂×C₅×SD₁₆	1	12	yes	—
161	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
163	C₂×C₄	C₂×C₅×#41	1	2	yes	—
164	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
165	C₈	—	2	6	no	—
166	C₈	—	2	2	no	—
167	C₅	#22738	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂²	C₅×#548	1	2	yes	—
170	C₂²	C₂²×C₅×SD₁₆	1	2	yes	—
171	C₂²	C₅×#576	1	1	yes	characteristic
172	C₂²	C₅×#578	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	6	no	—
176	C₄	C₂×C₅×#106	1	12	yes	—
177	C₄	C₄×C₅×#41	1	1	yes	characteristic
178	C₄	C₅×#519	1	1	yes	characteristic
179	C₄	C₅×#557	1	2	yes	—
180	C₂	C₅×#2351	1	1	yes	characteristic
181	C₂	C₂²×C₅×#106	1	1	yes	characteristic
182	C₂	C₅×#2913	1	1	yes	characteristic
183	C₂	C₅×#3359	1	2	yes	—
184	C₂	C₅×#3362	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₅×C₈×Q₈	C₄	semidirect
18. #22738	C₅	direct (N × H)
64. C₈×Q₈	C₄×C₅	semidirect
75. C₅×Q₈	#106	semidirect
149. Q₈	C₅×#106	semidirect
167. C₅	#22738	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×C₈×Q₈	2
C₅×Q₈×C₄⋊C₄	2
C₅×#1948	2
C₅×#1834	2
C₄×C₅×#106	2
#22738	5

Subgroups — C5×#22738 #11945458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22738 #11945458