Subgroups — C₅×#22277 #11944997

Order 1280.

826 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	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₅×#2327	C₂	1	2	yes	maximal
3	C₅×#1960	C₂	1	2	yes	maximal
4	C₅×#1878	C₂	1	8	yes	maximal
5	C₅×#1835	C₂	1	1	yes	maximal, characteristic
6	C₅×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
10	C₅×#494	C₄	1	8	yes	complemented
11	C₂×C₅×#103	C₂²	1	8	yes	—
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#391	C₂²	1	4	yes	—
15	C₅×#387	—	2	8	no	—
16	C₅×#382	C₂²	1	2	yes	—
17	C₅×#376	C₂²	1	1	yes	characteristic
18	C₅×#340	C₂²	1	4	yes	—
19	C₅×#336	C₂²	1	4	yes	—
20	#22277	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	C₂³	1	4	yes	—
22	C₂×C₅×Q₁₆	—	4	8	no	—
23	C₂×C₅×Q₁₆	C₂×C₄	1	16	yes	—
24	C₂²×C₅×C₈	C₂³	1	2	yes	—
25	C₂²×C₅×C₈	C₂³	1	2	yes	—
26	C₅×C₄⋊Q₈	—	2	4	no	—
27	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
28	C₄×C₅×Q₈	—	4	8	no	—
29	C₂×C₅×C₄⋊C₄	—	2	4	no	—
30	C₂×C₅×C₄⋊C₄	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
32	C₂×C₄²×C₅	—	2	4	no	—
33	C₂×C₄²×C₅	C₂³	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×#103	—	4	8	no	—
36	C₅×#97	—	2	2	no	—
37	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₅×#95	—	2	4	no	—
39	C₅×#95	—	2	8	no	—
40	C₅×#95	D₄	1	4	yes	—
41	C₂×#494	C₂×C₅	1	1	yes	characteristic
42	#2327	C₂×C₅	1	2	yes	—
43	#1960	C₂×C₅	1	2	yes	—
44	#1878	C₂×C₅	1	8	yes	—
45	#1835	C₂×C₅	1	1	yes	characteristic
46	#1746	C₂×C₅	1	1	yes	characteristic
47	C₂×C₅×Q₈	—	2	4	no	—
48	C₂×C₅×Q₈	—	2	8	no	—
49	C₂×C₅×Q₈	—	2	8	no	—
50	C₂×C₅×Q₈	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₅	—	2	4	no	—
52	C₂²×C₄×C₅	—	2	4	no	—
53	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
56	C₅×Q₁₆	—	4	16	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
61	C₅×C₄⋊C₄	—	2	16	no	—
62	C₅×C₄⋊C₄	—	2	8	no	—
63	C₅×C₄⋊C₄	—	4	8	no	—
64	C₄²×C₅	—	2	4	no	—
65	C₄²×C₅	—	4	8	no	—
66	C₄²×C₅	C₂²×C₄	1	2	yes	—
67	C₄²×C₅	#41	1	2	yes	—
68	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
69	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
70	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
71	#494	C₄×C₅	1	8	yes	complemented
72	C₂×#103	C₂²×C₅	1	8	yes	—
73	C₂×#97	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	#391	C₂²×C₅	1	4	yes	—
76	#387	—	2	8	no	—
77	#382	C₂²×C₅	1	2	yes	—
78	#376	C₂²×C₅	1	1	yes	characteristic
79	#340	C₂²×C₅	1	4	yes	—
80	#336	C₂²×C₅	1	4	yes	—
81	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
82	C₅×Q₈	—	2	16	no	—
83	C₅×Q₈	—	4	8	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	8	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	2	8	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	4	4	no	—
91	C₂×C₄×C₅	—	4	8	no	—
92	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
96	C₅×C₈	—	2	4	no	—
97	C₅×C₈	—	4	2	no	—
98	C₂²×Q₈	C₂³×C₅	1	4	yes	—
99	C₂×Q₁₆	—	4	8	no	—
100	C₂×Q₁₆	C₂×C₄×C₅	1	16	yes	—
101	C₂²×C₈	C₂³×C₅	1	2	yes	—
102	C₂²×C₈	C₂³×C₅	1	2	yes	—
103	C₄⋊Q₈	—	2	4	no	—
104	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
105	C₄×Q₈	—	4	8	no	—
106	C₂×C₄⋊C₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
109	C₂×C₄²	—	2	4	no	—
110	C₂×C₄²	C₂³×C₅	1	2	yes	—
111	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
112	#103	—	4	8	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
115	#95	—	2	4	no	—
116	#95	—	2	8	no	—
117	#95	C₅×D₄	1	4	yes	—
118	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₅	#535	1	2	yes	—
120	C₂²×C₅	#578	1	4	yes	—
121	C₄×C₅	—	2	4	no	—
122	C₄×C₅	—	4	8	no	—
123	C₄×C₅	—	4	4	no	—
124	C₄×C₅	—	8	4	no	—
125	C₄×C₅	#518	1	2	yes	—
126	C₄×C₅	#540	1	2	yes	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	8	no	—
129	C₂×Q₈	—	2	8	no	—
130	C₂×Q₈	C₂²×C₄×C₅	1	8	yes	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
134	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
135	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
136	Q₁₆	—	4	16	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
141	C₄⋊C₄	—	2	16	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	4	8	no	—
146	C₄²	C₂²×C₄×C₅	1	2	yes	—
147	C₄²	C₅×#41	1	2	yes	—
148	C₂×C₅	#2328	1	1	yes	characteristic
149	C₂×C₅	#2950	1	2	yes	—
150	C₂×C₅	#3223	1	4	yes	—
151	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
152	Q₈	—	2	16	no	—
153	Q₈	—	4	8	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
163	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
165	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₅	#22277	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
170	C₂²	C₅×#535	1	2	yes	—
171	C₂²	C₅×#578	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	4	8	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	4	no	—
176	C₄	C₅×#518	1	2	yes	—
177	C₄	C₅×#540	1	2	yes	—
178	C₂	C₅×#2328	1	1	yes	characteristic
179	C₂	C₅×#2950	1	2	yes	—
180	C₂	C₅×#3223	1	4	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 ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅×#494	C₄	semidirect
20. #22277	C₅	direct (N × H)
71. #494	C₄×C₅	semidirect
168. C₅	#22277	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×#494	2
C₅×#2327	2
C₅×#1960	2
C₅×#1878	2
C₅×#1835	2
C₅×#1746	2
#22277	5

Subgroups — C5×#22277 #11944997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22277 #11944997