Subgroups — C₂×C₅×C₂²⋊#168 #13102467

Order 1600.

640 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#168	C₂	1	2	yes	maximal, complemented
3	C₅×C₂²⋊#168	C₂	1	4	yes	maximal, direct product factor
4	C₂⁴×C₅²	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅²	—	2	4	no	—
6	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₄	1	2	yes	—
8	C₂²×C₄×C₅²	C₂²	1	2	yes	—
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂×C₅×#168	—	2	4	no	—
11	C₂×C₅×#168	C₂²	1	4	yes	complemented
12	C₂×C₅×C₂²⋊C₈	—	5	1	no	maximal
13	C₂×C₂²⋊#168	C₅	1	1	yes	maximal, characteristic, direct product factor
14	C₂³×C₅²	—	2	2	no	—
15	C₂³×C₅²	—	2	2	no	—
16	C₂³×C₅²	C₈	1	4	yes	complemented
17	C₂³×C₅²	C₂×C₄	1	2	yes	—
18	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	—	2	2	no	—
20	C₂×C₄×C₅²	—	2	4	no	—
21	C₂×C₄×C₅²	—	2	4	no	—
22	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₅²	D₄	1	4	yes	—
25	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
26	C₅×#168	—	2	4	no	—
27	C₂³×C₄×C₅	—	2	2	no	—
28	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
30	C₂²×C₅×C₈	—	5	2	no	—
31	C₂²×#168	C₂×C₅	1	2	yes	complemented
32	C₅×C₂²⋊C₈	—	5	4	no	—
33	C₂²⋊#168	C₂×C₅	1	4	yes	direct product factor
34	C₂²×C₅²	—	2	8	no	—
35	C₂²×C₅²	—	2	2	no	—
36	C₂²×C₅²	—	2	2	no	—
37	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₅²	C₂×C₈	1	2	yes	—
39	C₂²×C₅²	C₂×C₈	1	4	yes	complemented
40	C₂²×C₅²	M₁₆	1	2	yes	—
41	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₅²	—	2	2	no	—
43	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
44	C₄×C₅²	C₂×D₄	1	2	yes	—
45	C₂⁴×C₅	—	2	2	no	—
46	C₂⁴×C₅	Dic₅	1	1	yes	characteristic
47	C₂⁴×C₅	C₄×C₅	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	16	no	—
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	—	2	4	no	—
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	4	no	—
53	C₂²×C₄×C₅	—	2	4	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	Dic₅	1	1	yes	characteristic
56	C₂²×C₄×C₅	Dic₅	1	2	yes	—
57	C₂²×C₄×C₅	C₄×C₅	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₄×C₅	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₅×C₈	—	5	4	no	—
64	C₂×C₅×C₈	—	10	4	no	—
65	C₂×#168	—	2	4	no	—
66	C₂×#168	C₂²×C₅	1	4	yes	complemented
67	C₂×C₂²⋊C₈	—	5	1	no	Sylow
68	C₂×C₅²	—	2	4	no	—
69	C₂×C₅²	C₂²⋊C₈	1	4	yes	complemented
70	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
73	C₂³×C₅	—	2	8	no	—
74	C₂³×C₅	—	2	8	no	—
75	C₂³×C₅	—	2	2	no	—
76	C₂³×C₅	—	2	2	no	—
77	C₂³×C₅	—	2	2	no	—
78	C₂³×C₅	—	2	2	no	—
79	C₂³×C₅	—	2	8	no	—
80	C₂³×C₅	—	2	4	no	—
81	C₂³×C₅	—	2	2	no	—
82	C₂³×C₅	#168	1	4	yes	complemented
83	C₂³×C₅	C₅×C₈	1	4	yes	complemented
84	C₂³×C₅	C₂×Dic₅	1	2	yes	—
85	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
86	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
87	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	8	no	—
89	C₂×C₄×C₅	—	2	16	no	—
90	C₂×C₄×C₅	—	2	16	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
102	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
104	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
107	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
109	C₅×C₈	—	10	4	no	—
110	#168	—	2	4	no	—
111	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
112	C₂²×C₈	—	5	2	no	—
113	C₂²⋊C₈	—	5	4	no	—
114	C₅²	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₅	—	2	8	no	—
116	C₂²×C₅	—	2	8	no	—
117	C₂²×C₅	—	2	32	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	—	2	2	no	—
121	C₂²×C₅	—	2	8	no	—
122	C₂²×C₅	—	2	8	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	—	2	4	no	—
126	C₂²×C₅	—	2	8	no	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	C₂×#168	1	2	yes	—
130	C₂²×C₅	C₂×#168	1	4	yes	complemented
131	C₂²×C₅	#681	1	2	yes	—
132	C₂²×C₅	C₂²⋊Dic₅	1	2	yes	—
133	C₂²×C₅	C₅×C₂²⋊C₄	1	2	yes	—
134	C₂²×C₅	C₂×C₅×C₈	1	2	yes	—
135	C₂²×C₅	C₂×C₅×C₈	1	4	yes	complemented
136	C₂²×C₅	C₅×M₁₆	1	2	yes	—
137	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
138	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₄×C₅	—	2	8	no	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	2	4	no	—
143	C₄×C₅	—	2	4	no	—
144	C₄×C₅	C₂²⋊Dic₅	1	2	yes	—
145	C₄×C₅	C₅×C₂²⋊C₄	1	2	yes	—
146	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
147	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
148	C₂⁴	C₅×Dic₅	1	1	yes	characteristic
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₅×Dic₅	1	1	yes	characteristic
151	C₂²×C₄	C₅×Dic₅	1	2	yes	—
152	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
153	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
154	C₂×C₈	—	5	4	no	—
155	C₂×C₈	—	10	4	no	—
156	C₂×C₅	—	2	16	no	—
157	C₂×C₅	—	2	4	no	—
158	C₂×C₅	—	2	4	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	2	8	no	—
162	C₂×C₅	—	2	2	no	—
163	C₂×C₅	C₂²⋊#168	1	4	yes	direct product factor
164	C₂×C₅	C₅×C₂²⋊C₈	1	4	yes	complemented
165	C₂×C₅	C₂²×#168	1	1	yes	characteristic
166	C₂×C₅	C₂×#681	1	1	yes	characteristic
167	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
168	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₅	C₂²×C₅×C₈	1	1	yes	characteristic
170	C₂×C₅	C₂×C₅×M₁₆	1	1	yes	characteristic
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₅×#168	1	4	yes	complemented
174	C₂³	C₂×C₅×Dic₅	1	2	yes	—
175	C₂³	C₂×C₅×Dic₅	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
180	C₂×C₄	C₂×C₅×Dic₅	1	1	yes	characteristic
181	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
182	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
183	C₈	—	10	4	no	—
184	C₅	—	2	2	no	—
185	C₅	C₂×C₂²⋊#168	1	1	yes	characteristic, direct product factor
186	C₅	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic, complemented
187	C₂²	—	2	2	no	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	C₂×C₅×#168	1	2	yes	—
191	C₂²	C₂×C₅×#168	1	4	yes	complemented
192	C₂²	C₅×#681	1	2	yes	—
193	C₂²	C₅×C₂²⋊Dic₅	1	2	yes	—
194	C₂²	C₂²×C₅×Dic₅	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	C₅×C₂²⋊Dic₅	1	2	yes	—
197	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
198	C₂	—	2	4	no	—
199	C₂	C₅×C₂²⋊#168	1	4	yes	direct product factor
200	C₂	C₂²×C₅×#168	1	1	yes	characteristic
201	C₂	C₂×C₅×#681	1	1	yes	characteristic
202	C₂	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂²×C₅×C₈
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂×C₄
prime radical	C₂⁴×C₅²	C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#168	C₂	semidirect
3. C₅×C₂²⋊#168	C₂	direct (N × H)
11. C₂×C₅×#168	C₂²	semidirect
13. C₂×C₂²⋊#168	C₅	direct (N × H)
16. C₂³×C₅²	C₈	semidirect
31. C₂²×#168	C₂×C₅	semidirect
33. C₂²⋊#168	C₂×C₅	direct (N × H)
39. C₂²×C₅²	C₂×C₈	semidirect
66. C₂×#168	C₂²×C₅	semidirect
69. C₂×C₅²	C₂²⋊C₈	semidirect
82. C₂³×C₅	#168	semidirect
83. C₂³×C₅	C₅×C₈	semidirect
114. C₅²	C₂×C₂²⋊C₈	semidirect
130. C₂²×C₅	C₂×#168	semidirect
135. C₂²×C₅	C₂×C₅×C₈	semidirect
163. C₂×C₅	C₂²⋊#168	direct (N × H)
164. C₂×C₅	C₅×C₂²⋊C₈	semidirect
173. C₂³	C₅×#168	semidirect
185. C₅	C₂×C₂²⋊#168	direct (N × H)
186. C₅	C₂×C₅×C₂²⋊C₈	semidirect
191. C₂²	C₂×C₅×#168	semidirect
199. C₂	C₅×C₂²⋊#168	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×C₄×C₅²	2
C₂²×C₅×#168	2
C₅×C₂²⋊#168	2
C₂×C₅×C₂²⋊C₈	5
C₂×C₂²⋊#168	5

Subgroups — C2×C5×C22⋊#168 #13102467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₂²⋊#168 #13102467