Subgroups — C₂²×#64732 #12251953

Order 1280.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3947	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	C₂×#64732	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
7	C₂³×#168	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	C₄	1	6	yes	complemented
9	C₂×C₅×M₃₂	C₂²	1	6	yes	—
10	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
11	C₂×#3947	C₄	1	6	yes	—
12	C₂×#3947	C₂²	1	6	yes	—
13	C₂×#3944	C₄	1	6	yes	complemented
14	C₂×#3944	C₂²	1	6	yes	—
15	C₂²×#672	C₄	1	2	yes	complemented
16	#64732	C₂²	1	16	yes	direct product factor
17	C₂²×#347	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×Dic₅	—	2	1	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
25	C₂²×C₅×C₈	C₂³	1	3	yes	—
26	C₂×C₄×Dic₅	—	2	3	no	—
27	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
28	C₂²×#168	—	2	3	no	—
29	C₂²×#168	—	2	1	no	—
30	C₂²×#168	C₂×C₄	1	6	yes	—
31	C₅×M₃₂	C₂×C₄	1	12	yes	complemented
32	C₅×M₃₂	C₂³	1	4	yes	—
33	C₂×C₅×C₁₆	C₂×C₄	1	12	yes	complemented
34	#3947	C₂×C₄	1	12	yes	—
35	#3947	C₂³	1	4	yes	—
36	#3944	C₂×C₄	1	12	yes	complemented
37	#3944	C₂³	1	4	yes	—
38	C₂×#672	C₂×C₄	1	12	yes	complemented
39	C₂²×M₃₂	—	5	1	no	—
40	C₂²×M₃₂	D₅	1	1	yes	characteristic
41	C₂²×#97	—	5	1	no	—
42	C₂×#347	—	5	12	no	—
43	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₄²	1	3	yes	—
45	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅	M₁₆	1	3	yes	—
47	C₂²×C₄×C₅	M₁₆	1	3	yes	—
48	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₅	—	2	6	no	—
51	C₂²×Dic₅	—	4	4	no	—
52	C₂×C₅×C₈	C₄²	1	3	yes	—
53	C₂×C₅×C₈	C₄²	1	3	yes	—
54	C₂×C₅×C₈	C₄²	1	6	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
57	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
58	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
60	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
61	C₄×Dic₅	—	2	6	no	—
62	C₄×Dic₅	C₂²×C₄	1	4	yes	—
63	C₂×#168	—	2	6	no	—
64	C₂×#168	—	2	6	no	—
65	C₂×#168	C₂²×C₄	1	4	yes	—
66	C₅×C₁₆	C₂²×C₄	1	8	yes	complemented
67	#672	C₂²×C₄	1	8	yes	complemented
68	C₂³×C₈	—	5	1	no	—
69	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
70	C₂²×C₄²	—	5	1	no	—
71	C₂×M₃₂	—	5	6	no	—
72	C₂×M₃₂	—	5	6	no	—
73	C₂×M₃₂	Dic₅	1	6	yes	complemented
74	C₂×M₃₂	C₂×D₅	1	6	yes	—
75	C₂²×C₁₆	—	5	2	no	—
76	C₂²×C₁₆	Dic₅	1	2	yes	complemented
77	C₂×#97	—	5	6	no	—
78	C₂×#97	—	5	6	no	—
79	#347	—	5	16	no	—
80	C₂³×C₅	—	2	4	no	—
81	C₂³×C₅	#97	1	3	yes	—
82	C₂³×C₅	#97	1	1	yes	characteristic
83	C₂³×C₅	C₂×M₁₆	1	3	yes	—
84	C₂×C₄×C₅	#97	1	6	yes	—
85	C₂×C₄×C₅	#97	1	3	yes	—
86	C₂×C₄×C₅	#97	1	3	yes	—
87	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
88	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
89	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
90	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
91	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
92	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂×Dic₅	—	2	4	no	—
94	C₂×Dic₅	—	4	12	no	—
95	C₅×C₈	C₂×C₄²	1	3	yes	—
96	C₅×C₈	C₂×C₄²	1	3	yes	—
97	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
98	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
99	#168	—	2	4	no	—
100	C₂³×C₄	—	10	1	no	—
101	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
102	C₂²×C₈	—	5	6	no	—
103	C₂²×C₈	—	10	1	no	—
104	C₂²×C₈	—	10	3	no	—
105	C₂²×C₈	C₄×D₅	1	3	yes	—
106	C₂²×C₈	C₄×D₅	1	3	yes	—
107	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₅	1	3	yes	—
109	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅	1	3	yes	—
111	C₂×C₄²	—	5	6	no	—
112	C₂×C₄²	—	10	3	no	—
113	M₃₂	—	5	12	no	—
114	M₃₂	—	5	4	no	—
115	M₃₂	C₂×Dic₅	1	12	yes	complemented
116	M₃₂	C₂²×D₅	1	4	yes	—
117	C₂×C₁₆	—	5	12	no	—
118	C₂×C₁₆	C₂×Dic₅	1	12	yes	complemented
119	#97	—	5	12	no	—
120	#97	—	5	4	no	—
121	C₂²×C₅	—	2	6	no	—
122	C₂²×C₅	—	2	6	no	—
123	C₂²×C₅	#347	1	4	yes	complemented
124	C₂²×C₅	C₂×#97	1	3	yes	—
125	C₂²×C₅	C₂×#97	1	3	yes	—
126	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₅	C₂×#97	1	3	yes	—
128	C₄×C₅	C₂×#97	1	3	yes	—
129	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
130	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
131	Dic₅	—	4	4	no	—
132	C₂⁴	#676	1	1	yes	characteristic
133	C₂²×C₄	—	10	6	no	—
134	C₂²×C₄	—	20	4	no	—
135	C₂²×C₄	#676	1	3	yes	—
136	C₂²×C₄	#676	1	1	yes	characteristic
137	C₂²×C₄	#676	1	3	yes	—
138	C₂²×C₄	C₄×Dic₅	1	3	yes	—
139	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
141	C₂×C₈	—	5	4	no	—
142	C₂×C₈	—	10	6	no	—
143	C₂×C₈	—	10	6	no	—
144	C₂×C₈	C₄×Dic₅	1	6	yes	—
145	C₂×C₈	C₄×Dic₅	1	3	yes	—
146	C₂×C₈	C₄×Dic₅	1	3	yes	—
147	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
148	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
149	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
150	C₂×C₈	C₂²×Dic₅	1	3	yes	—
151	C₂×C₈	C₂²×Dic₅	1	3	yes	—
152	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
153	C₄²	—	5	4	no	—
154	C₄²	—	10	6	no	—
155	C₁₆	—	5	8	no	—
156	C₁₆	C₂²×Dic₅	1	8	yes	complemented
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	—	2	3	no	—
159	C₂×C₅	C₂×#347	1	6	yes	complemented
160	C₂×C₅	C₂²×#97	1	1	yes	characteristic
161	C₂³	—	2	4	no	—
162	C₂³	#3947	1	3	yes	—
163	C₂³	#3947	1	1	yes	characteristic
164	C₂³	C₂×#676	1	3	yes	—
165	C₂×C₄	—	10	4	no	—
166	C₂×C₄	—	20	12	no	—
167	C₂×C₄	#3947	1	6	yes	—
168	C₂×C₄	#3947	1	3	yes	—
169	C₂×C₄	#3947	1	3	yes	—
170	C₂×C₄	C₂×#676	1	3	yes	—
171	C₂×C₄	C₂×#676	1	3	yes	—
172	C₂×C₄	C₂×#676	1	3	yes	—
173	C₂×C₄	C₂×C₄×Dic₅	1	3	yes	—
174	C₂×C₄	C₂×C₄×Dic₅	1	3	yes	—
175	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
176	C₈	—	10	4	no	—
177	C₈	C₂×C₄×Dic₅	1	3	yes	—
178	C₈	C₂×C₄×Dic₅	1	3	yes	—
179	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
180	C₈	C₂³×Dic₅	1	1	yes	characteristic
181	C₅	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#64732	1	4	yes	direct product factor
185	C₂²	C₂×#3947	1	3	yes	—
186	C₂²	C₂×#3947	1	3	yes	—
187	C₂²	C₂²×#676	1	1	yes	characteristic
188	C₄	—	20	4	no	—
189	C₄	C₂×#3947	1	3	yes	—
190	C₄	C₂×#3947	1	3	yes	—
191	C₄	C₂²×#676	1	1	yes	characteristic
192	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#64732	1	6	yes	direct product factor
196	C₂	C₂²×#3947	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂³×C₅	#97
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	C₂²×#347
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₂×#64732	C₂	direct (N × H)
8. C₂×C₅×M₃₂	C₄	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
13. C₂×#3944	C₄	semidirect
15. C₂²×#672	C₄	semidirect
16. #64732	C₂²	direct (N × H)
31. C₅×M₃₂	C₂×C₄	semidirect
33. C₂×C₅×C₁₆	C₂×C₄	semidirect
36. #3944	C₂×C₄	semidirect
38. C₂×#672	C₂×C₄	semidirect
66. C₅×C₁₆	C₂²×C₄	semidirect
67. #672	C₂²×C₄	semidirect
73. C₂×M₃₂	Dic₅	semidirect
76. C₂²×C₁₆	Dic₅	semidirect
115. M₃₂	C₂×Dic₅	semidirect
118. C₂×C₁₆	C₂×Dic₅	semidirect
123. C₂²×C₅	#347	semidirect
156. C₁₆	C₂²×Dic₅	semidirect
159. C₂×C₅	C₂×#347	semidirect
181. C₅	C₂²×#347	semidirect
184. C₂²	#64732	direct (N × H)
195. C₂	C₂×#64732	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×M₃₂	2
C₂²×#3947	2
C₂²×#3944	2
C₂×#64732	2
C₂²×#347	5

Subgroups — C22×#64732 #12251953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64732 #12251953