Subgroups — C₂×#13099546 #72253201

Order 3200.

672 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11716682	C₂	1	1	yes	maximal, characteristic
3	C₂×#11716679	C₂	1	1	yes	maximal, characteristic
4	#13099546	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅²×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×#1039	C₄	1	1	yes	characteristic
7	C₂²×#6029	C₄	1	1	yes	characteristic
8	C₅²×M₃₂	C₄	1	2	yes	complemented
9	C₅²×M₃₂	C₂²	1	2	yes	—
10	C₂×C₅²×C₁₆	C₄	1	2	yes	complemented
11	#11716682	C₄	1	2	yes	—
12	#11716682	C₂²	1	2	yes	—
13	#11716679	C₄	1	2	yes	complemented
14	#11716679	C₂²	1	2	yes	—
15	C₂×#87756	C₄	1	2	yes	complemented
16	C₂×#64732	—	5	6	no	maximal
17	C₂²×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂²×#1039	—	2	1	no	—
19	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅²×C₈	C₂³	1	1	yes	characteristic
25	C₄×#1039	—	2	1	no	—
26	C₄×#1039	C₂×C₄	1	2	yes	—
27	C₂×#6029	—	2	1	no	—
28	C₂×#6029	—	2	1	no	—
29	C₂×#6029	C₂×C₄	1	2	yes	—
30	C₅²×C₁₆	C₂×C₄	1	4	yes	complemented
31	#87756	C₂×C₄	1	4	yes	complemented
32	C₂×C₅×M₃₂	D₅	1	6	yes	—
33	C₂×#3947	—	5	6	no	—
34	C₂×#3944	—	5	6	no	—
35	#64732	—	5	24	no	—
36	C₂³×C₅²	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅²	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₅²	M₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅²	M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅²	M₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₂×#1039	—	2	2	no	—
44	C₂×#1039	—	4	2	no	—
45	C₅²×C₈	C₄²	1	1	yes	characteristic
46	C₅²×C₈	C₄²	1	1	yes	characteristic
47	C₅²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₅²×C₈	C₂²×C₄	1	1	yes	characteristic
49	#6029	—	2	2	no	—
50	C₂²×C₅×C₈	C₂×D₅	1	6	yes	—
51	C₂×C₄×Dic₅	—	5	6	no	—
52	C₂²×#168	—	5	6	no	—
53	C₅×M₃₂	Dic₅	1	12	yes	complemented
54	C₅×M₃₂	C₂×D₅	1	12	yes	—
55	C₂×C₅×C₁₆	Dic₅	1	12	yes	complemented
56	#3947	—	5	12	no	—
57	#3947	—	5	12	no	—
58	#3944	—	5	12	no	—
59	#3944	—	5	12	no	—
60	C₂×#672	—	5	12	no	—
61	C₂×#347	—	25	1	no	Sylow
62	C₂²×C₅²	—	2	2	no	—
63	C₂²×C₅²	#97	1	1	yes	characteristic
64	C₂²×C₅²	#97	1	1	yes	characteristic
65	C₂²×C₅²	C₂×M₁₆	1	1	yes	characteristic
66	C₄×C₅²	#97	1	1	yes	characteristic
67	C₄×C₅²	#97	1	1	yes	characteristic
68	C₄×C₅²	C₂×C₄²	1	1	yes	characteristic
69	C₄×C₅²	C₂×M₁₆	1	1	yes	characteristic
70	#1039	—	4	2	no	—
71	C₂²×C₄×C₅	C₄×D₅	1	6	yes	—
72	C₂²×Dic₅	—	10	6	no	—
73	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
74	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
75	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
76	C₂×C₅×C₈	C₂×Dic₅	1	6	yes	—
77	C₂×C₅×C₈	C₂×Dic₅	1	6	yes	—
78	C₂×C₅×C₈	C₂²×D₅	1	6	yes	—
79	C₄×Dic₅	—	5	12	no	—
80	C₄×Dic₅	—	10	6	no	—
81	C₂×#168	—	5	12	no	—
82	C₂×#168	—	10	6	no	—
83	C₂×#168	—	10	6	no	—
84	C₅×C₁₆	C₂×Dic₅	1	24	yes	complemented
85	#672	—	5	24	no	—
86	C₂×M₃₂	—	25	1	no	—
87	C₂×M₃₂	C₅⋊D₅	1	1	yes	characteristic
88	C₂×#97	—	25	1	no	—
89	#347	—	25	4	no	—
90	C₂×C₅²	—	2	1	no	—
91	C₂×C₅²	—	2	1	no	—
92	C₂×C₅²	#347	1	2	yes	complemented
93	C₂×C₅²	C₂×#97	1	1	yes	characteristic
94	C₂³×C₅	#676	1	6	yes	—
95	C₂×C₄×C₅	#676	1	6	yes	—
96	C₂×C₄×C₅	#676	1	6	yes	—
97	C₂×C₄×C₅	#676	1	6	yes	—
98	C₂×C₄×C₅	C₄×Dic₅	1	6	yes	—
99	C₂×C₄×C₅	C₄×Dic₅	1	6	yes	—
100	C₂×C₄×C₅	C₂×C₄×D₅	1	6	yes	—
101	C₂×Dic₅	—	10	12	no	—
102	C₂×Dic₅	—	20	12	no	—
103	C₅×C₈	C₄×Dic₅	1	6	yes	—
104	C₅×C₈	C₄×Dic₅	1	6	yes	—
105	C₅×C₈	C₂×C₄×D₅	1	6	yes	—
106	C₅×C₈	C₂²×Dic₅	1	6	yes	—
107	#168	—	10	12	no	—
108	C₂²×C₈	—	25	1	no	—
109	C₂²×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
110	C₂×C₄²	—	25	1	no	—
111	M₃₂	—	25	2	no	—
112	M₃₂	—	25	2	no	—
113	M₃₂	#1039	1	2	yes	complemented
114	M₃₂	C₂×C₅⋊D₅	1	2	yes	—
115	C₂×C₁₆	—	25	2	no	—
116	C₂×C₁₆	#1039	1	2	yes	complemented
117	#97	—	25	2	no	—
118	#97	—	25	2	no	—
119	C₅²	C₂×#347	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₅	—	2	12	no	—
121	C₂²×C₅	#3947	1	6	yes	—
122	C₂²×C₅	#3947	1	6	yes	—
123	C₂²×C₅	C₂×#676	1	6	yes	—
124	C₄×C₅	#3947	1	6	yes	—
125	C₄×C₅	#3947	1	6	yes	—
126	C₄×C₅	C₂×#676	1	6	yes	—
127	C₄×C₅	C₂×C₄×Dic₅	1	6	yes	—
128	Dic₅	—	20	12	no	—
129	C₂²×C₄	—	50	1	no	—
130	C₂²×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
131	C₂×C₈	—	25	2	no	—
132	C₂×C₈	—	50	1	no	—
133	C₂×C₈	—	50	1	no	—
134	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
135	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
136	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
137	C₂×C₈	C₂×#1039	1	1	yes	characteristic
138	C₂×C₈	C₂×#1039	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₅⋊D₅	1	1	yes	characteristic
140	C₄²	—	25	2	no	—
141	C₄²	—	50	1	no	—
142	C₁₆	—	25	4	no	—
143	C₁₆	C₂×#1039	1	4	yes	complemented
144	C₂×C₅	—	2	6	no	—
145	C₂×C₅	—	2	6	no	—
146	C₂×C₅	#64732	1	12	yes	complemented
147	C₂×C₅	C₂×#3947	1	6	yes	—
148	C₂³	#87799	1	1	yes	characteristic
149	C₂×C₄	—	50	2	no	—
150	C₂×C₄	—	100	2	no	—
151	C₂×C₄	#87799	1	1	yes	characteristic
152	C₂×C₄	#87799	1	1	yes	characteristic
153	C₂×C₄	#87799	1	1	yes	characteristic
154	C₂×C₄	C₄×#1039	1	1	yes	characteristic
155	C₂×C₄	C₄×#1039	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
157	C₈	—	50	2	no	—
158	C₈	C₄×#1039	1	1	yes	characteristic
159	C₈	C₄×#1039	1	1	yes	characteristic
160	C₈	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
161	C₈	C₂²×#1039	1	1	yes	characteristic
162	C₅	C₂×#64732	1	6	yes	complemented
163	C₂²	—	2	2	no	—
164	C₂²	#11716682	1	1	yes	characteristic
165	C₂²	#11716682	1	1	yes	characteristic
166	C₂²	C₂×#87799	1	1	yes	characteristic
167	C₄	—	100	2	no	—
168	C₄	#11716682	1	1	yes	characteristic
169	C₄	#11716682	1	1	yes	characteristic
170	C₄	C₂×#87799	1	1	yes	characteristic
171	C₄	C₂×C₄×#1039	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#13099546	1	2	yes	direct product factor
175	C₂	C₂×#11716682	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#1039
commutator	C₄×C₅²	C₂×C₄²
Frattini	C₂×C₈	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×M₃₂	C₂
socle	C₂²×C₅²	#97
prime radical	C₂³×C₅²	M₁₆
cosocle	C₂×C₅²×C₈	C₂³
hypercenter	C₂×M₃₂	C₅⋊D₅
hypocenter	C₅²	C₂×#347
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
4. #13099546	C₂	direct (N × H)
8. C₅²×M₃₂	C₄	semidirect
10. C₂×C₅²×C₁₆	C₄	semidirect
13. #11716679	C₄	semidirect
15. C₂×#87756	C₄	semidirect
30. C₅²×C₁₆	C₂×C₄	semidirect
31. #87756	C₂×C₄	semidirect
53. C₅×M₃₂	Dic₅	semidirect
55. C₂×C₅×C₁₆	Dic₅	semidirect
84. C₅×C₁₆	C₂×Dic₅	semidirect
92. C₂×C₅²	#347	semidirect
113. M₃₂	#1039	semidirect
116. C₂×C₁₆	#1039	semidirect
119. C₅²	C₂×#347	semidirect
143. C₁₆	C₂×#1039	semidirect
146. C₂×C₅	#64732	semidirect
162. C₅	C₂×#64732	semidirect
174. C₂	#13099546	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×C₅²×M₃₂	2
C₂×#11716682	2
C₂×#11716679	2
#13099546	2
C₂×#64732	5

Subgroups — C2×#13099546 #72253201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13099546 #72253201