Subgroups — C₂×#13099521 #72253252

Order 3200.

608 conjugacy classes of subgroups.

The subgroup lattice has 171 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×#11716686	C₂	1	1	yes	maximal, characteristic
3	C₂×#11716679	C₂	1	1	yes	maximal, characteristic
4	#13099521	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅²×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₅²×C₄⋊C₄	—	2	1	no	—
7	C₂×#87804	—	2	1	no	—
8	C₅²×#106	—	2	1	no	—
9	C₅²×#106	C₂²	1	2	yes	—
10	#11716686	—	2	1	no	—
11	#11716686	C₂²	1	2	yes	—
12	#11716679	C₄	1	2	yes	complemented
13	#11716679	C₂²	1	2	yes	—
14	C₂×#87756	C₄	1	2	yes	complemented
15	C₂×#64707	—	5	6	no	maximal
16	C₂²×C₄×C₅²	—	4	1	no	—
17	C₂²×C₄×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅²×C₈	D₄	1	1	yes	characteristic
21	C₂×C₅²×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₅²×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₅²×C₈	C₂³	1	1	yes	characteristic
24	C₅²×C₄⋊C₄	—	2	2	no	—
25	C₅²×C₄⋊C₄	—	4	1	no	—
26	#87804	—	2	2	no	—
27	#87804	—	4	1	no	—
28	C₂×#6029	—	4	1	no	—
29	#87756	C₂×C₄	1	4	yes	complemented
30	C₂×C₅×#106	D₅	1	6	yes	—
31	C₂×#3951	—	5	6	no	—
32	C₂×#3944	—	5	6	no	—
33	#64707	—	5	24	no	—
34	C₂³×C₅²	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅²	—	4	2	no	—
36	C₂×C₄×C₅²	—	8	2	no	—
37	C₂×C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅²	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅²	SD₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅²	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
43	C₅²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₅²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₅²×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₅²×C₈	C₂×Q₈	1	1	yes	characteristic
47	#6029	—	4	2	no	—
48	C₂²×C₅×C₈	C₂×D₅	1	6	yes	—
49	C₂×C₅×C₄⋊C₄	—	2	6	no	—
50	C₂×#681	—	10	6	no	—
51	C₅×#106	—	2	6	no	—
52	C₅×#106	C₂×D₅	1	12	yes	—
53	#3951	—	5	12	no	—
54	#3951	—	10	6	no	—
55	#3944	—	5	12	no	—
56	#3944	—	5	12	no	—
57	C₂×#672	—	5	12	no	—
58	C₂×#365	—	25	1	no	Sylow
59	C₂²×C₅²	—	2	2	no	—
60	C₂²×C₅²	#106	1	1	yes	characteristic
61	C₂²×C₅²	#106	1	1	yes	characteristic
62	C₂²×C₅²	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₅²	—	8	2	no	—
64	C₄×C₅²	#106	1	1	yes	characteristic
65	C₄×C₅²	#106	1	1	yes	characteristic
66	C₄×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₅²	C₂×SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	4	6	no	—
69	C₂²×C₄×C₅	C₂²⋊D₅	1	6	yes	—
70	C₂×C₅×C₈	Dic₁₀	1	6	yes	—
71	C₂×C₅×C₈	Dic₁₀	1	6	yes	—
72	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
73	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
74	C₂×C₅×C₈	C₂²⋊D₅	1	6	yes	—
75	C₂×C₅×C₈	C₂²×D₅	1	6	yes	—
76	C₅×C₄⋊C₄	—	2	12	no	—
77	C₅×C₄⋊C₄	—	4	6	no	—
78	#681	—	10	12	no	—
79	#681	—	20	6	no	—
80	C₂×#168	—	20	6	no	—
81	#672	—	5	24	no	—
82	C₂×M₃₂	—	25	1	no	—
83	C₂×#108	—	25	1	no	—
84	C₂×#106	C₅⋊D₅	1	1	yes	characteristic
85	#365	—	25	4	no	—
86	C₂×C₅²	—	2	1	no	—
87	C₂×C₅²	—	2	1	no	—
88	C₂×C₅²	#365	1	2	yes	complemented
89	C₂×C₅²	C₂×#106	1	1	yes	characteristic
90	C₂³×C₅	#687	1	6	yes	—
91	C₂×C₄×C₅	—	4	12	no	—
92	C₂×C₄×C₅	—	8	12	no	—
93	C₂×C₄×C₅	#683	1	6	yes	—
94	C₂×C₄×C₅	#683	1	6	yes	—
95	C₂×C₄×C₅	#687	1	6	yes	—
96	C₂×C₄×C₅	#688	1	6	yes	—
97	C₂×C₄×C₅	#688	1	6	yes	—
98	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
99	C₅×C₈	#683	1	6	yes	—
100	C₅×C₈	#683	1	6	yes	—
101	C₅×C₈	C₂×Dic₁₀	1	6	yes	—
102	C₅×C₈	C₂×C₄×D₅	1	6	yes	—
103	#168	—	20	12	no	—
104	C₂×M₁₆	—	50	1	no	—
105	C₂²×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	1	no	—
107	M₃₂	—	25	2	no	—
108	M₃₂	—	25	2	no	—
109	C₂×C₁₆	—	25	2	no	—
110	#108	—	25	2	no	—
111	#108	—	50	1	no	—
112	#106	—	2	1	no	—
113	#106	C₂×C₅⋊D₅	1	2	yes	—
114	C₅²	C₂×#365	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₅	—	2	12	no	—
116	C₂²×C₅	#3940	1	6	yes	—
117	C₂²×C₅	#3940	1	6	yes	—
118	C₂²×C₅	C₂×#687	1	6	yes	—
119	C₄×C₅	—	8	12	no	—
120	C₄×C₅	#3940	1	6	yes	—
121	C₄×C₅	#3940	1	6	yes	—
122	C₄×C₅	C₂×#683	1	6	yes	—
123	C₄×C₅	C₂×#688	1	6	yes	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
126	M₁₆	—	50	2	no	—
127	M₁₆	—	100	1	no	—
128	C₂×C₈	—	100	1	no	—
129	C₂×C₈	#6045	1	1	yes	characteristic
130	C₂×C₈	#6045	1	1	yes	characteristic
131	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
132	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
133	C₂×C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
134	C₂×C₈	C₂²×C₅⋊D₅	1	1	yes	characteristic
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₁₆	—	25	4	no	—
138	C₂×C₅	—	2	6	no	—
139	C₂×C₅	—	2	6	no	—
140	C₂×C₅	#64707	1	12	yes	complemented
141	C₂×C₅	C₂×#3940	1	6	yes	—
142	C₂³	#87810	1	1	yes	characteristic
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#87806	1	1	yes	characteristic
146	C₂×C₄	#87806	1	1	yes	characteristic
147	C₂×C₄	#87810	1	1	yes	characteristic
148	C₂×C₄	#87811	1	1	yes	characteristic
149	C₂×C₄	#87811	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
151	C₈	—	100	2	no	—
152	C₈	#87806	1	1	yes	characteristic
153	C₈	#87806	1	1	yes	characteristic
154	C₈	C₂×#6045	1	1	yes	characteristic
155	C₈	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
156	C₅	C₂×#64707	1	6	yes	complemented
157	C₂²	—	2	2	no	—
158	C₂²	#11716675	1	1	yes	characteristic
159	C₂²	#11716675	1	1	yes	characteristic
160	C₂²	C₂×#87810	1	1	yes	characteristic
161	C₄	—	8	2	no	—
162	C₄	#11716675	1	1	yes	characteristic
163	C₄	#11716675	1	1	yes	characteristic
164	C₄	C₂×#87806	1	1	yes	characteristic
165	C₄	C₂×#87811	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#13099521	1	2	yes	direct product factor
169	C₂	C₂×#11716675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11716675
commutator	C₅²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×#106	C₂
socle	C₂²×C₅²	#106
prime radical	C₂³×C₅²	SD₁₆
cosocle	C₂×C₅²×C₈	C₂³
hypercenter	C₂×#106	C₅⋊D₅
hypocenter	C₅²	C₂×#365
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
4. #13099521	C₂	direct (N × H)
12. #11716679	C₄	semidirect
14. C₂×#87756	C₄	semidirect
29. #87756	C₂×C₄	semidirect
88. C₂×C₅²	#365	semidirect
114. C₅²	C₂×#365	semidirect
140. C₂×C₅	#64707	semidirect
156. C₅	C₂×#64707	semidirect
168. C₂	#13099521	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×#106	2
C₂×#11716686	2
C₂×#11716679	2
#13099521	2
C₂×#64707	5

Subgroups — C2×#13099521 #72253252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13099521 #72253252