Subgroups — C₂²×#64707 #12252306

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	C₂×#64707	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂²×#681	—	2	1	no	—
8	C₂×C₅×#106	—	2	3	no	—
9	C₂×C₅×#106	C₂²	1	6	yes	—
10	C₂×#3951	—	2	3	no	—
11	C₂×#3951	C₂²	1	6	yes	—
12	C₂×#3944	C₄	1	6	yes	complemented
13	C₂×#3944	C₂²	1	6	yes	—
14	C₂²×#672	C₄	1	2	yes	complemented
15	#64707	C₂²	1	16	yes	direct product factor
16	C₂²×#365	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	—	4	1	no	—
18	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₅×C₈	D₄	1	3	yes	—
22	C₂²×C₅×C₈	Q₈	1	3	yes	—
23	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	3	yes	—
25	C₂×C₅×C₄⋊C₄	—	2	6	no	—
26	C₂×C₅×C₄⋊C₄	—	4	3	no	—
27	C₂×#681	—	2	6	no	—
28	C₂×#681	—	4	3	no	—
29	C₂²×#168	—	4	1	no	—
30	C₅×#106	—	2	6	no	—
31	C₅×#106	C₂³	1	4	yes	—
32	#3951	—	2	6	no	—
33	#3951	C₂³	1	4	yes	—
34	#3944	C₂×C₄	1	12	yes	complemented
35	#3944	C₂³	1	4	yes	—
36	C₂×#672	C₂×C₄	1	12	yes	complemented
37	C₂²×M₃₂	—	5	1	no	—
38	C₂²×#108	—	5	1	no	—
39	C₂²×#106	D₅	1	1	yes	characteristic
40	C₂×#365	—	5	12	no	—
41	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	4	6	no	—
43	C₂²×C₄×C₅	—	8	4	no	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
47	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
49	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
50	C₂×C₅×C₈	C₄⋊C₄	1	3	yes	—
51	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
52	C₂×C₅×C₈	C₄⋊C₄	1	3	yes	—
53	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
54	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
55	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
56	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
57	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
58	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	—	4	6	no	—
61	#681	—	2	4	no	—
62	#681	—	4	6	no	—
63	C₂×#168	—	4	6	no	—
64	#672	C₂²×C₄	1	8	yes	complemented
65	C₂²×M₁₆	—	10	1	no	—
66	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
67	C₂²×C₄⋊C₄	—	2	1	no	—
68	C₂×M₃₂	—	5	6	no	—
69	C₂×M₃₂	—	5	6	no	—
70	C₂²×C₁₆	—	5	2	no	—
71	C₂×#108	—	5	6	no	—
72	C₂×#108	—	10	3	no	—
73	C₂×#106	—	2	3	no	—
74	C₂×#106	C₂×D₅	1	6	yes	—
75	#365	—	5	16	no	—
76	C₂³×C₅	—	2	4	no	—
77	C₂³×C₅	#106	1	3	yes	—
78	C₂³×C₅	#106	1	1	yes	characteristic
79	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	—	8	12	no	—
82	C₂×C₄×C₅	#106	1	6	yes	—
83	C₂×C₄×C₅	#106	1	3	yes	—
84	C₂×C₄×C₅	#106	1	3	yes	—
85	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
86	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
88	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
89	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
90	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
91	C₅×C₈	C₂×C₄⋊C₄	1	3	yes	—
92	C₅×C₈	C₂×C₄⋊C₄	1	3	yes	—
93	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
94	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
95	#168	—	4	4	no	—
96	C₂³×C₄	—	4	1	no	—
97	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
98	C₂×M₁₆	—	10	6	no	—
99	C₂×M₁₆	—	20	3	no	—
100	C₂²×C₈	—	20	1	no	—
101	C₂²×C₈	Dic₁₀	1	3	yes	—
102	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
103	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₅	1	3	yes	—
105	C₂²×C₈	C₂²⋊D₅	1	3	yes	—
106	C₂²×C₈	C₂²×D₅	1	3	yes	—
107	C₂×C₄⋊C₄	—	2	6	no	—
108	C₂×C₄⋊C₄	—	4	3	no	—
109	M₃₂	—	5	12	no	—
110	M₃₂	—	5	4	no	—
111	C₂×C₁₆	—	5	12	no	—
112	#108	—	5	4	no	—
113	#108	—	10	6	no	—
114	#106	—	2	6	no	—
115	#106	C₂²×D₅	1	4	yes	—
116	C₂²×C₅	—	2	6	no	—
117	C₂²×C₅	—	2	6	no	—
118	C₂²×C₅	#365	1	4	yes	complemented
119	C₂²×C₅	C₂×#106	1	3	yes	—
120	C₂²×C₅	C₂×#106	1	3	yes	—
121	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
122	C₄×C₅	—	8	4	no	—
123	C₄×C₅	C₂×#106	1	3	yes	—
124	C₄×C₅	C₂×#106	1	3	yes	—
125	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
127	C₂⁴	#687	1	1	yes	characteristic
128	C₂²×C₄	—	4	6	no	—
129	C₂²×C₄	—	8	4	no	—
130	C₂²×C₄	#683	1	3	yes	—
131	C₂²×C₄	#683	1	1	yes	characteristic
132	C₂²×C₄	#687	1	3	yes	—
133	C₂²×C₄	#688	1	3	yes	—
134	C₂²×C₄	#688	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
136	M₁₆	—	10	4	no	—
137	M₁₆	—	20	6	no	—
138	C₂×C₈	—	20	6	no	—
139	C₂×C₈	#683	1	3	yes	—
140	C₂×C₈	#683	1	6	yes	—
141	C₂×C₈	#683	1	3	yes	—
142	C₂×C₈	C₂×Dic₁₀	1	3	yes	—
143	C₂×C₈	C₂×Dic₁₀	1	3	yes	—
144	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
145	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
146	C₂×C₈	C₂×C₂²⋊D₅	1	3	yes	—
147	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	6	no	—
150	C₁₆	—	5	8	no	—
151	C₂×C₅	—	2	1	no	—
152	C₂×C₅	—	2	3	no	—
153	C₂×C₅	C₂×#365	1	6	yes	complemented
154	C₂×C₅	C₂²×#106	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	#3940	1	3	yes	—
157	C₂³	#3940	1	1	yes	characteristic
158	C₂³	C₂×#687	1	3	yes	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	12	no	—
161	C₂×C₄	#3940	1	6	yes	—
162	C₂×C₄	#3940	1	3	yes	—
163	C₂×C₄	#3940	1	3	yes	—
164	C₂×C₄	C₂×#683	1	3	yes	—
165	C₂×C₄	C₂×#683	1	3	yes	—
166	C₂×C₄	C₂×#687	1	3	yes	—
167	C₂×C₄	C₂×#688	1	3	yes	—
168	C₂×C₄	C₂×#688	1	3	yes	—
169	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
170	C₈	—	20	4	no	—
171	C₈	C₂×#683	1	3	yes	—
172	C₈	C₂×#683	1	3	yes	—
173	C₈	C₂²×Dic₁₀	1	1	yes	characteristic
174	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
175	C₅	C₂²×#365	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	#64707	1	4	yes	direct product factor
179	C₂²	C₂×#3940	1	3	yes	—
180	C₂²	C₂×#3940	1	3	yes	—
181	C₂²	C₂²×#687	1	1	yes	characteristic
182	C₄	—	8	4	no	—
183	C₄	C₂×#3940	1	3	yes	—
184	C₄	C₂×#3940	1	3	yes	—
185	C₄	C₂²×#683	1	1	yes	characteristic
186	C₄	C₂²×#688	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#64707	1	6	yes	direct product factor
190	C₂	C₂²×#3940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3940
commutator	C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂²×C₅×#106	C₂
socle	C₂³×C₅	#106
prime radical	C₂⁴×C₅	SD₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂²×#106	D₅
hypocenter	C₅	C₂²×#365
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
4. C₂×#64707	C₂	direct (N × H)
12. C₂×#3944	C₄	semidirect
14. C₂²×#672	C₄	semidirect
15. #64707	C₂²	direct (N × H)
34. #3944	C₂×C₄	semidirect
36. C₂×#672	C₂×C₄	semidirect
64. #672	C₂²×C₄	semidirect
118. C₂²×C₅	#365	semidirect
153. C₂×C₅	C₂×#365	semidirect
175. C₅	C₂²×#365	semidirect
178. C₂²	#64707	direct (N × H)
189. C₂	C₂×#64707	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×#106	2
C₂²×#3951	2
C₂²×#3944	2
C₂×#64707	2
C₂²×#365	5

Subgroups — C22×#64707 #12252306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64707 #12252306