Subgroups — C₂²×#65120 #12963654

Order 1280.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3946	C₂	1	2	yes	maximal
4	C₂²×#3947	C₂	1	2	yes	maximal
5	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
6	C₂×#65120	C₂	1	24	yes	maximal, direct product factor
7	C₂³×Dic₁₀	C₄	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₂²×#683	C₄	1	2	yes	—
12	C₂²×C₄×Dic₅	C₂²	1	2	yes	—
13	C₂³×#168	C₂²	1	1	yes	characteristic
14	C₂³×#168	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₀	C₄	1	12	yes	—
16	C₂×C₄×Dic₁₀	C₂²	1	12	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	12	yes	—
18	C₂×#3947	C₂²	1	24	yes	—
19	C₂×#3946	C₂²	1	24	yes	—
20	C₂×C₄×#168	C₂²	1	12	yes	—
21	#65120	C₂²	1	64	yes	direct product factor
22	C₂²×#446	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×Dic₅	C₂×C₄	1	2	yes	—
27	C₂³×Dic₅	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₀	—	2	8	no	—
29	C₂²×Dic₁₀	C₂×C₄	1	12	yes	—
30	C₂²×C₅×C₈	—	2	8	no	—
31	C₂²×C₅×C₈	C₂³	1	12	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
33	C₂×C₄²×C₅	C₂³	1	6	yes	—
34	C₂×C₄⋊Dic₅	C₂×C₄	1	12	yes	—
35	C₂×#683	C₂×C₄	1	24	yes	—
36	C₂×C₄×Dic₅	C₂×C₄	1	12	yes	—
37	C₂×C₄×Dic₅	C₂³	1	12	yes	—
38	C₂²×#168	—	2	4	no	—
39	C₂²×#168	Q₈	1	8	yes	complemented
40	C₂²×#168	C₂³	1	6	yes	—
41	C₂²×#168	C₂³	1	6	yes	—
42	C₄×Dic₁₀	C₂×C₄	1	48	yes	—
43	C₄×Dic₁₀	C₂³	1	16	yes	—
44	C₅×C₄⋊C₈	C₂³	1	16	yes	—
45	#3947	C₂³	1	32	yes	—
46	#3946	C₂³	1	32	yes	—
47	C₄×#168	C₂³	1	16	yes	—
48	C₂²×C₄×Q₈	—	5	1	no	—
49	C₂×#446	—	5	24	no	—
50	C₂²×C₄⋊C₈	—	5	2	no	—
51	C₂²×C₄⋊C₈	D₅	1	1	yes	characteristic
52	C₂²×#97	—	5	2	no	—
53	C₂²×C₄×C₈	—	5	1	no	—
54	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	M₁₆	1	8	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₅	#41	1	4	yes	—
62	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
63	C₂²×Dic₅	—	2	8	no	—
64	C₂²×Dic₅	—	2	8	no	—
65	C₂²×Dic₅	C₂²×C₄	1	12	yes	—
66	C₂²×Dic₅	C₂²×C₄	1	12	yes	—
67	C₂×Dic₁₀	—	2	48	no	—
68	C₂×Dic₁₀	C₂²×C₄	1	16	yes	—
69	C₂×C₅×C₈	—	2	24	no	—
70	C₂×C₅×C₈	C₂⁴	1	8	yes	—
71	C₄²×C₅	C₂²×C₄	1	12	yes	—
72	C₄²×C₅	C₂⁴	1	4	yes	—
73	C₄⋊Dic₅	C₂²×C₄	1	16	yes	—
74	#683	C₂²×C₄	1	32	yes	—
75	C₄×Dic₅	C₂²×C₄	1	24	yes	—
76	C₄×Dic₅	C₂⁴	1	8	yes	—
77	C₂×#168	—	2	12	no	—
78	C₂×#168	C₂×Q₈	1	24	yes	complemented
79	C₂×#168	C₂⁴	1	4	yes	—
80	C₂×#168	C₂⁴	1	4	yes	—
81	C₂³×Q₈	—	5	1	no	—
82	C₂³×C₈	—	5	1	no	—
83	C₂³×C₈	—	5	1	no	—
84	C₂³×C₈	C₂×D₅	1	2	yes	—
85	C₂×C₄×Q₈	—	5	12	no	—
86	C₂×C₄×Q₈	—	5	12	no	—
87	C₂²×C₄⋊C₄	—	5	2	no	—
88	C₂²×C₄⋊C₄	—	5	1	no	—
89	C₂²×C₄²	—	5	2	no	—
90	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
91	#446	—	5	64	no	—
92	C₂×C₄⋊C₈	—	5	24	no	—
93	C₂×C₄⋊C₈	C₂×D₅	1	12	yes	—
94	C₂×#97	—	5	24	no	—
95	C₂×C₄×C₈	—	5	12	no	—
96	C₂³×C₅	C₄×Q₈	1	4	yes	—
97	C₂³×C₅	C₂×M₁₆	1	4	yes	—
98	C₂³×C₅	#131	1	4	yes	—
99	C₂³×C₅	C₂³×C₄	1	3	yes	—
100	C₂×C₄×C₅	—	2	12	no	—
101	C₂×C₄×C₅	C₄×Q₈	1	12	yes	—
102	C₂×C₄×C₅	C₂×M₁₆	1	24	yes	—
103	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
105	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
107	C₂×C₄×C₅	C₂×#41	1	6	yes	—
108	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
109	C₂×Dic₅	—	2	24	no	—
110	C₂×Dic₅	—	2	24	no	—
111	C₂×Dic₅	C₂³×C₄	1	8	yes	—
112	C₂×Dic₅	C₂³×C₄	1	8	yes	—
113	Dic₁₀	—	2	32	no	—
114	C₅×C₈	—	2	8	no	—
115	#168	—	2	4	no	—
116	#168	C₂²×Q₈	1	8	yes	complemented
117	C₂²×Q₈	—	5	12	no	—
118	C₂²×Q₈	—	10	8	no	—
119	C₂³×C₄	—	5	2	no	—
120	C₂³×C₄	—	5	2	no	—
121	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
122	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
123	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
124	C₂²×C₈	—	2	8	no	—
125	C₂²×C₈	—	5	8	no	—
126	C₂²×C₈	—	5	6	no	—
127	C₂²×C₈	—	5	6	no	—
128	C₂²×C₈	—	10	4	no	—
129	C₂²×C₈	C₂²×D₅	1	12	yes	—
130	C₄×Q₈	—	5	48	no	—
131	C₄×Q₈	—	5	16	no	—
132	C₂×C₄⋊C₄	—	5	24	no	—
133	C₂×C₄⋊C₄	—	5	12	no	—
134	C₂×C₄²	—	5	12	no	—
135	C₂×C₄²	—	5	12	no	—
136	C₂×C₄²	C₄×D₅	1	6	yes	—
137	C₂×C₄²	C₂²×D₅	1	6	yes	—
138	C₄⋊C₈	—	5	32	no	—
139	C₄⋊C₈	C₂²×D₅	1	16	yes	—
140	#97	—	5	32	no	—
141	C₄×C₈	—	5	16	no	—
142	C₂²×C₅	#446	1	16	yes	complemented
143	C₂²×C₅	C₂×C₄×Q₈	1	6	yes	—
144	C₂²×C₅	C₂²×M₁₆	1	6	yes	—
145	C₂²×C₅	C₂×#131	1	6	yes	—
146	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	C₂×C₄×Q₈	1	6	yes	—
149	C₄×C₅	C₂²×M₁₆	1	8	yes	—
150	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
151	C₄×C₅	C₂²×#41	1	1	yes	characteristic
152	Dic₅	—	2	8	no	—
153	Dic₅	—	2	8	no	—
154	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
155	C₂×Q₈	—	5	16	no	—
156	C₂×Q₈	—	10	48	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	5	12	no	—
159	C₂²×C₄	—	5	12	no	—
160	C₂²×C₄	—	10	8	no	—
161	C₂²×C₄	—	10	8	no	—
162	C₂²×C₄	#676	1	8	yes	—
163	C₂²×C₄	C₂×C₄×D₅	1	6	yes	—
164	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
165	C₂²×C₄	C₂×C₄×D₅	1	6	yes	—
166	C₂²×C₄	#711	1	4	yes	—
167	C₂²×C₄	Q₈×D₅	1	4	yes	—
168	C₂²×C₄	C₂³×D₅	1	3	yes	—
169	C₂×C₈	—	2	24	no	—
170	C₂×C₈	—	5	24	no	—
171	C₂×C₈	—	5	4	no	—
172	C₂×C₈	—	5	4	no	—
173	C₂×C₈	—	10	12	no	—
174	C₂×C₈	C₂³×D₅	1	8	yes	—
175	C₄⋊C₄	—	5	32	no	—
176	C₄⋊C₄	—	5	16	no	—
177	C₄²	—	5	24	no	—
178	C₄²	—	5	8	no	—
179	C₄²	C₂×C₄×D₅	1	12	yes	—
180	C₄²	C₂³×D₅	1	4	yes	—
181	C₂×C₅	C₂×#446	1	12	yes	complemented
182	C₂×C₅	C₂²×C₄×Q₈	1	1	yes	characteristic
183	C₂×C₅	C₂³×M₁₆	1	1	yes	characteristic
184	C₂×C₅	C₂²×#131	1	1	yes	characteristic
185	C₂³	#4033	1	4	yes	—
186	C₂³	C₂×#676	1	4	yes	—
187	C₂³	#4053	1	4	yes	—
188	C₂³	C₂²×C₄×D₅	1	3	yes	—
189	Q₈	—	10	32	no	—
190	C₂×C₄	—	2	12	no	—
191	C₂×C₄	—	5	8	no	—
192	C₂×C₄	—	5	8	no	—
193	C₂×C₄	—	10	24	no	—
194	C₂×C₄	—	10	24	no	—
195	C₂×C₄	#4033	1	12	yes	—
196	C₂×C₄	C₂×#676	1	24	yes	—
197	C₂×C₄	C₂²×C₄×D₅	1	4	yes	—
198	C₂×C₄	C₂²×C₄×D₅	1	4	yes	—
199	C₂×C₄	C₂²×C₄×D₅	1	3	yes	—
200	C₂×C₄	C₂×#711	1	6	yes	—
201	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
202	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
203	C₈	—	2	8	no	—
204	C₈	—	5	8	no	—
205	C₈	—	10	4	no	—
206	C₅	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
207	C₂²	#65120	1	16	yes	direct product factor
208	C₂²	C₂×#4033	1	6	yes	—
209	C₂²	C₂²×#676	1	6	yes	—
210	C₂²	C₂×#4053	1	6	yes	—
211	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	10	8	no	—
214	C₄	—	10	8	no	—
215	C₄	C₂×#4033	1	6	yes	—
216	C₄	C₂²×#676	1	8	yes	—
217	C₄	C₂²×#711	1	1	yes	characteristic
218	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
219	C₂	C₂×#65120	1	12	yes	direct product factor
220	C₂	C₂²×#4033	1	1	yes	characteristic
221	C₂	C₂³×#676	1	1	yes	characteristic
222	C₂	C₂²×#4053	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂²×C₄⋊C₈	D₅
hypocenter	C₅	C₂²×#446
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
6. C₂×#65120	C₂	direct (N × H)
21. #65120	C₂²	direct (N × H)
39. C₂²×#168	Q₈	semidirect
78. C₂×#168	C₂×Q₈	semidirect
116. #168	C₂²×Q₈	semidirect
142. C₂²×C₅	#446	semidirect
181. C₂×C₅	C₂×#446	semidirect
206. C₅	C₂²×#446	semidirect
207. C₂²	#65120	direct (N × H)
219. C₂	C₂×#65120	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×C₄×Dic₁₀	2
C₂²×C₅×C₄⋊C₈	2
C₂²×#3946	2
C₂²×#3947	2
C₂²×C₄×#168	2
C₂×#65120	2
C₂²×#446	5

Subgroups — C22×#65120 #12963654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65120 #12963654