Subgroups — C₂²×#65144 #12968888

Order 1280.

2398 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×#4034	C₂	1	2	yes	maximal
2	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3949	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂²×#3940	C₂	1	2	yes	maximal
6	C₂×#65144	C₂	1	24	yes	maximal, direct product factor
7	C₂³×Dic₁₀	—	2	2	no	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂²×#683	—	2	2	no	—
12	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂³×#168	C₂²	1	1	yes	characteristic
14	C₂×#4034	—	2	12	no	—
15	C₂×#4034	C₂²	1	24	yes	—
16	C₂×C₅×#106	C₂²	1	12	yes	—
17	C₂×#3949	C₂²	1	12	yes	—
18	C₂×C₈×Dic₅	C₂²	1	12	yes	—
19	C₂×#3940	C₂²	1	24	yes	—
20	#65144	C₂²	1	64	yes	direct product factor
21	C₂²×#498	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	—	2	2	no	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂³×Dic₅	—	2	2	no	—
25	C₂³×Dic₅	D₄	1	2	yes	—
26	C₂²×Dic₁₀	—	2	24	no	—
27	C₂²×Dic₁₀	—	4	16	no	—
28	C₂²×C₅×C₈	Q₈	1	8	yes	complemented
29	C₂²×C₅×C₈	C₂³	1	6	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	6	no	—
31	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
32	C₂×C₄⋊Dic₅	—	2	6	no	—
33	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
34	C₂×#683	—	2	24	no	—
35	C₂×C₄×Dic₅	D₄	1	6	yes	—
36	C₂×C₄×Dic₅	C₂³	1	6	yes	—
37	C₂²×#168	Q₈	1	8	yes	complemented
38	C₂²×#168	C₂³	1	6	yes	—
39	#4034	—	2	48	no	—
40	#4034	C₂³	1	32	yes	—
41	C₅×#106	C₂³	1	16	yes	—
42	#3949	C₂³	1	16	yes	—
43	C₈×Dic₅	C₂³	1	16	yes	—
44	#3940	C₂³	1	32	yes	—
45	C₂²×C₄⋊Q₈	—	5	2	no	—
46	C₂×#498	—	5	24	no	—
47	C₂²×#106	—	5	1	no	—
48	C₂²×#106	—	5	2	no	—
49	C₂²×#106	D₅	1	1	yes	characteristic
50	C₂²×C₄×C₈	—	5	1	no	—
51	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	12	no	—
53	C₂²×C₄×C₅	—	4	8	no	—
54	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
58	C₂²×Dic₅	—	2	12	no	—
59	C₂²×Dic₅	—	4	8	no	—
60	C₂²×Dic₅	SD₁₆	1	16	yes	—
61	C₂²×Dic₅	C₂×D₄	1	12	yes	—
62	C₂×Dic₁₀	—	2	32	no	—
63	C₂×Dic₁₀	—	4	96	no	—
64	C₂×C₅×C₈	C₂×Q₈	1	24	yes	complemented
65	C₂×C₅×C₈	C₂⁴	1	4	yes	—
66	C₅×C₄⋊C₄	—	2	12	no	—
67	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
68	C₄⋊Dic₅	—	2	12	no	—
69	C₄⋊Dic₅	C₂⁴	1	8	yes	—
70	#683	—	2	32	no	—
71	C₄×Dic₅	C₂×D₄	1	12	yes	—
72	C₄×Dic₅	C₂⁴	1	4	yes	—
73	C₂×#168	C₂×Q₈	1	24	yes	complemented
74	C₂×#168	C₂⁴	1	4	yes	—
75	C₂³×Q₈	—	10	2	no	—
76	C₂³×C₈	—	5	1	no	—
77	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
78	C₂×C₄⋊Q₈	—	5	24	no	—
79	C₂×C₄⋊Q₈	—	10	12	no	—
80	C₂²×C₄⋊C₄	—	5	2	no	—
81	C₂²×C₄⋊C₄	—	10	2	no	—
82	C₂²×C₄⋊C₄	C₂×D₅	1	2	yes	—
83	C₂²×C₄²	—	5	1	no	—
84	#498	—	5	64	no	—
85	C₂×#106	—	5	12	no	—
86	C₂×#106	—	5	24	no	—
87	C₂×#106	C₂×D₅	1	12	yes	—
88	C₂×C₄×C₈	—	5	12	no	—
89	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
90	C₂³×C₅	C₂×SD₁₆	1	8	yes	—
91	C₂³×C₅	C₂²×D₄	1	3	yes	—
92	C₂×C₄×C₅	—	2	8	no	—
93	C₂×C₄×C₅	—	4	24	no	—
94	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
95	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
97	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
98	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
99	C₂×Dic₅	—	2	8	no	—
100	C₂×Dic₅	—	4	24	no	—
101	C₂×Dic₅	C₂×SD₁₆	1	48	yes	—
102	C₂×Dic₅	C₂²×D₄	1	8	yes	—
103	Dic₁₀	—	4	64	no	—
104	C₅×C₈	C₂²×Q₈	1	8	yes	complemented
105	#168	C₂²×Q₈	1	8	yes	complemented
106	C₂²×Q₈	—	10	24	no	—
107	C₂²×Q₈	—	20	16	no	—
108	C₂³×C₄	—	2	2	no	—
109	C₂³×C₄	—	5	2	no	—
110	C₂³×C₄	—	10	2	no	—
111	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
112	C₂²×C₈	—	5	8	no	—
113	C₂²×C₈	—	5	6	no	—
114	C₂²×C₈	Dic₁₀	1	8	yes	complemented
115	C₂²×C₈	C₂²×D₅	1	6	yes	—
116	C₄⋊Q₈	—	5	32	no	—
117	C₄⋊Q₈	—	10	48	no	—
118	C₂×C₄⋊C₄	—	2	6	no	—
119	C₂×C₄⋊C₄	—	5	12	no	—
120	C₂×C₄⋊C₄	—	10	24	no	—
121	C₂×C₄⋊C₄	—	10	6	no	—
122	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	—
123	C₂×C₄²	—	5	6	no	—
124	C₂×C₄²	—	5	6	no	—
125	#106	—	5	16	no	—
126	#106	—	5	32	no	—
127	#106	C₂²×D₅	1	16	yes	—
128	C₄×C₈	—	5	16	no	—
129	C₂²×C₅	#498	1	16	yes	complemented
130	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
131	C₂²×C₅	C₂²×SD₁₆	1	12	yes	—
132	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
133	C₄×C₅	—	4	8	no	—
134	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
135	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
136	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
137	Dic₅	—	4	8	no	—
138	Dic₅	C₂²×SD₁₆	1	16	yes	—
139	C₂⁴	D₄×D₅	1	1	yes	characteristic
140	C₂×Q₈	—	10	32	no	—
141	C₂×Q₈	—	20	96	no	—
142	C₂²×C₄	—	2	12	no	—
143	C₂²×C₄	—	4	8	no	—
144	C₂²×C₄	—	5	12	no	—
145	C₂²×C₄	—	5	16	no	—
146	C₂²×C₄	—	10	12	no	—
147	C₂²×C₄	—	20	8	no	—
148	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
149	C₂²×C₄	D₄×D₅	1	3	yes	—
150	C₂²×C₄	Q₈×D₅	1	4	yes	—
151	C₂²×C₄	C₂³×D₅	1	3	yes	—
152	C₂×C₈	—	5	24	no	—
153	C₂×C₈	—	5	4	no	—
154	C₂×C₈	C₂×Dic₁₀	1	24	yes	complemented
155	C₂×C₈	C₂³×D₅	1	4	yes	—
156	C₄⋊C₄	—	2	12	no	—
157	C₄⋊C₄	—	5	8	no	—
158	C₄⋊C₄	—	10	32	no	—
159	C₄⋊C₄	—	10	12	no	—
160	C₄⋊C₄	C₂³×D₅	1	8	yes	—
161	C₄²	—	5	12	no	—
162	C₄²	—	5	4	no	—
163	C₂×C₅	C₂×#498	1	12	yes	complemented
164	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₅	C₂³×SD₁₆	1	2	yes	—
166	C₂³	#4034	1	4	yes	—
167	C₂³	D₅×SD₁₆	1	8	yes	—
168	C₂³	C₂×D₄×D₅	1	3	yes	—
169	Q₈	—	20	64	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	24	no	—
172	C₂×C₄	—	5	48	no	—
173	C₂×C₄	—	5	8	no	—
174	C₂×C₄	—	10	8	no	—
175	C₂×C₄	—	20	24	no	—
176	C₂×C₄	#4034	1	12	yes	—
177	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
178	C₂×C₄	C₂×D₄×D₅	1	3	yes	—
179	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
180	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
181	C₈	—	5	8	no	—
182	C₈	C₂²×Dic₁₀	1	8	yes	complemented
183	C₅	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
184	C₂²	#65144	1	16	yes	direct product factor
185	C₂²	C₂×#4034	1	6	yes	—
186	C₂²	C₂×D₅×SD₁₆	1	12	yes	—
187	C₂²	C₂²×D₄×D₅	1	1	yes	characteristic
188	C₄	—	4	8	no	—
189	C₄	—	5	16	no	—
190	C₄	—	20	8	no	—
191	C₄	C₂×#4034	1	6	yes	—
192	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
193	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
194	C₂	C₂×#65144	1	12	yes	direct product factor
195	C₂	C₂²×#4034	1	1	yes	characteristic
196	C₂	C₂²×D₅×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×#106	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×#106	D₅
hypocenter	C₅	C₂²×#498
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
6. C₂×#65144	C₂	direct (N × H)
20. #65144	C₂²	direct (N × H)
28. C₂²×C₅×C₈	Q₈	semidirect
37. C₂²×#168	Q₈	semidirect
64. C₂×C₅×C₈	C₂×Q₈	semidirect
73. C₂×#168	C₂×Q₈	semidirect
104. C₅×C₈	C₂²×Q₈	semidirect
105. #168	C₂²×Q₈	semidirect
114. C₂²×C₈	Dic₁₀	semidirect
129. C₂²×C₅	#498	semidirect
154. C₂×C₈	C₂×Dic₁₀	semidirect
163. C₂×C₅	C₂×#498	semidirect
182. C₈	C₂²×Dic₁₀	semidirect
183. C₅	C₂²×#498	semidirect
184. C₂²	#65144	direct (N × H)
194. C₂	C₂×#65144	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×#4034	2
C₂²×C₅×#106	2
C₂²×#3949	2
C₂²×C₈×Dic₅	2
C₂²×#3940	2
C₂×#65144	2
C₂²×#498	5

Subgroups — C22×#65144 #12968888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65144 #12968888