Subgroups — C₂²×#65473 #12963339

Order 1280.

2790 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈×Dic₅	C₂	1	2	yes	maximal
3	C₂²×#3967	C₂	1	2	yes	maximal
4	C₂×#65473	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#3949	C₂	1	1	yes	maximal, characteristic
6	C₂²×#3947	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₅×Q₈	C₂²	1	2	yes	—
8	C₂²×C₅×Q₁₆	C₄	1	16	yes	complemented
9	C₂²×C₅×Q₁₆	C₂²	1	12	yes	—
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₈×Dic₅	—	2	12	no	—
12	C₂×Q₈×Dic₅	C₂²	1	24	yes	—
13	C₂²×C₄⋊Dic₅	—	2	2	no	—
14	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂²×C₄×Dic₅	—	2	2	no	—
16	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
17	C₂³×#168	C₂²	1	1	yes	characteristic
18	C₂×#3967	C₂²	1	24	yes	—
19	#65473	C₂²	1	64	yes	direct product factor
20	C₂×#3949	C₂²	1	12	yes	—
21	C₂×#3947	C₂²	1	12	yes	—
22	C₂²×#441	—	5	1	no	maximal, Sylow
23	C₂²×C₅×Q₈	—	2	6	no	—
24	C₂²×C₅×Q₈	—	2	8	no	—
25	C₂²×C₅×Q₈	C₂×C₄	1	16	yes	—
26	C₂²×C₅×Q₈	C₂³	1	12	yes	—
27	C₂³×C₄×C₅	—	2	2	no	—
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂³×Dic₅	—	2	2	no	—
30	C₂³×Dic₅	D₄	1	2	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	96	yes	complemented
32	C₂×C₅×Q₁₆	C₂³	1	16	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
34	C₂²×C₅×C₈	C₂³	1	6	yes	—
35	Q₈×Dic₅	—	2	48	no	—
36	Q₈×Dic₅	C₂³	1	32	yes	—
37	C₂×C₄⋊Dic₅	—	2	6	no	—
38	C₂×C₄⋊Dic₅	—	2	24	no	—
39	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
40	C₂×C₄×Dic₅	—	2	24	no	—
41	C₂×C₄×Dic₅	D₄	1	6	yes	—
42	C₂×C₄×Dic₅	C₂³	1	6	yes	—
43	C₂²×#168	—	2	4	no	—
44	C₂²×#168	C₂³	1	6	yes	—
45	#3967	C₂³	1	32	yes	—
46	#3949	C₂³	1	16	yes	—
47	#3947	C₂³	1	16	yes	—
48	C₂³×Q₁₆	D₅	1	1	yes	characteristic
49	C₂²×C₄×Q₈	—	5	2	no	—
50	C₂×#441	—	5	24	no	—
51	C₂²×#106	—	5	1	no	—
52	C₂²×#103	—	5	2	no	—
53	C₂²×#97	—	5	1	no	—
54	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×Q₈	—	2	48	no	—
56	C₂×C₅×Q₈	—	2	12	no	—
57	C₂×C₅×Q₈	—	2	24	no	—
58	C₂×C₅×Q₈	C₂²×C₄	1	48	yes	—
59	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₅	—	2	16	no	—
61	C₂²×C₄×C₅	—	2	12	no	—
62	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₅	#41	1	4	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
66	C₂²×Dic₅	—	2	8	no	—
67	C₂²×Dic₅	—	2	12	no	—
68	C₂²×Dic₅	—	4	8	no	—
69	C₂²×Dic₅	C₂×D₄	1	12	yes	—
70	C₅×Q₁₆	C₂²×C₄	1	64	yes	complemented
71	C₂×C₅×C₈	C₂²×C₄	1	24	yes	—
72	C₂×C₅×C₈	C₂⁴	1	4	yes	—
73	C₄⋊Dic₅	—	2	12	no	—
74	C₄⋊Dic₅	—	2	32	no	—
75	C₄⋊Dic₅	C₂⁴	1	8	yes	—
76	C₄×Dic₅	—	2	32	no	—
77	C₄×Dic₅	C₂×D₄	1	12	yes	—
78	C₄×Dic₅	C₂⁴	1	4	yes	—
79	C₂×#168	—	2	12	no	—
80	C₂×#168	C₂⁴	1	4	yes	—
81	C₂³×Q₈	C₂×D₅	1	2	yes	—
82	C₂²×Q₁₆	Dic₅	1	16	yes	complemented
83	C₂²×Q₁₆	C₂×D₅	1	12	yes	—
84	C₂³×C₈	—	5	1	no	—
85	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
86	C₂×C₄×Q₈	—	5	24	no	—
87	C₂×C₄×Q₈	—	10	12	no	—
88	C₂²×C₄⋊C₄	—	5	2	no	—
89	C₂²×C₄⋊C₄	—	10	2	no	—
90	C₂²×C₄²	—	5	1	no	—
91	C₂²×C₄²	—	10	2	no	—
92	#441	—	5	64	no	—
93	C₂×#106	—	5	12	no	—
94	C₂×#103	—	5	24	no	—
95	C₂×#97	—	5	12	no	—
96	C₂³×C₅	C₄×D₄	1	4	yes	—
97	C₂³×C₅	#137	1	8	yes	—
98	C₂³×C₅	C₂²×D₄	1	3	yes	—
99	C₅×Q₈	—	2	48	no	—
100	C₅×Q₈	—	2	8	no	—
101	C₅×Q₈	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₅	—	2	48	no	—
103	C₂×C₄×C₅	—	2	8	no	—
104	C₂×C₄×C₅	C₄×D₄	1	12	yes	—
105	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
106	C₂×C₄×C₅	C₂²×D₄	1	3	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	8	no	—
111	C₂×Dic₅	—	4	24	no	—
112	C₂×Dic₅	C₂²×D₄	1	8	yes	—
113	C₅×C₈	C₂³×C₄	1	8	yes	—
114	#168	—	2	4	no	—
115	C₂²×Q₈	—	2	6	no	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	C₂×Dic₅	1	16	yes	—
118	C₂²×Q₈	C₂²×D₅	1	12	yes	—
119	C₂³×C₄	—	2	2	no	—
120	C₂³×C₄	—	5	2	no	—
121	C₂³×C₄	—	10	2	no	—
122	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
123	C₂×Q₁₆	C₂×Dic₅	1	96	yes	complemented
124	C₂×Q₁₆	C₂²×D₅	1	16	yes	—
125	C₂²×C₈	—	5	6	no	—
126	C₂²×C₈	—	10	4	no	—
127	C₂²×C₈	C₂×Dic₅	1	8	yes	—
128	C₂²×C₈	C₂²×D₅	1	6	yes	—
129	C₄×Q₈	—	5	32	no	—
130	C₄×Q₈	—	10	48	no	—
131	C₂×C₄⋊C₄	—	5	12	no	—
132	C₂×C₄⋊C₄	—	10	24	no	—
133	C₂×C₄⋊C₄	—	10	6	no	—
134	C₂×C₄²	—	5	6	no	—
135	C₂×C₄²	—	5	6	no	—
136	C₂×C₄²	—	10	24	no	—
137	#106	—	5	16	no	—
138	#103	—	5	32	no	—
139	#97	—	5	16	no	—
140	C₂²×C₅	#441	1	16	yes	complemented
141	C₂²×C₅	C₂×C₄×D₄	1	6	yes	—
142	C₂²×C₅	C₂×#137	1	12	yes	—
143	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₅	—	2	16	no	—
145	C₄×C₅	C₂×C₄×D₄	1	6	yes	—
146	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
147	C₄×C₅	C₂²×#41	1	1	yes	characteristic
148	Dic₅	—	2	8	no	—
149	Dic₅	—	4	8	no	—
150	C₂⁴	D₄×D₅	1	1	yes	characteristic
151	C₂×Q₈	—	2	12	no	—
152	C₂×Q₈	—	2	24	no	—
153	C₂×Q₈	—	2	48	no	—
154	C₂×Q₈	C₂²×Dic₅	1	48	yes	—
155	C₂×Q₈	C₂³×D₅	1	8	yes	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	2	16	no	—
158	C₂²×C₄	—	5	12	no	—
159	C₂²×C₄	—	10	8	no	—
160	C₂²×C₄	—	10	12	no	—
161	C₂²×C₄	—	20	8	no	—
162	C₂²×C₄	D₄×D₅	1	3	yes	—
163	C₂²×C₄	#711	1	4	yes	—
164	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
165	C₂²×C₄	C₂³×D₅	1	3	yes	—
166	Q₁₆	C₂²×Dic₅	1	64	yes	complemented
167	C₂×C₈	—	5	4	no	—
168	C₂×C₈	—	10	12	no	—
169	C₂×C₈	C₂²×Dic₅	1	24	yes	—
170	C₂×C₈	C₂³×D₅	1	4	yes	—
171	C₄⋊C₄	—	5	8	no	—
172	C₄⋊C₄	—	10	32	no	—
173	C₄⋊C₄	—	10	12	no	—
174	C₄²	—	5	12	no	—
175	C₄²	—	5	4	no	—
176	C₄²	—	10	32	no	—
177	C₂×C₅	C₂×#441	1	12	yes	complemented
178	C₂×C₅	C₂²×C₄×D₄	1	1	yes	characteristic
179	C₂×C₅	C₂²×#137	1	2	yes	—
180	C₂³	#4064	1	8	yes	—
181	C₂³	D₄×Dic₅	1	4	yes	—
182	C₂³	C₂×D₄×D₅	1	3	yes	—
183	Q₈	—	2	48	no	—
184	Q₈	—	2	8	no	—
185	Q₈	C₂³×Dic₅	1	16	yes	—
186	C₂×C₄	—	2	48	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	5	8	no	—
189	C₂×C₄	—	10	24	no	—
190	C₂×C₄	—	10	8	no	—
191	C₂×C₄	—	20	24	no	—
192	C₂×C₄	D₄×Dic₅	1	12	yes	—
193	C₂×C₄	C₂×D₄×D₅	1	3	yes	—
194	C₂×C₄	C₂×#711	1	6	yes	—
195	C₂×C₄	C₂³×Dic₅	1	6	yes	—
196	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
197	C₈	—	10	4	no	—
198	C₈	C₂³×Dic₅	1	8	yes	—
199	C₅	C₂²×#441	1	1	yes	characteristic, complemented, Sylow
200	C₂²	#65473	1	16	yes	direct product factor
201	C₂²	C₂×#4064	1	12	yes	—
202	C₂²	C₂×D₄×Dic₅	1	6	yes	—
203	C₂²	C₂²×D₄×D₅	1	1	yes	characteristic
204	C₄	—	2	16	no	—
205	C₄	—	10	8	no	—
206	C₄	—	20	8	no	—
207	C₄	C₂×D₄×Dic₅	1	6	yes	—
208	C₄	C₂²×#711	1	1	yes	characteristic
209	C₄	C₂⁴×Dic₅	1	1	yes	characteristic
210	C₂	C₂×#65473	1	12	yes	direct product factor
211	C₂	C₂²×#4064	1	2	yes	—
212	C₂	C₂²×D₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅
commutator	C₄×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂³×C₅×Q₁₆	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂³×Q₁₆	D₅
hypocenter	C₅	C₂²×#441
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×#65473	C₂	direct (N × H)
8. C₂²×C₅×Q₁₆	C₄	semidirect
19. #65473	C₂²	direct (N × H)
31. C₂×C₅×Q₁₆	C₂×C₄	semidirect
70. C₅×Q₁₆	C₂²×C₄	semidirect
82. C₂²×Q₁₆	Dic₅	semidirect
123. C₂×Q₁₆	C₂×Dic₅	semidirect
140. C₂²×C₅	#441	semidirect
166. Q₁₆	C₂²×Dic₅	semidirect
177. C₂×C₅	C₂×#441	semidirect
199. C₅	C₂²×#441	semidirect
200. C₂²	#65473	direct (N × H)
210. C₂	C₂×#65473	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₅×Q₁₆	2
C₂²×Q₈×Dic₅	2
C₂²×#3967	2
C₂×#65473	2
C₂²×#3949	2
C₂²×#3947	2
C₂²×#441	5

Subgroups — C22×#65473 #12963339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65473 #12963339