Subgroups — C₂²×#65353 #12968891

Order 1280.

1966 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₂²×#4016	C₂	1	1	yes	maximal, characteristic
3	C₂×#65353	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#3939	C₂	1	2	yes	maximal
5	C₂²×#3940	C₂	1	2	yes	maximal
6	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅×#125	—	2	6	no	—
8	C₂×C₅×#125	C₂²	1	12	yes	—
9	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂²×#683	—	2	2	no	—
14	C₂³×#168	C₂²	1	1	yes	characteristic
15	C₂³×#168	C₂²	1	1	yes	characteristic
16	C₂×#4016	—	2	6	no	—
17	C₂×#4016	C₂²	1	12	yes	—
18	#65353	C₂²	1	64	yes	direct product factor
19	C₂×#3940	C₂²	1	24	yes	—
20	C₂×#3939	C₂²	1	24	yes	—
21	C₂×C₄×#168	C₂²	1	12	yes	—
22	C₂²×#499	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	—	2	2	no	—
24	C₂³×C₄×C₅	D₄	1	2	yes	—
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×Dic₅	—	2	2	no	—
27	C₅×#125	—	2	24	no	—
28	C₅×#125	C₂³	1	16	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	24	no	—
30	C₂×C₅×C₄⋊C₄	—	2	6	no	—
31	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
32	C₂×C₄²×C₅	D₄	1	6	yes	—
33	C₂×C₄²×C₅	C₂³	1	6	yes	—
34	C₂×C₄⋊Dic₅	—	2	6	no	—
35	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
36	C₂×#683	—	2	24	no	—
37	C₂²×#168	Q₈	1	8	yes	—
38	C₂²×#168	Q₈	1	8	yes	—
39	C₂²×#168	C₂³	1	6	yes	—
40	C₂²×#168	C₂³	1	6	yes	—
41	#4016	—	2	24	no	—
42	#4016	C₂³	1	16	yes	—
43	#3940	C₂³	1	32	yes	—
44	#3939	C₂³	1	32	yes	—
45	C₄×#168	C₂³	1	16	yes	—
46	C₂²×#125	—	5	1	no	—
47	C₂²×#125	D₅	1	1	yes	characteristic
48	C₂×#499	—	5	24	no	—
49	C₂²×#107	—	5	2	no	—
50	C₂²×#106	—	5	2	no	—
51	C₂²×C₄×C₈	—	5	1	no	—
52	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	2	8	no	—
54	C₂²×C₄×C₅	—	2	12	no	—
55	C₂²×C₄×C₅	—	4	8	no	—
56	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
61	C₂²×Dic₅	—	2	12	no	—
62	C₂²×Dic₅	—	4	8	no	—
63	C₅×C₄⋊C₄	—	2	32	no	—
64	C₅×C₄⋊C₄	—	2	12	no	—
65	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
66	C₄²×C₅	C₂×D₄	1	12	yes	—
67	C₄²×C₅	C₂⁴	1	4	yes	—
68	C₄⋊Dic₅	—	2	12	no	—
69	C₄⋊Dic₅	C₂⁴	1	8	yes	—
70	#683	—	2	32	no	—
71	C₂×#168	C₂×Q₈	1	24	yes	—
72	C₂×#168	C₂×Q₈	1	24	yes	—
73	C₂×#168	C₂⁴	1	4	yes	—
74	C₂×#168	C₂⁴	1	4	yes	—
75	C₂³×C₈	—	5	1	no	—
76	C₂³×C₈	—	5	1	no	—
77	C₂×#125	—	2	6	no	—
78	C₂×#125	—	5	12	no	—
79	C₂×#125	—	10	6	no	—
80	C₂×#125	C₂×D₅	1	12	yes	—
81	C₂²×C₄⋊C₄	—	2	2	no	—
82	C₂²×C₄⋊C₄	—	5	2	no	—
83	C₂²×C₄⋊C₄	—	10	2	no	—
84	C₂²×C₄⋊C₄	C₂×D₅	1	2	yes	—
85	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
86	#499	—	5	64	no	—
87	C₂×#107	—	5	24	no	—
88	C₂×#106	—	5	24	no	—
89	C₂×C₄×C₈	—	5	12	no	—
90	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
91	C₂³×C₅	#135	1	8	yes	—
92	C₂³×C₅	C₂²×D₄	1	3	yes	—
93	C₂×C₄×C₅	—	2	24	no	—
94	C₂×C₄×C₅	—	2	8	no	—
95	C₂×C₄×C₅	—	4	24	no	—
96	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
97	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
98	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
99	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
100	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
101	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₅	—	2	8	no	—
103	C₂×Dic₅	—	4	24	no	—
104	#168	C₂²×Q₈	1	8	yes	—
105	#168	C₂²×Q₈	1	8	yes	—
106	C₂³×C₄	—	2	2	no	—
107	C₂³×C₄	—	10	2	no	—
108	C₂³×C₄	C₂²⋊D₅	1	2	yes	—
109	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
110	C₂²×C₈	—	5	8	no	—
111	C₂²×C₈	—	5	8	no	—
112	C₂²×C₈	—	5	6	no	—
113	C₂²×C₈	—	5	6	no	—
114	#125	—	2	24	no	—
115	#125	—	5	16	no	—
116	#125	—	10	24	no	—
117	#125	C₂²×D₅	1	16	yes	—
118	C₂×C₄⋊C₄	—	2	24	no	—
119	C₂×C₄⋊C₄	—	2	6	no	—
120	C₂×C₄⋊C₄	—	5	12	no	—
121	C₂×C₄⋊C₄	—	10	6	no	—
122	C₂×C₄⋊C₄	—	10	24	no	—
123	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	—
124	C₂×C₄²	C₂²⋊D₅	1	6	yes	—
125	C₂×C₄²	C₂²×D₅	1	6	yes	—
126	#107	—	5	32	no	—
127	#106	—	5	32	no	—
128	C₄×C₈	—	5	16	no	—
129	C₂²×C₅	#499	1	16	yes	complemented
130	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
131	C₂²×C₅	C₂×#135	1	12	yes	—
132	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
133	C₄×C₅	—	2	8	no	—
134	C₄×C₅	—	4	8	no	—
135	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
136	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
137	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
138	Dic₅	—	4	8	no	—
139	C₂⁴	C₂×C₂²⋊D₅	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	2	12	no	—
142	C₂²×C₄	—	4	8	no	—
143	C₂²×C₄	—	10	12	no	—
144	C₂²×C₄	—	20	8	no	—
145	C₂²×C₄	Q₈×D₅	1	4	yes	—
146	C₂²×C₄	Q₈×D₅	1	4	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅	1	12	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
149	C₂²×C₄	C₂³×D₅	1	3	yes	—
150	C₂×C₈	—	5	24	no	—
151	C₂×C₈	—	5	24	no	—
152	C₂×C₈	—	5	4	no	—
153	C₂×C₈	—	5	4	no	—
154	C₄⋊C₄	—	2	32	no	—
155	C₄⋊C₄	—	2	12	no	—
156	C₄⋊C₄	—	5	8	no	—
157	C₄⋊C₄	—	10	12	no	—
158	C₄⋊C₄	—	10	32	no	—
159	C₄⋊C₄	C₂³×D₅	1	8	yes	—
160	C₄²	C₂×C₂²⋊D₅	1	12	yes	—
161	C₄²	C₂³×D₅	1	4	yes	—
162	C₂×C₅	C₂×#499	1	12	yes	complemented
163	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
164	C₂×C₅	C₂²×#135	1	2	yes	—
165	C₂³	#4090	1	4	yes	—
166	C₂³	#4096	1	8	yes	—
167	C₂³	C₂²×C₂²⋊D₅	1	3	yes	—
168	C₂×C₄	—	2	24	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	4	24	no	—
171	C₂×C₄	—	10	8	no	—
172	C₂×C₄	—	20	24	no	—
173	C₂×C₄	#4090	1	12	yes	—
174	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
175	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
176	C₂×C₄	C₂²×C₂²⋊D₅	1	8	yes	—
177	C₂×C₄	C₂²×C₂²⋊D₅	1	3	yes	—
178	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
179	C₈	—	5	8	no	—
180	C₈	—	5	8	no	—
181	C₅	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
182	C₂²	#65353	1	16	yes	direct product factor
183	C₂²	C₂×#4090	1	6	yes	—
184	C₂²	C₂×#4096	1	12	yes	—
185	C₂²	C₂³×C₂²⋊D₅	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	—	4	8	no	—
188	C₄	—	20	8	no	—
189	C₄	C₂×#4090	1	6	yes	—
190	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
191	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
192	C₂	C₂×#65353	1	12	yes	direct product factor
193	C₂	C₂²×#4090	1	1	yes	characteristic
194	C₂	C₂²×#4096	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×#125	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×#125	D₅
hypocenter	C₅	C₂²×#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
3. C₂×#65353	C₂	direct (N × H)
18. #65353	C₂²	direct (N × H)
129. C₂²×C₅	#499	semidirect
162. C₂×C₅	C₂×#499	semidirect
181. C₅	C₂²×#499	semidirect
182. C₂²	#65353	direct (N × H)
192. C₂	C₂×#65353	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×#125	2
C₂²×#4016	2
C₂×#65353	2
C₂²×#3939	2
C₂²×#3940	2
C₂²×C₄×#168	2
C₂²×#499	5

Subgroups — C22×#65353 #12968891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65353 #12968891