Subgroups — #12251941

Order 1280.

826 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	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3944	C₂	1	2	yes	maximal
3	#10600674	C₂	1	8	yes	maximal, complemented
4	C₄×#3944	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	C₄	1	1	yes	characteristic
8	C₅×#405	C₄	1	4	yes	—
9	C₅×#405	C₂²	1	4	yes	—
10	C₂×C₅×#97	C₄	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₂×#3944	C₄	1	16	yes	complemented
13	C₂×#3944	C₂²	1	4	yes	—
14	C₂×#3944	C₂²	1	4	yes	—
15	C₂²×#672	C₂²	1	4	yes	—
16	#64681	C₂²	1	8	yes	complemented
17	C₄×#672	C₂²	1	8	yes	complemented
18	#33324	—	5	1	no	maximal, Sylow
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×M₁₆	C₈	1	8	yes	—
21	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₅×#117	C₈	1	4	yes	—
29	C₅×#117	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
31	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₅×#97	C₂×C₄	1	8	yes	—
34	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
35	C₄×C₅×C₈	C₂³	1	4	yes	—
36	#3944	C₂×C₄	1	16	yes	—
37	#3944	C₂×C₄	1	16	yes	complemented
38	C₂×#672	C₂×C₄	1	16	yes	complemented
39	C₂×#672	C₂³	1	8	yes	—
40	C₂²×M₃₂	—	5	2	no	—
41	C₂×#405	D₅	1	1	yes	characteristic
42	#2108	—	5	8	no	—
43	C₄×M₃₂	—	5	4	no	—
44	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₄²	1	4	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
52	C₅×M₁₆	C₂×C₈	1	16	yes	—
53	C₂×C₅×C₈	C₄²	1	4	yes	—
54	C₂×C₅×C₈	C₄²	1	8	yes	—
55	C₂×C₅×C₈	C₂×C₈	1	8	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
57	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
58	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
62	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
63	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
64	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
65	C₄²×C₅	C₂×C₈	1	4	yes	—
66	C₄²×C₅	C₂²×C₄	1	4	yes	—
67	#672	C₂²×C₄	1	16	yes	—
68	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
69	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
70	C₂×#117	Dic₅	1	1	yes	characteristic
71	C₂×M₃₂	—	5	16	no	—
72	C₂×M₃₂	—	5	4	no	—
73	C₂×M₃₂	—	5	4	no	—
74	C₂²×C₁₆	—	5	4	no	—
75	#405	Dic₅	1	4	yes	—
76	#405	C₂×D₅	1	4	yes	—
77	C₂×#97	Dic₅	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
79	#346	—	5	8	no	—
80	C₄×C₁₆	—	5	8	no	—
81	C₂³×C₅	—	4	2	no	—
82	C₂³×C₅	C₄×C₈	1	4	yes	—
83	C₂³×C₅	C₂²×C₈	1	2	yes	—
84	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
87	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
88	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
91	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₅×C₈	C₂×C₄²	1	4	yes	—
97	C₅×C₈	C₂×C₄²	1	2	yes	—
98	C₅×C₈	C₂²×C₈	1	8	yes	—
99	C₅×C₈	C₂³×C₄	1	2	yes	—
100	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
101	C₂×M₁₆	#168	1	8	yes	—
102	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
103	C₂²×C₈	C₄×D₅	1	8	yes	—
104	C₂²×C₈	C₂×Dic₅	1	2	yes	—
105	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₅	1	2	yes	—
107	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅	1	2	yes	—
109	#117	#168	1	4	yes	—
110	#117	C₂×Dic₅	1	4	yes	—
111	C₂×C₄⋊C₄	#168	1	2	yes	—
112	C₂×C₂²⋊C₄	#168	1	2	yes	—
113	C₂×C₄²	C₂×Dic₅	1	2	yes	—
114	M₃₂	—	5	16	no	—
115	M₃₂	—	5	16	no	—
116	C₂×C₁₆	—	5	16	no	—
117	C₂×C₁₆	—	5	8	no	—
118	#97	C₂×Dic₅	1	8	yes	—
119	C₄×C₈	C₂×Dic₅	1	4	yes	—
120	C₄×C₈	C₂²×D₅	1	4	yes	—
121	C₂²×C₅	—	2	2	no	—
122	C₂²×C₅	—	2	4	no	—
123	C₂²×C₅	—	4	4	no	—
124	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
125	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
126	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₅	—	2	8	no	—
128	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
129	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
131	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
132	C₂⁴	C₂×#168	1	1	yes	characteristic
133	C₂²×C₄	C₂×#168	1	4	yes	—
134	C₂²×C₄	C₂×#168	1	1	yes	characteristic
135	C₂²×C₄	C₂×#168	1	2	yes	—
136	C₂²×C₄	C₂×#168	1	4	yes	—
137	C₂²×C₄	C₄×Dic₅	1	4	yes	—
138	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
140	M₁₆	C₂×#168	1	16	yes	—
141	C₂×C₈	C₂×#168	1	8	yes	—
142	C₂×C₈	C₄×Dic₅	1	4	yes	—
143	C₂×C₈	C₄×Dic₅	1	8	yes	—
144	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
145	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
146	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
147	C₂×C₈	C₂²×Dic₅	1	4	yes	—
148	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
149	C₂×C₈	C₂²×Dic₅	1	2	yes	—
150	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
151	C₄⋊C₄	C₂×#168	1	8	yes	—
152	C₂²⋊C₄	C₂×#168	1	8	yes	—
153	C₄²	C₂×#168	1	4	yes	—
154	C₄²	C₂²×Dic₅	1	4	yes	—
155	C₁₆	—	5	16	no	—
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	—	2	4	no	—
158	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
159	C₂×C₅	#3407	1	2	yes	—
160	C₂³	—	4	2	no	—
161	C₂³	C₄×#168	1	4	yes	—
162	C₂³	C₂²×#168	1	2	yes	—
163	C₂³	C₂²×#168	1	1	yes	characteristic
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	C₄×#168	1	4	yes	—
166	C₂×C₄	C₄×#168	1	8	yes	—
167	C₂×C₄	C₂²×#168	1	8	yes	—
168	C₂×C₄	C₂²×#168	1	2	yes	—
169	C₂×C₄	C₂²×#168	1	4	yes	—
170	C₂×C₄	C₂²×#168	1	1	yes	characteristic
171	C₂×C₄	C₂²×#168	1	2	yes	—
172	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
173	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
174	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
175	C₈	C₂²×#168	1	8	yes	—
176	C₈	C₂×C₄×Dic₅	1	4	yes	—
177	C₈	C₂×C₄×Dic₅	1	2	yes	—
178	C₈	C₂²×C₄×D₅	1	2	yes	—
179	C₅	#33324	1	1	yes	characteristic, complemented, Sylow
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#168	1	2	yes	—
184	C₂²	C₂×C₄×#168	1	4	yes	—
185	C₂²	C₂³×#168	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#168	1	2	yes	—
188	C₄	C₂×C₄×#168	1	4	yes	—
189	C₄	C₂³×#168	1	1	yes	characteristic
190	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#168	1	1	yes	characteristic
194	C₂	#10615511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#405	D₅
hypocenter	C₅	#33324
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #10600674	C₂	semidirect
4. C₄×#3944	C₂	semidirect
12. C₂×#3944	C₄	semidirect
16. #64681	C₂²	semidirect
17. C₄×#672	C₂²	semidirect
37. #3944	C₂×C₄	semidirect
38. C₂×#672	C₂×C₄	semidirect
179. C₅	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂²×#3944	2
#10600674	2
C₄×#3944	2
#33324	5