Subgroups — #13043526

Order 1280.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65791	C₂	1	2	yes	maximal, complemented
2	#10615006	C₂	1	1	yes	maximal, characteristic
3	#10605007	C₂	1	2	yes	maximal, complemented
4	#10604413	C₂	1	8	yes	maximal, complemented
5	#10604150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64859	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#532	C₄	1	1	yes	characteristic
8	C₂×#4094	C₂²	1	2	yes	—
9	#65791	—	2	8	no	—
10	C₂×#3998	C₂²	1	2	yes	—
11	C₂×#3998	C₂²	1	4	yes	—
12	C₂²×#674	C₂²	1	1	yes	characteristic
13	C₂²×#674	C₂²	1	4	yes	—
14	C₂²×#674	C₂²	1	1	yes	characteristic
15	#65523	C₄	1	2	yes	—
16	#65301	C₄	1	4	yes	—
17	D₄×#168	C₂²	1	4	yes	—
18	#65213	C₄	1	1	yes	characteristic
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	C₂²⋊#674	C₂²	1	8	yes	—
21	#64859	C₂²	1	4	yes	—
22	C₄⋊#674	C₂²	1	4	yes	—
23	#33435	—	5	1	no	maximal, Sylow
24	C₂×C₅×#41	C₂×C₄	1	2	yes	—
25	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
26	C₅×#127	C₈	1	1	yes	characteristic
27	C₅×#124	C₈	1	2	yes	—
28	C₅×#121	C₈	1	4	yes	—
29	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	#4094	—	2	4	no	—
32	#4094	—	2	4	no	—
33	C₂×#681	C₂×C₄	1	4	yes	—
34	C₂×#681	C₂×C₄	1	2	yes	—
35	C₂²×#168	C₂×C₄	1	1	yes	characteristic
36	C₂²×#168	C₂³	1	4	yes	—
37	C₂²×#168	C₂³	1	1	yes	characteristic
38	#3998	—	2	4	no	—
39	#3998	—	2	8	no	—
40	C₂×#674	—	2	2	no	—
41	C₂×#674	—	2	8	no	—
42	C₂×#674	D₄	1	4	yes	complemented
43	C₂×#674	C₂³	1	4	yes	—
44	C₂×#674	C₂³	1	2	yes	—
45	C₂×#674	C₂³	1	2	yes	—
46	C₂²⋊#168	C₂×C₄	1	8	yes	—
47	C₄⋊#168	C₂×C₄	1	4	yes	—
48	C₄×#168	C₂×C₄	1	2	yes	—
49	C₄×#168	C₂³	1	2	yes	—
50	C₂×#504	—	5	2	no	—
51	#2931	—	5	1	no	—
52	#2168	—	5	8	no	—
53	#2150	—	5	1	no	—
54	#2113	—	5	2	no	—
55	C₂×#346	—	5	1	no	—
56	C₅×#41	—	2	4	no	—
57	C₅×#41	—	2	4	no	—
58	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
59	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
60	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
61	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
67	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₅	C₂²×C₄	1	2	yes	—
70	#681	—	2	4	no	—
71	#681	—	2	8	no	—
72	C₂×#168	—	2	2	no	—
73	C₂×#168	—	2	4	no	—
74	C₂×#168	—	2	4	no	—
75	C₂×#168	C₂²×C₄	1	2	yes	—
76	C₂×#168	C₂²×C₄	1	4	yes	—
77	C₂×#168	C₂²×C₄	1	1	yes	characteristic
78	C₂×#168	C₂×D₄	1	2	yes	—
79	C₂×#168	#41	1	2	yes	—
80	C₂×#168	C₂⁴	1	1	yes	characteristic
81	#674	—	2	4	no	—
82	#674	—	2	2	no	—
83	#674	C₂×D₄	1	4	yes	—
84	C₂×#131	—	5	2	no	—
85	#532	AGL(1,𝔽₅)	1	1	yes	characteristic
86	#504	—	10	8	no	—
87	C₂×M₃₂	—	5	2	no	—
88	C₂×M₃₂	—	5	4	no	—
89	C₂²×C₁₆	—	5	1	no	—
90	C₂²×C₁₆	—	5	1	no	—
91	C₂²×C₁₆	—	5	4	no	—
92	#436	—	5	4	no	—
93	C₈×D₄	—	5	4	no	—
94	#423	—	5	1	no	—
95	#408	—	5	2	no	—
96	C₂×C₄×C₈	—	5	1	no	—
97	C₄⋊C₁₆	—	5	4	no	—
98	C₂²⋊C₁₆	—	5	8	no	—
99	#346	—	5	4	no	—
100	C₂³×C₅	C₂²×C₈	1	4	yes	—
101	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
102	C₅×Q₈	—	2	4	no	—
103	C₅×D₄	—	2	8	no	—
104	C₅×D₄	—	2	4	no	—
105	C₅×D₄	—	4	4	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	#131	1	2	yes	—
115	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
116	#168	—	2	4	no	—
117	#168	—	2	2	no	—
118	#168	C₄×D₄	1	2	yes	—
119	#168	C₂²×D₄	1	1	yes	characteristic
120	#168	C₂×#41	1	1	yes	characteristic
121	C₂×#41	C₂×AGL(1,𝔽₅)	1	2	yes	—
122	#131	—	10	4	no	—
123	#131	—	10	4	no	—
124	C₂×M₁₆	—	5	2	no	—
125	C₂×M₁₆	—	5	4	no	—
126	C₂²×C₈	—	5	1	no	—
127	C₂²×C₈	—	5	4	no	—
128	C₂²×C₈	—	5	1	no	—
129	C₄⋊Q₈	#170	1	1	yes	characteristic
130	#127	#170	1	1	yes	characteristic
131	#124	#170	1	2	yes	—
132	#121	#170	1	4	yes	—
133	C₄×D₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
134	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
135	M₃₂	—	10	8	no	—
136	M₃₂	—	10	4	no	—
137	C₂×C₁₆	—	5	2	no	—
138	C₂×C₁₆	—	5	4	no	—
139	C₂×C₁₆	—	5	2	no	—
140	C₂×C₁₆	—	5	4	no	—
141	C₂×C₁₆	—	10	8	no	—
142	C₂×C₁₆	—	10	2	no	—
143	C₄⋊C₈	—	5	4	no	—
144	C₂²⋊C₈	—	5	8	no	—
145	C₄×C₈	—	5	2	no	—
146	C₄×C₈	—	5	2	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	2	8	no	—
149	C₂²×C₅	—	2	4	no	—
150	C₂²×C₅	C₈×D₄	1	2	yes	—
151	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	C₈×D₄	1	2	yes	—
156	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₅	C₂×#131	1	1	yes	characteristic
158	#41	—	2	4	no	—
159	#41	—	2	4	no	—
160	C₂×Q₈	C₂×#170	1	2	yes	—
161	C₂×D₄	C₂×#170	1	4	yes	—
162	C₂×D₄	C₂×#170	1	2	yes	—
163	C₂×D₄	C₂×#170	1	4	yes	—
164	C₂²×C₄	C₂×#170	1	2	yes	—
165	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
166	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
167	M₁₆	—	10	8	no	—
168	M₁₆	—	10	4	no	—
169	C₂×C₈	—	5	2	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	5	1	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	2	no	—
174	C₂×C₈	—	5	4	no	—
175	C₂×C₈	—	10	4	no	—
176	C₂×C₈	—	10	4	no	—
177	C₂×C₈	—	10	2	no	—
178	C₄⋊C₄	C₂×#170	1	4	yes	—
179	C₂²⋊C₄	C₂×#170	1	8	yes	—
180	C₄²	C₂×#170	1	1	yes	characteristic
181	C₄²	C₂×#170	1	1	yes	characteristic
182	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
183	C₁₆	—	5	4	no	—
184	C₁₆	—	10	4	no	—
185	C₁₆	—	10	2	no	—
186	C₂×C₅	—	2	2	no	—
187	C₂×C₅	—	4	4	no	—
188	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
189	C₂×C₅	#3407	1	2	yes	—
190	C₂³	C₂²×#170	1	4	yes	—
191	C₂³	C₂²×#170	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	D₄	—	2	8	no	—
194	D₄	—	2	4	no	—
195	D₄	—	4	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#4131	1	2	yes	—
200	C₂×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
201	C₂×C₄	C₂²×#170	1	2	yes	—
202	C₂×C₄	C₂²×#170	1	2	yes	—
203	C₂×C₄	C₂²×#170	1	4	yes	—
204	C₂×C₄	C₂²×#170	1	1	yes	characteristic
205	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₈	—	5	1	no	—
207	C₈	—	5	1	no	—
208	C₈	—	5	2	no	—
209	C₈	—	10	2	no	—
210	C₈	—	10	4	no	—
211	C₅	#33435	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	D₄×#170	1	2	yes	—
216	C₂²	C₂³×#170	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	D₄×#170	1	2	yes	—
221	C₄	C₂×#4131	1	1	yes	characteristic
222	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	4	4	no	—
225	C₂	C₂×D₄×#170	1	1	yes	characteristic
226	C₂	#10617530	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting, prime radical	C₅×#532	C₄
cosocle	C₂×#168	C₂⁴
hypercenter	#532	AGL(1,𝔽₅)
hypocenter	C₅	#33435
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#65791	C₂	semidirect
3. #10605007	C₂	semidirect
4. #10604413	C₂	semidirect
5. #10604150	C₂	semidirect
6. C₂×#64859	C₂	semidirect
42. C₂×#674	D₄	semidirect
211. C₅	#33435	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65791	2
#10615006	2
#10605007	2
#10604413	2
#10604150	2
C₂×#64859	2
#33435	5