Subgroups — #11990536

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#64678	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10600270	C₂	1	2	yes	maximal
5	#10598110	C₂	1	2	yes	maximal
6	C₂×C₅×#107	C₄	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊#168	C₂²	1	2	yes	—
10	C₂×#3935	—	2	2	no	—
11	C₂×C₄×#168	C₂²	1	1	yes	characteristic
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	C₅×#340	C₄	1	2	yes	—
14	#64744	—	2	2	no	—
15	#64678	C₄	1	4	yes	complemented
16	C₈×#168	C₄	1	4	yes	complemented
17	#8282	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₅	—	2	2	no	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂²×#168	—	2	1	no	—
24	C₂²×#168	—	2	2	no	—
25	C₂²×#168	D₄	1	1	yes	characteristic
26	C₂²×#168	D₄	1	1	yes	characteristic
27	C₅×#107	—	2	4	no	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₅×#95	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	C₄⋊#168	C₂×C₄	1	4	yes	—
33	#3935	—	4	4	no	—
34	C₄×#168	—	2	1	no	—
35	C₄×#168	C₂×C₄	1	2	yes	—
36	C₄×#168	C₂×C₄	1	2	yes	—
37	C₄×#168	Q₈	1	2	yes	—
38	C₄×#107	D₅	1	1	yes	characteristic
39	#1770	—	5	2	no	—
40	C₂×#335	—	5	1	no	—
41	C₂×C₈²	—	5	1	no	—
42	#1279	—	5	2	no	—
43	C₂²×C₄×C₅	—	2	4	no	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	#41	1	1	yes	characteristic
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	C₄²	1	2	yes	—
49	C₂×C₅×C₈	C₄²	1	2	yes	—
50	C₂×C₅×C₈	M₁₆	1	4	yes	—
51	C₅×C₄⋊C₄	—	2	4	no	—
52	C₅×C₄⋊C₄	—	2	4	no	—
53	C₄²×C₅	—	4	2	no	—
54	C₄²×C₅	—	4	2	no	—
55	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅	#41	1	1	yes	characteristic
59	C₂×#168	—	2	4	no	—
60	C₂×#168	—	2	2	no	—
61	C₂×#168	—	2	1	no	—
62	C₂×#168	—	2	1	no	—
63	C₂×#168	—	2	4	no	—
64	C₂×#168	—	4	2	no	—
65	C₂×#168	—	4	2	no	—
66	C₂×#168	C₄⋊C₄	1	2	yes	—
67	C₂×#168	C₄⋊C₄	1	2	yes	—
68	C₂×#168	D₈	1	2	yes	—
69	C₂×#168	Q₁₆	1	2	yes	—
70	C₂×#107	Dic₅	1	2	yes	—
71	C₂×C₄⋊C₈	—	5	2	no	—
72	C₂×#97	—	10	2	no	—
73	C₂×C₄×C₈	—	5	1	no	—
74	C₂×C₄×C₈	—	5	1	no	—
75	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
76	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
77	#340	Dic₅	1	2	yes	—
78	#336	—	10	2	no	—
79	#335	—	5	4	no	—
80	C₈²	—	5	4	no	—
81	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	4	4	no	—
84	C₂×C₄×C₅	—	4	4	no	—
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	#117	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₅	#131	1	2	yes	—
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	#135	1	1	yes	characteristic
97	C₂×C₄×C₅	#135	1	1	yes	characteristic
98	C₅×C₈	—	2	2	no	—
99	C₅×C₈	#97	1	4	yes	complemented
100	#168	—	2	4	no	—
101	#168	—	2	2	no	—
102	#168	—	4	4	no	—
103	#168	#107	1	4	yes	complemented
104	C₂²×C₈	—	5	1	no	—
105	C₂²×C₈	—	5	1	no	—
106	C₂²×C₈	—	10	1	no	—
107	C₂²×C₈	—	10	2	no	—
108	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
113	#107	—	2	4	no	—
114	C₄⋊C₈	—	5	4	no	—
115	C₄⋊C₈	—	10	2	no	—
116	#97	—	20	4	no	—
117	C₄×C₈	—	5	2	no	—
118	C₄×C₈	—	5	2	no	—
119	C₄×C₈	—	5	2	no	—
120	C₄×C₈	—	10	1	no	—
121	C₄×C₈	C₄×D₅	1	2	yes	—
122	C₄×C₈	C₄×D₅	1	2	yes	—
123	#95	—	2	2	no	—
124	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₅	#436	1	2	yes	—
126	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
127	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
128	C₂²×C₅	#443	1	2	yes	—
129	C₄×C₅	—	2	1	no	—
130	C₄×C₅	—	2	1	no	—
131	C₄×C₅	—	8	4	no	—
132	C₄×C₅	C₂×#97	1	1	yes	characteristic
133	C₄×C₅	#405	1	1	yes	characteristic
134	C₄×C₅	C₂×#107	1	1	yes	characteristic
135	C₄×C₅	#427	1	1	yes	characteristic
136	C₄×C₅	C₄×D₈	1	1	yes	characteristic
137	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
138	C₄×C₅	#446	1	2	yes	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
141	C₂²×C₄	#711	1	1	yes	characteristic
142	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	5	2	no	—
145	C₂×C₈	—	5	2	no	—
146	C₂×C₈	—	5	2	no	—
147	C₂×C₈	—	5	2	no	—
148	C₂×C₈	—	10	4	no	—
149	C₂×C₈	—	10	2	no	—
150	C₂×C₈	—	10	1	no	—
151	C₂×C₈	—	10	1	no	—
152	C₂×C₈	—	10	4	no	—
153	C₂×C₈	—	20	2	no	—
154	C₂×C₈	—	20	2	no	—
155	C₂×C₈	#681	1	4	yes	—
156	C₂×C₈	C₄×Dic₅	1	2	yes	—
157	C₂×C₈	C₄×Dic₅	1	2	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄²	—	4	2	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
163	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
164	C₄²	Q₈×D₅	1	1	yes	characteristic
165	C₄²	#713	1	1	yes	characteristic
166	C₂×C₅	#1589	1	2	yes	—
167	C₂×C₅	#1591	1	2	yes	—
168	C₂×C₅	#1770	1	1	yes	characteristic
169	C₂×C₅	#1772	1	1	yes	characteristic
170	C₂×C₅	C₄×#107	1	1	yes	characteristic
171	C₂³	D₄×Dic₅	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	#4033	1	1	yes	characteristic
177	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄	#4038	1	1	yes	characteristic
179	C₂×C₄	#4039	1	1	yes	characteristic
180	C₂×C₄	D₅×D₈	1	1	yes	characteristic
181	C₂×C₄	#4058	1	1	yes	characteristic
182	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
183	C₂×C₄	#4065	1	1	yes	characteristic
184	C₂×C₄	C₂×#681	1	2	yes	—
185	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
186	C₂×C₄	Q₈×Dic₅	1	1	yes	characteristic
187	C₂×C₄	#4094	1	2	yes	—
188	C₈	—	2	2	no	—
189	C₈	—	5	4	no	—
190	C₈	—	10	4	no	—
191	C₈	—	10	2	no	—
192	C₈	—	20	4	no	—
193	C₈	#3935	1	4	yes	complemented
194	C₅	#8282	1	1	yes	characteristic, complemented, Sylow
195	C₂²	C₄⋊C₄×Dic₅	1	1	yes	characteristic
196	C₂²	#65301	1	2	yes	—
197	C₂²	D₈×Dic₅	1	1	yes	characteristic
198	C₂²	Q₁₆×Dic₅	1	1	yes	characteristic
199	C₂²	#65485	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	#65161	1	1	yes	characteristic
204	C₄	#65162	1	1	yes	characteristic
205	C₄	D₅×#107	1	1	yes	characteristic
206	C₄	#65169	1	1	yes	characteristic
207	C₄	C₂×#3935	1	1	yes	characteristic
208	C₄	#65286	1	1	yes	characteristic
209	C₄	#65313	1	2	yes	—
210	C₂	#10600270	1	1	yes	characteristic
211	C₂	Dic₅×#107	1	1	yes	characteristic
212	C₂	#10601452	1	1	yes	characteristic
213	C₂	#10601674	1	2	yes	—
214	C₂	#10601698	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#107	C₂
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#107	D₅
hypocenter	C₅	#8282
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
15. #64678	C₄	semidirect
16. C₈×#168	C₄	semidirect
99. C₅×C₈	#97	semidirect
103. #168	#107	semidirect
193. C₈	#3935	semidirect
194. C₅	#8282	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#107	2
C₂×#64678	2
C₂×C₈×#168	2
#10600270	2
#10598110	2
#8282	5