Subgroups — #11999314

Order 1280.

458 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4016	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	#10600181	C₂	1	1	yes	maximal, characteristic
4	#10598195	C₂	1	4	yes	maximal
5	C₂×#4016	C₄	1	1	yes	characteristic
6	C₂×C₅×C₄⋊C₈	—	2	2	no	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	#65225	C₄	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
11	#65222	—	2	2	no	—
12	C₄×#683	—	2	2	no	—
13	#65217	C₄	1	2	yes	—
14	C₂×C₄×#168	C₂²	1	2	yes	—
15	#64744	—	2	2	no	—
16	#9872	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	D₄	1	2	yes	—
21	C₂×C₄²×C₅	D₄	1	2	yes	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
25	C₂×#683	—	2	2	no	—
26	C₂×#683	—	2	2	no	—
27	C₂×C₄×Dic₅	—	2	2	no	—
28	C₂²×#168	—	2	2	no	—
29	#4016	—	2	2	no	—
30	#4016	C₈	1	4	yes	complemented
31	C₅×C₄⋊C₈	—	2	4	no	—
32	C₅×C₄⋊C₈	—	4	4	no	—
33	#3963	—	2	2	no	—
34	#3963	—	2	2	no	—
35	#3963	—	2	2	no	—
36	C₄×#168	—	2	2	no	—
37	C₄×#168	Q₈	1	4	yes	—
38	C₄×#125	—	5	1	no	—
39	#1842	D₅	1	1	yes	characteristic
40	#1744	—	5	1	no	—
41	#1279	—	5	4	no	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	2	2	no	—
48	C₂²×Dic₅	—	2	2	no	—
49	C₂×C₅×C₈	—	4	2	no	—
50	C₂×C₅×C₈	—	4	4	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	—	2	2	no	—
54	C₄²×C₅	—	4	2	no	—
55	C₄²×C₅	C₂×C₈	1	2	yes	—
56	C₄²×C₅	M₁₆	1	2	yes	—
57	C₄²×C₅	C₂×Q₈	1	2	yes	—
58	C₄²×C₅	#41	1	2	yes	—
59	C₄⋊Dic₅	—	2	2	no	—
60	C₄⋊Dic₅	—	2	4	no	—
61	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
62	#683	—	2	8	no	—
63	#683	—	4	4	no	—
64	C₄×Dic₅	—	4	2	no	—
65	C₄×Dic₅	—	4	2	no	—
66	C₂×#168	—	2	4	no	—
67	C₂×#168	—	4	4	no	—
68	C₂×#125	—	5	1	no	—
69	C₂×C₄⋊C₈	—	2	2	no	—
70	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
71	C₂×C₄×C₈	—	5	2	no	—
72	#389	—	5	2	no	—
73	#387	—	10	2	no	—
74	#382	—	5	1	no	—
75	C₄×C₄⋊C₄	—	5	2	no	—
76	C₄×C₄⋊C₄	—	10	2	no	—
77	C₄³	C₂×D₅	1	1	yes	characteristic
78	#336	—	10	2	no	—
79	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
87	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅	#124	1	1	yes	characteristic
90	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	#131	1	2	yes	—
94	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₄×C₅	#137	1	2	yes	—
96	C₂×Dic₅	—	2	4	no	—
97	C₂×Dic₅	—	4	2	no	—
98	C₂×Dic₅	—	4	2	no	—
99	C₂×Dic₅	—	4	2	no	—
100	C₂×Dic₅	—	4	2	no	—
101	C₂×Dic₅	—	4	2	no	—
102	C₅×C₈	—	8	2	no	—
103	#168	—	4	4	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	10	2	no	—
106	#125	—	5	4	no	—
107	#125	—	10	2	no	—
108	C₂×C₄⋊C₄	—	5	2	no	—
109	C₂×C₄⋊C₄	—	5	2	no	—
110	C₂×C₄⋊C₄	—	10	2	no	—
111	C₂×C₄⋊C₄	—	10	2	no	—
112	C₂×C₄²	—	10	2	no	—
113	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
114	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
115	C₂×C₄²	D₂₀	1	2	yes	—
116	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
117	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	4	4	no	—
120	C₄×C₈	—	5	4	no	—
121	C₄×C₈	—	10	2	no	—
122	#95	—	10	2	no	—
123	#95	—	10	2	no	—
124	#95	—	10	2	no	—
125	C₂²×C₅	#391	1	1	yes	characteristic
126	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₅	#408	1	1	yes	characteristic
128	C₂²×C₅	#417	1	2	yes	—
129	C₂²×C₅	#421	1	2	yes	—
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	—	2	2	no	—
132	C₄×C₅	—	4	2	no	—
133	C₄×C₅	C₈×Q₈	1	2	yes	—
134	C₄×C₅	#446	1	2	yes	—
135	C₄×C₅	#490	1	1	yes	characteristic
136	C₄×C₅	#491	1	1	yes	characteristic
137	C₄×C₅	#501	1	2	yes	—
138	Dic₅	—	4	4	no	—
139	Dic₅	—	8	2	no	—
140	C₂²×C₄	—	10	2	no	—
141	C₂²×C₄	—	10	2	no	—
142	C₂²×C₄	#685	1	2	yes	—
143	C₂²×C₄	#685	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	10	4	no	—
150	C₂×C₈	—	20	4	no	—
151	C₄⋊C₄	—	5	4	no	—
152	C₄⋊C₄	—	10	8	no	—
153	C₄⋊C₄	—	10	2	no	—
154	C₄⋊C₄	—	10	4	no	—
155	C₄⋊C₄	—	20	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	—	20	2	no	—
161	C₄²	—	20	2	no	—
162	C₄²	C₈×D₅	1	2	yes	—
163	C₄²	#676	1	2	yes	—
164	C₄²	#711	1	2	yes	—
165	C₄²	Q₈×D₅	1	2	yes	—
166	C₂×C₅	#1477	1	2	yes	—
167	C₂×C₅	#1495	1	2	yes	—
168	C₂×C₅	#1984	1	1	yes	characteristic
169	C₂×C₅	#1992	1	1	yes	characteristic
170	C₂×C₅	#1995	1	1	yes	characteristic
171	C₂³	C₂×#685	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	10	4	no	—
179	C₂×C₄	—	20	2	no	—
180	C₂×C₄	—	20	2	no	—
181	C₂×C₄	—	20	2	no	—
182	C₂×C₄	—	20	2	no	—
183	C₂×C₄	—	20	2	no	—
184	C₂×C₄	#3952	1	4	yes	—
185	C₂×C₄	#4033	1	2	yes	—
186	C₂×C₄	#4043	1	2	yes	—
187	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
188	C₂×C₄	C₂×#676	1	1	yes	characteristic
189	C₂×C₄	#4053	1	2	yes	—
190	C₂×C₄	#4078	1	2	yes	—
191	C₂×C₄	#4082	1	1	yes	characteristic
192	C₂×C₄	#4088	1	2	yes	—
193	C₂×C₄	#4090	1	1	yes	characteristic
194	C₈	—	8	2	no	—
195	C₈	—	20	4	no	—
196	C₅	#9872	1	1	yes	characteristic, complemented, Sylow
197	C₂²	#65255	1	2	yes	—
198	C₂²	#65263	1	1	yes	characteristic
199	C₂²	C₂×#3952	1	1	yes	characteristic
200	C₂²	#65418	1	1	yes	characteristic
201	C₂²	#65428	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	20	4	no	—
206	C₄	—	40	2	no	—
207	C₄	#65119	1	2	yes	—
208	C₄	#65120	1	2	yes	—
209	C₄	#65360	1	1	yes	characteristic
210	C₄	#65369	1	2	yes	—
211	C₄	#65371	1	1	yes	characteristic
212	C₂	#10600376	1	1	yes	characteristic
213	C₂	#10601218	1	2	yes	—
214	C₂	#10601265	1	1	yes	characteristic
215	C₂	#10601292	1	2	yes	—
216	C₂	#10601308	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1842	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1842	D₅
hypocenter	C₅	#9872
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
30. #4016	C₈	semidirect
196. C₅	#9872	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4016	2
C₅×#1842	2
#10600181	2
#10598195	2
#9872	5