Subgroups — #11998331

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂×#64766	C₂	1	4	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#384	—	2	1	no	—
8	C₅×#384	—	2	1	no	—
9	C₅×#384	—	2	2	no	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#168	—	2	2	no	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₂×C₄×#168	—	2	1	no	—
15	#64766	—	2	8	no	—
16	#64744	—	2	1	no	—
17	#9691	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	C₂×C₄	1	4	yes	—
19	C₅×C₄⋊Q₈	—	2	4	no	—
20	C₅×C₄⋊Q₈	—	2	4	no	—
21	C₅×C₄⋊Q₈	—	4	8	no	—
22	C₂×C₅×C₄⋊C₄	—	2	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	4	no	—
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂²×#168	—	2	2	no	—
32	C₂²×#168	—	2	2	no	—
33	C₄⋊#168	—	2	2	no	—
34	C₄⋊#168	—	2	4	no	—
35	C₄⋊#168	—	4	4	no	—
36	C₄⋊#168	D₄	1	4	yes	—
37	C₄×#168	—	4	2	no	—
38	#2868	D₅	1	1	yes	characteristic
39	#1842	—	5	1	no	—
40	C₄×C₄⋊C₈	—	5	1	no	—
41	C₂×#333	—	5	4	no	—
42	C₂×C₅×Q₈	—	2	8	no	—
43	C₂×C₅×Q₈	—	4	8	no	—
44	C₂²×C₄×C₅	—	2	4	no	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₅×C₄⋊C₄	—	4	8	no	—
51	C₅×C₄⋊C₄	—	4	8	no	—
52	C₅×C₄⋊C₄	—	4	4	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₅	—	2	4	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	SD₁₆	1	2	yes	—
58	C₄²×C₅	SD₁₆	1	4	yes	—
59	C₄²×C₅	Q₁₆	1	4	yes	—
60	C₄²×C₅	Q₁₆	1	2	yes	—
61	C₄²×C₅	C₂×D₄	1	2	yes	—
62	C₄²×C₅	#41	1	2	yes	—
63	C₂×#168	—	2	4	no	—
64	C₂×#168	—	4	2	no	—
65	C₂×#168	—	4	2	no	—
66	C₂×#168	—	4	2	no	—
67	C₂×#168	—	4	4	no	—
68	C₂×C₄⋊Q₈	Dic₅	1	4	yes	—
69	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
70	C₂×C₄⋊C₈	—	5	2	no	—
71	C₂×C₄⋊C₈	—	5	2	no	—
72	C₂×C₄⋊C₈	—	10	2	no	—
73	C₂×C₄×C₈	—	10	1	no	—
74	#384	—	2	1	no	—
75	#384	—	2	1	no	—
76	#384	—	2	2	no	—
77	C₄³	C₂×D₅	1	1	yes	characteristic
78	#336	—	10	1	no	—
79	#333	—	10	8	no	—
80	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₅×Q₈	—	8	8	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	—	4	4	no	—
86	C₂×C₄×C₅	—	4	8	no	—
87	C₂×C₄×C₅	#100	1	4	yes	—
88	C₂×C₄×C₅	#103	1	4	yes	—
89	C₂×C₄×C₅	#103	1	4	yes	—
90	C₂×C₄×C₅	#103	1	4	yes	—
91	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	#124	1	1	yes	characteristic
94	C₂×C₄×C₅	#127	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅	#137	1	2	yes	—
100	#168	—	4	4	no	—
101	#168	—	8	2	no	—
102	C₂²×Q₈	C₂×Dic₅	1	4	yes	—
103	C₂²×C₈	—	10	2	no	—
104	C₂²×C₈	—	10	2	no	—
105	C₄⋊Q₈	—	2	4	no	—
106	C₄⋊Q₈	—	2	4	no	—
107	C₄⋊Q₈	—	4	8	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄²	C₂×Dic₅	1	2	yes	—
113	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
114	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
115	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
116	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
117	C₄⋊C₈	—	5	4	no	—
118	C₄⋊C₈	—	10	4	no	—
119	C₄⋊C₈	—	10	2	no	—
120	C₄⋊C₈	—	20	4	no	—
121	C₄×C₈	—	20	2	no	—
122	C₂²×C₅	#390	1	1	yes	characteristic
123	C₂²×C₅	C₂×#100	1	2	yes	—
124	C₂²×C₅	C₂×#103	1	1	yes	characteristic
125	C₂²×C₅	C₂×#103	1	2	yes	—
126	C₂²×C₅	#419	1	1	yes	characteristic
127	C₄×C₅	—	2	4	no	—
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	8	4	no	—
130	C₄×C₅	#333	1	8	yes	complemented
131	C₄×C₅	#461	1	2	yes	—
132	C₄×C₅	#462	1	2	yes	—
133	C₄×C₅	#487	1	2	yes	—
134	C₄×C₅	#492	1	1	yes	characteristic
135	C₄×C₅	#494	1	1	yes	characteristic
136	C₂×Q₈	—	2	8	no	—
137	C₂×Q₈	—	4	8	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
140	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₈	—	10	4	no	—
145	C₂×C₈	—	20	2	no	—
146	C₂×C₈	—	20	2	no	—
147	C₂×C₈	—	20	2	no	—
148	C₂×C₈	—	20	4	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	#688	1	2	yes	—
157	C₄²	#688	1	4	yes	—
158	C₄²	#689	1	4	yes	—
159	C₄²	#689	1	2	yes	—
160	C₄²	D₄×D₅	1	2	yes	—
161	C₄²	#711	1	2	yes	—
162	C₂×C₅	C₂×#333	1	2	yes	—
163	C₂×C₅	#1548	1	2	yes	—
164	C₂×C₅	#1956	1	1	yes	characteristic
165	C₂×C₅	#1965	1	1	yes	characteristic
166	C₂×C₅	#1981	1	1	yes	characteristic
167	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
168	Q₈	—	8	8	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	#3965	1	4	yes	—
175	C₂×C₄	#3967	1	4	yes	—
176	C₂×C₄	#3967	1	4	yes	—
177	C₂×C₄	#3967	1	4	yes	—
178	C₂×C₄	D₄×Dic₅	1	2	yes	—
179	C₂×C₄	#4082	1	1	yes	characteristic
180	C₂×C₄	#4084	1	2	yes	—
181	C₂×C₄	#4086	1	1	yes	characteristic
182	C₂×C₄	C₂×#688	1	2	yes	—
183	C₂×C₄	C₂×#688	1	1	yes	characteristic
184	C₂×C₄	#4088	1	2	yes	—
185	C₂×C₄	C₂×#689	1	2	yes	—
186	C₂×C₄	C₂×#689	1	1	yes	characteristic
187	C₈	—	20	4	no	—
188	C₈	—	40	2	no	—
189	C₅	#9691	1	1	yes	characteristic, complemented, Sylow
190	C₂²	C₂×#3965	1	2	yes	—
191	C₂²	#65510	1	1	yes	characteristic
192	C₂²	C₂×#3967	1	1	yes	characteristic
193	C₂²	C₂×#3967	1	2	yes	—
194	C₂²	#65514	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	8	4	no	—
198	C₄	#64766	1	8	yes	complemented
199	C₄	#65370	1	2	yes	—
200	C₄	#65372	1	2	yes	—
201	C₄	#65374	1	1	yes	characteristic
202	C₄	#65379	1	2	yes	—
203	C₄	#65381	1	1	yes	characteristic
204	C₂	#10602559	1	1	yes	characteristic
205	C₂	C₂×#64766	1	2	yes	—
206	C₂	#10602567	1	2	yes	—
207	C₂	#10602571	1	1	yes	characteristic
208	C₂	#10602578	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2868	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2868	D₅
hypocenter	C₅	#9691
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
130. C₄×C₅	#333	semidirect
189. C₅	#9691	semidirect
198. C₄	#64766	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2868	2
C₂×#64766	2
#10600187	2
C₄×C₄⋊#168	2
#9691	5