Subgroups — #11993068

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1845	C₂	1	1	yes	maximal, characteristic
2	#10600558	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10600369	C₂	1	2	yes	maximal
5	#10600367	C₂	1	2	yes	maximal
6	C₂×C₅×#107	—	2	2	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#384	C₂²	1	2	yes	—
9	C₂×#3950	—	2	2	no	—
10	C₂×#3940	—	2	4	no	—
11	#65224	C₂²	1	2	yes	—
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	#64749	—	2	4	no	—
15	C₈×#168	C₄	1	4	yes	complemented
16	#8814	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	2	yes	—
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅	—	2	2	no	—
23	C₂×C₄⋊Dic₅	—	2	2	no	—
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂²×#168	D₄	1	2	yes	—
26	C₂²×#168	D₄	1	2	yes	—
27	C₅×#107	—	4	4	no	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	Q₈	1	2	yes	—
30	#3950	—	4	4	no	—
31	#3940	—	4	8	no	—
32	C₄×#168	C₂×C₄	1	2	yes	—
33	C₄×#168	C₂×C₄	1	2	yes	—
34	C₄×#168	Q₈	1	2	yes	—
35	C₄×#168	Q₈	1	2	yes	—
36	#1845	—	5	1	no	—
37	#1845	D₅	1	1	yes	characteristic
38	#1844	—	5	2	no	—
39	#1834	—	5	2	no	—
40	C₂×C₈²	—	5	1	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅	—	4	2	no	—
46	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
47	C₂×C₅×C₈	SD₁₆	1	4	yes	—
48	C₂×C₅×C₈	SD₁₆	1	4	yes	—
49	C₅×C₄⋊C₄	—	4	4	no	—
50	C₅×C₄⋊C₄	—	4	4	no	—
51	C₅×C₄⋊C₄	—	4	4	no	—
52	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₄⋊Dic₅	—	4	4	no	—
57	C₄⋊Dic₅	—	4	4	no	—
58	C₄⋊Dic₅	—	4	4	no	—
59	C₂×#168	—	2	2	no	—
60	C₂×#168	—	2	2	no	—
61	C₂×#168	C₄⋊C₄	1	4	yes	—
62	C₂×#168	C₄⋊C₄	1	4	yes	—
63	C₂×#168	D₈	1	4	yes	—
64	C₂×#168	Q₁₆	1	4	yes	—
65	C₂×#107	—	2	2	no	—
66	C₂×#107	—	10	2	no	—
67	C₂×#106	—	10	4	no	—
68	C₂×C₄×C₈	—	5	1	no	—
69	C₂×C₄×C₈	—	5	1	no	—
70	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	#384	—	5	2	no	—
72	#384	C₂×D₅	1	2	yes	—
73	#340	—	10	4	no	—
74	C₈²	—	5	4	no	—
75	C₂³×C₅	#127	1	1	yes	characteristic
76	C₂×C₄×C₅	—	8	2	no	—
77	C₂×C₄×C₅	—	8	2	no	—
78	C₂×C₄×C₅	—	8	2	no	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₅	#135	1	2	yes	—
90	C₂×C₄×C₅	#135	1	2	yes	—
91	C₂×Dic₅	—	8	2	no	—
92	C₂×Dic₅	—	8	2	no	—
93	C₂×Dic₅	—	8	2	no	—
94	C₅×C₈	#106	1	8	yes	complemented
95	#168	—	2	4	no	—
96	#168	#107	1	8	yes	complemented
97	C₂²×C₈	—	5	2	no	—
98	C₂²×C₈	—	5	2	no	—
99	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	10	2	no	—
104	C₂×C₄⋊C₄	—	10	2	no	—
105	C₂×C₄⋊C₄	—	10	2	no	—
106	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
107	#107	—	4	4	no	—
108	#107	—	20	4	no	—
109	#106	—	20	8	no	—
110	C₄×C₈	—	5	2	no	—
111	C₄×C₈	—	5	2	no	—
112	C₄×C₈	—	5	2	no	—
113	C₄×C₈	—	5	2	no	—
114	C₄×C₈	Dic₁₀	1	2	yes	—
115	C₄×C₈	C₄×D₅	1	2	yes	—
116	C₂²×C₅	#384	1	1	yes	characteristic
117	C₂²×C₅	#492	1	1	yes	characteristic
118	C₂²×C₅	#492	1	1	yes	characteristic
119	C₂²×C₅	#493	1	1	yes	characteristic
120	C₂²×C₅	#494	1	1	yes	characteristic
121	C₂²×C₅	#495	1	1	yes	characteristic
122	C₂²×C₅	#495	1	1	yes	characteristic
123	C₄×C₅	—	16	2	no	—
124	C₄×C₅	C₂×#106	1	2	yes	—
125	C₄×C₅	C₂×#107	1	2	yes	—
126	C₄×C₅	#427	1	2	yes	—
127	C₄×C₅	#498	1	2	yes	—
128	C₄×C₅	#499	1	2	yes	—
129	C₄×C₅	#500	1	2	yes	—
130	Dic₅	—	16	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	20	2	no	—
133	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
136	C₂×C₈	—	5	4	no	—
137	C₂×C₈	—	5	4	no	—
138	C₂×C₈	—	5	4	no	—
139	C₂×C₈	—	5	4	no	—
140	C₂×C₈	—	10	2	no	—
141	C₂×C₈	—	10	2	no	—
142	C₂×C₈	#683	1	4	yes	—
143	C₂×C₈	#687	1	4	yes	—
144	C₂×C₈	#688	1	4	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	20	4	no	—
149	C₄⋊C₄	—	20	4	no	—
150	C₄⋊C₄	—	20	4	no	—
151	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
153	C₄²	Q₈×D₅	1	1	yes	characteristic
154	C₄²	Q₈×D₅	1	1	yes	characteristic
155	C₂×C₅	#1706	1	1	yes	characteristic
156	C₂×C₅	#1706	1	1	yes	characteristic
157	C₂×C₅	#1707	1	1	yes	characteristic
158	C₂×C₅	#1707	1	1	yes	characteristic
159	C₂×C₅	#1844	1	1	yes	characteristic
160	C₂×C₅	#1845	1	1	yes	characteristic
161	C₂×C₅	#1846	1	1	yes	characteristic
162	C₂³	#4086	1	1	yes	characteristic
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	40	2	no	—
167	C₂×C₄	—	40	2	no	—
168	C₂×C₄	—	40	2	no	—
169	C₂×C₄	#4034	1	1	yes	characteristic
170	C₂×C₄	#4034	1	1	yes	characteristic
171	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄	D₅×D₈	1	2	yes	—
174	C₂×C₄	#4058	1	2	yes	—
175	C₂×C₄	D₅×Q₁₆	1	2	yes	—
176	C₂×C₄	#4065	1	2	yes	—
177	C₂×C₄	C₂×#683	1	1	yes	characteristic
178	C₂×C₄	C₂×#687	1	2	yes	—
179	C₂×C₄	C₂×#688	1	2	yes	—
180	C₂×C₄	#4090	1	1	yes	characteristic
181	C₈	—	5	8	no	—
182	C₈	—	10	4	no	—
183	C₈	#3940	1	8	yes	complemented
184	C₅	#8814	1	1	yes	characteristic, complemented, Sylow
185	C₂²	#65265	1	1	yes	characteristic
186	C₂²	#65365	1	1	yes	characteristic
187	C₂²	#65374	1	1	yes	characteristic
188	C₂²	#65440	1	1	yes	characteristic
189	C₂²	#65444	1	1	yes	characteristic
190	C₂²	#65470	1	1	yes	characteristic
191	C₂²	#65480	1	1	yes	characteristic
192	C₄	—	16	2	no	—
193	C₄	—	80	2	no	—
194	C₄	#65163	1	2	yes	—
195	C₄	#65167	1	2	yes	—
196	C₄	D₅×#107	1	2	yes	—
197	C₄	#65169	1	2	yes	—
198	C₄	C₂×#3940	1	2	yes	—
199	C₄	#65368	1	2	yes	—
200	C₂	#10600369	1	1	yes	characteristic
201	C₂	#10601399	1	1	yes	characteristic
202	C₂	#10601400	1	1	yes	characteristic
203	C₂	#10602067	1	1	yes	characteristic
204	C₂	#10602089	1	1	yes	characteristic
205	C₂	#10602152	1	1	yes	characteristic
206	C₂	#10602153	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1845	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1845	D₅
hypocenter	C₅	#8814
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
15. C₈×#168	C₄	semidirect
94. C₅×C₈	#106	semidirect
96. #168	#107	semidirect
183. C₈	#3940	semidirect
184. C₅	#8814	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1845	2
#10600558	2
C₂×C₈×#168	2
#10600369	2
#10600367	2
#8814	5