Subgroups — #11986243

Order 1280.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1772	C₂	1	1	yes	maximal, characteristic
2	C₂×#64683	C₂	1	1	yes	maximal, characteristic
3	#10600517	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#108	C₄	1	1	yes	characteristic
5	C₅×#427	C₄	1	1	yes	characteristic
6	C₅×#405	—	2	1	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3944	—	2	1	no	—
9	#65214	—	2	1	no	—
10	C₅×#339	—	2	1	no	—
11	#64683	—	2	1	no	—
12	#64683	C₄	1	2	yes	complemented
13	#64675	—	2	1	no	—
14	C₈×#168	C₄	1	2	yes	complemented
15	#7180	—	5	1	no	maximal, Sylow
16	C₂×C₅×M₁₆	—	2	1	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
19	C₅×#117	—	2	1	no	—
20	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
21	C₅×#108	—	2	1	no	—
22	C₅×#108	C₈	1	2	yes	complemented
23	C₅×#107	—	2	1	no	—
24	C₅×#106	C₈	1	2	yes	complemented
25	C₅×#97	—	4	1	no	—
26	C₄×C₅×C₈	—	4	1	no	—
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
31	C₂²⋊#168	—	2	2	no	—
32	#3944	—	2	2	no	—
33	#3944	—	4	1	no	—
34	C₂×#672	—	4	1	no	—
35	C₄⋊#168	—	2	2	no	—
36	C₄×#168	—	2	2	no	—
37	C₂×#364	—	5	1	no	—
38	#1772	D₅	1	1	yes	characteristic
39	#1450	—	5	1	no	—
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
42	C₅×M₁₆	—	2	2	no	—
43	C₅×M₁₆	—	4	1	no	—
44	C₂×C₅×C₈	—	4	1	no	—
45	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
51	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
52	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
53	C₂×C₅×C₈	D₈	1	1	yes	characteristic
54	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
55	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
56	C₂×C₅×C₈	Q₁₆	1	1	yes	characteristic
57	C₅×C₄⋊C₄	—	2	2	no	—
58	C₅×C₂²⋊C₄	—	4	1	no	—
59	C₄²×C₅	—	2	1	no	—
60	C₄²×C₅	—	4	1	no	—
61	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×#168	—	2	4	no	—
64	#672	—	4	2	no	—
65	C₂×M₃₂	—	10	1	no	—
66	#431	—	10	1	no	—
67	C₂×#108	Dic₅	1	1	yes	characteristic
68	#427	Dic₅	1	1	yes	characteristic
69	#405	—	2	1	no	—
70	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	#364	—	5	2	no	—
72	#364	—	10	1	no	—
73	#339	—	2	1	no	—
74	#322	—	10	1	no	—
75	C₈²	—	5	2	no	—
76	C₂³×C₅	#104	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	#95	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅	#102	1	1	yes	characteristic
85	C₂×C₄×C₅	#103	1	1	yes	characteristic
86	C₂×C₄×C₅	#104	1	1	yes	characteristic
87	C₅×C₈	—	4	2	no	—
88	C₅×C₈	C₄×C₈	1	1	yes	characteristic
89	C₅×C₈	#97	1	1	yes	characteristic
90	C₅×C₈	#102	1	1	yes	characteristic
91	C₅×C₈	#103	1	1	yes	characteristic
92	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
93	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
94	C₅×C₈	#106	1	1	yes	characteristic
95	C₅×C₈	#107	1	1	yes	characteristic
96	#168	—	4	4	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
102	M₃₂	—	10	2	no	—
103	M₃₂	—	20	1	no	—
104	C₂×C₁₆	—	20	1	no	—
105	#108	—	2	1	no	—
106	#108	#168	1	2	yes	complemented
107	#107	—	2	1	no	—
108	#106	#168	1	2	yes	complemented
109	C₄⋊C₈	—	10	2	no	—
110	C₂²⋊C₈	—	10	2	no	—
111	#97	—	4	1	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	—	10	2	no	—
114	C₄×C₈	Dic₁₀	1	1	yes	characteristic
115	C₄×C₈	C₄×D₅	1	1	yes	characteristic
116	C₄×C₈	C₄×D₅	1	1	yes	characteristic
117	C₄×C₈	D₂₀	1	1	yes	characteristic
118	C₂²×C₅	—	4	1	no	—
119	C₂²×C₅	D₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₅	Q₈⋊C₈	1	1	yes	characteristic
121	C₂²×C₅	#339	1	1	yes	characteristic
122	C₄×C₅	—	2	2	no	—
123	C₄×C₅	—	4	1	no	—
124	C₄×C₅	—	8	2	no	—
125	C₄×C₅	D₄⋊C₈	1	1	yes	characteristic
126	C₄×C₅	Q₈⋊C₈	1	1	yes	characteristic
127	C₄×C₅	#336	1	1	yes	characteristic
128	C₄×C₅	#340	1	1	yes	characteristic
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
131	M₁₆	—	2	2	no	—
132	M₁₆	—	4	1	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	10	4	no	—
135	C₂×C₈	C₂×#168	1	1	yes	characteristic
136	C₂×C₈	C₂×#168	1	1	yes	characteristic
137	C₂×C₈	#681	1	1	yes	characteristic
138	C₂×C₈	#681	1	1	yes	characteristic
139	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
140	C₂×C₈	#683	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
142	C₂×C₈	#685	1	1	yes	characteristic
143	C₂×C₈	#686	1	1	yes	characteristic
144	C₂×C₈	#687	1	1	yes	characteristic
145	C₂×C₈	#688	1	1	yes	characteristic
146	C₂×C₈	#689	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	4	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	#683	1	1	yes	characteristic
152	C₄²	#685	1	1	yes	characteristic
153	C₁₆	—	20	2	no	—
154	C₂×C₅	—	2	1	no	—
155	C₂×C₅	—	2	1	no	—
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	#1279	1	1	yes	characteristic
158	C₂³	#3969	1	1	yes	characteristic
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
164	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
165	C₂×C₄	#3963	1	1	yes	characteristic
166	C₂×C₄	#3964	1	1	yes	characteristic
167	C₂×C₄	#3967	1	1	yes	characteristic
168	C₂×C₄	#3969	1	1	yes	characteristic
169	C₈	—	4	2	no	—
170	C₈	—	20	4	no	—
171	C₈	C₄×#168	1	1	yes	characteristic
172	C₈	#3935	1	1	yes	characteristic
173	C₈	C₄⋊#168	1	1	yes	characteristic
174	C₈	C₄⋊#168	1	1	yes	characteristic
175	C₈	#3939	1	1	yes	characteristic
176	C₈	#3940	1	1	yes	characteristic
177	C₈	#3941	1	1	yes	characteristic
178	C₈	#3942	1	1	yes	characteristic
179	C₅	#7180	1	1	yes	characteristic, complemented, Sylow
180	C₂²	—	4	1	no	—
181	C₂²	#64752	1	1	yes	characteristic
182	C₂²	D₄⋊#168	1	1	yes	characteristic
183	C₂²	Q₈⋊#168	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	#64744	1	1	yes	characteristic
188	C₄	#64749	1	1	yes	characteristic
189	C₄	D₄⋊#168	1	1	yes	characteristic
190	C₄	Q₈⋊#168	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#10598110	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#3940
commutator	C₅×C₈	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₅×#1772	C₂
socle	C₂×C₅	#1279
prime radical	C₂³×C₅	#104
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	#1772	D₅
hypocenter	C₅	#7180
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
12. #64683	C₄	semidirect
14. C₈×#168	C₄	semidirect
22. C₅×#108	C₈	semidirect
24. C₅×#106	C₈	semidirect
106. #108	#168	semidirect
108. #106	#168	semidirect
179. C₅	#7180	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1772	2
C₂×#64683	2
#10600517	2
#7180	5