Subgroups — #11986237

Order 1280.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 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×#64681	C₂	1	1	yes	maximal, characteristic
3	C₂×#64677	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#107	C₄	1	2	yes	—
5	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₄×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂²×#672	—	2	1	no	—
8	C₂×C₄⋊#168	—	2	1	no	—
9	C₅×#340	—	2	1	no	—
10	#64681	—	2	1	no	—
11	#64681	C₄	1	2	yes	complemented
12	#64677	—	2	1	no	—
13	#64677	C₄	1	2	yes	complemented
14	#7178	—	5	1	no	maximal, Sylow
15	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄²×C₅	—	4	1	no	—
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂²×#168	—	4	1	no	—
21	C₅×#107	—	2	2	no	—
22	C₅×#107	C₈	1	4	yes	complemented
23	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	D₄	1	1	yes	characteristic
26	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₅×#95	—	4	1	no	—
28	C₂×#672	—	2	4	no	—
29	C₂×#672	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	4	1	no	—
32	C₂×#346	—	5	1	no	—
33	C₄×#107	D₅	1	1	yes	characteristic
34	C₂×#334	—	5	1	no	—
35	C₂²×C₄×C₅	—	4	2	no	—
36	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
39	C₂×C₅×C₈	—	2	2	no	—
40	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
45	C₂×C₅×C₈	M₁₆	1	2	yes	—
46	C₅×C₄⋊C₄	—	2	4	no	—
47	C₅×C₄⋊C₄	—	4	2	no	—
48	C₄²×C₅	—	8	2	no	—
49	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅	SD₁₆	1	1	yes	characteristic
52	C₄²×C₅	Q₁₆	1	1	yes	characteristic
53	C₂×#168	—	4	2	no	—
54	C₂×#168	—	8	2	no	—
55	#672	—	2	4	no	—
56	C₂²×C₁₆	—	10	1	no	—
57	C₂×#107	Dic₅	1	2	yes	—
58	C₂×C₄⋊C₈	—	10	1	no	—
59	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
60	C₄×C₄⋊C₄	—	2	1	no	—
61	#346	—	5	2	no	—
62	#346	—	10	1	no	—
63	#340	—	2	1	no	—
64	#334	—	5	2	no	—
65	#334	—	10	1	no	—
66	C₂³×C₅	#102	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	1	no	—
69	C₂×C₄×C₅	—	2	1	no	—
70	C₂×C₄×C₅	—	4	4	no	—
71	C₂×C₄×C₅	—	8	2	no	—
72	C₂×C₄×C₅	—	8	2	no	—
73	C₂×C₄×C₅	#95	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₅	#102	1	1	yes	characteristic
76	C₂×C₄×C₅	#103	1	1	yes	characteristic
77	C₂×C₄×C₅	#103	1	1	yes	characteristic
78	C₂×C₄×C₅	#104	1	2	yes	—
79	C₂×C₄×C₅	#106	1	1	yes	characteristic
80	C₂×C₄×C₅	#107	1	1	yes	characteristic
81	C₅×C₈	—	2	1	no	—
82	C₅×C₈	—	2	1	no	—
83	C₅×C₈	C₄×C₈	1	1	yes	characteristic
84	C₅×C₈	#97	1	1	yes	characteristic
85	C₅×C₈	C₄⋊C₈	1	2	yes	—
86	#168	—	8	2	no	—
87	C₂²×C₈	—	20	1	no	—
88	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
89	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄²	—	4	1	no	—
92	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
93	C₂×C₁₆	—	10	4	no	—
94	C₂×C₁₆	—	10	2	no	—
95	#107	—	2	2	no	—
96	#107	#168	1	4	yes	complemented
97	C₄⋊C₈	—	10	2	no	—
98	C₄⋊C₈	—	20	1	no	—
99	C₄×C₈	Dic₁₀	1	1	yes	characteristic
100	C₄×C₈	C₄×D₅	1	1	yes	characteristic
101	C₄×C₈	C₄×D₅	1	1	yes	characteristic
102	C₄×C₈	D₂₀	1	1	yes	characteristic
103	#95	—	4	1	no	—
104	C₂²×C₅	D₄⋊C₈	1	2	yes	—
105	C₂²×C₅	#340	1	1	yes	characteristic
106	C₂²×C₅	#360	1	2	yes	—
107	C₂²×C₅	#361	1	1	yes	characteristic
108	C₂²×C₅	#362	1	1	yes	characteristic
109	C₄×C₅	—	2	1	no	—
110	C₄×C₅	—	2	1	no	—
111	C₄×C₅	—	2	1	no	—
112	C₄×C₅	—	2	1	no	—
113	C₄×C₅	—	8	4	no	—
114	C₄×C₅	Q₈⋊C₈	1	2	yes	—
115	C₄×C₅	#336	1	1	yes	characteristic
116	C₄×C₅	#339	1	1	yes	characteristic
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	#686	1	1	yes	characteristic
119	C₂²×C₄	#687	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	20	2	no	—
123	C₂×C₈	—	40	2	no	—
124	C₂×C₈	C₂×#168	1	2	yes	—
125	C₂×C₈	#681	1	2	yes	—
126	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
127	C₂×C₈	#683	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
129	C₂×C₈	#685	1	1	yes	characteristic
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄²	—	8	2	no	—
133	C₄²	#683	1	1	yes	characteristic
134	C₄²	#685	1	1	yes	characteristic
135	C₄²	#688	1	1	yes	characteristic
136	C₄²	#689	1	1	yes	characteristic
137	C₁₆	—	10	4	no	—
138	C₂×C₅	#1279	1	1	yes	characteristic
139	C₂×C₅	#1333	1	2	yes	—
140	C₂×C₅	#1334	1	2	yes	—
141	C₂×C₅	#1386	1	1	yes	characteristic
142	C₂×C₅	#1387	1	1	yes	characteristic
143	C₂³	#3964	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	#3939	1	1	yes	characteristic
151	C₂×C₄	#3940	1	1	yes	characteristic
152	C₂×C₄	#3941	1	1	yes	characteristic
153	C₂×C₄	#3942	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊#168	1	2	yes	—
155	C₂×C₄	#3963	1	1	yes	characteristic
156	C₂×C₄	#3967	1	1	yes	characteristic
157	C₂×C₄	#3969	1	2	yes	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	40	2	no	—
161	C₈	C₄×#168	1	1	yes	characteristic
162	C₈	#3935	1	1	yes	characteristic
163	C₈	C₄⋊#168	1	2	yes	—
164	C₅	#7178	1	1	yes	characteristic, complemented, Sylow
165	C₂²	#64749	1	1	yes	characteristic
166	C₂²	D₄⋊#168	1	2	yes	—
167	C₂²	#64783	1	1	yes	characteristic
168	C₂²	#64785	1	1	yes	characteristic
169	C₂²	#64786	1	2	yes	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	8	4	no	—
175	C₄	#64744	1	1	yes	characteristic
176	C₄	#64752	1	1	yes	characteristic
177	C₄	Q₈⋊#168	1	2	yes	—
178	C₂	#10598110	1	1	yes	characteristic
179	C₂	#10598214	1	1	yes	characteristic
180	C₂	#10598217	1	1	yes	characteristic
181	C₂	#10598253	1	2	yes	—
182	C₂	#10598255	1	2	yes	—

	subgroup	G/H
center	C₂³	#3964
commutator	C₅×C₈	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₄×C₅×#107	C₂
socle, prime radical	C₂³×C₅	#102
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	C₄×#107	D₅
hypocenter	C₅	#7178
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #64681	C₄	semidirect
13. #64677	C₄	semidirect
22. C₅×#107	C₈	semidirect
96. #107	#168	semidirect
164. C₅	#7178	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×#64681	2
C₂×#64677	2
#7178	5