Subgroups — #11993075

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1834	C₂	1	1	yes	maximal, characteristic
2	#10600555	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10600370	C₂	1	2	yes	maximal
5	#10600367	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#384	C₂²	1	2	yes	—
8	C₂×#3939	—	2	4	no	—
9	#65224	C₂²	1	2	yes	—
10	C₂×C₄×#168	C₂²	1	1	yes	characteristic
11	C₂×C₄×#168	C₂²	1	1	yes	characteristic
12	C₅×#340	—	2	2	no	—
13	#64771	—	2	2	no	—
14	#64749	—	2	4	no	—
15	C₈×#168	C₄	1	4	yes	complemented
16	#8816	—	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	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×C₈	Q₈	1	2	yes	—
29	#3939	—	4	8	no	—
30	C₄×#168	C₂×C₄	1	2	yes	—
31	C₄×#168	C₂×C₄	1	2	yes	—
32	C₄×#168	Q₈	1	2	yes	—
33	C₄×#168	Q₈	1	2	yes	—
34	#1845	—	5	2	no	—
35	#1834	—	5	1	no	—
36	#1834	—	5	2	no	—
37	#1834	D₅	1	1	yes	characteristic
38	C₂×C₈²	—	5	1	no	—
39	C₂²×C₄×C₅	—	4	2	no	—
40	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅	—	4	2	no	—
44	C₂×C₅×C₈	—	2	4	no	—
45	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
46	C₅×C₄⋊C₄	—	4	4	no	—
47	C₅×C₄⋊C₄	—	4	4	no	—
48	C₅×C₄⋊C₄	—	4	4	no	—
49	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₄⋊Dic₅	—	4	4	no	—
54	C₄⋊Dic₅	—	4	4	no	—
55	C₄⋊Dic₅	—	4	4	no	—
56	C₂×#168	—	2	4	no	—
57	C₂×#168	C₄⋊C₄	1	4	yes	—
58	C₂×#168	C₄⋊C₄	1	4	yes	—
59	C₂×#168	D₈	1	4	yes	—
60	C₂×#168	Q₁₆	1	4	yes	—
61	C₂×#107	—	10	4	no	—
62	C₂×C₄×C₈	—	5	1	no	—
63	C₂×C₄×C₈	—	5	1	no	—
64	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
65	#384	—	5	2	no	—
66	#384	C₂×D₅	1	2	yes	—
67	#340	—	2	2	no	—
68	#340	—	10	4	no	—
69	#340	—	10	2	no	—
70	C₈²	—	5	4	no	—
71	C₂³×C₅	#127	1	1	yes	characteristic
72	C₂×C₄×C₅	—	8	2	no	—
73	C₂×C₄×C₅	—	8	2	no	—
74	C₂×C₄×C₅	—	8	2	no	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₅	#135	1	4	yes	—
84	C₂×C₄×C₅	#135	1	4	yes	—
85	C₂×Dic₅	—	8	2	no	—
86	C₂×Dic₅	—	8	2	no	—
87	C₂×Dic₅	—	8	2	no	—
88	C₅×C₈	—	2	4	no	—
89	#168	—	2	4	no	—
90	#168	#107	1	8	yes	—
91	C₂²×C₈	—	5	2	no	—
92	C₂²×C₈	—	5	2	no	—
93	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	10	2	no	—
98	C₂×C₄⋊C₄	—	10	2	no	—
99	C₂×C₄⋊C₄	—	10	2	no	—
100	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
101	#107	—	20	8	no	—
102	C₄×C₈	—	5	2	no	—
103	C₄×C₈	—	5	2	no	—
104	C₄×C₈	—	5	2	no	—
105	C₄×C₈	—	5	2	no	—
106	C₄×C₈	Dic₁₀	1	2	yes	—
107	C₄×C₈	C₄×D₅	1	2	yes	—
108	C₂²×C₅	#384	1	1	yes	characteristic
109	C₂²×C₅	#493	1	1	yes	characteristic
110	C₂²×C₅	#494	1	1	yes	characteristic
111	C₂²×C₅	#495	1	2	yes	—
112	C₂²×C₅	#495	1	2	yes	—
113	C₄×C₅	—	16	2	no	—
114	C₄×C₅	C₂×#107	1	2	yes	—
115	C₄×C₅	#427	1	2	yes	—
116	C₄×C₅	#427	1	2	yes	—
117	C₄×C₅	#499	1	2	yes	—
118	C₄×C₅	#499	1	2	yes	—
119	C₄×C₅	#500	1	2	yes	—
120	Dic₅	—	16	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	20	2	no	—
123	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
124	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	5	4	no	—
128	C₂×C₈	—	5	4	no	—
129	C₂×C₈	—	5	4	no	—
130	C₂×C₈	—	5	4	no	—
131	C₂×C₈	—	10	4	no	—
132	C₂×C₈	#683	1	4	yes	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	20	4	no	—
137	C₄⋊C₄	—	20	4	no	—
138	C₄⋊C₄	—	20	4	no	—
139	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
140	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
141	C₄²	Q₈×D₅	1	1	yes	characteristic
142	C₄²	Q₈×D₅	1	1	yes	characteristic
143	C₂×C₅	#1709	1	2	yes	—
144	C₂×C₅	#1710	1	2	yes	—
145	C₂×C₅	#1845	1	1	yes	characteristic
146	C₂×C₅	#1846	1	1	yes	characteristic
147	C₂×C₅	#1846	1	1	yes	characteristic
148	C₂³	#4086	1	1	yes	characteristic
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	40	2	no	—
153	C₂×C₄	—	40	2	no	—
154	C₂×C₄	—	40	2	no	—
155	C₂×C₄	#4034	1	1	yes	characteristic
156	C₂×C₄	#4034	1	1	yes	characteristic
157	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄	D₅×D₈	1	2	yes	—
160	C₂×C₄	#4062	1	4	yes	—
161	C₂×C₄	D₅×Q₁₆	1	2	yes	—
162	C₂×C₄	C₂×#683	1	1	yes	characteristic
163	C₂×C₄	#4090	1	1	yes	characteristic
164	C₂×C₄	#4096	1	4	yes	—
165	C₈	—	2	4	no	—
166	C₈	—	5	8	no	—
167	C₈	—	10	4	no	—
168	C₅	#8816	1	1	yes	characteristic, complemented, Sylow
169	C₂²	#65265	1	1	yes	characteristic
170	C₂²	#65348	1	2	yes	—
171	C₂²	#65440	1	1	yes	characteristic
172	C₂²	#65457	1	2	yes	—
173	C₂²	#65470	1	1	yes	characteristic
174	C₄	—	16	2	no	—
175	C₄	—	80	2	no	—
176	C₄	#65147	1	2	yes	—
177	C₄	#65149	1	2	yes	—
178	C₄	#65163	1	2	yes	—
179	C₄	D₅×#107	1	2	yes	—
180	C₄	#65287	1	2	yes	—
181	C₄	#65353	1	2	yes	—
182	C₂	#10600342	1	1	yes	characteristic
183	C₂	#10601348	1	1	yes	characteristic
184	C₂	#10601399	1	1	yes	characteristic
185	C₂	#10601975	1	2	yes	—
186	C₂	#10602007	1	2	yes	—

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1834	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1834	D₅
hypocenter	C₅	#8816
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
15. C₈×#168	C₄	semidirect
168. C₅	#8816	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1834	2
#10600555	2
C₂×C₈×#168	2
#10600370	2
#10600367	2
#8816	5