Subgroups — #11993040

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1842	C₂	1	1	yes	maximal, characteristic
2	C₂×#64699	C₂	1	4	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	C₄²×#168	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊C₈	—	2	2	no	—
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#168	—	2	2	no	—
9	C₂×C₄⋊#168	C₂²	1	2	yes	—
10	C₂×C₄×#168	C₄	1	2	yes	—
11	C₂×C₄×#168	C₄	1	1	yes	characteristic
12	C₂×C₄×#168	C₄	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	2	yes	—
14	#64699	—	2	8	no	—
15	#8806	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	—	2	2	no	—
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	2	yes	—
20	C₂×C₄²×C₅	Q₈	1	2	yes	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
22	C₂²×#168	—	2	2	no	—
23	C₂²×#168	C₂×C₄	1	2	yes	—
24	C₂²×#168	C₂×C₄	1	2	yes	—
25	C₅×C₄⋊C₈	—	2	4	no	—
26	C₅×C₄⋊C₈	—	4	4	no	—
27	C₄⋊#168	—	2	4	no	—
28	C₄⋊#168	—	4	4	no	—
29	C₄×#168	—	2	2	no	—
30	C₄×#168	—	2	4	no	—
31	C₄×#168	C₈	1	4	yes	complemented
32	C₄×#168	D₄	1	4	yes	—
33	C₄×#168	Q₈	1	4	yes	—
34	#1842	—	5	1	no	—
35	#1842	D₅	1	1	yes	characteristic
36	C₄²×C₈	—	5	1	no	—
37	C₂×#335	—	5	4	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₅×C₈	—	4	2	no	—
44	C₂×C₅×C₈	—	4	4	no	—
45	C₄²×C₅	—	2	4	no	—
46	C₄²×C₅	—	2	2	no	—
47	C₄²×C₅	C₂×C₈	1	2	yes	—
48	C₄²×C₅	M₁₆	1	2	yes	—
49	C₄²×C₅	D₈	1	4	yes	—
50	C₄²×C₅	Q₁₆	1	4	yes	—
51	C₄²×C₅	C₂×D₄	1	2	yes	—
52	C₄²×C₅	C₂×Q₈	1	2	yes	—
53	C₂×#168	—	2	2	no	—
54	C₂×#168	—	2	2	no	—
55	C₂×#168	—	4	2	no	—
56	C₂×#168	—	4	4	no	—
57	C₂×#168	C₄⋊C₄	1	4	yes	—
58	C₂×#168	C₄⋊C₄	1	4	yes	—
59	C₂×#168	C₂×C₈	1	4	yes	—
60	C₂×#168	M₁₆	1	4	yes	—
61	C₂×C₄⋊C₈	—	2	2	no	—
62	C₂×C₄⋊C₈	—	5	2	no	—
63	C₂×C₄⋊C₈	—	10	2	no	—
64	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
65	C₂×C₄×C₈	—	5	1	no	—
66	C₂×C₄×C₈	—	5	2	no	—
67	C₂×C₄×C₈	—	5	2	no	—
68	C₂×C₄×C₈	—	5	1	no	—
69	C₄³	C₂×D₅	1	1	yes	characteristic
70	#335	—	10	8	no	—
71	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
76	C₂×C₄×C₅	#107	1	4	yes	—
77	C₂×C₄×C₅	#107	1	4	yes	—
78	C₂×C₄×C₅	#108	1	4	yes	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅	#127	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
88	C₅×C₈	—	8	2	no	—
89	#168	—	2	4	no	—
90	#168	—	8	2	no	—
91	#168	C₄⋊C₈	1	8	yes	complemented
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	5	2	no	—
94	C₂²×C₈	—	5	2	no	—
95	C₂²×C₈	—	10	2	no	—
96	C₂×C₄²	Dic₁₀	1	2	yes	—
97	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
100	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
101	C₄⋊C₈	—	2	4	no	—
102	C₄⋊C₈	—	4	4	no	—
103	C₄⋊C₈	—	10	4	no	—
104	C₄⋊C₈	—	20	4	no	—
105	C₄×C₈	—	5	4	no	—
106	C₄×C₈	—	5	4	no	—
107	C₄×C₈	—	5	4	no	—
108	C₄×C₈	—	10	4	no	—
109	C₄×C₈	—	10	2	no	—
110	C₂²×C₅	#384	1	1	yes	characteristic
111	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₅	#423	1	1	yes	characteristic
113	C₂²×C₅	C₂×#107	1	2	yes	—
114	C₂²×C₅	C₂×#108	1	2	yes	—
115	C₄×C₅	—	2	4	no	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	#335	1	8	yes	complemented
118	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
119	C₄×C₅	#423	1	2	yes	—
120	C₄×C₅	#493	1	1	yes	characteristic
121	C₄×C₅	#494	1	1	yes	characteristic
122	C₄×C₅	#500	1	2	yes	—
123	C₂²×C₄	#683	1	2	yes	—
124	C₂²×C₄	#683	1	2	yes	—
125	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
128	C₂×C₈	—	4	4	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	5	4	no	—
131	C₂×C₈	—	5	4	no	—
132	C₂×C₈	—	5	4	no	—
133	C₂×C₈	—	5	4	no	—
134	C₂×C₈	—	10	2	no	—
135	C₂×C₈	—	10	2	no	—
136	C₂×C₈	—	20	4	no	—
137	C₂×C₈	—	20	2	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	C₈×D₅	1	2	yes	—
141	C₄²	#676	1	2	yes	—
142	C₄²	#686	1	4	yes	—
143	C₄²	#689	1	4	yes	—
144	C₄²	D₄×D₅	1	2	yes	—
145	C₄²	Q₈×D₅	1	2	yes	—
146	C₂×C₅	C₂×#335	1	2	yes	—
147	C₂×C₅	#1567	1	2	yes	—
148	C₂×C₅	#1842	1	1	yes	characteristic
149	C₂×C₅	#1845	1	1	yes	characteristic
150	C₂×C₅	#1848	1	1	yes	characteristic
151	C₂³	C₂×#683	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	#3939	1	4	yes	—
156	C₂×C₄	#3939	1	4	yes	—
157	C₂×C₄	#3946	1	4	yes	—
158	C₂×C₄	#3954	1	4	yes	—
159	C₂×C₄	#4034	1	2	yes	—
160	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
162	C₂×C₄	C₂×#676	1	1	yes	characteristic
163	C₂×C₄	D₅×M₁₆	1	2	yes	—
164	C₂×C₄	C₂×#686	1	2	yes	—
165	C₂×C₄	#4086	1	1	yes	characteristic
166	C₂×C₄	C₂×#689	1	2	yes	—
167	C₂×C₄	#4090	1	1	yes	characteristic
168	C₈	—	5	8	no	—
169	C₈	—	8	2	no	—
170	C₈	—	10	4	no	—
171	C₈	—	40	2	no	—
172	C₅	#8806	1	1	yes	characteristic, complemented, Sylow
173	C₂²	C₂×#3939	1	2	yes	—
174	C₂²	#65265	1	1	yes	characteristic
175	C₂²	C₂×#3946	1	1	yes	characteristic
176	C₂²	#65407	1	1	yes	characteristic
177	C₂²	C₂×#3954	1	2	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	#64699	1	8	yes	complemented
181	C₄	D₅×C₄⋊C₈	1	2	yes	—
182	C₄	#65126	1	2	yes	—
183	C₄	#65362	1	1	yes	characteristic
184	C₄	#65367	1	2	yes	—
185	C₄	#65381	1	1	yes	characteristic
186	C₂	#10600370	1	1	yes	characteristic
187	C₂	C₂×#64699	1	2	yes	—
188	C₂	#10601238	1	1	yes	characteristic
189	C₂	#10601284	1	2	yes	—
190	C₂	#10601311	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1842	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1842	D₅
hypocenter	C₅	#8806
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
31. C₄×#168	C₈	semidirect
91. #168	C₄⋊C₈	semidirect
117. C₄×C₅	#335	semidirect
172. C₅	#8806	semidirect
180. C₄	#64699	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1842	2
C₂×#64699	2
#10600187	2
C₄²×#168	2
#8806	5