Subgroups — #11998335

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂×#64767	C₂	1	4	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊Q₈	—	2	2	no	—
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#384	—	2	1	no	—
8	C₅×#384	—	2	1	no	—
9	C₅×#384	C₄	1	4	yes	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#168	—	2	2	no	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₂×C₄×#168	—	2	1	no	—
15	#64767	—	2	8	no	—
16	#64744	—	2	1	no	—
17	#9692	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	—	2	2	no	—
19	C₅×C₄⋊Q₈	—	2	4	no	—
20	C₅×C₄⋊Q₈	—	2	4	no	—
21	C₅×C₄⋊Q₈	—	4	4	no	—
22	C₅×C₄⋊Q₈	—	4	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	4	no	—
24	C₂×C₅×C₄⋊C₄	—	2	4	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₂²×#168	—	2	2	no	—
33	C₂²×#168	—	2	2	no	—
34	C₄⋊#168	—	2	2	no	—
35	C₄⋊#168	—	2	4	no	—
36	C₄⋊#168	—	4	4	no	—
37	C₄⋊#168	D₄	1	4	yes	—
38	C₄×#168	—	4	2	no	—
39	#2868	D₅	1	1	yes	characteristic
40	#1842	—	5	1	no	—
41	C₄×C₄⋊C₈	—	5	1	no	—
42	C₂×#332	—	5	4	no	—
43	C₂×C₅×Q₈	—	4	2	no	—
44	C₂×C₅×Q₈	—	4	2	no	—
45	C₂×C₅×Q₈	—	4	4	no	—
46	C₂×C₅×Q₈	—	4	4	no	—
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₅×C₄⋊C₄	—	2	8	no	—
54	C₅×C₄⋊C₄	—	4	8	no	—
55	C₅×C₄⋊C₄	—	4	8	no	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₄²×C₅	—	2	4	no	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	D₈	1	4	yes	—
60	C₄²×C₅	SD₁₆	1	4	yes	—
61	C₄²×C₅	SD₁₆	1	2	yes	—
62	C₄²×C₅	Q₁₆	1	2	yes	—
63	C₄²×C₅	C₂×D₄	1	2	yes	—
64	C₄²×C₅	#41	1	2	yes	—
65	C₂×#168	—	2	4	no	—
66	C₂×#168	—	4	2	no	—
67	C₂×#168	—	4	2	no	—
68	C₂×#168	—	4	2	no	—
69	C₂×#168	—	4	4	no	—
70	C₂×C₄⋊Q₈	—	2	2	no	—
71	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
72	C₂×C₄⋊C₈	—	5	2	no	—
73	C₂×C₄⋊C₈	—	5	2	no	—
74	C₂×C₄⋊C₈	—	10	2	no	—
75	C₂×C₄×C₈	—	10	1	no	—
76	#384	—	2	1	no	—
77	#384	—	2	1	no	—
78	#384	Dic₅	1	4	yes	—
79	C₄³	C₂×D₅	1	1	yes	characteristic
80	#336	—	10	1	no	—
81	#332	—	10	8	no	—
82	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₅×Q₈	—	8	4	no	—
84	C₅×Q₈	—	8	2	no	—
85	C₅×Q₈	—	8	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	4	8	no	—
91	C₂×C₄×C₅	#101	1	4	yes	—
92	C₂×C₄×C₅	#102	1	4	yes	—
93	C₂×C₄×C₅	#102	1	4	yes	—
94	C₂×C₄×C₅	#103	1	4	yes	—
95	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅	#121	1	2	yes	—
97	C₂×C₄×C₅	#124	1	1	yes	characteristic
98	C₂×C₄×C₅	#127	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅	#137	1	2	yes	—
104	#168	—	4	4	no	—
105	#168	—	8	2	no	—
106	C₂²×Q₈	—	2	2	no	—
107	C₂²×C₈	—	10	2	no	—
108	C₂²×C₈	—	10	2	no	—
109	C₄⋊Q₈	—	2	4	no	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₄⋊Q₈	—	4	4	no	—
112	C₄⋊Q₈	—	4	4	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
117	C₂×C₄²	C₂×Dic₅	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
122	C₄⋊C₈	—	5	4	no	—
123	C₄⋊C₈	—	10	4	no	—
124	C₄⋊C₈	—	10	2	no	—
125	C₄⋊C₈	—	20	4	no	—
126	C₄×C₈	—	20	2	no	—
127	C₂²×C₅	#390	1	1	yes	characteristic
128	C₂²×C₅	C₂×#101	1	2	yes	—
129	C₂²×C₅	C₂×#102	1	2	yes	—
130	C₂²×C₅	C₂×#103	1	1	yes	characteristic
131	C₂²×C₅	#419	1	1	yes	characteristic
132	C₄×C₅	—	2	4	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	8	4	no	—
135	C₄×C₅	#332	1	8	yes	complemented
136	C₄×C₅	#461	1	2	yes	—
137	C₄×C₅	#462	1	2	yes	—
138	C₄×C₅	#486	1	2	yes	—
139	C₄×C₅	#492	1	1	yes	characteristic
140	C₄×C₅	#493	1	1	yes	characteristic
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
147	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₂×C₈	—	10	4	no	—
152	C₂×C₈	—	20	2	no	—
153	C₂×C₈	—	20	2	no	—
154	C₂×C₈	—	20	4	no	—
155	C₂×C₈	—	20	2	no	—
156	C₄⋊C₄	—	2	8	no	—
157	C₄⋊C₄	—	4	8	no	—
158	C₄⋊C₄	—	4	8	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	#686	1	4	yes	—
163	C₄²	#687	1	4	yes	—
164	C₄²	#688	1	2	yes	—
165	C₄²	#689	1	2	yes	—
166	C₄²	D₄×D₅	1	2	yes	—
167	C₄²	#711	1	2	yes	—
168	C₂×C₅	C₂×#332	1	2	yes	—
169	C₂×C₅	#1546	1	2	yes	—
170	C₂×C₅	#1957	1	1	yes	characteristic
171	C₂×C₅	#1966	1	1	yes	characteristic
172	C₂×C₅	#1981	1	1	yes	characteristic
173	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
174	Q₈	—	8	4	no	—
175	Q₈	—	8	2	no	—
176	Q₈	—	8	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	#3964	1	4	yes	—
183	C₂×C₄	#3964	1	4	yes	—
184	C₂×C₄	#3967	1	4	yes	—
185	C₂×C₄	#3968	1	4	yes	—
186	C₂×C₄	C₂×#686	1	2	yes	—
187	C₂×C₄	C₂×#687	1	2	yes	—
188	C₂×C₄	D₄×Dic₅	1	2	yes	—
189	C₂×C₄	#4082	1	1	yes	characteristic
190	C₂×C₄	#4084	1	2	yes	—
191	C₂×C₄	#4086	1	1	yes	characteristic
192	C₂×C₄	C₂×#688	1	1	yes	characteristic
193	C₂×C₄	#4088	1	2	yes	—
194	C₂×C₄	C₂×#689	1	1	yes	characteristic
195	C₈	—	20	4	no	—
196	C₈	—	40	2	no	—
197	C₅	#9692	1	1	yes	characteristic, complemented, Sylow
198	C₂²	C₂×#3964	1	2	yes	—
199	C₂²	#65510	1	1	yes	characteristic
200	C₂²	C₂×#3967	1	1	yes	characteristic
201	C₂²	#65514	1	1	yes	characteristic
202	C₂²	C₂×#3968	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	#64767	1	8	yes	complemented
207	C₄	#65359	1	2	yes	—
208	C₄	#65362	1	1	yes	characteristic
209	C₄	#65365	1	1	yes	characteristic
210	C₄	#65372	1	2	yes	—
211	C₄	#65379	1	2	yes	—
212	C₂	#10602557	1	1	yes	characteristic
213	C₂	C₂×#64767	1	2	yes	—
214	C₂	#10602566	1	2	yes	—
215	C₂	#10602571	1	1	yes	characteristic
216	C₂	#10602582	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2868	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2868	D₅
hypocenter	C₅	#9692
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
135. C₄×C₅	#332	semidirect
197. C₅	#9692	semidirect
206. C₄	#64767	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2868	2
C₂×#64767	2
#10600187	2
C₄×C₄⋊#168	2
#9692	5