Subgroups — #27282645

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275434	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810376	—	2	2	no	—
12	#16810344	—	2	2	no	—
13	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#340	C₄	1	2	yes	—
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	D₄	1	1	yes	characteristic
21	C₂²×#88322	D₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	4	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#106×C₅₃	—	2	2	no	—
36	#106×C₅₃	C₈	1	4	yes	complemented
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	#95×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₄²×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₅₃	#41	1	1	yes	characteristic
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
88	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₅₃	#137	1	2	yes	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
92	#88322	—	4	2	no	—
93	#88322	—	4	1	no	—
94	#88322	—	4	1	no	—
95	#88322	—	8	2	no	—
96	#9596	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	#389	1	1	yes	characteristic
98	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
99	C₂²×C₅₃	#436	1	1	yes	characteristic
100	C₂²×C₅₃	#440	1	2	yes	—
101	C₂²×C₅₃	#444	1	2	yes	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	4	4	no	—
105	C₄×C₅₃	—	8	2	no	—
106	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₅₃	#424	1	1	yes	characteristic
108	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
109	C₄×C₅₃	#446	1	1	yes	characteristic
110	C₄×C₅₃	#469	1	1	yes	characteristic
111	C₄×C₅₃	#470	1	1	yes	characteristic
112	C₄×C₅₃	#501	1	2	yes	—
113	#1939	—	53	2	no	—
114	C₄×#106	D₅₃	1	1	yes	characteristic
115	C₂×#335	—	53	1	no	—
116	C₂×#322	—	53	1	no	—
117	#1279	—	53	2	no	—
118	C₂×C₅₃	SD₁₆⋊C₈	1	2	yes	—
119	C₂×C₅₃	#1592	1	2	yes	—
120	C₂×C₅₃	#1939	1	1	yes	characteristic
121	C₂×C₅₃	#1944	1	1	yes	characteristic
122	C₂×C₅₃	#1946	1	1	yes	characteristic
123	C₂×#106	Dic₅₃	1	1	yes	characteristic
124	C₂×#106	Dic₅₃	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	53	2	no	—
126	C₂×C₄⋊C₈	—	106	2	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
131	#340	Dic₅₃	1	2	yes	—
132	#336	—	106	2	no	—
133	#335	—	106	2	no	—
134	#322	—	106	2	no	—
135	C₅₃	#9596	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	106	1	no	—
139	C₂²×C₈	—	106	2	no	—
140	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
146	#106	—	2	2	no	—
147	#106	#88322	1	4	yes	complemented
148	C₄⋊C₈	—	106	2	no	—
149	C₄⋊C₈	—	106	2	no	—
150	C₄⋊C₈	—	212	4	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	106	1	no	—
153	C₄×C₈	—	106	1	no	—
154	C₄×C₈	—	106	1	no	—
155	C₄×C₈	Dic₁₀₆	1	2	yes	—
156	C₄×C₈	D₂₁₂	1	2	yes	—
157	#95	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	#11719947	1	1	yes	characteristic
162	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	212	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	C₂×#88322	1	2	yes	—
175	C₂×C₈	#11719917	1	2	yes	—
176	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
177	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂×#88322	1	4	yes	—
182	C₄²	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
185	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
186	C₄²	Q₈×D₅₃	1	1	yes	characteristic
187	C₄²	#11719949	1	1	yes	characteristic
188	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	#13133192	1	1	yes	characteristic
199	C₂×C₄	#13133197	1	1	yes	characteristic
200	C₂×C₄	#13133208	1	1	yes	characteristic
201	C₂×C₄	#13133216	1	1	yes	characteristic
202	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
203	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
204	C₂×C₄	#13133224	1	1	yes	characteristic
205	C₂×C₄	#13133224	1	1	yes	characteristic
206	C₂×C₄	#13133244	1	2	yes	—
207	C₂×C₄	#13133246	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	424	2	no	—
210	C₈	—	212	2	no	—
211	C₈	—	212	1	no	—
212	C₈	—	212	1	no	—
213	C₈	C₄⋊#88322	1	4	yes	complemented
214	C₂²	#16810522	1	1	yes	characteristic
215	C₂²	D₄×#88322	1	1	yes	characteristic
216	C₂²	#16810701	1	1	yes	characteristic
217	C₂²	#16810724	1	2	yes	—
218	C₂²	#16810745	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
224	C₄	#16810650	1	1	yes	characteristic
225	C₄	Q₈×#88322	1	1	yes	characteristic
226	C₄	#16810754	1	1	yes	characteristic
227	C₄	#16810877	1	1	yes	characteristic
228	C₄	#16810882	1	1	yes	characteristic
229	C₄	#16811030	1	2	yes	—
230	C₂	SD₁₆⋊#88322	1	2	yes	—
231	C₂	#20273618	1	2	yes	—
232	C₂	#20275434	1	1	yes	characteristic
233	C₂	#20275459	1	1	yes	characteristic
234	C₂	#20275469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#106×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#106	D₅₃
hypocenter	C₅₃	#9596
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
36. #106×C₅₃	C₈	semidirect
91. C₈×C₅₃	C₄⋊C₈	semidirect
135. C₅₃	#9596	semidirect
147. #106	#88322	semidirect
213. C₈	C₄⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275434	2
C₂×#16810376	2
C₂×#16810344	2
#20272556	2
C₄×#106×C₅₃	2
#9596	53