Subgroups — #27282709

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273467	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20272998	C₂	1	1	yes	maximal, characteristic
4	#20272566	C₂	1	2	yes	maximal
5	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	#16810676	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	#16810376	—	2	2	no	—
10	#16810376	—	2	1	no	—
11	#16810376	—	2	1	no	—
12	#16810374	—	2	2	no	—
13	#16810344	—	2	1	no	—
14	C₈×#88322	—	2	1	no	—
15	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
17	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#341	C₄	1	2	yes	—
20	C₂²×#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	1	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	D₄	1	1	yes	characteristic
25	C₄⋊#88322	D₄	1	1	yes	characteristic
26	C₂²⋊#88322	—	2	1	no	—
27	C₂²⋊#88322	Q₈	1	2	yes	—
28	C₄×#88322	—	2	1	no	—
29	C₄×#88322	—	2	1	no	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#108×C₅₃	—	2	2	no	—
36	#108×C₅₃	C₈	1	4	yes	complemented
37	#97×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
46	M₁₆×C₅₃	—	2	2	no	—
47	M₁₆×C₅₃	—	2	2	no	—
48	M₁₆×C₅₃	—	2	2	no	—
49	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₃	#41	1	1	yes	characteristic
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	8	2	no	—
69	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	1	no	—
85	C₈×C₅₃	—	2	4	no	—
86	C₈×C₅₃	—	4	2	no	—
87	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
88	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
89	#88322	—	4	2	no	—
90	#88322	—	4	1	no	—
91	#88322	—	4	1	no	—
92	#88322	—	8	2	no	—
93	#9608	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₅₃	#424	1	1	yes	characteristic
104	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
105	C₄×C₅₃	#436	1	1	yes	characteristic
106	C₄×C₅₃	#472	1	1	yes	characteristic
107	C₄×C₅₃	#473	1	1	yes	characteristic
108	C₄×#108	D₅₃	1	1	yes	characteristic
109	#1565	—	53	2	no	—
110	C₂×#335	—	53	1	no	—
111	#1450	—	53	1	no	—
112	#1283	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1939	1	1	yes	characteristic
116	C₂×C₅₃	#1952	1	1	yes	characteristic
117	C₂×C₅₃	#1953	1	1	yes	characteristic
118	#431	—	53	1	no	—
119	#431	—	53	1	no	—
120	C₂×#108	Dic₅₃	1	1	yes	characteristic
121	C₂×#108	Dic₅₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	#341	Dic₅₃	1	2	yes	—
126	#335	—	106	2	no	—
127	#335	—	106	1	no	—
128	#335	—	106	1	no	—
129	#334	—	106	2	no	—
130	#322	—	106	1	no	—
131	C₈²	—	106	1	no	—
132	C₅₃	#9608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
134	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
135	C₂²×C₈	—	106	2	no	—
136	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
139	#108	—	2	2	no	—
140	#108	#88322	1	4	yes	complemented
141	C₄⋊C₈	—	53	1	no	—
142	C₄⋊C₈	—	53	1	no	—
143	C₄⋊C₈	—	106	2	no	—
144	C₄⋊C₈	—	106	2	no	—
145	C₄⋊C₈	—	106	1	no	—
146	C₂²⋊C₈	—	53	2	no	—
147	C₂²⋊C₈	—	106	1	no	—
148	#97	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	106	1	no	—
151	C₄×C₈	—	106	1	no	—
152	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
153	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
154	C₄×C₈	D₂₁₂	1	1	yes	characteristic
155	C₄×C₈	D₂₁₂	1	1	yes	characteristic
156	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	#11719949	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	—	2	2	no	—
162	M₁₆	C₂×#88322	1	4	yes	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	106	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	424	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	C₂×#88322	1	2	yes	—
173	C₂×C₈	#11719917	1	2	yes	—
174	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
176	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
178	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
180	C₄²	D₄×D₅₃	1	1	yes	characteristic
181	C₄²	#11719947	1	1	yes	characteristic
182	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄	#13133192	1	1	yes	characteristic
189	C₂×C₄	#13133197	1	1	yes	characteristic
190	C₂×C₄	#13133208	1	1	yes	characteristic
191	C₂×C₄	#13133216	1	1	yes	characteristic
192	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
194	C₂×C₄	#13133224	1	1	yes	characteristic
195	C₂×C₄	#13133224	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	424	2	no	—
201	C₈	—	212	2	no	—
202	C₈	—	212	1	no	—
203	C₈	—	212	1	no	—
204	C₈	C₄⋊#88322	1	2	yes	—
205	C₈	C₄⋊#88322	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16810522	1	1	yes	characteristic
209	C₂²	Q₈×#88322	1	1	yes	characteristic
210	C₂²	#16810754	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
215	C₄	#16810650	1	1	yes	characteristic
216	C₄	D₄×#88322	1	1	yes	characteristic
217	C₄	#16810701	1	1	yes	characteristic
218	C₄	#16810894	1	1	yes	characteristic
219	C₄	#16810899	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#20275434	1	1	yes	characteristic
223	C₂	#20275501	1	1	yes	characteristic
224	C₂	#20275506	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#9608
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
36. #108×C₅₃	C₈	semidirect
132. C₅₃	#9608	semidirect
140. #108	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273467	2
C₂×#16810376	2
#20272998	2
#20272566	2
C₄×#108×C₅₃	2
#9608	53