Subgroups — #27282704

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 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×#16810374	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	#16810374	—	2	2	no	—
11	#16810374	—	2	1	no	—
12	#16810374	—	2	1	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	2	yes	—
25	C₂²⋊#88322	—	2	1	no	—
26	C₂²⋊#88322	Q₈	1	2	yes	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#108×C₅₃	—	2	2	no	—
35	#108×C₅₃	C₈	1	4	yes	complemented
36	#97×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	M₁₆×C₅₃	—	2	2	no	—
46	M₁₆×C₅₃	—	2	2	no	—
47	M₁₆×C₅₃	—	2	2	no	—
48	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
59	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₅₃	#41	1	1	yes	characteristic
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	8	2	no	—
67	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	4	1	no	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	—	2	4	no	—
83	C₈×C₅₃	—	4	2	no	—
84	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
86	#88322	—	4	2	no	—
87	#88322	—	4	2	no	—
88	#88322	—	8	2	no	—
89	#9607	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	#389	1	1	yes	characteristic
93	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
94	C₂²×C₅₃	#446	1	1	yes	characteristic
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₅₃	#424	1	1	yes	characteristic
100	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
101	C₄×C₅₃	#436	1	1	yes	characteristic
102	C₄×C₅₃	#471	1	2	yes	—
103	C₄×#108	D₅₃	1	1	yes	characteristic
104	#1565	—	53	2	no	—
105	C₂×#334	—	53	1	no	—
106	#1450	—	53	1	no	—
107	#1283	—	53	2	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	#1939	1	1	yes	characteristic
111	C₂×C₅₃	#1951	1	1	yes	characteristic
112	C₂×C₅₃	#1953	1	1	yes	characteristic
113	#431	—	53	1	no	—
114	#431	—	53	1	no	—
115	C₂×#108	Dic₅₃	1	1	yes	characteristic
116	C₂×#108	Dic₅₃	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	53	2	no	—
118	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	#341	Dic₅₃	1	2	yes	—
121	#335	—	106	2	no	—
122	#334	—	106	2	no	—
123	#334	—	106	1	no	—
124	#334	—	106	1	no	—
125	#322	—	106	1	no	—
126	C₈²	—	106	1	no	—
127	C₅₃	#9607	1	1	yes	characteristic, complemented, Sylow
128	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
129	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
130	C₂²×C₈	—	106	2	no	—
131	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
132	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	#108	—	2	2	no	—
135	#108	#88322	1	4	yes	complemented
136	C₄⋊C₈	—	53	2	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₄⋊C₈	—	106	2	no	—
139	C₄⋊C₈	—	106	1	no	—
140	C₂²⋊C₈	—	53	2	no	—
141	C₂²⋊C₈	—	106	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	106	1	no	—
145	C₄×C₈	—	106	1	no	—
146	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
147	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
148	C₄×C₈	D₂₁₂	1	1	yes	characteristic
149	C₄×C₈	D₂₁₂	1	1	yes	characteristic
150	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
151	C₂²×C₄	#11719949	1	1	yes	characteristic
152	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	C₂×#88322	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	424	2	no	—
164	C₂×C₈	—	212	2	no	—
165	C₂×C₈	C₂×#88322	1	2	yes	—
166	C₂×C₈	#11719917	1	2	yes	—
167	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
169	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
172	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
173	C₄²	D₄×D₅₃	1	1	yes	characteristic
174	C₄²	#11719947	1	1	yes	characteristic
175	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
180	C₂×C₄	#13133192	1	1	yes	characteristic
181	C₂×C₄	#13133197	1	1	yes	characteristic
182	C₂×C₄	#13133208	1	1	yes	characteristic
183	C₂×C₄	#13133216	1	1	yes	characteristic
184	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
186	C₂×C₄	#13133224	1	1	yes	characteristic
187	C₂×C₄	#13133224	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	4	no	—
191	C₈	—	4	2	no	—
192	C₈	—	424	2	no	—
193	C₈	—	212	2	no	—
194	C₈	—	212	2	no	—
195	C₈	C₄⋊#88322	1	2	yes	—
196	C₈	C₄⋊#88322	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#16810522	1	1	yes	characteristic
200	C₂²	Q₈×#88322	1	1	yes	characteristic
201	C₂²	#16810754	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
206	C₄	#16810650	1	1	yes	characteristic
207	C₄	D₄×#88322	1	1	yes	characteristic
208	C₄	#16810701	1	1	yes	characteristic
209	C₄	#16810887	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#20275434	1	1	yes	characteristic
213	C₂	#20275496	1	1	yes	characteristic
214	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	#9607
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
35. #108×C₅₃	C₈	semidirect
127. C₅₃	#9607	semidirect
135. #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×#16810374	2
#20272998	2
#20272566	2
C₄×#108×C₅₃	2
#9607	53