Subgroups — #27294036

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20276514	C₂	1	1	yes	maximal, characteristic
3	#20272695	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₈×Dic₁₀₆	—	2	1	no	—
7	C₈×Dic₁₀₆	—	2	1	no	—
8	C₈×Dic₁₀₆	C₄	1	2	yes	—
9	C₈×Dic₁₀₆	C₂²	1	2	yes	—
10	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
11	C₂×#13133146	—	2	1	no	—
12	C₂×C₈×Dic₅₃	—	2	1	no	—
13	C₄⋊#11719908	C₂²	1	2	yes	—
14	C₂²⋊#11719908	—	2	1	no	—
15	C₄×#11719908	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	—	2	1	no	—
18	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
19	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
21	C₂²×#88322	—	2	1	no	—
22	C₄×Dic₁₀₆	—	2	1	no	—
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	C₈	1	2	yes	—
25	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
27	C₂×#11719919	—	2	1	no	—
28	C₂×C₄×Dic₅₃	—	2	1	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₄⋊#88322	—	2	1	no	—
31	C₄⋊#88322	C₂×C₄	1	2	yes	—
32	#13133146	—	2	2	no	—
33	#13133146	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	2	1	no	—
48	C₂²×Dic₅₃	—	2	1	no	—
49	C₂×Dic₁₀₆	—	2	1	no	—
50	C₂×Dic₁₀₆	—	2	2	no	—
51	C₂×Dic₁₀₆	C₁₆	1	4	yes	complemented
52	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄⋊Dic₅₃	—	2	1	no	—
64	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
65	#11719919	—	2	2	no	—
66	#11719919	—	2	2	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	C₄×Dic₅₃	—	2	2	no	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₁₆×C₅₃	—	4	2	no	—
72	#11719908	—	4	2	no	—
73	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₅₃	—	2	4	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	Dic₁₀₆	—	2	4	no	—
90	Dic₁₀₆	—	2	2	no	—
91	Dic₁₀₆	C₂×C₁₆	1	4	yes	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	#137	1	2	yes	—
96	#88322	—	4	2	no	—
97	#11637	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
101	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
106	C₄×C₅₃	#419	1	1	yes	characteristic
107	C₄×C₅₃	#421	1	1	yes	characteristic
108	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
109	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
110	Dic₅₃	—	2	4	no	—
111	Dic₅₃	—	4	2	no	—
112	C₂×C₈×Q₈	—	53	1	no	—
113	#2162	—	53	1	no	—
114	#2162	D₅₃	1	1	yes	characteristic
115	Q₈⋊C₁₆	—	53	4	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	#1479	1	1	yes	characteristic
118	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₅₃	#2117	1	1	yes	characteristic
120	C₂×C₄×Q₈	—	53	1	no	—
121	C₈×Q₈	—	53	2	no	—
122	C₈×Q₈	—	53	2	no	—
123	C₈×Q₈	—	106	1	no	—
124	C₈×Q₈	—	106	1	no	—
125	C₂×C₄⋊C₈	—	53	1	no	—
126	C₂×C₄⋊C₈	—	106	1	no	—
127	C₂×C₄×C₈	—	106	1	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	—	53	2	no	—
131	C₂²⋊C₁₆	—	2	1	no	—
132	C₂²⋊C₁₆	—	106	1	no	—
133	C₄×C₁₆	—	106	1	no	—
134	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
135	C₅₃	#11637	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	53	1	no	—
137	C₂²×C₈	—	106	1	no	—
138	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄×Q₈	—	53	2	no	—
141	C₄×Q₈	—	53	2	no	—
142	C₄×Q₈	—	106	1	no	—
143	C₄×Q₈	—	106	1	no	—
144	C₂×C₄⋊C₄	—	53	1	no	—
145	C₂×C₄⋊C₄	—	106	1	no	—
146	C₂×C₄²	—	106	1	no	—
147	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	106	2	no	—
150	C₄⋊C₈	—	53	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₄⋊C₈	—	106	1	no	—
154	C₄×C₈	—	106	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
157	C₄×C₈	D₂₁₂	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	C₂×Q₈	—	53	4	no	—
161	C₂×Q₈	—	53	2	no	—
162	C₂×Q₈	—	106	1	no	—
163	C₂×Q₈	—	106	2	no	—
164	C₂²×C₄	—	106	1	no	—
165	C₂²×C₄	—	106	1	no	—
166	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
167	C₂²×C₄	#11719912	1	1	yes	characteristic
168	C₂²×C₄	#11719921	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	#11719921	1	1	yes	characteristic
174	C₂×C₈	#11719921	1	1	yes	characteristic
175	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	53	2	no	—
178	C₄⋊C₄	—	106	1	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄⋊C₄	—	106	2	no	—
181	C₄²	—	106	2	no	—
182	C₄²	—	106	2	no	—
183	C₄²	C₈×D₅₃	1	1	yes	characteristic
184	C₄²	#11719912	1	1	yes	characteristic
185	C₄²	#11719921	1	1	yes	characteristic
186	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₁₆	—	212	2	no	—
189	C₂³	#13133129	1	1	yes	characteristic
190	Q₈	—	53	4	no	—
191	Q₈	—	106	2	no	—
192	Q₈	—	106	4	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	106	4	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	#13133129	1	1	yes	characteristic
201	C₂×C₄	#13133129	1	1	yes	characteristic
202	C₂×C₄	#13133129	1	1	yes	characteristic
203	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
204	C₂×C₄	#13133156	1	2	yes	—
205	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	212	2	no	—
212	C₈	#13133250	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#16810402	1	2	yes	—
216	C₂²	C₂×#13133129	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	106	4	no	—
221	C₄	—	212	2	no	—
222	C₄	#16810402	1	2	yes	—
223	C₄	#16810633	1	1	yes	characteristic
224	C₄	#16810641	1	1	yes	characteristic
225	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
226	C₄	C₂×#13133156	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#20273111	1	1	yes	characteristic
229	C₂	C₂×#16810402	1	1	yes	characteristic
230	C₂	#20276342	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#11637
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #20272695	C₂	semidirect
51. C₂×Dic₁₀₆	C₁₆	semidirect
135. C₅₃	#11637	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
#20276514	2
#20272695	2
C₅₃×#2162	2
#11637	53