Subgroups — #27537732

Order 13568.

526 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	C₁₆×Dic₁₀₆	C₂	1	4	yes	maximal, complemented
3	#20276617	C₂	1	4	yes	maximal, complemented
4	#20276514	C₂	1	1	yes	maximal, characteristic
5	#20276511	C₂	1	4	yes	maximal
6	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
8	C₈×Dic₁₀₆	C₄	1	2	yes	—
9	C₈×Dic₁₀₆	C₄	1	2	yes	—
10	C₈×Dic₁₀₆	C₂²	1	2	yes	—
11	C₈×Dic₁₀₆	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
13	C₂×#13133146	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₄⋊#11719908	C₂²	1	2	yes	—
16	#16810441	C₂²	1	8	yes	—
17	C₂²⋊#11719908	C₂²	1	2	yes	—
18	C₄×#11719908	C₂²	1	2	yes	—
19	C₁₆×Dic₅₃	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
22	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₆	C₈	1	2	yes	—
27	C₄×Dic₁₀₆	C₈	1	2	yes	—
28	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
31	C₂×#11719919	C₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
33	C₂×#11719908	C₂³	1	4	yes	—
34	C₄⋊#88322	C₂×C₄	1	2	yes	—
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133146	C₂×C₄	1	4	yes	—
37	#13133146	C₂×C₄	1	4	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
39	C₈×Dic₅₃	C₂³	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
52	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
53	C₂×Dic₁₀₆	C₁₆	1	8	yes	complemented
54	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
55	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
68	#11719919	C₂×C₈	1	4	yes	—
69	#11719919	C₂×C₈	1	4	yes	—
70	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
71	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
72	C₂×#88322	—	4	2	no	—
73	C₂×#88322	C₂²×C₄	1	4	yes	—
74	C₁₆×C₅₃	—	2	4	no	—
75	#11719908	—	2	4	no	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	—
89	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
90	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
91	Dic₁₀₆	C₂×C₁₆	1	8	yes	—
92	Dic₁₀₆	C₂×C₁₆	1	8	yes	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
97	#88322	—	2	4	no	—
98	#33496	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
102	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
107	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
108	C₄×C₅₃	#519	1	1	yes	characteristic
109	C₄×C₅₃	#568	1	1	yes	characteristic
110	Dic₅₃	—	2	4	no	—
111	Dic₅₃	C₂²×C₁₆	1	8	yes	—
112	C₂×C₈×Q₈	—	53	1	no	—
113	Q₈×C₁₆	—	53	4	no	—
114	Q₈×C₁₆	—	53	4	no	—
115	#2162	—	53	1	no	—
116	#2162	—	53	4	no	—
117	#2162	D₅₃	1	1	yes	characteristic
118	C₂×C₅₃	—	2	2	no	—
119	C₂×C₅₃	#2982	1	1	yes	characteristic
120	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
121	C₂×C₅₃	#3407	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	53	1	no	—
123	C₈×Q₈	—	53	2	no	—
124	C₈×Q₈	—	53	2	no	—
125	C₈×Q₈	—	53	2	no	—
126	C₈×Q₈	—	53	2	no	—
127	C₂×C₄⋊C₈	—	53	2	no	—
128	C₂×C₄⋊C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	2	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	53	8	no	—
132	C₄⋊C₁₆	—	53	2	no	—
133	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
134	C₂²⋊C₁₆	—	53	2	no	—
135	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
136	C₄×C₁₆	—	53	8	no	—
137	C₄×C₁₆	—	53	2	no	—
138	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
139	C₅₃	#33496	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	—	53	1	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₄×Q₈	—	53	2	no	—
145	C₄×Q₈	—	53	2	no	—
146	C₄×Q₈	—	53	2	no	—
147	C₄×Q₈	—	53	2	no	—
148	C₂×C₄⋊C₄	—	53	2	no	—
149	C₂×C₄⋊C₄	—	53	1	no	—
150	C₂×C₄²	—	53	2	no	—
151	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₁₆	—	53	4	no	—
153	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
154	C₄⋊C₈	—	53	4	no	—
155	C₄⋊C₈	—	53	4	no	—
156	C₄⋊C₈	—	53	2	no	—
157	C₄⋊C₈	—	53	2	no	—
158	C₄×C₈	—	53	4	no	—
159	C₄×C₈	—	53	4	no	—
160	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
164	C₂×Q₈	—	53	8	no	—
165	C₂×Q₈	—	53	2	no	—
166	C₂×Q₈	—	53	2	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	53	4	no	—
175	C₂×C₈	—	212	2	no	—
176	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
180	C₄⋊C₄	—	53	4	no	—
181	C₄⋊C₄	—	53	4	no	—
182	C₄⋊C₄	—	53	2	no	—
183	C₄⋊C₄	—	53	2	no	—
184	C₄²	—	53	4	no	—
185	C₄²	—	53	4	no	—
186	C₄²	C₈×D₅₃	1	1	yes	characteristic
187	C₄²	C₈×D₅₃	1	1	yes	characteristic
188	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
190	C₁₆	—	2	4	no	—
191	C₁₆	—	106	4	no	—
192	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
193	Q₈	—	53	8	no	—
194	Q₈	—	53	8	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	53	8	no	—
199	C₂×C₄	—	53	4	no	—
200	C₂×C₄	—	53	4	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
203	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	106	4	no	—
213	C₈	#13133264	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
217	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	53	8	no	—
222	C₄	—	106	4	no	—
223	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
224	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
225	C₄	#16811100	1	1	yes	characteristic
226	C₄	#16811389	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#20283795	1	1	yes	characteristic
229	C₂	C₂²×C₁₆×D₅₃	1	1	yes	characteristic
230	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#33496
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₁₀₆	C₂	semidirect
3. #20276617	C₂	semidirect
53. C₂×Dic₁₀₆	C₁₆	semidirect
139. C₅₃	#33496	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
C₁₆×Dic₁₀₆	2
#20276617	2
#20276514	2
#20276511	2
C₅₃×#2162	2
#33496	53