Subgroups — #27537599

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283649	C₂	1	1	yes	maximal, characteristic
2	#20276623	C₂	1	2	yes	maximal, complemented
3	#20276621	C₂	1	4	yes	maximal, complemented
4	C₁₆×Dic₁₀₆	C₂	1	2	yes	maximal, complemented
5	#20276475	C₂	1	2	yes	maximal
6	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
7	#20276314	C₂	1	2	yes	maximal
8	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₄	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	2	yes	—
11	#16810753	C₄	1	2	yes	—
12	#16810753	C₄	1	2	yes	—
13	C₈×Dic₁₀₆	C₂²	1	2	yes	—
14	C₈×Dic₁₀₆	C₂²	1	2	yes	—
15	#16810649	C₄	1	2	yes	—
16	#16810647	C₄	1	1	yes	characteristic
17	#16810578	C₂²	1	2	yes	—
18	C₄⋊#11719908	C₂²	1	1	yes	characteristic
19	C₄⋊#11719908	C₂²	1	2	yes	—
20	C₄⋊#11719908	C₂²	1	1	yes	characteristic
21	#16810441	C₂²	1	4	yes	—
22	#16810441	C₂²	1	4	yes	—
23	#16810398	C₂²	1	4	yes	—
24	C₁₆×Dic₅₃	C₂²	1	4	yes	—
25	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#346	C₂²	1	2	yes	—
29	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
30	C₂×#11719917	C₂×C₄	1	2	yes	—
31	#13133217	C₈	1	1	yes	characteristic
32	#13133217	C₈	1	1	yes	characteristic
33	#13133209	C₈	1	2	yes	—
34	#13133196	C₈	1	2	yes	—
35	#13133196	C₈	1	2	yes	—
36	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
38	#13133176	C₂×C₄	1	2	yes	—
39	C₂×#11719908	—	2	4	no	—
40	C₂×#11719908	C₂³	1	2	yes	—
41	C₂×#11719908	C₂³	1	2	yes	—
42	C₄⋊#88322	C₂×C₄	1	2	yes	—
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	#13133146	C₂×C₄	1	4	yes	—
45	#13133146	C₂×C₄	1	4	yes	—
46	#13133127	C₂×C₄	1	4	yes	—
47	C₈×Dic₅₃	C₂³	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₃	—	2	2	no	—
52	M₃₂×C₅₃	Q₈	1	4	yes	complemented
53	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
54	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
63	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₅₃	#41	1	2	yes	—
70	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
76	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
77	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
78	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
79	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
80	#11719919	C₂×C₈	1	4	yes	—
81	#11719919	C₂×C₈	1	4	yes	—
82	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
83	#11719917	—	2	4	no	—
84	C₂×#88322	C₂²×C₄	1	4	yes	—
85	C₁₆×C₅₃	—	2	2	no	—
86	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
87	#11719908	—	2	2	no	—
88	#11719908	—	2	2	no	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#131	1	2	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
101	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
102	Dic₁₀₆	—	2	4	no	—
103	Dic₁₀₆	—	4	2	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	C₄×Q₈	1	2	yes	—
107	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
109	#88322	—	2	4	no	—
110	#33476	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
114	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	C₈×Q₈	1	2	yes	—
119	C₄×C₅₃	#504	1	2	yes	—
120	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
122	Dic₅₃	—	2	4	no	—
123	Dic₅₃	—	2	4	no	—
124	#2960	—	53	1	no	—
125	#2183	—	53	4	no	—
126	#2183	—	53	2	no	—
127	Q₈×C₁₆	—	53	2	no	—
128	#2151	—	53	2	no	—
129	C₂×C₄⋊C₁₆	—	53	1	no	—
130	#2108	—	53	2	no	—
131	C₄×M₃₂	D₅₃	1	1	yes	characteristic
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
135	C₂×C₅₃	#3407	1	1	yes	characteristic
136	#533	—	53	1	no	—
137	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
138	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
139	C₂²×C₁₆	—	53	2	no	—
140	#446	—	53	2	no	—
141	#446	—	53	2	no	—
142	C₈×Q₈	—	53	2	no	—
143	C₈×Q₈	—	53	2	no	—
144	#424	—	53	2	no	—
145	#423	—	53	1	no	—
146	#405	—	53	2	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₄⋊C₁₆	—	53	4	no	—
149	C₄⋊C₁₆	—	53	4	no	—
150	C₄⋊C₁₆	—	53	1	no	—
151	C₄⋊C₁₆	—	53	1	no	—
152	C₄⋊C₁₆	—	53	2	no	—
153	#346	—	53	4	no	—
154	#346	C₂×D₅₃	1	2	yes	—
155	C₄×C₁₆	—	53	4	no	—
156	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
157	C₅₃	#33476	1	1	yes	characteristic, complemented, Sylow
158	C₂×M₁₆	—	53	2	no	—
159	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₄⋊Q₈	—	53	1	no	—
162	C₄⋊Q₈	—	53	1	no	—
163	#125	—	53	2	no	—
164	C₂²⋊Q₈	—	53	2	no	—
165	C₂²⋊Q₈	—	53	2	no	—
166	C₄×Q₈	—	53	2	no	—
167	C₄×Q₈	—	53	2	no	—
168	#117	—	53	2	no	—
169	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	Dic₁₀₆	1	4	yes	complemented
172	C₂×C₁₆	—	53	2	no	—
173	C₂×C₁₆	—	53	2	no	—
174	C₂×C₁₆	—	106	4	no	—
175	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
176	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
177	C₄⋊C₈	—	53	4	no	—
178	C₄⋊C₈	—	53	4	no	—
179	C₄⋊C₈	—	53	2	no	—
180	C₄⋊C₈	—	53	2	no	—
181	#97	—	53	4	no	—
182	C₄×C₈	—	53	4	no	—
183	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	C₂×Q₈	—	53	2	no	—
188	C₂×Q₈	—	53	2	no	—
189	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
192	M₁₆	—	106	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	53	4	no	—
195	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	#11719945	1	2	yes	—
200	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	—	53	2	no	—
202	C₄⋊C₄	—	53	2	no	—
203	C₄⋊C₄	—	53	4	no	—
204	C₄⋊C₄	—	53	4	no	—
205	C₄⋊C₄	—	53	2	no	—
206	C₄⋊C₄	—	53	2	no	—
207	C₂²⋊C₄	—	53	4	no	—
208	C₄²	—	53	4	no	—
209	C₄²	C₈×D₅₃	1	1	yes	characteristic
210	C₄²	C₈×D₅₃	1	1	yes	characteristic
211	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₁₆	—	2	2	no	—
214	C₁₆	—	106	2	no	—
215	C₁₆	—	106	2	no	—
216	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
217	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
218	Q₈	—	106	4	no	—
219	Q₈	—	212	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	53	4	no	—
223	C₂×C₄	—	53	4	no	—
224	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
225	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
230	C₂×C₄	#13133223	1	2	yes	—
231	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	106	4	no	—
235	C₈	C₄×Dic₁₀₆	1	2	yes	—
236	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
237	C₈	C₂×#11719945	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₈×Dic₁₀₆	1	2	yes	—
241	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	106	4	no	—
246	C₄	—	106	4	no	—
247	C₄	C₈×Dic₁₀₆	1	2	yes	—
248	C₄	#16811041	1	2	yes	—
249	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
250	C₄	C₂×#13133223	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×C₈×Dic₁₀₆	1	1	yes	characteristic
253	C₂	C₂×#16811041	1	1	yes	characteristic
254	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×M₃₂×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33476
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20276623	C₂	semidirect
3. #20276621	C₂	semidirect
4. C₁₆×Dic₁₀₆	C₂	semidirect
52. M₃₂×C₅₃	Q₈	semidirect
157. C₅₃	#33476	semidirect
171. M₃₂	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20283649	2
#20276623	2
#20276621	2
C₁₆×Dic₁₀₆	2
#20276475	2
C₂×C₄⋊#11719908	2
#20276314	2
C₄×M₃₂×C₅₃	2
#33476	53