Subgroups — #27537601

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	#20276624	C₂	1	2	yes	maximal, complemented
2	#20276624	C₂	1	4	yes	maximal, complemented
3	Q₈×#11719908	C₂	1	2	yes	maximal, complemented
4	#20276474	C₂	1	2	yes	maximal
5	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
6	#20276312	C₂	1	2	yes	maximal
7	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂²×#11719908	C₂²	1	2	yes	—
12	C₄⋊#11719908	C₂²	1	1	yes	characteristic
13	C₄⋊#11719908	C₂²	1	4	yes	—
14	C₄⋊#11719908	C₂²	1	1	yes	characteristic
15	C₄⋊#11719908	C₂²	1	2	yes	—
16	C₄⋊#11719908	C₂²	1	4	yes	—
17	#16810399	C₂²	1	2	yes	—
18	#16810399	C₂²	1	4	yes	—
19	C₄×#11719908	C₂²	1	2	yes	—
20	C₄×#11719908	C₂²	1	4	yes	—
21	C₅₃×#533	C₄	1	1	yes	characteristic
22	C₅₃×#446	C₄	1	2	yes	—
23	C₅₃×#446	C₄	1	2	yes	—
24	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#424	C₄	1	2	yes	—
27	C₅₃×#423	C₄	1	1	yes	characteristic
28	C₅₃×#405	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	#13133157	—	2	2	no	—
31	#13133157	Q₈	1	4	yes	complemented
32	C₂×#11719908	—	2	4	no	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	C₂×#11719908	C₂³	1	2	yes	—
35	C₂×#11719908	C₂³	1	2	yes	—
36	C₂×#11719908	C₂³	1	2	yes	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
42	#125×C₅₃	C₈	1	2	yes	—
43	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
44	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
45	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
47	#117×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
53	#97×C₅₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂³	1	4	yes	—
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
60	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
74	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
75	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
76	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
77	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
80	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
85	#11719908	—	2	2	no	—
86	#11719908	—	2	2	no	—
87	#11719908	—	2	2	no	—
88	#11719908	C₂×Q₈	1	4	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	Q₈×C₅₃	—	2	4	no	—
91	Q₈×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	1	no	—
94	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
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	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#131	1	2	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	C₄×Q₈	1	2	yes	—
108	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
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	4	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	C₈×Q₈	1	2	yes	—
121	C₄×C₅₃	#504	1	2	yes	—
122	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
124	#2960	D₅₃	1	1	yes	characteristic
125	#2183	—	53	2	no	—
126	#2183	—	53	4	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	—	53	1	no	—
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	Dic₅₃	1	1	yes	characteristic
137	C₂×M₃₂	—	53	1	no	—
138	C₂×M₃₂	—	53	1	no	—
139	C₂²×C₁₆	—	53	2	no	—
140	#446	Dic₅₃	1	2	yes	—
141	#446	Dic₅₃	1	2	yes	—
142	C₈×Q₈	C₂×D₅₃	1	2	yes	—
143	C₈×Q₈	C₂×D₅₃	1	2	yes	—
144	#424	Dic₅₃	1	2	yes	—
145	#423	Dic₅₃	1	1	yes	characteristic
146	#405	C₂×D₅₃	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₄⋊C₁₆	—	53	1	no	—
149	C₄⋊C₁₆	—	53	1	no	—
150	C₄⋊C₁₆	—	53	2	no	—
151	C₄⋊C₁₆	—	53	4	no	—
152	C₄⋊C₁₆	—	53	4	no	—
153	#346	—	53	2	no	—
154	#346	—	53	4	no	—
155	C₄×C₁₆	—	53	2	no	—
156	C₄×C₁₆	—	53	4	no	—
157	C₅₃	#33476	1	1	yes	characteristic, complemented, Sylow
158	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₄⋊Q₈	#88322	1	1	yes	characteristic
162	C₄⋊Q₈	#88322	1	1	yes	characteristic
163	#125	#88322	1	2	yes	—
164	C₂²⋊Q₈	#88322	1	2	yes	—
165	C₂²⋊Q₈	#88322	1	2	yes	—
166	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
167	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
168	#117	C₂×Dic₅₃	1	2	yes	—
169	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
170	M₃₂	—	53	4	no	—
171	M₃₂	—	106	2	no	—
172	C₂×C₁₆	—	53	2	no	—
173	C₂×C₁₆	—	53	2	no	—
174	C₂×C₁₆	—	53	2	no	—
175	C₂×C₁₆	—	53	2	no	—
176	C₂×C₁₆	—	106	4	no	—
177	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
178	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
179	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
180	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
181	#97	C₂×Dic₅₃	1	4	yes	—
182	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
183	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₅₃	1	4	yes	—
186	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	C₂×Q₈	C₂×#88322	1	2	yes	—
188	C₂×Q₈	C₂×#88322	1	2	yes	—
189	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
190	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	Q₈×D₅₃	1	2	yes	—
195	C₂×C₈	#11719949	1	2	yes	—
196	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
199	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂×#88322	1	2	yes	—
202	C₄⋊C₄	C₂×#88322	1	2	yes	—
203	C₄⋊C₄	C₂×#88322	1	4	yes	—
204	C₄⋊C₄	C₂×#88322	1	4	yes	—
205	C₄⋊C₄	C₂×#88322	1	2	yes	—
206	C₄⋊C₄	C₂×#88322	1	2	yes	—
207	C₂²⋊C₄	C₂×#88322	1	4	yes	—
208	C₄²	C₂×#88322	1	1	yes	characteristic
209	C₄²	C₂×#88322	1	1	yes	characteristic
210	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
212	C₄²	C₂²×Dic₅₃	1	4	yes	—
213	C₁₆	—	53	4	no	—
214	C₁₆	—	106	2	no	—
215	C₁₆	—	106	2	no	—
216	C₁₆	—	106	2	no	—
217	C₂³	C₂²×#88322	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
223	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
224	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
225	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
226	C₂×C₄	C₂²×#88322	1	4	yes	—
227	C₂×C₄	C₂²×#88322	1	4	yes	—
228	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
229	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
230	C₂×C₄	#13133224	1	2	yes	—
231	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	Q₈×Dic₅₃	1	2	yes	—
236	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
237	C₈	C₂×#11719949	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×#88322	1	2	yes	—
241	C₂²	C₂³×#88322	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×#88322	1	2	yes	—
248	C₄	#16811040	1	2	yes	—
249	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
250	C₄	C₂×#13133224	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×#88322	1	1	yes	characteristic
253	C₂	C₂×#16811040	1	1	yes	characteristic
254	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2960	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#33476
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. #20276624	C₂	semidirect
3. Q₈×#11719908	C₂	semidirect
31. #13133157	Q₈	semidirect
157. C₅₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
Q₈×#11719908	2
#20276474	2
C₂×C₄⋊#11719908	2
#20276312	2
C₄×#13133157	2
C₅₃×#2960	2
#33476	53