Subgroups — #27274724

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	#20272556	C₂	1	2	yes	maximal
5	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	—	2	1	no	—
8	C₂×C₄×#88322	—	2	2	no	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	#16810378	—	2	1	no	—
11	Q₈⋊#88322	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#391	C₄	1	2	yes	—
14	C₅₃×#389	C₄	1	2	yes	—
15	C₅₃×#387	—	2	2	no	—
16	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#376	—	2	1	no	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	C₂×C₄	1	4	yes	—
24	#13133128	—	4	2	no	—
25	C₄×#88322	—	2	8	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	—	2	1	no	—
36	C₂×C₄²×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#95×C₅₃	—	2	2	no	—
43	#95×C₅₃	—	2	2	no	—
44	#95×C₅₃	—	2	2	no	—
45	#95×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	1	no	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
62	C₄²×C₅₃	—	2	4	no	—
63	C₄²×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	SD₁₆	1	2	yes	—
68	C₄²×C₅₃	Q₁₆	1	2	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	#41	1	2	yes	—
71	C₂×#88322	—	2	8	no	—
72	C₂×#88322	—	2	4	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₅₃	—	2	4	no	—
78	Q₈×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	#103	1	4	yes	—
92	C₂×C₄×C₅₃	#104	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	#131	1	4	yes	—
98	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	#88322	—	2	8	no	—
102	#88322	—	4	4	no	—
103	#8176	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₅₃	#408	1	2	yes	—
106	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
107	C₂²×C₅₃	#416	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#104	1	2	yes	—
109	C₄×C₅₃	—	2	4	no	—
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	—	8	2	no	—
115	C₄×C₅₃	#405	1	2	yes	—
116	C₄×C₅₃	#433	1	2	yes	—
117	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
118	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
119	#2295	D₅₃	1	1	yes	characteristic
120	#1767	—	53	1	no	—
121	C₄²×C₈	—	53	1	no	—
122	C₂×Q₈⋊C₈	—	53	2	no	—
123	#1279	—	53	2	no	—
124	C₂×C₅₃	#1493	1	2	yes	—
125	C₂×C₅₃	#1494	1	2	yes	—
126	C₂×C₅₃	#1750	1	1	yes	characteristic
127	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
128	C₂×C₅₃	C₄×#103	1	1	yes	characteristic
129	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	53	2	no	—
131	C₂×#97	—	106	1	no	—
132	C₂×C₄×C₈	—	53	2	no	—
133	C₂×C₄×C₈	—	106	2	no	—
134	#391	Dic₅₃	1	2	yes	—
135	#389	Dic₅₃	1	2	yes	—
136	#387	—	2	2	no	—
137	C₄×C₄⋊C₄	—	2	1	no	—
138	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
139	#376	—	2	1	no	—
140	C₄³	C₂×D₅₃	1	1	yes	characteristic
141	#336	—	106	1	no	—
142	Q₈⋊C₈	—	53	8	no	—
143	C₅₃	#8176	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₈	—	106	2	no	—
146	C₂²×C₈	—	106	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	C₄×D₅₃	1	4	yes	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
156	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
160	C₄⋊C₈	—	53	4	no	—
161	C₄⋊C₈	—	106	2	no	—
162	#97	—	212	2	no	—
163	C₄×C₈	—	53	4	no	—
164	C₄×C₈	—	106	8	no	—
165	C₄×C₈	—	106	2	no	—
166	#95	—	2	1	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₄×Dic₅₃	1	2	yes	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
177	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
178	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×C₈	—	106	8	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	#11719924	1	2	yes	—
197	C₄²	#11719925	1	2	yes	—
198	C₄²	C₂×C₄×D₅₃	1	2	yes	—
199	C₄²	#11719945	1	2	yes	—
200	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	#13133141	1	4	yes	—
216	C₂×C₄	#13133144	1	4	yes	—
217	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
218	C₂×C₄	#13133176	1	1	yes	characteristic
219	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
220	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
221	C₂×C₄	#13133224	1	4	yes	—
222	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
224	C₂×C₄	#13133237	1	2	yes	—
225	C₈	—	106	8	no	—
226	C₈	—	212	4	no	—
227	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
228	C₂²	#16810590	1	2	yes	—
229	C₂²	C₂×#13133141	1	1	yes	characteristic
230	C₂²	#16810619	1	1	yes	characteristic
231	C₂²	C₂×#13133144	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	8	2	no	—
238	C₄	#16810579	1	2	yes	—
239	C₄	#16810684	1	2	yes	—
240	C₄	C₄×#11719924	1	2	yes	—
241	C₄	C₄×#11719925	1	2	yes	—
242	C₂	#20273170	1	2	yes	—
243	C₂	#20273175	1	2	yes	—
244	C₂	#20274481	1	1	yes	characteristic
245	C₂	C₄×#13133144	1	1	yes	characteristic
246	C₂	C₄×#13133141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2295	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2295	D₅₃
hypocenter	C₅₃	#8176
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
11. Q₈⋊#88322	C₄	semidirect
143. C₅₃	#8176	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₄²×#88322	2
C₂×Q₈⋊#88322	2
#20272556	2
C₅₃×#2295	2
#8176	53