Subgroups — #27276961

Order 13568.

368 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	2	yes	maximal, complemented
2	#20273029	C₂	1	2	yes	maximal
3	#20272546	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	4	yes	—
6	#16810376	—	2	2	no	—
7	#16810374	—	2	2	no	—
8	#16810349	—	2	2	no	—
9	#16810347	—	2	2	no	—
10	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#433	—	2	2	no	—
12	C₅₃×#433	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	—	2	4	no	—
20	C₂²⋊#88322	—	2	4	no	—
21	C₄×#88322	—	2	4	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	#117×C₅₃	—	2	2	no	—
28	#117×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	1	no	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₂²⋊C₈×C₅₃	—	2	1	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	2	1	no	—
41	#97×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	C₄²×C₅₃	#41	1	2	yes	—
72	C₂×#88322	—	4	4	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	4	1	no	—
76	C₂³×C₅₃	—	4	1	no	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₅₃	#131	1	2	yes	—
94	C₂×C₄×C₅₃	#131	1	2	yes	—
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	4	1	no	—
99	C₈×C₅₃	—	4	1	no	—
100	#88322	—	8	4	no	—
101	#8603	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	4	1	no	—
105	C₂²×C₅₃	—	4	1	no	—
106	C₂²×C₅₃	—	4	1	no	—
107	C₂²×C₅₃	—	8	2	no	—
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#424	1	2	yes	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	4	1	no	—
113	C₄×C₅₃	—	4	1	no	—
114	C₄×C₅₃	—	4	1	no	—
115	C₄×C₅₃	#408	1	1	yes	characteristic
116	C₄×C₅₃	#408	1	1	yes	characteristic
117	C₄×C₅₃	#433	1	2	yes	—
118	C₂×#433	D₅₃	1	1	yes	characteristic
119	#1564	—	53	2	no	—
120	#1458	—	53	2	no	—
121	#1275	—	53	2	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	—	8	1	no	—
124	C₂×C₅₃	#1798	1	1	yes	characteristic
125	C₂×C₅₃	#1812	1	1	yes	characteristic
126	C₂×C₅₃	#1813	1	1	yes	characteristic
127	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
128	#433	—	2	2	no	—
129	#433	—	2	2	no	—
130	#431	—	53	4	no	—
131	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
134	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
135	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#335	—	106	2	no	—
138	#334	—	106	2	no	—
139	#324	—	106	2	no	—
140	#323	—	106	2	no	—
141	C₅₃	#8603	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
147	#117	—	2	2	no	—
148	#117	—	2	2	no	—
149	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
150	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	Dic₁₀₆	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	106	4	no	—
159	C₂²⋊C₈	—	2	1	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₂²⋊C₈	—	106	4	no	—
163	#97	—	2	2	no	—
164	C₄×C₈	—	106	4	no	—
165	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
166	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
167	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	212	4	no	—
185	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	#11719945	1	2	yes	—
195	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133176	1	1	yes	characteristic
214	C₂×C₄	#13133191	1	2	yes	—
215	C₂×C₄	#13133196	1	2	yes	—
216	C₂×C₄	#13133224	1	2	yes	—
217	C₂×C₄	#13133224	1	2	yes	—
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	1	no	—
222	C₈	—	4	1	no	—
223	C₈	—	424	4	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	#16810483	1	1	yes	characteristic
231	C₂²	#16810650	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	#16810590	1	1	yes	characteristic
238	C₄	#16810590	1	1	yes	characteristic
239	C₄	#16810684	1	2	yes	—
240	C₂	—	2	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#20274699	1	1	yes	characteristic
243	C₂	#20274761	1	1	yes	characteristic
244	C₂	#20274765	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#433	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#433	D₅₃
hypocenter	C₅₃	#8603
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
1. #20273463	C₂	semidirect
3. #20272546	C₂	semidirect
141. C₅₃	#8603	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
#20273029	2
#20272546	2
C₂×C₅₃×#433	2
#8603	53