Subgroups — #28311411

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289754	C₂	1	2	yes	maximal
2	#20289754	C₂	1	2	yes	maximal
3	#20289753	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×#11719910	—	2	4	no	—
14	C₂×#11719910	—	2	4	no	—
15	C₂×#11719910	—	2	4	no	—
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#88322	—	2	2	no	—
25	C₈×Dic₅₃	—	2	2	no	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₄²×C₅₃	—	2	1	no	—
47	C₄²×C₅₃	—	2	1	no	—
48	C₄×Dic₅₃	—	2	2	no	—
49	C₄×Dic₅₃	—	2	1	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	2	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	C₁₆	1	4	yes	complemented
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	M₁₆	1	2	yes	—
59	C₂×#88322	#41	1	2	yes	—
60	C₂×#88322	#41	1	2	yes	—
61	C₂×#88322	#41	1	2	yes	—
62	#11719910	—	4	4	no	—
63	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
85	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
86	C₂×Dic₅₃	M₃₂	1	2	yes	—
87	C₂×Dic₅₃	M₃₂	1	2	yes	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	2	2	no	—
90	#88322	—	2	2	no	—
91	#88322	#126	1	4	yes	—
92	#12894	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	#431	1	1	yes	characteristic
94	C₂²×C₅₃	#431	1	1	yes	characteristic
95	C₂²×C₅₃	#431	1	1	yes	characteristic
96	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	#383	1	1	yes	characteristic
103	C₄×C₅₃	#432	1	1	yes	characteristic
104	C₄×C₅₃	#432	1	1	yes	characteristic
105	C₄×C₅₃	#432	1	1	yes	characteristic
106	Dic₅₃	—	2	2	no	—
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	2	no	—
109	Dic₅₃	—	2	2	no	—
110	C₄²×C₈	—	53	1	no	—
111	#1374	—	53	2	no	—
112	#1374	—	53	2	no	—
113	#1374	—	53	2	no	—
114	C₂×C₅₃	#1839	1	1	yes	characteristic
115	C₂×C₅₃	#2162	1	1	yes	characteristic
116	C₂×C₅₃	#2162	1	1	yes	characteristic
117	C₂×C₅₃	#2162	1	1	yes	characteristic
118	C₂×C₅₃	#2163	1	1	yes	characteristic
119	C₂×C₅₃	#2163	1	1	yes	characteristic
120	C₂×C₅₃	#2163	1	1	yes	characteristic
121	C₂²×C₁₆	—	53	4	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	#6157	1	1	yes	characteristic
128	C₄³	—	53	1	no	—
129	C₅₃	#12894	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
133	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
134	C₂×C₄²	—	53	1	no	—
135	C₂×C₄²	—	53	1	no	—
136	C₂×C₄²	—	53	1	no	—
137	C₂×C₄²	—	53	1	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
141	C₂×C₁₆	—	106	4	no	—
142	C₂×C₁₆	—	106	4	no	—
143	C₂×C₁₆	—	106	4	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	106	2	no	—
146	C₄×C₈	—	106	2	no	—
147	C₄×C₈	—	106	2	no	—
148	C₄×C₈	—	106	2	no	—
149	C₄×C₈	—	106	2	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	—	53	1	no	—
152	C₂²×C₄	—	53	1	no	—
153	C₂²×C₄	—	53	1	no	—
154	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
155	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	53	2	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	#11719936	1	2	yes	—
168	C₂×C₈	#11719940	1	2	yes	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	53	4	no	—
172	C₄²	—	106	2	no	—
173	C₄²	—	106	2	no	—
174	C₄²	—	106	2	no	—
175	C₄²	—	106	1	no	—
176	C₄²	—	106	1	no	—
177	C₄²	—	106	1	no	—
178	C₄²	—	106	1	no	—
179	C₁₆	—	212	4	no	—
180	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	2	no	—
189	C₂×C₄	—	106	1	no	—
190	C₂×C₄	—	106	1	no	—
191	C₂×C₄	—	106	1	no	—
192	C₂×C₄	—	106	1	no	—
193	C₂×C₄	—	106	1	no	—
194	C₂×C₄	—	106	1	no	—
195	C₂×C₄	—	106	1	no	—
196	C₂×C₄	—	106	1	no	—
197	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
198	C₂×C₄	#13133292	1	2	yes	—
199	C₂×C₄	#13133296	1	1	yes	characteristic
200	C₂×C₄	#13133296	1	1	yes	characteristic
201	C₂×C₄	#13133296	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
203	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	53	4	no	—
208	C₈	—	106	2	no	—
209	C₂²	#16811640	1	1	yes	characteristic
210	C₂²	#16811641	1	1	yes	characteristic
211	C₂²	#16811642	1	1	yes	characteristic
212	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
213	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
214	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
215	C₂²	C₂×#13133292	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	106	2	no	—
219	C₄	—	106	2	no	—
220	C₄	—	106	2	no	—
221	C₄	—	106	2	no	—
222	C₄	#16811568	1	1	yes	characteristic
223	C₄	#16811643	1	1	yes	characteristic
224	C₄	#16811644	1	1	yes	characteristic
225	C₄	#16811645	1	1	yes	characteristic
226	C₂	#20290643	1	1	yes	characteristic
227	C₂	#20291073	1	1	yes	characteristic
228	C₂	#20291074	1	1	yes	characteristic
229	C₂	#20291075	1	1	yes	characteristic
230	C₂	#20291077	1	1	yes	characteristic
231	C₂	#20291078	1	1	yes	characteristic
232	C₂	#20291079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12894
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
55. C₄×Dic₅₃	C₁₆	semidirect
129. C₅₃	#12894	semidirect

Maximal subgroups

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

subgroup	index
#20289754	2
#20289753	2
C₄×C₈×Dic₅₃	2
#12894	53