Subgroups — #27302632

Order 13568.

350 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	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	#20272799	C₂	1	4	yes	maximal
4	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	—	2	1	no	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₅₃×#446	—	2	1	no	—
13	C₅₃×#446	—	2	1	no	—
14	C₅₃×#446	—	2	1	no	—
15	C₅₃×#446	—	2	1	no	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	2	2	no	—
21	#13133157	—	2	4	no	—
22	#13133157	Q₈	1	4	yes	complemented
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₅₃	—	2	1	no	—
30	C₄×Q₈×C₅₃	—	2	1	no	—
31	C₄×Q₈×C₅₃	—	2	1	no	—
32	C₄×Q₈×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	1	no	—
40	C₄⋊C₈×C₅₃	—	2	1	no	—
41	#97×C₅₃	—	2	2	no	—
42	#97×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	—	2	1	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
73	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
74	#11719908	—	2	4	no	—
75	#11719908	—	4	2	no	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	Q₈×C₅₃	—	4	2	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	—	4	1	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#131	1	2	yes	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	4	2	no	—
96	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
97	C₈×C₅₃	#124	1	1	yes	characteristic
98	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
99	#13268	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	4	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	#349	1	2	yes	—
110	C₄×C₅₃	#391	1	1	yes	characteristic
111	C₄×C₅₃	#408	1	1	yes	characteristic
112	C₄×C₅₃	#446	1	2	yes	—
113	C₂×#446	D₅₃	1	1	yes	characteristic
114	#2150	—	53	1	no	—
115	C₄×M₃₂	—	53	1	no	—
116	#1377	—	53	4	no	—
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	#1984	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
120	C₂×C₅₃	#2115	1	1	yes	characteristic
121	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
122	C₂×M₃₂	—	53	2	no	—
123	C₂×M₃₂	—	53	2	no	—
124	#446	—	2	1	no	—
125	#446	—	2	1	no	—
126	#446	—	2	1	no	—
127	#446	—	2	1	no	—
128	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
129	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
130	C₂×#97	C₂×D₅₃	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	106	1	no	—
133	C₄⋊C₁₆	—	106	1	no	—
134	#346	—	106	1	no	—
135	C₄×C₁₆	—	106	1	no	—
136	C₅₃	#13268	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	#88322	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
139	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	C₄×Q₈	—	2	1	no	—
145	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
146	C₂×C₄⋊C₄	#88322	1	2	yes	—
147	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
148	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
149	M₃₂	—	53	4	no	—
150	M₃₂	—	106	2	no	—
151	M₃₂	—	106	4	no	—
152	C₂×C₁₆	—	106	2	no	—
153	C₂×C₁₆	—	106	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	#97	—	2	2	no	—
159	#97	—	2	2	no	—
160	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
168	C₂²×C₄	C₂×#88322	1	2	yes	—
169	C₂²×C₄	C₂×#88322	1	2	yes	—
170	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₅₃	1	2	yes	—
178	C₂×C₈	#11719949	1	2	yes	—
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	#11719917	1	1	yes	characteristic
188	C₄²	#11719917	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₁₆	—	106	4	no	—
192	C₁₆	—	212	2	no	—
193	C₂³	C₂²×#88322	1	1	yes	characteristic
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
205	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
208	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
209	C₂×C₄	#13133224	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	#13133193	1	2	yes	—
214	C₈	#13133206	1	1	yes	characteristic
215	C₈	#13133215	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
219	C₂²	Q₈×#88322	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	#16810405	1	2	yes	—
226	C₄	#16810530	1	1	yes	characteristic
227	C₄	#16810590	1	1	yes	characteristic
228	C₄	#16810754	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	#20275652	1	1	yes	characteristic
231	C₂	C₂×#16810405	1	1	yes	characteristic
232	C₂	#20276336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator, prime radical	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#13268
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
22. #13133157	Q₈	semidirect
136. C₅₃	#13268	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₄×#13133157	2
#20272799	2
C₂×C₅₃×#446	2
#13268	53