Subgroups — #27383130

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20274580	C₂	1	3	yes	maximal
4	C₈×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
5	C₄⋊C₈×Dic₅₃	C₂	1	3	yes	maximal
6	#20274460	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₃	C₄	1	2	yes	—
9	Q₈×#88322	C₄	1	8	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	3	yes	—
11	C₂×C₄×#88322	C₂²	1	3	yes	—
12	C₂×C₈×Dic₅₃	C₂²	1	6	yes	—
13	C₄⋊C₄×Dic₅₃	C₄	1	6	yes	—
14	C₄×C₄⋊Dic₅₃	C₂²	1	3	yes	—
15	C₄²×Dic₅₃	C₂²	1	3	yes	—
16	#16810377	C₂²	1	1	yes	characteristic
17	#16810377	C₂²	1	3	yes	—
18	#16810377	C₂²	1	3	yes	—
19	#16810377	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂²×#88322	C₂³	1	3	yes	—
25	C₂²×#88322	C₂³	1	1	yes	characteristic
26	Q₈×Dic₅₃	C₈	1	16	yes	complemented
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	6	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊#88322	C₂×C₄	1	12	yes	—
33	C₄×#88322	C₂×C₄	1	12	yes	—
34	C₈×Dic₅₃	—	4	6	no	—
35	#13133124	C₂×C₄	1	6	yes	—
36	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
43	C₄⋊C₈×C₅₃	C₂×C₄	1	12	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
45	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
46	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
51	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₃	—	2	3	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	3	no	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
58	C₄⋊C₄×C₅₃	C₄²	1	12	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
60	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
61	C₄⋊Dic₅₃	C₂×C₈	1	24	yes	—
62	C₄×Dic₅₃	—	2	3	no	—
63	C₄×Dic₅₃	—	2	1	no	—
64	C₄×Dic₅₃	—	2	1	no	—
65	C₄×Dic₅₃	—	2	3	no	—
66	C₄×Dic₅₃	—	4	6	no	—
67	C₄×Dic₅₃	C₂×C₈	1	24	yes	—
68	C₂×#88322	—	2	3	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	3	no	—
72	C₂×#88322	C₂²×C₄	1	2	yes	—
73	C₂×#88322	C₂²×C₄	1	6	yes	—
74	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₅₃	C₄×C₈	1	16	yes	complemented
76	C₂×C₄×C₅₃	—	2	6	no	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	2	6	no	—
88	C₂×Dic₅₃	—	2	6	no	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
91	C₂×Dic₅₃	C₂²×C₈	1	12	yes	—
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	—	2	6	no	—
94	#88322	—	2	2	no	—
95	#88322	—	2	6	no	—
96	#20569	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
98	C₂²×C₅₃	#519	1	1	yes	characteristic
99	C₂²×C₅₃	#520	1	1	yes	characteristic
100	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
101	C₂²×C₅₃	#568	1	2	yes	—
102	C₄×C₅₃	—	2	6	no	—
103	C₄×C₅₃	C₂×C₄×C₈	1	12	yes	—
104	C₄×C₅₃	#519	1	1	yes	characteristic
105	C₄×C₅₃	#519	1	1	yes	characteristic
106	C₄×C₅₃	#520	1	1	yes	characteristic
107	C₄×C₅₃	#520	1	1	yes	characteristic
108	Dic₅₃	—	2	4	no	—
109	Dic₅₃	—	2	12	no	—
110	C₂×C₈×Q₈	—	53	1	no	—
111	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
112	C₄²×Q₈	—	53	1	no	—
113	C₈×C₄⋊C₄	—	53	3	no	—
114	C₈×C₄⋊C₄	—	53	3	no	—
115	C₄×C₄⋊C₈	—	53	3	no	—
116	#1744	—	53	3	no	—
117	C₂×C₅₃	#2276	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
119	C₂×C₅₃	#2875	1	1	yes	characteristic
120	C₂×C₅₃	#2982	1	2	yes	—
121	C₂×C₅₃	#2988	1	2	yes	—
122	C₂×C₄×Q₈	—	53	2	no	—
123	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
124	C₈×Q₈	—	53	8	no	—
125	C₈×Q₈	Dic₅₃	1	8	yes	complemented
126	C₂×C₄⋊C₈	—	53	3	no	—
127	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
128	C₂×C₄×C₈	—	53	3	no	—
129	C₂×C₄×C₈	—	53	6	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
131	C₄×C₄⋊C₄	—	53	6	no	—
132	C₄×C₄⋊C₄	—	53	3	no	—
133	C₄³	—	53	3	no	—
134	#336	—	53	1	no	—
135	#336	—	53	3	no	—
136	#336	—	53	3	no	—
137	#336	—	53	3	no	—
138	C₅₃	#20569	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	53	3	no	—
142	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
144	C₄×Q₈	—	53	16	no	—
145	C₄×Q₈	C₄×D₅₃	1	4	yes	—
146	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
147	C₂×C₄⋊C₄	—	53	6	no	—
148	C₂×C₄⋊C₄	C₄×D₅₃	1	3	yes	—
149	C₂×C₄²	—	53	6	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	3	no	—
152	C₂×C₄²	—	53	6	no	—
153	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
154	C₄⋊C₈	—	53	12	no	—
155	C₄⋊C₈	C₂×Dic₅₃	1	12	yes	—
156	C₄×C₈	—	53	12	no	—
157	C₄×C₈	—	212	6	no	—
158	C₄×C₈	C₂×Dic₅₃	1	12	yes	—
159	#95	—	53	6	no	—
160	C₂×Q₈	C₈×D₅₃	1	8	yes	—
161	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
162	C₂²×C₄	—	53	6	no	—
163	C₂²×C₄	—	53	2	no	—
164	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
165	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
166	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
167	C₂×C₈	—	2	3	no	—
168	C₂×C₈	—	2	3	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	53	6	no	—
173	C₂×C₈	—	106	3	no	—
174	C₂×C₈	—	106	1	no	—
175	C₂×C₈	—	106	1	no	—
176	C₂×C₈	—	106	3	no	—
177	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
178	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
179	C₄⋊C₄	—	53	24	no	—
180	C₄⋊C₄	C₄×Dic₅₃	1	12	yes	—
181	C₄²	—	53	24	no	—
182	C₄²	—	106	3	no	—
183	C₄²	—	106	1	no	—
184	C₄²	—	106	1	no	—
185	C₄²	—	106	3	no	—
186	C₄²	—	212	6	no	—
187	C₄²	C₂×C₄×D₅₃	1	6	yes	—
188	C₄²	C₂²×Dic₅₃	1	6	yes	—
189	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
190	Q₈	C₈×Dic₅₃	1	16	yes	complemented
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	53	4	no	—
193	C₂×C₄	—	53	12	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	6	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	106	6	no	—
198	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
199	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
200	C₂×C₄	C₂×C₈×D₅₃	1	12	yes	—
201	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄	#13133263	1	1	yes	characteristic
204	C₂×C₄	#13133264	1	1	yes	characteristic
205	C₂×C₄	#13133265	1	1	yes	characteristic
206	C₂×C₄	#13133265	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	6	no	—
209	C₈	—	106	2	no	—
210	C₈	—	106	6	no	—
211	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
212	C₂²	#16811099	1	1	yes	characteristic
213	C₂²	#16811105	1	1	yes	characteristic
214	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
215	C₂²	#16811388	1	2	yes	—
216	C₄	—	2	6	no	—
217	C₄	—	106	4	no	—
218	C₄	—	106	12	no	—
219	C₄	C₂×C₈×Dic₅₃	1	12	yes	—
220	C₄	#16811099	1	1	yes	characteristic
221	C₄	#16811101	1	1	yes	characteristic
222	C₄	#16811105	1	1	yes	characteristic
223	C₄	#16811106	1	1	yes	characteristic
224	C₂	#20277080	1	1	yes	characteristic
225	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
226	C₂	#20283141	1	1	yes	characteristic
227	C₂	#20283794	1	2	yes	—
228	C₂	#20283825	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#20569
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×#88322	C₄	semidirect
21. C₈×Q₈×C₅₃	C₄	semidirect
26. Q₈×Dic₅₃	C₈	semidirect
75. Q₈×C₅₃	C₄×C₈	semidirect
125. C₈×Q₈	Dic₅₃	semidirect
138. C₅₃	#20569	semidirect
190. Q₈	C₈×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88322	2
C₄×Q₈×Dic₅₃	2
#20274580	2
C₈×C₄⋊Dic₅₃	2
C₄⋊C₈×Dic₅₃	2
#20274460	2
C₂×C₈×Q₈×C₅₃	2
#20569	53