Subgroups — #27544189

Order 13568.

2858 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×#16811101	C₂	1	2	yes	maximal
2	#20277636	C₂	1	8	yes	maximal, complemented
3	Q₈×C₄⋊Dic₅₃	C₂	1	8	yes	maximal, complemented
4	#20277333	C₂	1	6	yes	maximal
5	#20277122	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811101	—	2	16	no	—
8	C₂×Q₈×Dic₅₃	C₂²	1	16	yes	—
9	C₂×#13133176	C₂²	1	12	yes	—
10	#16810522	C₂²	1	48	yes	—
11	#16810493	C₂²	1	12	yes	—
12	#16810489	C₂²	1	12	yes	—
13	#16810483	C₂²	1	12	yes	—
14	#16810483	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	12	yes	—
16	C₄×C₄⋊Dic₅₃	C₂²	1	12	yes	—
17	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₄	1	16	yes	complemented
19	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
23	Q₈×Dic₅₃	—	2	64	no	—
24	#13133176	—	2	48	no	—
25	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
26	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
28	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
29	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
30	#13133124	C₂³	1	24	yes	—
31	#13133124	C₂³	1	8	yes	—
32	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
33	C₂²×Q₈×C₅₃	D₄	1	8	yes	—
34	C₂²×Q₈×C₅₃	Q₈	1	8	yes	—
35	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
39	C₄×Q₈×C₅₃	C₂×C₄	1	32	yes	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	32	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	24	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
46	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	C₄⋊C₄	1	64	yes	complemented
48	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
49	C₂×Q₈×C₅₃	C₂×D₄	1	16	yes	—
50	C₂×Q₈×C₅₃	C₂×Q₈	1	16	yes	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	—	4	6	no	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
63	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
66	C₂²⋊Dic₅₃	—	2	16	no	—
67	C₄⋊Dic₅₃	—	2	48	no	—
68	C₄⋊Dic₅₃	—	2	16	no	—
69	C₄⋊Dic₅₃	—	4	24	no	—
70	C₄×Dic₅₃	—	2	48	no	—
71	C₂³×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	—	2	2	no	—
73	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
74	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
75	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
76	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
77	Q₈×C₅₃	C₂×C₄⋊C₄	1	64	yes	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	12	no	—
80	C₂×C₄×C₅₃	—	2	8	no	—
81	C₂×C₄×C₅₃	—	4	12	no	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
85	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
88	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
89	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
90	C₂×Dic₅₃	—	2	16	no	—
91	C₂×Dic₅₃	—	2	16	no	—
92	C₂×Dic₅₃	—	2	16	no	—
93	#34182	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	4	no	—
97	C₂²×C₅₃	—	2	4	no	—
98	C₂²×C₅₃	C₂²×C₄⋊C₄	1	4	yes	—
99	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
100	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
101	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
103	C₂²×C₅₃	#585	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	12	no	—
106	C₄×C₅₃	C₂²×C₄⋊C₄	1	24	yes	—
107	C₄×C₅₃	#519	1	4	yes	complemented
108	Dic₅₃	—	4	16	no	—
109	C₂×#519	—	53	2	no	—
110	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic
111	#2351	—	53	8	no	—
112	Q₈×C₄⋊C₄	—	53	8	no	—
113	#2316	—	53	6	no	—
114	#2285	—	53	6	no	—
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#519	1	1	yes	characteristic
118	C₂×C₅₃	#3428	1	1	yes	characteristic
119	C₂×C₅₃	#3497	1	2	yes	—
120	C₂×C₅₃	#3502	1	2	yes	—
121	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
122	#519	—	106	16	no	—
123	C₂×C₄×Q₈	—	53	16	no	—
124	C₂×C₄×Q₈	Dic₅₃	1	16	yes	complemented
125	C₂×C₄×Q₈	C₂×D₅₃	1	4	yes	—
126	C₂×C₄×Q₈	C₂×D₅₃	1	4	yes	—
127	C₂×#117	—	53	12	no	—
128	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
129	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
130	#389	—	53	48	no	—
131	#384	—	53	12	no	—
132	#382	—	53	12	no	—
133	#380	—	53	12	no	—
134	#380	—	53	4	no	—
135	C₄×C₄⋊C₄	—	53	12	no	—
136	C₄×C₄⋊C₄	—	53	12	no	—
137	C₅₃	#34182	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	Dic₁₀₆	1	8	yes	—
139	C₂²×Q₈	D₂₁₂	1	8	yes	—
140	C₂²×Q₈	C₂×Dic₅₃	1	8	yes	—
141	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
142	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
144	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
145	C₄×Q₈	—	106	64	no	—
146	C₄×Q₈	C₂×Dic₅₃	1	32	yes	—
147	C₄×Q₈	C₂×Dic₅₃	1	32	yes	—
148	#117	—	106	48	no	—
149	C₂×C₄⋊C₄	—	53	24	no	—
150	C₂×C₄⋊C₄	—	53	8	no	—
151	C₂×C₄⋊C₄	—	53	24	no	—
152	C₂×C₄⋊C₄	C₂×Dic₅₃	1	24	yes	—
153	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
154	C₂×C₂²⋊C₄	—	53	8	no	—
155	C₂×C₄²	—	53	24	no	—
156	C₂×C₄²	C₂×Dic₅₃	1	24	yes	—
157	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
158	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
159	#95	—	53	8	no	—
160	#95	—	53	24	no	—
161	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
162	C₂×Q₈	C₄⋊Dic₅₃	1	64	yes	complemented
163	C₂×Q₈	C₂×Dic₁₀₆	1	16	yes	—
164	C₂×Q₈	C₂×D₂₁₂	1	16	yes	—
165	C₂×Q₈	C₂²×Dic₅₃	1	16	yes	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	6	no	—
168	C₂²×C₄	—	53	16	no	—
169	C₂²×C₄	C₂×Dic₁₀₆	1	12	yes	—
170	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
173	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
174	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
176	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
177	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
178	C₄⋊C₄	—	106	48	no	—
179	C₄⋊C₄	—	106	16	no	—
180	C₄⋊C₄	—	212	24	no	—
181	C₄⋊C₄	C₂²×Dic₅₃	1	48	yes	—
182	C₂²⋊C₄	—	106	16	no	—
183	C₄²	—	106	48	no	—
184	C₄²	C₂²×Dic₅₃	1	24	yes	—
185	C₄²	C₂²×Dic₅₃	1	24	yes	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂³×Dic₅₃	1	2	yes	—
189	C₂³	C₂²×D₂₁₂	1	2	yes	—
190	C₂³	C₂²×Dic₁₀₆	1	2	yes	—
191	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
192	Q₈	C₂×C₄⋊Dic₅₃	1	64	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	4	12	no	—
197	C₂×C₄	—	106	16	no	—
198	C₂×C₄	—	106	16	no	—
199	C₂×C₄	—	106	16	no	—
200	C₂×C₄	C₂×C₄⋊Dic₅₃	1	48	yes	—
201	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
202	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
203	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
204	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
205	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
206	C₂×C₄	C₂²×Dic₁₀₆	1	12	yes	—
207	C₂×C₄	#13133265	1	4	yes	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	C₂²×C₄⋊Dic₅₃	1	4	yes	—
213	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
214	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
215	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
216	C₂²	C₂×#13133265	1	2	yes	—
217	C₂²	#16811459	1	2	yes	—
218	C₄	—	2	2	no	—
219	C₄	—	2	12	no	—
220	C₄	—	212	16	no	—
221	C₄	C₂²×C₄⋊Dic₅₃	1	24	yes	—
222	C₄	#16811101	1	4	yes	complemented
223	C₂	—	2	4	no	—
224	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂	C₂×#16811101	1	1	yes	characteristic
226	C₂	#20287918	1	1	yes	characteristic
227	C₂	#20288486	1	2	yes	—
228	C₂	#20288525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	#34182
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #20277636	C₂	semidirect
3. Q₈×C₄⋊Dic₅₃	C₂	semidirect
18. C₂×C₄×Q₈×C₅₃	C₄	semidirect
47. C₂×Q₈×C₅₃	C₄⋊C₄	semidirect
107. C₄×C₅₃	#519	semidirect
124. C₂×C₄×Q₈	Dic₅₃	semidirect
137. C₅₃	#34182	semidirect
162. C₂×Q₈	C₄⋊Dic₅₃	semidirect
222. C₄	#16811101	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811101	2
#20277636	2
Q₈×C₄⋊Dic₅₃	2
#20277333	2
#20277122	2
C₂²×C₄×Q₈×C₅₃	2
#34182	53