Subgroups — #27537471

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#20283584	C₂	1	2	yes	maximal
3	#20276587	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₄₂₄	C₂	1	4	yes	maximal, complemented
5	#20276499	C₂	1	1	yes	maximal, characteristic
6	#20276436	C₂	1	2	yes	maximal
7	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
9	#16811100	—	2	2	no	—
10	C₂×Dic₄₂₄	C₄	1	8	yes	complemented
11	C₂×Dic₄₂₄	C₂²	1	2	yes	—
12	C₂×Dic₄₂₄	C₂²	1	2	yes	—
13	#16810731	—	2	4	no	—
14	C₄×Dic₂₁₂	C₂²	1	8	yes	—
15	#16810661	C₂²	1	2	yes	—
16	#16810633	—	2	2	no	—
17	#16810451	C₂²	1	2	yes	—
18	#16810449	C₂²	1	2	yes	—
19	#16810428	C₂²	1	8	yes	—
20	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#405	C₂²	1	1	yes	characteristic
23	C₅₃×#346	C₂²	1	2	yes	—
24	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂²×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₂₁₂	—	2	2	no	—
27	C₂×Dic₂₁₂	—	2	4	no	—
28	C₂×Dic₂₁₂	C₂×C₄	1	8	yes	—
29	C₂×Dic₂₁₂	C₂³	1	4	yes	—
30	C₄×Dic₁₀₆	—	2	8	no	—
31	#13133182	—	4	4	no	—
32	#13133176	—	2	2	no	—
33	#13133173	—	4	4	no	—
34	Dic₄₂₄	C₂×C₄	1	8	yes	—
35	Dic₄₂₄	C₂×C₄	1	8	yes	—
36	#13133151	C₂³	1	4	yes	—
37	#13133149	—	2	2	no	—
38	#13133142	—	2	8	no	—
39	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#117×C₅₃	D₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
45	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
47	#97×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	4	2	no	—
51	C₂×Dic₁₀₆	—	2	8	no	—
52	C₂×Dic₁₀₆	—	2	4	no	—
53	C₂×Dic₁₀₆	—	4	2	no	—
54	C₂×Dic₁₀₆	—	4	4	no	—
55	M₁₆×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
62	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	2	4	no	—
66	#11719919	—	4	8	no	—
67	C₄×Dic₅₃	—	4	8	no	—
68	C₄×Dic₅₃	—	4	2	no	—
69	Dic₂₁₂	—	2	8	no	—
70	Dic₂₁₂	—	2	4	no	—
71	Dic₂₁₂	C₂²×C₄	1	8	yes	—
72	C₁₆×C₅₃	—	2	2	no	—
73	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
74	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#135	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	4	8	no	—
83	C₂×Dic₅₃	—	4	4	no	—
84	Dic₁₀₆	—	2	8	no	—
85	Dic₁₀₆	—	4	4	no	—
86	Dic₁₀₆	—	4	8	no	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	C₄×D₄	1	2	yes	—
89	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
91	#33458	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	C₄×D₈	1	2	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	C₄×D₈	1	2	yes	—
97	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
99	Dic₅₃	—	4	8	no	—
100	Dic₅₃	—	8	4	no	—
101	C₂²×Q₃₂	—	53	1	no	—
102	#2950	—	53	2	no	—
103	#2176	—	53	4	no	—
104	C₄×Q₃₂	—	53	4	no	—
105	#2158	—	53	1	no	—
106	#2141	—	53	2	no	—
107	#2108	D₅₃	1	1	yes	characteristic
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₅₃	#3417	1	2	yes	—
111	C₂²×Q₁₆	—	53	2	no	—
112	#519	—	106	2	no	—
113	C₂×Q₃₂	—	53	8	no	—
114	C₂×Q₃₂	—	53	2	no	—
115	C₂×Q₃₂	—	53	2	no	—
116	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
117	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
118	#441	—	106	4	no	—
119	C₄×Q₁₆	—	53	8	no	—
120	#427	—	53	2	no	—
121	#419	—	106	2	no	—
122	#405	C₂×D₅₃	1	1	yes	characteristic
123	#367	—	53	2	no	—
124	#366	—	53	2	no	—
125	#358	—	53	8	no	—
126	#346	C₂×D₅₃	1	2	yes	—
127	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
128	C₅₃	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	106	2	no	—
130	C₂×Q₁₆	—	53	8	no	—
131	C₂×Q₁₆	—	53	4	no	—
132	C₂×Q₁₆	—	106	2	no	—
133	C₂×Q₁₆	—	106	4	no	—
134	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	C₄×Q₈	—	106	8	no	—
137	C₄×Q₈	—	212	4	no	—
138	#117	—	106	2	no	—
139	#117	—	212	4	no	—
140	#117	D₂₁₂	1	1	yes	characteristic
141	Q₃₂	—	53	8	no	—
142	Q₃₂	—	53	8	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
145	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
148	#107	—	53	4	no	—
149	#106	—	106	2	no	—
150	#103	—	106	8	no	—
151	#97	D₂₁₂	1	2	yes	—
152	C₄×C₈	C₂²×D₅₃	1	2	yes	—
153	C₂×Q₈	—	106	8	no	—
154	C₂×Q₈	—	106	4	no	—
155	C₂×Q₈	—	212	2	no	—
156	C₂×Q₈	—	212	4	no	—
157	C₂²×C₄	—	212	2	no	—
158	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
159	Q₁₆	—	53	8	no	—
160	Q₁₆	—	106	8	no	—
161	Q₁₆	—	106	4	no	—
162	M₁₆	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
164	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
165	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
166	C₂×C₈	#11719945	1	2	yes	—
167	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	106	4	no	—
169	C₄⋊C₄	—	212	8	no	—
170	C₄⋊C₄	D₄₂₄	1	2	yes	—
171	C₂²⋊C₄	—	212	2	no	—
172	C₂²⋊C₄	D₄₂₄	1	2	yes	—
173	C₄²	—	212	2	no	—
174	C₄²	—	212	8	no	—
175	C₄²	C₂×D₂₁₂	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
178	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
179	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
180	Q₈	—	106	8	no	—
181	Q₈	—	212	4	no	—
182	Q₈	—	212	8	no	—
183	C₂×C₄	—	212	4	no	—
184	C₂×C₄	—	212	4	no	—
185	C₂×C₄	—	212	8	no	—
186	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
187	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
188	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
189	C₂×C₄	#13133239	1	2	yes	—
190	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	C₄×D₂₁₂	1	2	yes	—
193	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
194	C₈	C₂×#11719945	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	C₄×D₄₂₄	1	2	yes	—
197	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	424	4	no	—
200	C₄	—	212	8	no	—
201	C₄	C₄×D₄₂₄	1	2	yes	—
202	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
203	C₄	C₂×#13133239	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
206	C₂	#20287868	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33458
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #20276587	C₂	semidirect
4. C₄×Dic₄₂₄	C₂	semidirect
10. C₂×Dic₄₂₄	C₄	semidirect
128. C₅₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₄	2
#20283584	2
#20276587	2
C₄×Dic₄₂₄	2
#20276499	2
#20276436	2
C₅₃×#2108	2
#33458	53