Subgroups — #27539622

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
3	#20277001	C₂	1	4	yes	maximal, complemented
4	#20276994	C₂	1	4	yes	maximal, complemented
5	#20276503	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811180	—	2	1	no	—
8	C₂×#13133209	—	2	1	no	—
9	#16811028	C₂²	1	2	yes	—
10	#16811025	—	2	1	no	—
11	#16811025	—	2	1	no	—
12	#16811025	—	2	1	no	—
13	#16811025	C₂²	1	2	yes	—
14	#16811025	C₂²	1	2	yes	—
15	#16811025	C₂²	1	2	yes	—
16	#16811021	—	2	1	no	—
17	#16810661	—	2	1	no	—
18	#16810661	C₂²	1	2	yes	—
19	C₂×#13133151	C₂²	1	2	yes	—
20	C₂×#13133149	—	2	1	no	—
21	#16810451	C₂²	1	8	yes	—
22	#16810449	C₂²	1	8	yes	—
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#346	C₂²	1	2	yes	—
26	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
27	#13133217	—	2	2	no	—
28	#13133209	—	2	2	no	—
29	#13133209	—	2	2	no	—
30	#13133209	—	2	2	no	—
31	#13133209	—	4	2	no	—
32	#13133196	—	4	2	no	—
33	C₄×Dic₁₀₆	—	4	2	no	—
34	#13133176	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	C₂×#11719919	—	4	2	no	—
37	#13133151	—	2	2	no	—
38	#13133151	—	2	2	no	—
39	#13133151	C₂³	1	4	yes	—
40	#13133151	C₂³	1	4	yes	—
41	#13133149	—	2	2	no	—
42	#13133149	—	2	2	no	—
43	#13133149	—	2	2	no	—
44	#13133149	—	2	2	no	—
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	M₃₂×C₅₃	Q₈	1	8	yes	complemented
49	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	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	4	yes	—
63	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₄²×C₅₃	D₈	1	2	yes	—
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	2	4	no	—
69	C₄⋊Dic₅₃	—	2	4	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	#11719919	—	4	4	no	—
73	#11719919	—	4	4	no	—
74	#11719919	—	4	4	no	—
75	C₄×Dic₅₃	—	4	2	no	—
76	C₁₆×C₅₃	C₂×Q₈	1	8	yes	—
77	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₅₃	—	4	4	no	—
86	C₂×Dic₅₃	—	4	4	no	—
87	C₂×Dic₅₃	—	8	4	no	—
88	Dic₁₀₆	—	8	4	no	—
89	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
90	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
91	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
93	#33732	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	#500	1	2	yes	—
96	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	#500	1	2	yes	—
100	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₅₃	—	8	4	no	—
103	Dic₅₃	—	8	4	no	—
104	#3160	—	53	1	no	—
105	C₂×#499	—	53	1	no	—
106	#2255	—	53	4	no	—
107	#2253	—	53	4	no	—
108	#2159	—	53	4	no	—
109	C₄×M₃₂	D₅₃	1	1	yes	characteristic
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
112	C₂×C₅₃	#3414	1	2	yes	—
113	#533	—	106	1	no	—
114	C₂×#125	—	106	1	no	—
115	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
116	#500	—	53	2	no	—
117	#499	—	53	2	no	—
118	#499	—	53	2	no	—
119	#499	—	53	2	no	—
120	#499	—	106	1	no	—
121	#499	—	106	1	no	—
122	#499	—	106	1	no	—
123	#498	—	106	1	no	—
124	#427	—	53	2	no	—
125	#427	—	106	1	no	—
126	C₂×#107	—	53	2	no	—
127	C₂×#106	—	106	1	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	#367	—	53	8	no	—
130	#366	—	53	8	no	—
131	#346	C₂×D₅₃	1	2	yes	—
132	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
133	C₅₃	#33732	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	106	2	no	—
137	#125	—	106	2	no	—
138	#125	—	106	2	no	—
139	#125	—	106	2	no	—
140	#125	—	212	2	no	—
141	C₂²⋊Q₈	—	212	2	no	—
142	C₄×Q₈	—	212	2	no	—
143	#117	—	106	2	no	—
144	C₂×C₄⋊C₄	—	106	2	no	—
145	C₂×C₄⋊C₄	—	212	2	no	—
146	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
147	M₃₂	Dic₁₀₆	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
149	#107	—	53	4	no	—
150	#107	—	53	4	no	—
151	#107	—	106	2	no	—
152	#107	—	106	2	no	—
153	#106	—	106	2	no	—
154	#106	—	106	2	no	—
155	#106	—	106	2	no	—
156	#106	—	106	2	no	—
157	C₄×C₈	D₂₁₂	1	1	yes	characteristic
158	C₄×C₈	D₂₁₂	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₂×Q₈	—	212	2	no	—
162	C₂²×C₄	—	212	2	no	—
163	C₂²×C₄	D₄₂₄	1	2	yes	—
164	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂×C₈	Dic₂₁₂	1	4	yes	—
166	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
167	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
168	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
170	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
171	C₄⋊C₄	—	106	4	no	—
172	C₄⋊C₄	—	106	4	no	—
173	C₄⋊C₄	—	212	2	no	—
174	C₄⋊C₄	—	212	4	no	—
175	C₄⋊C₄	—	212	4	no	—
176	C₄⋊C₄	—	212	4	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₂²⋊C₄	—	212	2	no	—
179	C₄²	—	212	2	no	—
180	C₄²	D₄₂₄	1	2	yes	—
181	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
182	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
183	C₁₆	C₂×Dic₁₀₆	1	8	yes	—
184	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
185	Q₈	—	424	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	424	4	no	—
188	C₂×C₄	—	212	4	no	—
189	C₂×C₄	—	212	4	no	—
190	C₂×C₄	#13133217	1	2	yes	—
191	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
192	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
193	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
195	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
196	C₈	#13133217	1	2	yes	—
197	C₈	C₂×Dic₂₁₂	1	2	yes	—
198	C₈	C₂×Dic₂₁₂	1	2	yes	—
199	C₈	C₂²×Dic₁₀₆	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#16811028	1	2	yes	—
202	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	424	4	no	—
206	C₄	—	424	4	no	—
207	C₄	#16811028	1	2	yes	—
208	C₄	C₂×#13133217	1	1	yes	characteristic
209	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#16811028	1	1	yes	characteristic
212	C₂	#20287851	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33732
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20277001	C₂	semidirect
4. #20276994	C₂	semidirect
48. M₃₂×C₅₃	Q₈	semidirect
133. C₅₃	#33732	semidirect
147. M₃₂	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₂×#16811025	2
#20277001	2
#20276994	2
#20276503	2
C₄×M₃₂×C₅₃	2
#33732	53