Subgroups — #27302725

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272824	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	—	2	1	no	—
6	#16811028	—	2	1	no	—
7	#16811028	—	2	1	no	—
8	#16811028	C₄	1	2	yes	—
9	#16811028	C₄	1	2	yes	—
10	C₂×#13133153	—	2	1	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133151	—	2	1	no	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	#16810647	—	2	1	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	—	2	1	no	—
18	C₄×C₁₆×C₅₃	—	2	1	no	—
19	C₂²×Dic₁₀₆	—	4	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	#13133217	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	4	1	no	—
24	#13133217	—	4	1	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	4	1	no	—
27	#13133153	—	2	4	no	—
28	#13133153	—	4	2	no	—
29	#13133151	—	2	2	no	—
30	#13133151	—	2	2	no	—
31	#13133151	—	2	2	no	—
32	#13133151	C₂×C₄	1	4	yes	—
33	C₄⋊#88322	—	4	1	no	—
34	C₄⋊#88322	—	4	1	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	M₃₂×C₅₃	—	2	2	no	—
39	M₃₂×C₅₃	Q₈	1	4	yes	complemented
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×Dic₁₀₆	—	4	2	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×Dic₁₀₆	—	8	4	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	D₈	1	1	yes	characteristic
63	C₄²×C₅₃	D₈	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	2	4	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	#11719917	—	4	2	no	—
69	#11719917	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₁₆×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×Dic₅₃	—	4	4	no	—
86	C₂×Dic₅₃	—	8	4	no	—
87	Dic₁₀₆	—	8	4	no	—
88	Dic₁₀₆	—	8	4	no	—
89	C₈×C₅₃	#103	1	2	yes	—
90	C₈×C₅₃	#103	1	2	yes	—
91	C₈×C₅₃	C₄×Q₈	1	2	yes	—
92	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	#88322	—	8	2	no	—
94	#13290	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
98	C₂²×C₅₃	#487	1	1	yes	characteristic
99	C₂²×C₅₃	#498	1	1	yes	characteristic
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	#361	1	2	yes	—
103	C₄×C₅₃	#362	1	2	yes	—
104	C₄×C₅₃	#391	1	1	yes	characteristic
105	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
106	C₄×C₅₃	#486	1	1	yes	characteristic
107	C₄×C₅₃	#500	1	1	yes	characteristic
108	Dic₅₃	—	8	4	no	—
109	C₂×#500	—	53	1	no	—
110	C₄×M₃₂	D₅₃	1	1	yes	characteristic
111	#1848	—	53	1	no	—
112	#1387	—	53	4	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1990	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	106	1	no	—
119	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
120	#500	—	53	2	no	—
121	#500	—	53	2	no	—
122	#500	—	106	1	no	—
123	#500	—	106	1	no	—
124	C₂×#108	—	53	2	no	—
125	C₂×#108	—	106	1	no	—
126	C₂×#107	—	53	2	no	—
127	C₂×#107	—	106	1	no	—
128	#423	—	106	1	no	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#346	—	2	1	no	—
131	C₄×C₁₆	—	2	1	no	—
132	C₅₃	#13290	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	212	1	no	—
134	C₂×M₁₆	—	106	2	no	—
135	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	106	2	no	—
138	C₄⋊Q₈	—	106	2	no	—
139	C₄⋊Q₈	—	212	1	no	—
140	C₄⋊Q₈	—	212	1	no	—
141	C₂×C₄⋊C₄	—	106	2	no	—
142	C₂×C₄⋊C₄	—	212	1	no	—
143	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
144	M₃₂	—	2	2	no	—
145	M₃₂	Dic₁₀₆	1	4	yes	complemented
146	C₂×C₁₆	—	2	2	no	—
147	#108	—	106	4	no	—
148	#108	—	212	2	no	—
149	#107	—	53	4	no	—
150	#107	—	106	2	no	—
151	#107	—	106	2	no	—
152	#107	—	106	2	no	—
153	C₄⋊C₈	—	212	1	no	—
154	C₄⋊C₈	—	212	1	no	—
155	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×Q₈	—	424	4	no	—
160	C₂×Q₈	—	212	2	no	—
161	C₂×Q₈	—	212	2	no	—
162	C₂²×C₄	—	212	2	no	—
163	C₂²×C₄	#11719913	1	1	yes	characteristic
164	C₂²×C₄	#11719913	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
166	M₁₆	—	212	2	no	—
167	M₁₆	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	#11719913	1	2	yes	—
170	C₂×C₈	Dic₂₁₂	1	2	yes	—
171	C₂×C₈	#11719921	1	1	yes	characteristic
172	C₂×C₈	#11719921	1	1	yes	characteristic
173	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	#11719945	1	2	yes	—
176	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	106	4	no	—
178	C₄⋊C₄	—	212	2	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₄²	D₄₂₄	1	1	yes	characteristic
182	C₄²	D₄₂₄	1	1	yes	characteristic
183	C₄²	#11719921	1	1	yes	characteristic
184	C₄²	#11719921	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₂³	C₂×#11719913	1	1	yes	characteristic
187	Q₈	—	424	4	no	—
188	Q₈	—	424	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	424	4	no	—
192	C₂×C₄	—	212	4	no	—
193	C₂×C₄	#13133139	1	1	yes	characteristic
194	C₂×C₄	#13133139	1	1	yes	characteristic
195	C₂×C₄	#13133139	1	1	yes	characteristic
196	C₂×C₄	#13133139	1	1	yes	characteristic
197	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
198	C₂×C₄	#13133205	1	1	yes	characteristic
199	C₂×C₄	#13133217	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
203	C₈	—	424	2	no	—
204	C₈	#13133142	1	2	yes	—
205	C₈	#13133142	1	2	yes	—
206	C₈	C₄×Dic₁₀₆	1	2	yes	—
207	C₈	#13133196	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#13133139	1	1	yes	characteristic
211	C₂²	#16810974	1	1	yes	characteristic
212	C₂²	#16811021	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	424	4	no	—
216	C₄	#16810437	1	2	yes	—
217	C₄	#16810440	1	2	yes	—
218	C₄	#16810529	1	1	yes	characteristic
219	C₄	C₂×#13133142	1	1	yes	characteristic
220	C₄	#16810970	1	1	yes	characteristic
221	C₄	#16811028	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#20275676	1	1	yes	characteristic
225	C₂	C₂×#16810437	1	1	yes	characteristic
226	C₂	C₂×#16810440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#13290
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
39. M₃₂×C₅₃	Q₈	semidirect
132. C₅₃	#13290	semidirect
145. M₃₂	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
#20274887	2
#20272824	2
C₄×M₃₂×C₅₃	2
#13290	53