Subgroups — #28328549

Order 13568.

452 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291132	C₂	1	4	yes	maximal, complemented
2	#20291131	C₂	1	2	yes	maximal, complemented
3	#20291129	C₂	1	2	yes	maximal, complemented
4	#20291081	C₂	1	4	yes	maximal
5	#20291076	C₂	1	1	yes	maximal, characteristic
6	#20291072	C₂	1	1	yes	maximal, characteristic
7	#20283645	C₂	1	1	yes	maximal, characteristic
8	#16811544	C₂²	1	4	yes	—
9	#16811544	C₂²	1	4	yes	—
10	C₄⋊#11719910	C₂²	1	2	yes	—
11	C₄⋊#11719910	C₂²	1	2	yes	—
12	C₂²⋊#11719910	C₂²	1	2	yes	—
13	C₂²⋊#11719910	C₂²	1	2	yes	—
14	#16811512	C₂²	1	4	yes	—
15	#16811511	C₂²	1	2	yes	—
16	#16811509	C₂²	1	4	yes	—
17	C₄×#11719910	C₂²	1	1	yes	characteristic
18	C₄×#11719910	C₂²	1	1	yes	characteristic
19	#16811180	C₄	1	1	yes	characteristic
20	#16810750	C₄	1	2	yes	—
21	#16810750	C₄	1	2	yes	—
22	#16810747	C₂²	1	2	yes	—
23	#16810747	C₂²	1	2	yes	—
24	#16810651	C₄	1	2	yes	—
25	#16810581	C₂²	1	2	yes	—
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	C₅₃×#423	C₄	1	1	yes	characteristic
28	C₂×#11719910	C₂³	1	2	yes	—
29	C₂×#11719910	C₂³	1	2	yes	—
30	C₂×#11719910	C₂³	1	4	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂³	1	1	yes	characteristic
33	#13133217	C₈	1	1	yes	characteristic
34	#13133217	C₈	1	1	yes	characteristic
35	#13133209	C₈	1	2	yes	—
36	#13133196	C₈	1	2	yes	—
37	#13133196	C₈	1	2	yes	—
38	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
39	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
40	#13133176	C₂×C₄	1	2	yes	—
41	#13133146	C₂×C₄	1	4	yes	—
42	#13133146	C₂×C₄	1	4	yes	—
43	#13133127	C₂×C₄	1	4	yes	—
44	C₄×#88322	C₂×C₄	1	1	yes	characteristic
45	C₄×#88322	C₂×C₄	1	1	yes	characteristic
46	C₄×#88322	C₂³	1	1	yes	characteristic
47	C₄×#88322	C₂³	1	1	yes	characteristic
48	C₈×Dic₅₃	C₂³	1	4	yes	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
57	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
58	M₁₆×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
60	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
65	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
66	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
68	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
69	#11719919	C₂×C₈	1	4	yes	—
70	#11719919	C₂×C₈	1	4	yes	—
71	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
75	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
76	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
77	C₂×#88322	C₂⁴	1	1	yes	characteristic
78	#11719910	—	2	2	no	—
79	#11719910	—	2	2	no	—
80	#11719910	—	2	4	no	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
93	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
94	Dic₁₀₆	—	2	4	no	—
95	Dic₁₀₆	—	4	2	no	—
96	C₈×C₅₃	—	2	4	no	—
97	#88322	—	2	1	no	—
98	#88322	—	2	1	no	—
99	#88322	—	2	1	no	—
100	#88322	2₋¹⁺⁴	1	2	yes	—
101	#33499	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	#504	1	2	yes	—
111	C₄×C₅₃	#519	1	1	yes	characteristic
112	C₄×C₅₃	#568	1	1	yes	characteristic
113	Dic₅₃	—	2	4	no	—
114	Dic₅₃	—	2	4	no	—
115	#2960	—	53	1	no	—
116	#2183	—	53	2	no	—
117	#2183	—	53	4	no	—
118	Q₈×C₁₆	—	53	2	no	—
119	#2164	—	53	4	no	—
120	#2163	—	53	1	no	—
121	#2162	—	53	1	no	—
122	C₂×C₅₃	—	4	1	no	—
123	C₂×C₅₃	#2982	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
125	C₂×C₅₃	#3407	1	1	yes	characteristic
126	#533	—	53	1	no	—
127	#446	—	53	2	no	—
128	#446	—	53	2	no	—
129	C₈×Q₈	—	53	2	no	—
130	C₈×Q₈	—	53	2	no	—
131	#424	—	53	2	no	—
132	#423	#6157	1	1	yes	characteristic
133	#405	—	53	2	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₄⋊C₁₆	—	53	2	no	—
136	C₄⋊C₁₆	—	53	2	no	—
137	C₄⋊C₁₆	—	53	4	no	—
138	C₄⋊C₁₆	—	53	4	no	—
139	C₂²⋊C₁₆	—	53	2	no	—
140	C₂²⋊C₁₆	—	53	2	no	—
141	#346	—	53	2	no	—
142	#346	—	53	4	no	—
143	C₄×C₁₆	—	53	1	no	—
144	C₄×C₁₆	—	53	1	no	—
145	C₄×C₁₆	—	53	4	no	—
146	C₅₃	#33499	1	1	yes	characteristic, complemented, Sylow
147	C₂×M₁₆	C₂×#6157	1	2	yes	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	53	1	no	—
150	C₄⋊Q₈	—	53	1	no	—
151	C₄⋊Q₈	—	53	1	no	—
152	#125	—	53	2	no	—
153	C₂²⋊Q₈	—	53	2	no	—
154	C₂²⋊Q₈	—	53	2	no	—
155	C₄×Q₈	—	53	2	no	—
156	C₄×Q₈	—	53	2	no	—
157	#117	—	53	2	no	—
158	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
159	C₂×C₁₆	—	53	2	no	—
160	C₂×C₁₆	—	53	2	no	—
161	C₂×C₁₆	—	53	4	no	—
162	C₄⋊C₈	—	53	4	no	—
163	C₄⋊C₈	—	53	4	no	—
164	C₄⋊C₈	C₂×#6157	1	2	yes	—
165	C₄⋊C₈	C₂×#6157	1	2	yes	—
166	#97	—	53	4	no	—
167	C₄×C₈	—	53	4	no	—
168	C₄×C₈	—	53	1	no	—
169	C₄×C₈	—	53	1	no	—
170	C₄×C₈	—	53	1	no	—
171	C₄×C₈	—	53	1	no	—
172	C₂×Q₈	—	53	2	no	—
173	C₂×Q₈	—	53	2	no	—
174	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
175	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	53	1	no	—
179	C₂×C₈	—	53	1	no	—
180	C₂×C₈	—	53	1	no	—
181	C₂×C₈	—	53	1	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	212	1	no	—
184	C₂×C₈	C₂²×#6157	1	4	yes	—
185	C₄⋊C₄	—	53	2	no	—
186	C₄⋊C₄	—	53	2	no	—
187	C₄⋊C₄	—	53	4	no	—
188	C₄⋊C₄	—	53	4	no	—
189	C₄⋊C₄	—	53	2	no	—
190	C₄⋊C₄	—	53	2	no	—
191	C₂²⋊C₄	—	53	4	no	—
192	C₄²	—	53	4	no	—
193	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
194	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
195	C₄²	C₂²×#6157	1	1	yes	characteristic
196	C₄²	C₂²×#6157	1	1	yes	characteristic
197	C₁₆	—	106	2	no	—
198	C₁₆	—	106	2	no	—
199	C₁₆	—	106	4	no	—
200	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
201	Q₈	—	106	4	no	—
202	Q₈	—	212	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	53	4	no	—
208	C₂×C₄	—	53	4	no	—
209	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
210	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
211	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
212	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
214	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	53	2	no	—
217	C₈	—	106	1	no	—
218	C₈	—	106	1	no	—
219	C₈	—	106	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	106	4	no	—
229	C₄	—	106	4	no	—
230	C₄	#16811682	1	2	yes	—
231	C₄	#16811691	1	1	yes	characteristic
232	C₄	#16811701	1	1	yes	characteristic
233	C₂	—	4	1	no	—
234	C₂	#20291610	1	1	yes	characteristic
235	C₂	C₂×#16811682	1	1	yes	characteristic
236	C₂	#20291684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#423	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#423	#6157
hypocenter	C₅₃	#33499
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291131	C₂	semidirect
3. #20291129	C₂	semidirect
146. C₅₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291131	2
#20291129	2
#20291081	2
#20291076	2
#20291072	2
#20283645	2
#33499	53