Subgroups — #27302353

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810436	C₂	1	4	yes	maximal, complemented
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	#16810436	—	2	8	no	—
10	#16810399	—	2	1	no	—
11	C₄×#11719908	—	2	1	no	—
12	C₂²×D₈×C₅₃	—	2	1	no	—
13	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
14	C₂×#127×C₅₃	—	2	1	no	—
15	C₅₃×#493	—	2	1	no	—
16	C₅₃×#493	—	2	1	no	—
17	C₅₃×#493	C₄	1	2	yes	—
18	C₅₃×#493	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	#13133157	—	2	2	no	—
22	#13133157	D₄	1	4	yes	complemented
23	C₂×#11719908	—	2	2	no	—
24	#13133153	—	2	4	no	—
25	#13133153	—	4	2	no	—
26	C₄⋊#88322	—	4	2	no	—
27	C₂²×D₄×C₅₃	—	2	2	no	—
28	C₂²×D₄×C₅₃	—	4	2	no	—
29	C₂×D₈×C₅₃	—	2	2	no	—
30	C₂×D₈×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	2	no	—
32	C₂×D₈×C₅₃	—	2	4	no	—
33	C₂×D₈×C₅₃	—	2	4	no	—
34	C₂×D₈×C₅₃	—	4	2	no	—
35	C₂×D₈×C₅₃	—	4	2	no	—
36	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	#127×C₅₃	—	2	2	no	—
40	#127×C₅₃	—	2	2	no	—
41	#127×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂⁴×C₅₃	—	4	2	no	—
48	C₂×D₄×C₅₃	—	2	4	no	—
49	C₂×D₄×C₅₃	—	4	4	no	—
50	C₂×D₄×C₅₃	—	4	4	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	8	8	no	—
57	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	D₈×C₅₃	—	2	8	no	—
60	D₈×C₅₃	—	4	4	no	—
61	D₈×C₅₃	—	4	4	no	—
62	D₈×C₅₃	—	4	4	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	D₈	1	2	yes	—
66	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	2	yes	—
71	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₅₃	D₈	1	2	yes	—
74	#11719917	—	4	2	no	—
75	#11719917	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	#11719908	—	2	4	no	—
78	C₂³×C₅₃	—	4	4	no	—
79	C₂³×C₅₃	—	8	8	no	—
80	C₂³×C₅₃	—	8	2	no	—
81	C₂³×C₅₃	—	8	2	no	—
82	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	D₄×C₅₃	—	4	4	no	—
84	D₄×C₅₃	—	4	4	no	—
85	D₄×C₅₃	—	8	8	no	—
86	D₄×C₅₃	—	8	8	no	—
87	D₄×C₅₃	—	8	2	no	—
88	D₄×C₅₃	—	8	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₅₃	#102	1	2	yes	—
99	C₈×C₅₃	#102	1	2	yes	—
100	C₈×C₅₃	C₄×D₄	1	2	yes	—
101	C₈×C₅₃	#121	1	2	yes	—
102	#88322	—	8	2	no	—
103	#13213	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	8	8	no	—
106	C₂²×C₅₃	—	8	4	no	—
107	C₂²×C₅₃	—	16	4	no	—
108	C₂²×C₅₃	—	16	4	no	—
109	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
110	C₂²×C₅₃	#486	1	1	yes	characteristic
111	C₂²×C₅₃	#492	1	1	yes	characteristic
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	#361	1	4	yes	complemented
114	C₄×C₅₃	#390	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
116	C₄×C₅₃	#486	1	1	yes	characteristic
117	C₄×C₅₃	#493	1	1	yes	characteristic
118	C₂×#493	D₅₃	1	1	yes	characteristic
119	C₂×#361	—	53	4	no	—
120	C₄×M₃₂	—	53	1	no	—
121	#1848	—	53	1	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	—	2	1	no	—
124	C₂×C₅₃	—	16	4	no	—
125	C₂×C₅₃	#1957	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#361	1	2	yes	—
127	C₂²×D₈	—	2	1	no	—
128	C₂²×D₈	C₂×D₅₃	1	2	yes	—
129	C₂×#127	—	2	1	no	—
130	C₂×M₃₂	—	53	2	no	—
131	#493	—	2	1	no	—
132	#493	—	2	1	no	—
133	#493	Dic₅₃	1	2	yes	—
134	#493	Dic₅₃	1	2	yes	—
135	C₂×#108	—	53	2	no	—
136	C₂×#108	—	106	1	no	—
137	#423	—	106	1	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	#361	—	106	8	no	—
140	#346	—	106	1	no	—
141	C₄×C₁₆	—	106	1	no	—
142	C₅₃	#13213	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	—	4	2	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	4	no	—
147	C₂×D₈	—	2	2	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	4	2	no	—
151	C₂×D₈	—	4	2	no	—
152	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
153	C₂×M₁₆	—	106	2	no	—
154	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	4	2	no	—
159	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
160	M₃₂	—	53	4	no	—
161	M₃₂	—	106	2	no	—
162	C₂×C₁₆	—	106	2	no	—
163	#108	—	106	4	no	—
164	#108	—	212	2	no	—
165	C₄⋊C₈	—	212	2	no	—
166	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
167	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂⁴	—	4	2	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	8	8	no	—
180	C₂²×C₄	#11719923	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	D₈	—	2	8	no	—
183	D₈	—	4	4	no	—
184	D₈	—	4	4	no	—
185	D₈	—	4	4	no	—
186	M₁₆	—	212	2	no	—
187	M₁₆	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	#11719922	1	2	yes	—
190	C₂×C₈	#11719923	1	2	yes	—
191	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	D₄×D₅₃	1	2	yes	—
194	C₂×C₈	#11719947	1	2	yes	—
195	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₄²	#11719922	1	2	yes	—
198	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
200	C₁₆	—	106	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	8	8	no	—
203	C₂³	—	8	2	no	—
204	C₂³	—	8	2	no	—
205	C₂³	C₂×#11719923	1	1	yes	characteristic
206	D₄	—	4	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	8	8	no	—
209	D₄	—	8	8	no	—
210	D₄	—	8	2	no	—
211	D₄	—	8	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	#13133138	1	2	yes	—
214	C₂×C₄	#13133138	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	#13133204	1	1	yes	characteristic
217	C₂×C₄	#13133213	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
219	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
221	C₈	—	424	2	no	—
222	C₈	#13133138	1	2	yes	—
223	C₈	#13133138	1	2	yes	—
224	C₈	D₄×Dic₅₃	1	2	yes	—
225	C₈	#13133188	1	2	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	8	no	—
229	C₂²	—	16	4	no	—
230	C₂²	—	16	4	no	—
231	C₂²	C₂×#13133138	1	1	yes	characteristic
232	C₂²	#16810969	1	1	yes	characteristic
233	C₂²	#16810995	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	#16810436	1	4	yes	complemented
236	C₄	#16810526	1	1	yes	characteristic
237	C₄	C₂×#13133138	1	1	yes	characteristic
238	C₄	#16810969	1	1	yes	characteristic
239	C₄	#16810999	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	16	4	no	—
243	C₂	#20275522	1	1	yes	characteristic
244	C₂	C₂×#16810436	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	#13213
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16810436	C₂	semidirect
2. C₄×#13133157	C₂	semidirect
3. #20274887	C₂	semidirect
22. #13133157	D₄	semidirect
113. C₄×C₅₃	#361	semidirect
142. C₅₃	#13213	semidirect
235. C₄	#16810436	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810436	2
C₄×#13133157	2
#20274887	2
C₂×C₅₃×#493	2
#13213	53