Subgroups — #27282708

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20273468	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272564	C₂	1	2	yes	maximal
5	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₄	1	1	yes	characteristic
7	C₂×#13133153	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₄×#11719917	C₂²	1	2	yes	—
10	#16810388	C₄	1	2	yes	—
11	#16810376	—	2	1	no	—
12	#16810376	—	2	1	no	—
13	#16810375	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	C₅₃×#431	C₂²	1	1	yes	characteristic
16	C₅₃×#431	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#322	—	2	1	no	—
19	C₈²×C₅₃	—	2	1	no	—
20	C₂×#11719917	C₂×C₄	1	2	yes	—
21	C₂×#11719917	C₂×C₄	1	2	yes	—
22	C₂²×#88322	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133153	C₈	1	4	yes	complemented
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	#13133128	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²⋊C₈×C₅₃	—	2	1	no	—
36	C₂²⋊C₈×C₅₃	Q₈	1	2	yes	—
37	C₄×C₈×C₅₃	—	2	1	no	—
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₅₃	#41	1	1	yes	characteristic
61	#11719917	—	2	2	no	—
62	#11719917	—	2	2	no	—
63	#11719917	—	2	2	no	—
64	#11719917	C₂×C₈	1	4	yes	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	8	2	no	—
69	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	1	no	—
85	C₈×C₅₃	—	4	2	no	—
86	C₈×C₅₃	—	4	1	no	—
87	C₈×C₅₃	—	4	1	no	—
88	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
89	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
90	#88322	—	2	4	no	—
91	#88322	—	4	2	no	—
92	#88322	—	8	2	no	—
93	#9608	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₅₃	#424	1	1	yes	characteristic
104	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
105	C₄×C₅₃	#436	1	1	yes	characteristic
106	C₄×C₅₃	#472	1	1	yes	characteristic
107	C₄×C₅₃	#473	1	1	yes	characteristic
108	C₄×#108	—	53	1	no	—
109	#1565	—	53	2	no	—
110	C₂×#335	—	53	1	no	—
111	#1450	D₅₃	1	1	yes	characteristic
112	#1283	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1939	1	1	yes	characteristic
116	C₂×C₅₃	#1952	1	1	yes	characteristic
117	C₂×C₅₃	#1953	1	1	yes	characteristic
118	#431	C₂×D₅₃	1	1	yes	characteristic
119	#431	C₂×D₅₃	1	1	yes	characteristic
120	C₂×#108	—	53	1	no	—
121	C₂×#108	—	53	1	no	—
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₄×M₁₆	—	53	2	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	#341	—	53	2	no	—
126	#335	—	106	2	no	—
127	#335	—	106	1	no	—
128	#335	—	106	1	no	—
129	#334	—	106	2	no	—
130	#322	—	2	1	no	—
131	C₈²	—	2	1	no	—
132	C₅₃	#9608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	53	2	no	—
134	C₂×M₁₆	—	53	2	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
137	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
139	#108	—	53	4	no	—
140	#108	—	106	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	106	2	no	—
143	C₄⋊C₈	—	106	2	no	—
144	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
145	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
146	C₂²⋊C₈	—	2	1	no	—
147	C₂²⋊C₈	Dic₁₀₆	1	2	yes	—
148	#97	—	106	2	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	106	2	no	—
152	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
153	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	#11719945	1	1	yes	characteristic
159	M₁₆	—	53	4	no	—
160	M₁₆	—	106	2	no	—
161	M₁₆	—	106	2	no	—
162	M₁₆	—	106	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	424	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	C₈×D₅₃	1	2	yes	—
173	C₂×C₈	#11719912	1	2	yes	—
174	C₂×C₈	#11719919	1	1	yes	characteristic
175	C₂×C₈	#11719919	1	1	yes	characteristic
176	C₂×C₈	#11719919	1	1	yes	characteristic
177	C₂×C₈	#11719919	1	1	yes	characteristic
178	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
180	C₄²	#11719945	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
187	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
188	C₂×C₄	#13133191	1	1	yes	characteristic
189	C₂×C₄	#13133196	1	1	yes	characteristic
190	C₂×C₄	#13133209	1	1	yes	characteristic
191	C₂×C₄	#13133217	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
194	C₂×C₄	#13133223	1	1	yes	characteristic
195	C₂×C₄	#13133223	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	4	2	no	—
199	C₈	—	4	1	no	—
200	C₈	—	4	1	no	—
201	C₈	—	106	4	no	—
202	C₈	—	424	2	no	—
203	C₈	—	212	2	no	—
204	C₈	#13133146	1	2	yes	—
205	C₈	#13133146	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16810521	1	1	yes	characteristic
209	C₂²	C₈×Dic₁₀₆	1	1	yes	characteristic
210	C₂²	#16810753	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×#13133146	1	1	yes	characteristic
215	C₄	#16810649	1	1	yes	characteristic
216	C₄	C₈×D₂₁₂	1	1	yes	characteristic
217	C₄	#16810702	1	1	yes	characteristic
218	C₄	#16810893	1	1	yes	characteristic
219	C₄	#16810898	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#20275433	1	1	yes	characteristic
223	C₂	#20275500	1	1	yes	characteristic
224	C₂	#20275505	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#9608
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
24. #13133153	C₈	semidirect
132. C₅₃	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133153	2
#20273468	2
C₂×#16810376	2
#20272564	2
C₅₃×#1450	2
#9608	53