Subgroups — #27353567

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283795	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273535	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₁₂	C₂	1	4	yes	maximal, complemented
5	#20273463	C₂	1	1	yes	maximal, characteristic
6	#20273111	C₂	1	2	yes	maximal
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
10	C₈×Dic₁₀₆	C₂²	1	8	yes	—
11	#16810747	—	2	4	no	—
12	C₄×Dic₂₁₂	C₄	1	2	yes	—
13	C₄×Dic₂₁₂	C₄	1	2	yes	—
14	C₄×Dic₂₁₂	C₂²	1	2	yes	—
15	C₄×Dic₂₁₂	C₂²	1	2	yes	—
16	#16810676	C₂²	1	2	yes	—
17	#16810673	—	2	4	no	—
18	C₂×#13133151	C₄	1	1	yes	characteristic
19	C₂×#13133142	C₄	1	2	yes	—
20	#16810376	C₂²	1	2	yes	—
21	#16810374	C₂²	1	2	yes	—
22	#16810355	C₂²	1	8	yes	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₅₃×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#322	C₂²	1	2	yes	—
27	C₈²×C₅₃	C₂²	1	2	yes	—
28	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₁₂	C₈	1	8	yes	complemented
30	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₀₆	—	2	2	no	—
33	C₄×Dic₁₀₆	—	2	2	no	—
34	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₀₆	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
37	C₂×#11719919	—	2	2	no	—
38	C₂×C₄×Dic₅₃	—	2	2	no	—
39	#13133151	C₂×C₄	1	2	yes	—
40	#13133151	C₂×C₄	1	2	yes	—
41	C₄⋊#88322	C₂³	1	4	yes	—
42	#13133146	—	2	8	no	—
43	#13133142	C₂×C₄	1	4	yes	—
44	#13133142	C₂×C₄	1	4	yes	—
45	C₂²⋊#88322	—	2	2	no	—
46	C₄×#88322	—	2	2	no	—
47	C₈×Dic₅₃	—	2	8	no	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	—	2	1	no	—
52	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
53	C₂²⋊C₈×C₅₃	—	2	1	no	—
54	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×Dic₅₃	—	2	2	no	—
65	C₂²×Dic₅₃	—	2	2	no	—
66	C₂×Dic₁₀₆	—	2	2	no	—
67	C₂×Dic₁₀₆	—	2	4	no	—
68	C₂×Dic₁₀₆	C₂×C₈	1	8	yes	—
69	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	#41	1	1	yes	characteristic
81	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊Dic₅₃	—	2	2	no	—
83	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
84	#11719919	—	2	4	no	—
85	#11719919	—	2	4	no	—
86	C₄×Dic₅₃	—	2	4	no	—
87	C₄×Dic₅₃	—	2	4	no	—
88	C₂×#88322	—	2	4	no	—
89	Dic₂₁₂	C₂×C₈	1	8	yes	—
90	Dic₂₁₂	C₂×C₈	1	8	yes	—
91	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	1	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₅₃	#131	1	2	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	4	no	—
103	C₂×Dic₅₃	—	2	8	no	—
104	C₂×Dic₅₃	—	2	4	no	—
105	C₂×Dic₅₃	—	8	2	no	—
106	Dic₁₀₆	—	2	4	no	—
107	Dic₁₀₆	—	2	8	no	—
108	Dic₁₀₆	C₂²×C₈	1	8	yes	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	—	2	4	no	—
112	C₈×C₅₃	—	4	2	no	—
113	C₈×C₅₃	C₂²×C₈	1	2	yes	—
114	C₈×C₅₃	C₂²×C₈	1	2	yes	—
115	#88322	—	4	4	no	—
116	#17856	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	C₈×D₄	1	2	yes	—
120	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	C₈×D₄	1	2	yes	—
125	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
127	C₄×C₅₃	#578	1	2	yes	—
128	Dic₅₃	—	2	8	no	—
129	Dic₅₃	—	4	4	no	—
130	#2982	—	53	2	no	—
131	C₂×C₄×Q₁₆	—	53	1	no	—
132	#1579	—	53	4	no	—
133	C₈×Q₁₆	—	53	4	no	—
134	#1564	—	53	1	no	—
135	#1479	—	53	2	no	—
136	#1450	D₅₃	1	1	yes	characteristic
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₅₃	#2950	1	1	yes	characteristic
140	C₂×C₅₃	#2957	1	1	yes	characteristic
141	C₂²×Q₁₆	—	53	1	no	—
142	C₂×C₄×Q₈	—	53	2	no	—
143	C₈×Q₈	—	53	8	no	—
144	C₈×Q₈	—	106	4	no	—
145	C₄×Q₁₆	—	53	2	no	—
146	C₄×Q₁₆	—	53	2	no	—
147	C₄×Q₁₆	—	53	2	no	—
148	C₄×Q₁₆	—	53	2	no	—
149	#431	—	53	2	no	—
150	#431	—	106	4	no	—
151	#431	C₂×D₅₃	1	1	yes	characteristic
152	#431	C₂×D₅₃	1	1	yes	characteristic
153	C₂×#107	—	53	1	no	—
154	C₂×#103	—	53	2	no	—
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	#335	—	53	2	no	—
157	#334	—	53	2	no	—
158	Q₈⋊C₈	—	53	8	no	—
159	#322	C₂×D₅₃	1	2	yes	—
160	C₈²	C₂×D₅₃	1	2	yes	—
161	C₅₃	#17856	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	53	2	no	—
163	C₂×Q₁₆	—	53	2	no	—
164	C₂×Q₁₆	—	53	2	no	—
165	C₂×Q₁₆	—	53	8	no	—
166	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
168	C₄×Q₈	—	53	4	no	—
169	C₄×Q₈	—	53	4	no	—
170	C₄×Q₈	—	106	2	no	—
171	C₄×Q₈	—	106	2	no	—
172	C₂×C₄⋊C₄	—	53	2	no	—
173	C₂×C₄⋊C₄	—	106	2	no	—
174	C₂×C₄²	—	106	2	no	—
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	#107	—	53	2	no	—
177	#107	—	53	2	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	53	4	no	—
180	C₄⋊C₈	—	106	8	no	—
181	C₄⋊C₈	D₂₁₂	1	2	yes	—
182	#103	—	53	4	no	—
183	#103	—	53	4	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	106	2	no	—
186	C₂²⋊C₈	D₂₁₂	1	2	yes	—
187	C₄×C₈	—	106	8	no	—
188	C₄×C₈	—	106	2	no	—
189	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
190	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅₃	1	2	yes	—
192	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₅₃	1	2	yes	—
195	C₂×Q₈	—	53	8	no	—
196	C₂×Q₈	—	53	4	no	—
197	C₂×Q₈	—	106	2	no	—
198	C₂×Q₈	—	106	4	no	—
199	C₂²×C₄	—	106	2	no	—
200	C₂²×C₄	—	106	2	no	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
203	C₂²×C₄	#11719945	1	1	yes	characteristic
204	Q₁₆	—	53	8	no	—
205	Q₁₆	—	53	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	C₈×D₅₃	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
215	C₄⋊C₄	—	53	4	no	—
216	C₄⋊C₄	—	106	2	no	—
217	C₄⋊C₄	—	106	4	no	—
218	C₄⋊C₄	—	106	4	no	—
219	C₄²	—	106	4	no	—
220	C₄²	—	106	4	no	—
221	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
223	C₄²	#11719945	1	1	yes	characteristic
224	C₄²	C₂³×D₅₃	1	1	yes	characteristic
225	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
226	Q₈	—	53	8	no	—
227	Q₈	—	106	4	no	—
228	Q₈	—	106	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	106	4	no	—
232	C₂×C₄	—	106	8	no	—
233	C₂×C₄	—	106	4	no	—
234	C₂×C₄	—	424	2	no	—
235	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
238	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
239	C₂×C₄	#13133223	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	212	4	no	—
248	C₈	C₂×C₈×D₅₃	1	2	yes	—
249	C₈	C₂×C₈×D₅₃	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₈×D₂₁₂	1	2	yes	—
253	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	106	8	no	—
258	C₄	—	212	4	no	—
259	C₄	C₈×D₂₁₂	1	2	yes	—
260	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
261	C₄	C₂×#13133223	1	1	yes	characteristic
262	C₄	#16811440	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₈×D₂₁₂	1	1	yes	characteristic
265	C₂	#20283584	1	1	yes	characteristic
266	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17856
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20273535	C₂	semidirect
4. C₈×Dic₂₁₂	C₂	semidirect
29. C₂×Dic₂₁₂	C₈	semidirect
161. C₅₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#20283795	2
C₂×C₄×Dic₂₁₂	2
#20273535	2
C₈×Dic₂₁₂	2
#20273463	2
#20273111	2
C₅₃×#1450	2
#17856	53