Subgroups — #27402981

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283666	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20275488	C₂	1	2	yes	maximal
4	#20275487	C₂	1	4	yes	maximal
5	#20274885	C₂	1	2	yes	maximal
6	#108×Dic₅₃	C₂	1	2	yes	maximal
7	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16810752	—	2	4	no	—
11	#16810750	—	2	2	no	—
12	C₈×Dic₁₀₆	C₄	1	4	yes	—
13	C₈×Dic₁₀₆	C₄	1	2	yes	—
14	C₈×Dic₁₀₆	C₄	1	2	yes	—
15	#16810747	—	2	2	no	—
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	C₂×#13133153	C₂²	1	1	yes	characteristic
18	C₂×#13133152	C₂²	1	4	yes	—
19	#16810646	C₂²	1	4	yes	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×#13133146	C₂²	1	2	yes	—
22	C₄×#11719917	C₂²	1	2	yes	—
23	M₁₆×Dic₅₃	C₂²	1	4	yes	—
24	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
25	#16810388	C₂²	1	2	yes	—
26	#16810387	C₂²	1	4	yes	—
27	C₂×#108×C₅₃	C₂²	1	2	yes	—
28	C₂×#108×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#423	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
32	C₂×#11719917	C₂³	1	2	yes	—
33	C₂×#11719917	C₂³	1	2	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₀₆	D₄	1	2	yes	—
39	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
40	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
42	C₂×#11719919	Q₈	1	2	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
44	#13133153	—	2	2	no	—
45	#13133153	Q₈	1	4	yes	complemented
46	#13133152	—	4	4	no	—
47	C₄⋊#88322	C₂×C₄	1	2	yes	—
48	C₄⋊#88322	C₂×C₄	1	2	yes	—
49	#13133146	—	2	4	no	—
50	#13133146	—	2	4	no	—
51	#13133146	C₂×C₄	1	4	yes	—
52	#13133146	C₂×C₄	1	4	yes	—
53	#13133128	—	2	2	no	—
54	#13133127	—	2	4	no	—
55	C₄×#88322	—	2	2	no	—
56	C₈×Dic₅₃	—	2	4	no	—
57	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
58	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
59	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#108×C₅₃	—	2	4	no	—
64	#108×C₅₃	—	4	2	no	—
65	C₄⋊C₈×C₅₃	—	2	2	no	—
66	C₄⋊C₈×C₅₃	—	2	2	no	—
67	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
74	C₂×Dic₁₀₆	—	2	4	no	—
75	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
76	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
77	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
78	M₁₆×C₅₃	—	2	4	no	—
79	M₁₆×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₅₃	#41	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
92	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
94	#11719919	C₄⋊C₄	1	2	yes	—
95	#11719919	C₄⋊C₄	1	2	yes	—
96	#11719919	C₄⋊C₄	1	2	yes	—
97	#11719919	C₄⋊C₄	1	2	yes	—
98	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
99	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
101	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
102	#11719917	—	2	2	no	—
103	#11719917	—	2	2	no	—
104	#11719917	—	2	2	no	—
105	#11719917	C₂×Q₈	1	4	yes	—
106	C₂×#88322	—	2	2	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	C₂²×C₄	1	4	yes	—
111	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	Dic₁₀₆	—	2	2	no	—
131	Dic₁₀₆	—	2	4	no	—
132	Dic₁₀₆	#108	1	4	yes	complemented
133	C₈×C₅₃	—	2	2	no	—
134	C₈×C₅₃	—	4	4	no	—
135	C₈×C₅₃	C₄×Q₈	1	4	yes	—
136	#88322	—	2	4	no	—
137	#88322	—	2	4	no	—
138	#88322	—	4	2	no	—
139	Q₈×#108	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₅₃	#557	1	1	yes	characteristic
144	C₂²×C₅₃	Q₈²	1	1	yes	characteristic
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	C₂×#108	1	2	yes	—
149	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₅₃	#520	1	1	yes	characteristic
151	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
152	C₄×C₅₃	#550	1	1	yes	characteristic
153	Dic₅₃	—	2	2	no	—
154	Dic₅₃	—	2	2	no	—
155	Dic₅₃	—	2	2	no	—
156	Dic₅₃	C₂×#108	1	4	yes	—
157	Q₈×M₁₆	—	53	2	no	—
158	C₂×C₈×Q₈	—	53	1	no	—
159	#1950	—	53	4	no	—
160	#1950	—	53	2	no	—
161	#1848	—	53	2	no	—
162	#1848	D₅₃	1	1	yes	characteristic
163	C₄×#108	—	53	2	no	—
164	C₄×#108	—	53	1	no	—
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
169	C₂×C₅₃	#2916	1	1	yes	characteristic
170	C₂×C₄×Q₈	—	53	1	no	—
171	#446	—	106	4	no	—
172	#446	—	106	2	no	—
173	C₈×Q₈	—	53	2	no	—
174	C₈×Q₈	—	53	4	no	—
175	C₈×Q₈	—	53	2	no	—
176	C₈×Q₈	—	106	2	no	—
177	C₂×#108	—	53	1	no	—
178	C₂×#108	—	53	1	no	—
179	C₂×#108	—	53	4	no	—
180	C₂×#108	C₂×D₅₃	1	2	yes	—
181	C₂×#108	C₂×D₅₃	1	2	yes	—
182	#423	—	53	4	no	—
183	#423	C₂×D₅₃	1	2	yes	—
184	C₂×C₄⋊C₈	—	53	2	no	—
185	C₂×C₄⋊C₈	—	53	1	no	—
186	C₄×M₁₆	—	53	4	no	—
187	C₄×M₁₆	—	53	2	no	—
188	C₂×C₄×C₈	—	53	2	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#341	—	53	4	no	—
191	#341	—	53	2	no	—
192	C₅₃	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₈	—	53	1	no	—
194	C₂×M₁₆	—	53	2	no	—
195	C₂×M₁₆	—	53	2	no	—
196	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
200	C₄×Q₈	—	53	1	no	—
201	C₄×Q₈	—	53	2	no	—
202	C₄×Q₈	—	53	1	no	—
203	C₄×Q₈	—	53	1	no	—
204	C₄×Q₈	—	53	2	no	—
205	C₄×Q₈	—	53	1	no	—
206	C₂×C₄⋊C₄	—	53	2	no	—
207	C₂×C₄⋊C₄	—	53	1	no	—
208	C₂×C₄²	—	53	2	no	—
209	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
210	#108	—	2	4	no	—
211	#108	—	4	2	no	—
212	#108	—	53	4	no	—
213	#108	—	106	2	no	—
214	#108	—	212	4	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	53	4	no	—
218	C₄⋊C₈	—	53	4	no	—
219	C₄⋊C₈	—	53	2	no	—
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	—	106	4	no	—
222	C₄⋊C₈	—	106	4	no	—
223	#97	—	106	4	no	—
224	#97	—	106	2	no	—
225	C₄×C₈	—	53	4	no	—
226	C₄×C₈	—	53	4	no	—
227	C₄×C₈	—	106	4	no	—
228	C₄×C₈	—	106	2	no	—
229	C₄×C₈	C₄×D₅₃	1	2	yes	—
230	C₄×C₈	C₄×D₅₃	1	2	yes	—
231	C₂×Q₈	—	53	1	no	—
232	C₂×Q₈	—	53	2	no	—
233	C₂×Q₈	—	53	1	no	—
234	C₂×Q₈	—	106	4	no	—
235	C₂²×C₄	—	53	2	no	—
236	C₂²×C₄	—	53	2	no	—
237	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	53	4	no	—
243	M₁₆	—	106	2	no	—
244	M₁₆	—	106	2	no	—
245	M₁₆	—	106	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	53	4	no	—
250	C₂×C₈	—	106	2	no	—
251	C₂×C₈	—	106	2	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
256	C₂×C₈	#11719947	1	2	yes	—
257	C₂×C₈	Q₈×D₅₃	1	2	yes	—
258	C₄⋊C₄	—	53	2	no	—
259	C₄⋊C₄	—	53	2	no	—
260	C₄⋊C₄	—	53	2	no	—
261	C₄⋊C₄	—	53	2	no	—
262	C₄⋊C₄	—	53	2	no	—
263	C₄⋊C₄	—	53	1	no	—
264	C₄⋊C₄	—	53	1	no	—
265	C₄²	—	53	2	no	—
266	C₄²	—	53	2	no	—
267	C₄²	—	53	2	no	—
268	C₄²	—	53	2	no	—
269	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₄²	D₄×D₅₃	1	1	yes	characteristic
272	C₄²	D₄×D₅₃	1	1	yes	characteristic
273	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
274	Q₈	—	53	4	no	—
275	Q₈	—	106	4	no	—
276	Q₈	—	106	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	53	2	no	—
280	C₂×C₄	—	53	2	no	—
281	C₂×C₄	—	53	2	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	106	4	no	—
284	C₂×C₄	—	212	2	no	—
285	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
286	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
287	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
293	C₂×C₄	#13133262	1	1	yes	characteristic
294	C₂×C₄	#13133265	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	4	no	—
297	C₈	—	106	4	no	—
298	C₈	—	106	4	no	—
299	C₈	—	212	2	no	—
300	C₈	#13133182	1	4	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
304	C₂²	#16811348	1	1	yes	characteristic
305	C₂²	#16811353	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	53	4	no	—
310	C₄	—	106	2	no	—
311	C₄	—	106	2	no	—
312	C₄	—	106	2	no	—
313	C₄	#108×D₅₃	1	2	yes	—
314	C₄	C₂×#13133182	1	1	yes	characteristic
315	C₄	#16811104	1	1	yes	characteristic
316	C₄	#16811288	1	1	yes	characteristic
317	C₄	#16811293	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	#20277627	1	1	yes	characteristic
321	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
322	C₂	#20283323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1848	D₅₃
hypocenter	C₅₃	Q₈×#108
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
45. #13133153	Q₈	semidirect
132. Dic₁₀₆	#108	semidirect
192. C₅₃	Q₈×#108	semidirect

Maximal subgroups

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

subgroup	index
#20283666	2
C₂×C₈×Dic₁₀₆	2
#20275488	2
#20275487	2
#20274885	2
#108×Dic₅₃	2
C₄×#13133153	2
C₅₃×#1848	2
Q₈×#108	53