Subgroups — #27402991

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719917	C₂	1	2	yes	maximal
2	#20275491	C₂	1	2	yes	maximal
3	#20275491	C₂	1	4	yes	maximal
4	#20274890	C₂	1	2	yes	maximal
5	#20274887	C₂	1	1	yes	maximal, characteristic
6	#20274621	C₂	1	2	yes	maximal
7	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	4	no	—
10	#16810754	—	2	2	no	—
11	Q₈×#88322	—	2	2	no	—
12	C₂×#13133153	C₂²	1	1	yes	characteristic
13	C₂×#13133153	C₂²	1	2	yes	—
14	C₂×#13133153	C₂²	1	2	yes	—
15	C₂×#13133153	C₂²	1	4	yes	—
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	#16810647	C₂²	1	2	yes	—
18	#16810647	C₂²	1	4	yes	—
19	C₄×#11719917	C₂²	1	2	yes	—
20	C₄×#11719917	C₂²	1	4	yes	—
21	#16810388	C₂²	1	2	yes	—
22	#16810388	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#446	C₄	1	4	yes	—
25	C₅₃×#446	C₄	1	4	yes	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×#97×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719917	C₂³	1	2	yes	—
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂×#11719917	C₂³	1	2	yes	—
33	#13133153	—	2	2	no	—
34	#13133153	—	2	4	no	—
35	#13133153	—	4	2	no	—
36	#13133153	—	4	4	no	—
37	#13133153	Q₈	1	4	yes	complemented
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	—	2	2	no	—
41	C₄⋊#88322	—	2	4	no	—
42	#13133128	—	2	4	no	—
43	#13133128	—	2	2	no	—
44	C₄×#88322	—	2	2	no	—
45	C₄×#88322	—	2	4	no	—
46	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₅₃	D₄	1	2	yes	—
53	C₄×Q₈×C₅₃	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
61	#97×C₅₃	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×Q₈×C₅₃	—	4	2	no	—
65	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	2	yes	—
81	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	4	yes	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	#11719917	—	2	2	no	—
92	#11719917	—	2	2	no	—
93	#11719917	—	2	4	no	—
94	#11719917	—	2	4	no	—
95	#11719917	—	2	2	no	—
96	#11719917	C₂×Q₈	1	4	yes	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	4	no	—
100	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
101	Q₈×C₅₃	—	2	4	no	—
102	Q₈×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	C₄×Q₈	1	4	yes	—
122	#88322	—	2	4	no	—
123	#88322	—	4	2	no	—
124	#88322	—	4	4	no	—
125	#22733	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅₃	#557	1	1	yes	characteristic
130	C₂²×C₅₃	Q₈²	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	4	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	#430	1	2	yes	—
137	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₅₃	#520	1	1	yes	characteristic
139	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
140	C₄×C₅₃	#550	1	1	yes	characteristic
141	Q₈×M₁₆	—	53	2	no	—
142	C₂×#446	D₅₃	1	1	yes	characteristic
143	#1950	—	53	4	no	—
144	#1950	—	53	2	no	—
145	#1849	—	53	2	no	—
146	#1848	—	53	1	no	—
147	#1778	—	53	2	no	—
148	C₄×#108	—	53	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
153	C₂×C₅₃	#2916	1	1	yes	characteristic
154	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
155	#446	—	106	2	no	—
156	#446	—	106	4	no	—
157	#446	Dic₅₃	1	4	yes	—
158	#446	Dic₅₃	1	4	yes	—
159	C₈×Q₈	—	106	2	no	—
160	C₂×#108	—	53	1	no	—
161	C₂×#108	—	53	2	no	—
162	C₂×#108	—	53	1	no	—
163	C₂×#108	—	53	2	no	—
164	C₂×#108	—	53	4	no	—
165	#423	—	53	2	no	—
166	#423	—	53	4	no	—
167	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
169	C₄×M₁₆	—	53	2	no	—
170	C₄×M₁₆	—	53	4	no	—
171	C₂×#97	C₂×D₅₃	1	2	yes	—
172	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
173	#341	—	53	2	no	—
174	#341	—	53	4	no	—
175	C₅₃	#22733	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
177	C₂×M₁₆	—	53	2	no	—
178	C₂×M₁₆	—	53	2	no	—
179	C₂×M₁₆	—	53	4	no	—
180	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₄×Q₈	D₂₁₂	1	2	yes	—
184	C₄×Q₈	D₂₁₂	1	2	yes	—
185	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
186	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
187	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
188	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
191	#108	—	53	4	no	—
192	#108	—	106	4	no	—
193	#108	—	106	2	no	—
194	#108	—	212	2	no	—
195	#108	—	212	4	no	—
196	C₄⋊C₈	—	106	2	no	—
197	C₄⋊C₈	—	106	4	no	—
198	C₄⋊C₈	—	106	4	no	—
199	C₄⋊C₈	—	106	2	no	—
200	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
201	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
202	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
203	#97	—	106	4	no	—
204	#97	—	106	2	no	—
205	#97	C₂×Dic₅₃	1	8	yes	—
206	C₄×C₈	—	106	2	no	—
207	C₄×C₈	—	106	4	no	—
208	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
209	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
212	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
213	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
216	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
217	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	M₁₆	—	53	4	no	—
219	M₁₆	—	106	2	no	—
220	M₁₆	—	106	4	no	—
221	M₁₆	—	106	2	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	Q₈×D₅₃	1	2	yes	—
232	C₂×C₈	#11719949	1	2	yes	—
233	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
235	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
236	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
237	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
238	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
239	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
240	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
241	C₄²	C₂×D₂₁₂	1	4	yes	—
242	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
243	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
244	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
245	C₄²	C₂²×Dic₅₃	1	4	yes	—
246	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
263	C₂×C₄	#13133263	1	1	yes	characteristic
264	C₂×C₄	#13133264	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	106	4	no	—
268	C₈	—	212	2	no	—
269	C₈	—	212	4	no	—
270	C₈	Q₈×Dic₅₃	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
274	C₂²	#16811352	1	1	yes	characteristic
275	C₂²	Q₈×Dic₁₀₆	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	1	no	—
281	C₄	#16810672	1	2	yes	—
282	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
283	C₄	#16811106	1	1	yes	characteristic
284	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
285	C₄	#16811295	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
289	C₂	C₂×#16810672	1	1	yes	characteristic
290	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#22733
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
37. #13133153	Q₈	semidirect
175. C₅₃	#22733	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719917	2
#20275491	2
#20274890	2
#20274887	2
#20274621	2
C₄×#13133153	2
C₂×C₅₃×#446	2
#22733	53