Subgroups — #27400895

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811352	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20275638	C₂	1	3	yes	maximal
4	#20275516	C₂	1	3	yes	maximal
5	#20274717	C₂	1	3	yes	maximal
6	C₄×#13133140	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811352	—	2	4	no	—
9	#16811352	C₄	1	8	yes	complemented
10	C₂×#13133217	C₂²	1	3	yes	—
11	C₂×#13133208	—	2	3	no	—
12	C₂×Q₈×Dic₅₃	—	2	1	no	—
13	C₂×C₄×Dic₁₀₆	C₂²	1	3	yes	—
14	Q₈×#88322	—	2	4	no	—
15	C₂×C₄⋊#88322	C₂²	1	3	yes	—
16	C₂×#13133140	C₂²	1	3	yes	—
17	C₂×#13133140	C₂²	1	3	yes	—
18	C₂×#13133140	C₂²	1	6	yes	—
19	C₂×C₄×#88322	C₂²	1	3	yes	—
20	#16810385	C₂²	1	3	yes	—
21	#16810385	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#389	—	2	3	no	—
25	C₅₃×#384	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
27	C₂²×Dic₁₀₆	C₂³	1	3	yes	—
28	C₂²×#88322	C₂³	1	3	yes	—
29	C₂²×#88322	C₂³	1	1	yes	characteristic
30	#13133217	—	2	6	no	—
31	#13133217	C₂×C₄	1	12	yes	—
32	#13133208	—	2	24	no	—
33	Q₈×Dic₅₃	—	2	8	no	—
34	C₄×Dic₁₀₆	—	2	6	no	—
35	C₄×Dic₁₀₆	C₂×C₄	1	12	yes	—
36	C₂×C₄⋊Dic₅₃	—	2	3	no	—
37	C₂×C₄⋊Dic₅₃	—	2	3	no	—
38	C₂×C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂×#11719919	—	2	3	no	—
40	C₂×C₄×Dic₅₃	—	2	3	no	—
41	C₄⋊#88322	—	2	6	no	—
42	#13133140	—	2	6	no	—
43	#13133140	—	2	6	no	—
44	#13133140	—	4	6	no	—
45	C₄×#88322	—	2	6	no	—
46	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₅₃	—	4	4	no	—
48	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₅₃	D₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	—	2	3	no	—
56	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
57	#95×C₅₃	—	2	3	no	—
58	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
59	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
60	C₂×Q₈×C₅₃	Q₁₆	1	4	yes	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	3	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₅₃	—	2	1	no	—
67	C₂²×Dic₅₃	—	2	3	no	—
68	C₂×Dic₁₀₆	—	2	3	no	—
69	C₂×Dic₁₀₆	—	2	3	no	—
70	C₂×Dic₁₀₆	—	2	3	no	—
71	C₂×Dic₁₀₆	—	2	3	no	—
72	C₂×Dic₁₀₆	—	2	3	no	—
73	C₂×Dic₁₀₆	C₂²×C₄	1	6	yes	—
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₄⋊C₄×C₅₃	—	2	3	no	—
76	C₄⋊C₄×C₅₃	—	2	3	no	—
77	C₄⋊C₄×C₅₃	—	4	6	no	—
78	C₄⋊C₄×C₅₃	—	8	3	no	—
79	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
80	C₄⋊C₄×C₅₃	#41	1	2	yes	—
81	C₄²×C₅₃	—	4	6	no	—
82	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
83	C₄²×C₅₃	C₂×D₄	1	6	yes	—
84	C₄⋊Dic₅₃	—	2	12	no	—
85	C₄⋊Dic₅₃	—	2	12	no	—
86	C₄⋊Dic₅₃	—	2	1	no	—
87	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
88	#11719919	—	2	12	no	—
89	C₄×Dic₅₃	—	2	12	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	2	3	no	—
93	C₂×#88322	—	2	3	no	—
94	C₂×#88322	—	2	3	no	—
95	C₂×#88322	#41	1	2	yes	—
96	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₅₃	#103	1	16	yes	complemented
98	C₂×C₄×C₅₃	—	2	3	no	—
99	C₂×C₄×C₅₃	—	2	3	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	3	no	—
104	C₂×C₄×C₅₃	—	4	3	no	—
105	C₂×C₄×C₅₃	—	4	3	no	—
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
108	C₂×C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
109	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×Dic₅₃	—	2	6	no	—
117	C₂×Dic₅₃	—	2	2	no	—
118	C₂×Dic₅₃	—	4	3	no	—
119	C₂×Dic₅₃	—	4	3	no	—
120	C₂×Dic₅₃	—	4	1	no	—
121	C₂×Dic₅₃	—	4	1	no	—
122	Dic₁₀₆	—	2	6	no	—
123	Dic₁₀₆	—	2	6	no	—
124	Dic₁₀₆	—	4	3	no	—
125	Dic₁₀₆	—	4	3	no	—
126	#88322	—	2	2	no	—
127	#88322	—	4	3	no	—
128	#22542	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅₃	#548	1	1	yes	characteristic
131	C₂²×C₅₃	#550	1	1	yes	characteristic
132	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
133	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
134	C₂²×C₅₃	#577	1	1	yes	characteristic
135	C₂²×C₅₃	#578	1	1	yes	characteristic
136	C₄×C₅₃	—	2	6	no	—
137	C₄×C₅₃	—	4	2	no	—
138	C₄×C₅₃	—	8	3	no	—
139	C₄×C₅₃	C₂×#103	1	12	yes	—
140	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
141	C₄×C₅₃	#520	1	1	yes	characteristic
142	C₄×C₅₃	#548	1	1	yes	characteristic
143	C₄×C₅₃	#550	1	1	yes	characteristic
144	Dic₅₃	—	4	6	no	—
145	Dic₅₃	—	4	2	no	—
146	C₂×#557	—	53	1	no	—
147	C₂×C₈×Q₈	—	53	1	no	—
148	Q₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
149	#1981	—	53	3	no	—
150	#1956	—	53	3	no	—
151	#1803	—	53	3	no	—
152	C₄×#103	—	53	3	no	—
153	C₂×C₅₃	#2341	1	1	yes	characteristic
154	C₂×C₅₃	C₂²×#103	1	1	yes	characteristic
155	C₂×C₅₃	#2898	1	1	yes	characteristic
156	C₂×C₅₃	#3359	1	1	yes	characteristic
157	C₂×C₅₃	#3363	1	1	yes	characteristic
158	C₂×C₅₃	#3392	1	1	yes	characteristic
159	C₂×C₅₃	#3395	1	1	yes	characteristic
160	#557	—	53	8	no	—
161	#557	—	106	4	no	—
162	C₂×C₄⋊Q₈	—	53	3	no	—
163	C₂×#125	—	106	3	no	—
164	C₂×C₄×Q₈	—	2	1	no	—
165	C₂×C₄×Q₈	—	53	3	no	—
166	C₂×C₄×Q₈	—	106	1	no	—
167	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
168	C₈×Q₈	—	106	4	no	—
169	C₂×C₄⋊C₈	—	53	3	no	—
170	C₂×#103	—	53	3	no	—
171	C₂×#103	—	53	3	no	—
172	C₂×#103	—	53	6	no	—
173	C₂×C₄×C₈	—	53	3	no	—
174	#389	—	2	3	no	—
175	#384	C₂×D₅₃	1	3	yes	—
176	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
177	#340	—	53	3	no	—
178	#340	—	53	1	no	—
179	C₅₃	#22542	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	53	3	no	—
181	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	3	no	—
184	C₄⋊Q₈	—	53	12	no	—
185	C₄⋊Q₈	—	106	6	no	—
186	#125	—	106	24	no	—
187	C₄×Q₈	—	4	4	no	—
188	C₄×Q₈	—	53	12	no	—
189	C₄×Q₈	—	106	8	no	—
190	C₄×Q₈	—	106	6	no	—
191	C₄×Q₈	C₄×D₅₃	1	4	yes	—
192	C₄×Q₈	D₂₁₂	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	3	no	—
194	C₂×C₄⋊C₄	—	2	3	no	—
195	C₂×C₄⋊C₄	—	53	1	no	—
196	C₂×C₄⋊C₄	—	106	3	no	—
197	C₂×C₄⋊C₄	—	106	3	no	—
198	C₂×C₄⋊C₄	—	106	3	no	—
199	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
200	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
201	C₂×C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
202	C₂×C₄²	—	2	3	no	—
203	C₂×C₄²	—	106	3	no	—
204	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
205	C₄⋊C₈	—	106	6	no	—
206	#103	—	106	6	no	—
207	#103	—	106	6	no	—
208	#103	—	212	6	no	—
209	C₄×C₈	—	106	6	no	—
210	#95	—	2	3	no	—
211	C₂×Q₈	—	53	6	no	—
212	C₂×Q₈	—	106	3	no	—
213	C₂×Q₈	—	106	3	no	—
214	C₂×Q₈	—	106	3	no	—
215	C₂×Q₈	—	106	3	no	—
216	C₂×Q₈	—	106	3	no	—
217	C₂×Q₈	#11719921	1	4	yes	—
218	C₂×Q₈	#11719923	1	4	yes	—
219	C₂×Q₈	#11719925	1	4	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	106	1	no	—
223	C₂²×C₄	—	106	3	no	—
224	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
226	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	3	no	—
231	C₂×C₈	—	106	3	no	—
232	C₂×C₈	—	106	3	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	3	no	—
235	C₄⋊C₄	—	2	3	no	—
236	C₄⋊C₄	—	4	6	no	—
237	C₄⋊C₄	—	8	3	no	—
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	106	12	no	—
240	C₄⋊C₄	—	106	12	no	—
241	C₄⋊C₄	—	106	12	no	—
242	C₄⋊C₄	—	106	1	no	—
243	C₄⋊C₄	#11719921	1	12	yes	—
244	C₄⋊C₄	#11719949	1	2	yes	—
245	C₄²	—	4	6	no	—
246	C₄²	—	106	12	no	—
247	C₄²	C₂×C₄×D₅₃	1	6	yes	—
248	C₄²	C₂×D₂₁₂	1	6	yes	—
249	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
250	Q₈	—	106	6	no	—
251	Q₈	—	106	6	no	—
252	Q₈	—	212	3	no	—
253	Q₈	—	212	3	no	—
254	Q₈	#13133140	1	16	yes	complemented
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	3	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	—	106	2	no	—
264	C₂×C₄	—	106	6	no	—
265	C₂×C₄	—	212	3	no	—
266	C₂×C₄	—	212	3	no	—
267	C₂×C₄	—	212	1	no	—
268	C₂×C₄	—	212	1	no	—
269	C₂×C₄	C₂×#11719921	1	6	yes	—
270	C₂×C₄	C₂×#11719921	1	6	yes	—
271	C₂×C₄	C₂×#11719923	1	6	yes	—
272	C₂×C₄	C₂×#11719925	1	6	yes	—
273	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
275	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
277	C₂×C₄	#13133263	1	1	yes	characteristic
278	C₂×C₄	#13133265	1	1	yes	characteristic
279	C₈	—	106	2	no	—
280	C₈	—	212	3	no	—
281	C₂²	C₂²×#11719921	1	1	yes	characteristic
282	C₂²	#16811284	1	1	yes	characteristic
283	C₂²	#16811292	1	1	yes	characteristic
284	C₂²	C₂²×#11719923	1	1	yes	characteristic
285	C₂²	C₂²×#11719925	1	1	yes	characteristic
286	C₂²	#16811434	1	1	yes	characteristic
287	C₂²	#16811438	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	4	2	no	—
290	C₄	—	8	3	no	—
291	C₄	—	212	6	no	—
292	C₄	—	212	2	no	—
293	C₄	C₂×#13133140	1	12	yes	—
294	C₄	C₄×#11719949	1	1	yes	characteristic
295	C₄	#16811105	1	1	yes	characteristic
296	C₄	#16811285	1	1	yes	characteristic
297	C₄	#16811295	1	1	yes	characteristic
298	C₂	#20277553	1	1	yes	characteristic
299	C₂	C₂²×#13133140	1	1	yes	characteristic
300	C₂	#20283245	1	1	yes	characteristic
301	C₂	#20287357	1	1	yes	characteristic
302	C₂	#20287393	1	1	yes	characteristic
303	C₂	#20287696	1	1	yes	characteristic
304	C₂	#20287718	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	Q₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#22542
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. #16811352	C₄	semidirect
97. Q₈×C₅₃	#103	semidirect
179. C₅₃	#22542	semidirect
254. Q₈	#13133140	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811352	2
C₂×Q₈×#88322	2
#20275638	2
#20275516	2
#20274717	2
C₄×#13133140	2
Q₈×C₄⋊C₄×C₅₃	2
#22542	53