Subgroups — C₄×#16810368 #27275041

Order 13568.

540 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810368	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	—	2	1	no	—
10	#16810529	—	2	1	no	—
11	#16810521	—	2	1	no	—
12	#16810521	C₄	1	2	yes	—
13	#16810493	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Dic₅₃	—	2	1	no	—
15	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
16	#16810378	—	2	1	no	—
17	#16810368	C₄	1	8	yes	direct product factor
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	—	2	1	no	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#336	—	2	1	no	—
22	C₂²×Dic₁₀₆	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₀₆	—	4	2	no	—
27	C₄×Dic₁₀₆	—	4	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
33	C₂×#11719919	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	C₂×C₄	1	4	yes	—
37	C₄×#88322	—	4	2	no	—
38	#13133124	—	2	2	no	—
39	#13133124	—	2	2	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	2	no	—
49	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	2	no	—
59	C₂²×Dic₅₃	—	2	2	no	—
60	C₂×Dic₁₀₆	—	2	4	no	—
61	C₂×Dic₁₀₆	—	4	1	no	—
62	C₂×Dic₁₀₆	—	4	1	no	—
63	C₂×Dic₁₀₆	—	4	2	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	4	1	no	—
71	C₄²×C₅₃	D₈	1	2	yes	—
72	C₄²×C₅₃	SD₁₆	1	2	yes	—
73	C₄²×C₅₃	SD₁₆	1	2	yes	—
74	C₄²×C₅₃	Q₁₆	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
77	C₄⋊Dic₅₃	—	2	2	no	—
78	C₄⋊Dic₅₃	—	2	4	no	—
79	C₄⋊Dic₅₃	—	4	1	no	—
80	C₄⋊Dic₅₃	—	4	1	no	—
81	C₄⋊Dic₅₃	—	4	2	no	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	C₄⋊Dic₅₃	C₄²	1	4	yes	—
84	#11719919	—	4	4	no	—
85	C₄×Dic₅₃	—	4	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	#101	1	2	yes	—
100	C₂×C₄×C₅₃	#102	1	2	yes	—
101	C₂×C₄×C₅₃	#102	1	2	yes	—
102	C₂×C₄×C₅₃	#103	1	2	yes	—
103	C₂×C₄×C₅₃	#103	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×C₄×C₅₃	#135	1	2	yes	—
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₂×Dic₅₃	—	4	2	no	—
120	Dic₁₀₆	—	4	4	no	—
121	Dic₁₀₆	—	4	4	no	—
122	C₈×C₅₃	—	4	4	no	—
123	#88322	—	4	4	no	—
124	C₄×#332	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
127	C₂²×C₅₃	#413	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
130	C₂²×C₅₃	#416	1	1	yes	characteristic
131	C₂²×C₅₃	#416	1	1	yes	characteristic
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	4	1	no	—
137	C₄×C₅₃	#332	1	4	yes	complemented
138	C₄×C₅₃	C₄×D₈	1	2	yes	—
139	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
140	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
141	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
142	Dic₅₃	—	4	4	no	—
143	Dic₅₃	—	8	2	no	—
144	C₄×C₄⋊Q₈	—	53	1	no	—
145	C₄×C₄⋊C₈	—	53	1	no	—
146	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
147	C₂×#332	—	53	2	no	—
148	#1296	—	53	2	no	—
149	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
150	C₂×C₅₃	#1540	1	1	yes	characteristic
151	C₂×C₅₃	#1542	1	1	yes	characteristic
152	C₂×C₅₃	#1544	1	1	yes	characteristic
153	C₂×C₅₃	C₄×#101	1	1	yes	characteristic
154	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
155	C₂×C₅₃	C₄×#103	1	1	yes	characteristic
156	C₂×C₄⋊Q₈	—	53	1	no	—
157	C₂×C₄×Q₈	—	106	1	no	—
158	C₂×C₄⋊C₈	—	53	2	no	—
159	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	#391	—	106	1	no	—
163	#389	—	53	2	no	—
164	#389	—	106	1	no	—
165	#384	—	53	1	no	—
166	C₄×C₄⋊C₄	—	53	2	no	—
167	C₄×C₄⋊C₄	—	106	1	no	—
168	C₄³	C₂×D₅₃	1	1	yes	characteristic
169	#336	—	2	1	no	—
170	#336	—	106	1	no	—
171	#332	—	53	8	no	—
172	C₅₃	C₄×#332	1	1	yes	characteristic, complemented, Sylow
173	C₂²×Q₈	—	106	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₄⋊Q₈	—	53	4	no	—
177	C₄⋊Q₈	—	106	2	no	—
178	C₄×Q₈	—	212	2	no	—
179	C₄×Q₈	—	212	2	no	—
180	C₂×C₄⋊C₄	—	53	2	no	—
181	C₂×C₄⋊C₄	—	53	2	no	—
182	C₂×C₄⋊C₄	—	106	1	no	—
183	C₂×C₄⋊C₄	—	106	1	no	—
184	C₂×C₄⋊C₄	—	106	1	no	—
185	C₂×C₄⋊C₄	—	106	2	no	—
186	C₂×C₄²	—	106	2	no	—
187	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
189	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
190	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	53	4	no	—
196	C₄⋊C₈	—	106	2	no	—
197	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
198	C₄×C₈	—	4	2	no	—
199	C₄×C₈	—	212	2	no	—
200	#95	—	106	2	no	—
201	#95	—	106	2	no	—
202	C₂×Q₈	—	106	4	no	—
203	C₂×Q₈	—	212	1	no	—
204	C₂×Q₈	—	212	1	no	—
205	C₂×Q₈	—	212	2	no	—
206	C₂²×C₄	—	106	2	no	—
207	C₂²×C₄	—	106	2	no	—
208	C₂²×C₄	#11719921	1	1	yes	characteristic
209	C₂²×C₄	#11719921	1	1	yes	characteristic
210	C₂²×C₄	#11719921	1	1	yes	characteristic
211	C₂²×C₄	#11719921	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	212	2	no	—
220	C₂×C₈	—	212	2	no	—
221	C₄⋊C₄	—	53	4	no	—
222	C₄⋊C₄	—	106	2	no	—
223	C₄⋊C₄	—	106	4	no	—
224	C₄⋊C₄	—	212	2	no	—
225	C₄⋊C₄	—	212	4	no	—
226	C₄⋊C₄	—	212	1	no	—
227	C₄⋊C₄	—	212	1	no	—
228	C₄⋊C₄	—	212	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	212	4	no	—
234	C₄²	#11719913	1	2	yes	—
235	C₄²	D₄₂₄	1	2	yes	—
236	C₄²	#11719923	1	2	yes	—
237	C₄²	#11719925	1	2	yes	—
238	C₄²	C₂×C₄×D₅₃	1	2	yes	—
239	C₄²	#11719945	1	2	yes	—
240	C₂³	C₂×#11719921	1	1	yes	characteristic
241	Q₈	—	212	4	no	—
242	Q₈	—	212	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	106	4	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	—	212	2	no	—
255	C₂×C₄	—	212	2	no	—
256	C₂×C₄	—	212	2	no	—
257	C₂×C₄	—	212	2	no	—
258	C₂×C₄	#13133135	1	2	yes	—
259	C₂×C₄	#13133139	1	2	yes	—
260	C₂×C₄	#13133139	1	2	yes	—
261	C₂×C₄	#13133140	1	2	yes	—
262	C₂×C₄	#13133140	1	2	yes	—
263	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
264	C₂×C₄	#13133174	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
266	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
267	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
268	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
271	C₂×C₄	#13133237	1	2	yes	—
272	C₂×C₄	#13133239	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	212	4	no	—
275	C₂²	C₄×#11719921	1	1	yes	characteristic
276	C₂²	C₂×#13133135	1	1	yes	characteristic
277	C₂²	#16810605	1	1	yes	characteristic
278	C₂²	C₂×#13133139	1	1	yes	characteristic
279	C₂²	C₂×#13133140	1	1	yes	characteristic
280	C₂²	#16810617	1	1	yes	characteristic
281	C₂²	#16810620	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	424	2	no	—
288	C₄	—	212	4	no	—
289	C₄	#16810368	1	4	yes	direct product factor
290	C₄	C₄×D₄₂₄	1	2	yes	—
291	C₄	C₄×#11719923	1	2	yes	—
292	C₄	C₄×#11719913	1	2	yes	—
293	C₄	C₄×#11719925	1	2	yes	—
294	C₂	C₂×#16810368	1	1	yes	characteristic
295	C₂	#20273362	1	1	yes	characteristic
296	C₂	#20273373	1	1	yes	characteristic
297	C₂	#20273379	1	1	yes	characteristic
298	C₂	C₄×#13133135	1	1	yes	characteristic
299	C₂	C₄×#13133139	1	1	yes	characteristic
300	C₂	C₄×#13133140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₄×#332
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. #16810368	C₄	direct (N × H)
137. C₄×C₅₃	#332	semidirect
172. C₅₃	C₄×#332	semidirect
289. C₄	#16810368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
C₄×C₄⋊#88322	2
C₂×#16810368	2
#20272599	2
C₄×C₄⋊C₈×C₅₃	2
C₄×#332	53

Subgroups — C4×#16810368 #27275041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810368 #27275041