Subgroups — C₄×#16810369 #27275040

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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810369	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	1	no	—
9	C₂×C₄×#88322	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810369	C₄	1	8	yes	direct product factor
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#389	C₄	1	2	yes	—
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	C₂×C₄	1	4	yes	—
27	C₄⋊#88322	C₂×C₄	1	4	yes	—
28	C₄×#88322	—	4	2	no	—
29	C₄×#88322	—	4	2	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₄×Q₈×C₅₃	—	4	2	no	—
34	C₄×Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	4	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	4	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	D₈	1	2	yes	—
78	C₄²×C₅₃	SD₁₆	1	2	yes	—
79	C₄²×C₅₃	SD₁₆	1	2	yes	—
80	C₄²×C₅₃	Q₁₆	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₅₃	—	4	4	no	—
91	Q₈×C₅₃	—	4	4	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	4	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	#101	1	2	yes	—
108	C₂×C₄×C₅₃	#102	1	2	yes	—
109	C₂×C₄×C₅₃	#102	1	2	yes	—
110	C₂×C₄×C₅₃	#103	1	2	yes	—
111	C₂×C₄×C₅₃	#103	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#135	1	2	yes	—
121	C₂×C₄×C₅₃	#135	1	2	yes	—
122	#88322	—	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	—	4	4	no	—
138	C₄×C₅₃	—	8	2	no	—
139	C₄×C₅₃	#332	1	4	yes	complemented
140	C₄×C₅₃	C₄×D₈	1	2	yes	—
141	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
142	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
143	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
144	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
145	C₄×C₄⋊C₈	—	53	1	no	—
146	C₄×C₄⋊C₈	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
157	C₂×C₄×Q₈	—	2	1	no	—
158	C₂×C₄⋊C₈	—	53	2	no	—
159	C₂×C₄⋊C₈	—	53	2	no	—
160	C₂×C₄×C₈	—	106	1	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	#391	—	2	1	no	—
163	#389	—	2	1	no	—
164	#389	Dic₅₃	1	2	yes	—
165	#384	C₂×D₅₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	—	2	1	no	—
167	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
168	C₄³	C₂×D₅₃	1	1	yes	characteristic
169	#336	—	106	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	—	2	1	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₄×D₅₃	1	4	yes	—
178	C₄×Q₈	—	4	2	no	—
179	C₄×Q₈	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅₃	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	—	53	4	no	—
195	C₄⋊C₈	—	53	4	no	—
196	C₄⋊C₈	—	106	2	no	—
197	C₄⋊C₈	—	106	2	no	—
198	C₄×C₈	—	212	2	no	—
199	C₄×C₈	—	212	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	—	212	2	no	—
219	C₂×C₈	—	212	2	no	—
220	C₂×C₈	—	212	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	4	no	—
234	C₄²	#11719922	1	2	yes	—
235	C₄²	#11719923	1	2	yes	—
236	C₄²	#11719924	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×C₂²⋊Dic₅₃	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	4	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	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	#13133136	1	2	yes	—
259	C₂×C₄	#13133138	1	2	yes	—
260	C₂×C₄	#13133138	1	2	yes	—
261	C₂×C₄	#13133141	1	2	yes	—
262	C₂×C₄	#13133141	1	2	yes	—
263	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
264	C₂×C₄	#13133176	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
266	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
267	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
268	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
271	C₂×C₄	#13133237	1	2	yes	—
272	C₂×C₄	#13133237	1	2	yes	—
273	C₈	—	212	4	no	—
274	C₈	—	212	4	no	—
275	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
276	C₂²	C₂×#13133136	1	1	yes	characteristic
277	C₂²	#16810607	1	1	yes	characteristic
278	C₂²	C₂×#13133138	1	1	yes	characteristic
279	C₂²	C₂×#13133141	1	1	yes	characteristic
280	C₂²	#16810619	1	1	yes	characteristic
281	C₂²	#16810619	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	4	no	—
287	C₄	—	4	1	no	—
288	C₄	—	8	2	no	—
289	C₄	#16810369	1	4	yes	direct product factor
290	C₄	C₄×#11719922	1	2	yes	—
291	C₄	C₄×#11719923	1	2	yes	—
292	C₄	C₄×#11719924	1	2	yes	—
293	C₄	C₄×#11719925	1	2	yes	—
294	C₂	C₂×#16810369	1	1	yes	characteristic
295	C₂	#20273364	1	1	yes	characteristic
296	C₂	#20273372	1	1	yes	characteristic
297	C₂	#20273380	1	1	yes	characteristic
298	C₂	C₄×#13133136	1	1	yes	characteristic
299	C₂	C₄×#13133138	1	1	yes	characteristic
300	C₂	C₄×#13133141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₄×#332
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
12. #16810369	C₄	direct (N × H)
139. C₄×C₅₃	#332	semidirect
172. C₅₃	C₄×#332	semidirect
289. C₄	#16810369	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×C₄⋊#88322	2
C₂×#16810369	2
#20272598	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#332	53

Subgroups — C4×#16810369 #27275040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810369 #27275040