Subgroups — C₄×#16810379 #27289256

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133176	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810379	C₂	1	4	yes	maximal
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂×#13133176	C₂²	1	2	yes	—
7	C₄×#11719917	—	2	2	no	—
8	C₄×#11719917	—	2	2	no	—
9	C₂×#13133128	C₄	1	2	yes	complemented
10	C₂×C₄×#88322	C₄	1	2	yes	complemented
11	C₄×C₄⋊Dic₅₃	—	2	2	no	—
12	C₄×C₂²⋊Dic₅₃	—	2	2	no	—
13	#16810469	C₄	1	2	yes	complemented
14	C₄²×Dic₅₃	C₄	1	2	yes	complemented
15	#16810379	C₄	1	16	yes	direct product factor
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
20	C₅₃×#376	C₄	1	2	yes	complemented
21	C₄³×C₅₃	C₄	1	2	yes	complemented
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	4	no	—
25	C₂×#11719917	C₂×C₄	1	8	yes	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	#13133176	—	2	4	no	—
28	#13133176	C₂×C₄	1	8	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	4	no	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
33	#13133128	—	2	4	no	—
34	C₄×#88322	—	2	4	no	—
35	#13133124	—	2	4	no	—
36	C₂³×C₄×C₅₃	D₄	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	#117×C₅₃	—	2	4	no	—
39	#117×C₅₃	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	—	2	4	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	D₄	1	2	yes	—
52	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
53	#95×C₅₃	—	2	4	no	—
54	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	4	4	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	4	yes	—
71	C₂²×Dic₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	2	8	no	—
73	C₂²⋊C₄×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	4	8	no	—
76	C₄²×C₅₃	C₄²	1	1	yes	characteristic
77	C₄²×C₅₃	C₄²	1	1	yes	characteristic
78	C₄²×C₅₃	C₄²	1	1	yes	characteristic
79	C₄²×C₅₃	C₄²	1	1	yes	characteristic
80	C₄²×C₅₃	C₄²	1	8	yes	complemented
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂²⋊Dic₅₃	—	2	8	no	—
88	C₄⋊Dic₅₃	—	2	8	no	—
89	C₄×Dic₅₃	—	2	4	no	—
90	C₄×Dic₅₃	—	4	8	no	—
91	C₄×Dic₅₃	C₄²	1	8	yes	complemented
92	#11719917	—	2	8	no	—
93	#11719917	—	2	8	no	—
94	C₂×#88322	—	2	4	no	—
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	C₄²	1	8	yes	complemented
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	#95	1	2	yes	—
111	C₂×C₄×C₅₃	#95	1	2	yes	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	#95	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#117	1	2	yes	—
122	C₂×C₄×C₅₃	#117	1	2	yes	—
123	C₂×C₄×C₅₃	#117	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
127	C₂×Dic₅₃	—	2	8	no	—
128	C₂×Dic₅₃	—	4	8	no	—
129	#88322	—	2	8	no	—
130	C₄×H(3,ℤ/4)	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	4	4	no	—
135	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
136	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	4	8	no	—
143	C₄×C₅₃	H(3,ℤ/4)	1	4	yes	complemented
144	C₄×C₅₃	C₄³	1	1	yes	characteristic
145	C₄×C₅₃	#376	1	1	yes	characteristic
146	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
147	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
148	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
149	Dic₅₃	—	4	8	no	—
150	C₂×C₄×M₁₆	—	53	1	no	—
151	C₄×#117	—	53	1	no	—
152	C₄×#117	D₅₃	1	1	yes	characteristic
153	C₂×H(3,ℤ/4)	—	53	4	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
158	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
159	C₂²×M₁₆	—	53	2	no	—
160	C₂×#117	—	53	2	no	—
161	C₂×#117	C₂×D₅₃	1	2	yes	—
162	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
163	C₄×M₁₆	—	106	2	no	—
164	C₄×M₁₆	—	106	2	no	—
165	C₂×#97	—	53	2	no	—
166	C₂×C₄×C₈	—	53	2	no	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄×C₄⋊C₄	—	106	2	no	—
169	C₄×C₂²⋊C₄	—	2	2	no	—
170	C₄×C₂²⋊C₄	—	106	2	no	—
171	#376	—	53	2	no	—
172	#376	Dic₅₃	1	2	yes	complemented
173	C₄³	—	53	2	no	—
174	C₄³	Dic₅₃	1	2	yes	complemented
175	H(3,ℤ/4)	—	53	16	no	—
176	C₅₃	C₄×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₂²⋊D₅₃	1	2	yes	—
178	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×M₁₆	—	53	8	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	4	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂²×C₈	—	53	4	no	—
184	#117	—	2	4	no	—
185	#117	—	53	8	no	—
186	#117	—	106	4	no	—
187	#117	C₄×D₅₃	1	8	yes	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	106	2	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	106	2	no	—
192	C₂×C₄²	—	2	4	no	—
193	C₂×C₄²	—	53	4	no	—
194	C₂×C₄²	—	106	4	no	—
195	C₂×C₄²	Dic₁₀₆	1	2	yes	—
196	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄²	D₂₁₂	1	2	yes	—
201	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
202	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
205	#97	—	106	4	no	—
206	C₄×C₈	—	106	4	no	—
207	#95	—	2	4	no	—
208	#95	—	106	4	no	—
209	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	106	4	no	—
213	C₂²×C₄	#11719919	1	2	yes	—
214	C₂²×C₄	#11719919	1	2	yes	—
215	C₂²×C₄	#11719921	1	2	yes	—
216	C₂²×C₄	#11719921	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
218	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
219	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
223	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
224	C₂²×C₄	#11719945	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	M₁₆	—	106	8	no	—
228	M₁₆	—	106	8	no	—
229	C₂×C₈	—	53	8	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	4	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	106	8	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	106	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	4	8	no	—
238	C₄²	—	53	8	no	—
239	C₄²	—	106	4	no	—
240	C₄²	—	212	8	no	—
241	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
242	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
243	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
244	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
245	C₄²	C₄×Dic₅₃	1	8	yes	complemented
246	C₄²	#11719919	1	2	yes	—
247	C₄²	#11719919	1	2	yes	—
248	C₄²	C₄⋊Dic₅₃	1	2	yes	—
249	C₄²	C₄⋊Dic₅₃	1	2	yes	—
250	C₄²	#11719921	1	2	yes	—
251	C₄²	#11719921	1	2	yes	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂×#11719921	1	1	yes	characteristic
254	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
255	C₂³	C₂×#11719919	1	1	yes	characteristic
256	C₂³	C₄×C₂²⋊D₅₃	1	4	yes	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	106	8	no	—
266	C₂×C₄	—	212	8	no	—
267	C₂×C₄	#13133124	1	2	yes	—
268	C₂×C₄	#13133124	1	2	yes	—
269	C₂×C₄	#13133124	1	2	yes	—
270	C₂×C₄	#13133124	1	2	yes	—
271	C₂×C₄	C₄²×D₅₃	1	2	yes	—
272	C₂×C₄	C₄²×D₅₃	1	2	yes	—
273	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
275	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂×C₄	#13133174	1	2	yes	—
279	C₂×C₄	#13133174	1	2	yes	—
280	C₂×C₄	#13133176	1	2	yes	—
281	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
282	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
283	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
284	C₈	—	106	8	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×#13133124	1	1	yes	characteristic
290	C₂²	C₄×C₂²⋊Dic₅₃	1	2	yes	—
291	C₂²	C₄×#11719921	1	2	yes	—
292	C₂²	C₄×#11719919	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	212	8	no	—
298	C₄	#16810379	1	4	yes	direct product factor
299	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
300	C₄	#16810469	1	1	yes	characteristic
301	C₄	C₄×#11719921	1	2	yes	—
302	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
303	C₄	C₄×#11719919	1	2	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	C₄×#13133124	1	1	yes	characteristic
308	C₂	C₂×#16810379	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	C₄×H(3,ℤ/4)
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
9. C₂×#13133128	C₄	semidirect
10. C₂×C₄×#88322	C₄	semidirect
13. #16810469	C₄	semidirect
14. C₄²×Dic₅₃	C₄	semidirect
15. #16810379	C₄	direct (N × H)
20. C₅₃×#376	C₄	semidirect
21. C₄³×C₅₃	C₄	semidirect
80. C₄²×C₅₃	C₄²	semidirect
91. C₄×Dic₅₃	C₄²	semidirect
96. C₂×#88322	C₄²	semidirect
143. C₄×C₅₃	H(3,ℤ/4)	semidirect
172. #376	Dic₅₃	semidirect
174. C₄³	Dic₅₃	semidirect
176. C₅₃	C₄×H(3,ℤ/4)	semidirect
245. C₄²	C₄×Dic₅₃	semidirect
298. C₄	#16810379	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719917	2
C₄×#13133176	2
C₂×#16810379	2
C₄×#117×C₅₃	2
C₄×H(3,ℤ/4)	53

Subgroups — C4×#16810379 #27289256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810379 #27289256