Subgroups — C₄×#16810741 #27382649

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810741	C₂	1	2	yes	maximal
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133223	C₂	1	2	yes	maximal, complemented
4	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274594	C₂	1	2	yes	maximal, complemented
6	#20274552	C₂	1	4	yes	maximal, complemented
7	C₄×#13133145	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂×#13133223	C₂²	1	4	yes	—
11	C₄×#11719945	C₂²	1	2	yes	—
12	#16810741	C₄	1	16	yes	direct product factor
13	C₄×Dic₂₁₂	C₄	1	2	yes	complemented
14	C₄×#11719913	C₄	1	4	yes	complemented
15	C₄×D₄₂₄	C₄	1	2	yes	complemented
16	C₂×#13133153	C₂²	1	2	yes	—
17	#16810661	C₂²	1	1	yes	characteristic
18	C₂×#13133145	C₂²	1	4	yes	—
19	#16810620	C₂²	1	2	yes	—
20	#16810578	C₂²	1	4	yes	—
21	#16810577	—	2	4	no	—
22	C₄×#11719917	C₂²	1	2	yes	—
23	C₄×#11719912	—	2	2	no	—
24	C₄×C₈×D₅₃	—	2	2	no	—
25	#16810388	C₂²	1	2	yes	—
26	#16810383	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₄³×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#11719945	C₂³	1	2	yes	—
33	#13133239	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×#11719913	C₂×C₄	1	2	yes	—
36	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
37	#13133223	—	2	8	no	—
38	#13133223	C₂×C₄	1	16	yes	complemented
39	C₂×#11719917	C₂³	1	4	yes	—
40	C₂×#11719912	—	2	4	no	—
41	C₂×C₈×D₅₃	—	2	4	no	—
42	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₅₃	—	2	4	no	—
45	#13133176	C₂³	1	2	yes	—
46	#13133174	—	2	2	no	—
47	C₄²×D₅₃	—	2	2	no	—
48	#13133153	C₂×C₄	1	8	yes	complemented
49	#13133151	C₂×C₄	1	2	yes	—
50	#13133149	C₂×C₄	1	2	yes	—
51	#13133145	C₂×C₄	1	16	yes	complemented
52	#13133142	C₂×C₄	1	4	yes	—
53	#13133139	C₂×C₄	1	4	yes	—
54	#13133128	—	2	2	no	—
55	#13133127	—	2	4	no	—
56	C₄×#88322	—	2	2	no	—
57	C₈×Dic₅₃	—	2	4	no	—
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₅₃	—	2	2	no	—
62	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₅₃	—	2	4	no	—
65	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
68	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₅₃	D₄	1	2	yes	—
70	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	—	2	2	no	—
72	C₂²×C₄×C₅₃	#41	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₃	—	2	2	no	—
75	#11719945	—	2	4	no	—
76	#11719945	C₂²×C₄	1	8	yes	—
77	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₅₃	—	2	2	no	—
79	C₂×C₄×D₅₃	—	2	4	no	—
80	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	#41	1	4	yes	—
90	C₄²×C₅₃	—	2	8	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₅₃	—	2	2	no	—
97	#11719921	—	2	4	no	—
98	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
99	#11719919	—	2	4	no	—
100	C₄×Dic₅₃	—	2	4	no	—
101	C₄×Dic₅₃	—	2	2	no	—
102	#11719917	—	2	4	no	—
103	#11719917	C₂²×C₄	1	8	yes	—
104	C₂×#88322	—	2	4	no	—
105	Dic₂₁₂	C₄²	1	4	yes	complemented
106	D₄₂₄	C₄²	1	4	yes	complemented
107	#11719913	C₄²	1	8	yes	complemented
108	#11719912	—	2	16	no	—
109	C₈×D₅₃	—	2	16	no	—
110	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂²×D₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	8	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
121	C₂²⋊D₅₃	—	2	8	no	—
122	C₂×Dic₅₃	—	2	2	no	—
123	C₂×Dic₅₃	—	2	4	no	—
124	D₂₁₂	—	2	2	no	—
125	D₂₁₂	C₂×C₄²	1	4	yes	—
126	C₄×D₅₃	—	2	8	no	—
127	C₄×D₅₃	—	4	8	no	—
128	Dic₁₀₆	—	2	2	no	—
129	Dic₁₀₆	C₂×C₄²	1	4	yes	—
130	C₈×C₅₃	—	2	2	no	—
131	C₈×C₅₃	C₂×C₄²	1	2	yes	—
132	C₈×C₅₃	C₂×C₄²	1	2	yes	—
133	C₈×C₅₃	C₄×D₄	1	4	yes	—
134	C₈×C₅₃	C₄×D₄	1	4	yes	—
135	#88322	—	2	8	no	—
136	C₄×#443	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₅₃	C₄×#41	1	2	yes	—
140	C₂×D₅₃	—	2	4	no	—
141	C₂×D₅₃	—	4	4	no	—
142	C₄×C₅₃	—	2	8	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	#443	1	4	yes	complemented
145	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
147	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
148	Dic₅₃	—	2	4	no	—
149	Dic₅₃	—	4	4	no	—
150	C₂×#443	—	53	2	no	—
151	C₄×#135	—	53	1	no	—
152	C₄×#131	—	53	2	no	—
153	C₄×#108	—	53	1	no	—
154	#1772	—	53	2	no	—
155	#1764	—	53	4	no	—
156	C₄×#104	—	53	2	no	—
157	C₄²×C₈	D₅₃	1	1	yes	characteristic, complemented
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
160	C₂×C₅₃	C₂×#443	1	2	yes	—
161	D₅₃	—	4	4	no	—
162	C₂×#135	—	53	1	no	—
163	C₂×#131	—	53	4	no	—
164	C₄×#41	—	53	2	no	—
165	#443	—	53	16	no	—
166	C₄×Q₁₆	—	53	2	no	—
167	C₄×SD₁₆	—	53	4	no	—
168	C₄×D₈	—	53	2	no	—
169	C₂×#108	—	53	2	no	—
170	#427	—	53	1	no	—
171	C₂×#104	—	53	4	no	—
172	#416	—	53	2	no	—
173	#405	—	53	4	no	—
174	#405	—	106	4	no	—
175	C₄×M₁₆	—	53	2	no	—
176	C₄×M₁₆	—	106	2	no	—
177	C₂×C₄×C₈	—	106	2	no	—
178	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
181	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
182	C₄³	C₂×D₅₃	1	1	yes	characteristic
183	#341	—	53	2	no	—
184	#339	—	53	4	no	—
185	C₅₃	C₄×#443	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	53	2	no	—
187	#135	—	53	8	no	—
188	C₂×Q₁₆	—	53	1	no	—
189	C₂×SD₁₆	—	53	2	no	—
190	C₂×D₈	—	53	1	no	—
191	#131	—	53	16	no	—
192	#131	—	106	8	no	—
193	C₂×M₁₆	—	53	4	no	—
194	C₂×M₁₆	—	106	4	no	—
195	C₂²×C₈	—	106	4	no	—
196	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
199	C₄×Q₈	—	53	4	no	—
200	C₄×D₄	—	53	4	no	—
201	C₄×D₄	—	106	4	no	—
202	#117	—	53	2	no	—
203	#117	—	106	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	106	2	no	—
206	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
208	#108	—	53	8	no	—
209	#107	—	53	2	no	—
210	#106	—	53	2	no	—
211	#104	—	53	16	no	—
212	#103	—	53	4	no	—
213	#102	—	53	4	no	—
214	#97	—	106	4	no	—
215	#97	—	106	2	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	106	4	no	—
218	C₄×C₈	—	106	2	no	—
219	C₄×C₈	C₄×D₅₃	1	2	yes	—
220	C₄×C₈	C₄×D₅₃	1	2	yes	—
221	C₄×C₈	C₄×D₅₃	1	8	yes	complemented
222	C₄×C₈	D₂₁₂	1	1	yes	characteristic
223	C₄×C₈	D₂₁₂	1	2	yes	—
224	C₄×C₈	D₂₁₂	1	1	yes	characteristic
225	#41	—	53	8	no	—
226	#41	—	106	4	no	—
227	C₂×Q₈	—	53	2	no	—
228	C₂×D₄	—	53	2	no	—
229	C₂×D₄	—	106	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	106	2	no	—
232	C₂²×C₄	—	106	4	no	—
233	C₂²×C₄	#11719945	1	2	yes	—
234	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
235	Q₁₆	—	53	4	no	—
236	SD₁₆	—	53	8	no	—
237	D₈	—	53	4	no	—
238	M₁₆	—	53	8	no	—
239	M₁₆	—	106	4	no	—
240	M₁₆	—	106	16	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	106	16	no	—
244	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
248	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
249	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
250	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
251	C₂×C₈	#11719945	1	4	yes	—
252	C₄⋊C₄	—	53	4	no	—
253	C₄⋊C₄	—	106	4	no	—
254	C₂²⋊C₄	—	106	2	no	—
255	C₂²⋊C₄	—	106	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	106	4	no	—
259	C₄²	—	106	2	no	—
260	C₄²	C₂×C₄×D₅₃	1	4	yes	—
261	C₄²	C₂×C₄×D₅₃	1	2	yes	—
262	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
263	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
264	C₂³	—	106	2	no	—
265	C₂³	C₂×#11719945	1	1	yes	characteristic
266	Q₈	—	53	4	no	—
267	Q₈	—	106	2	no	—
268	D₄	—	53	4	no	—
269	D₄	—	106	8	no	—
270	D₄	—	106	2	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	106	2	no	—
275	C₂×C₄	—	106	8	no	—
276	C₂×C₄	—	106	4	no	—
277	C₂×C₄	—	212	8	no	—
278	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
279	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
281	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
283	C₂×C₄	C₂×#11719945	1	2	yes	—
284	C₈	—	2	2	no	—
285	C₈	—	106	8	no	—
286	C₈	C₄²×D₅₃	1	2	yes	—
287	C₈	C₄²×D₅₃	1	2	yes	—
288	C₈	C₄×D₂₁₂	1	4	yes	—
289	C₈	C₄×D₂₁₂	1	4	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	106	4	no	—
292	C₂²	—	212	4	no	—
293	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
294	C₂²	C₄×#11719945	1	2	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	2	2	no	—
297	C₄	—	106	4	no	—
298	C₄	—	212	4	no	—
299	C₄	#16810741	1	4	yes	direct product factor
300	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
301	C₄	C₂×C₄×D₂₁₂	1	2	yes	—
302	C₄	C₄×#11719945	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	212	4	no	—
305	C₂	C₄²×D₂₁₂	1	1	yes	characteristic
306	C₂	C₂×#16810741	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	C₄×#443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#13133223	C₂	semidirect
4. C₄×#13133153	C₂	semidirect
5. #20274594	C₂	semidirect
6. #20274552	C₂	semidirect
7. C₄×#13133145	C₂	semidirect
8. C₄²×C₈×C₅₃	C₂	semidirect
12. #16810741	C₄	direct (N × H)
13. C₄×Dic₂₁₂	C₄	semidirect
14. C₄×#11719913	C₄	semidirect
15. C₄×D₄₂₄	C₄	semidirect
38. #13133223	C₂×C₄	semidirect
48. #13133153	C₂×C₄	semidirect
51. #13133145	C₂×C₄	semidirect
67. C₄×C₈×C₅₃	C₂×C₄	semidirect
105. Dic₂₁₂	C₄²	semidirect
106. D₄₂₄	C₄²	semidirect
107. #11719913	C₄²	semidirect
144. C₄×C₅₃	#443	semidirect
157. C₄²×C₈	D₅₃	semidirect
185. C₅₃	C₄×#443	semidirect
221. C₄×C₈	C₄×D₅₃	semidirect
299. C₄	#16810741	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×#16810741	2
C₄×#13133239	2
C₄×#13133223	2
C₄×#13133153	2
#20274594	2
#20274552	2
C₄×#13133145	2
C₄²×C₈×C₅₃	2
C₄×#443	53

Subgroups — C4×#16810741 #27382649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810741 #27382649