Subgroups — C₄×#13133237 #20283485

Order 6784.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133237	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#11719925	C₂	1	2	yes	maximal, complemented
4	C₄×#11719924	C₂	1	2	yes	maximal, complemented
5	C₄×#11719923	C₂	1	2	yes	maximal, complemented
6	C₄×#11719922	C₂	1	2	yes	maximal, complemented
7	#16810659	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810619	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810617	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133237	C₄	1	8	yes	direct product factor
14	C₂×#11719925	C₂²	1	1	yes	characteristic
15	C₂×#11719923	C₂²	1	1	yes	characteristic
16	C₂×#11719924	C₂²	1	1	yes	characteristic
17	C₂×#11719922	C₂²	1	1	yes	characteristic
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂²×#88322	C₂²	1	1	yes	characteristic
20	C₄×Dic₁₀₆	C₂²	1	2	yes	—
21	C₄×D₂₁₂	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₅₃	—	2	2	no	—
23	#13133176	C₂²	1	1	yes	characteristic
24	#13133174	—	2	1	no	—
25	C₄²×D₅₃	—	2	1	no	—
26	#13133150	C₂²	1	2	yes	—
27	#13133148	C₂²	1	2	yes	—
28	#13133141	C₂²	1	2	yes	—
29	#13133140	C₂²	1	2	yes	—
30	#13133138	C₂²	1	2	yes	—
31	#13133137	C₂²	1	2	yes	—
32	C₄×#88322	C₂²	1	2	yes	—
33	C₄×#88322	C₂²	1	2	yes	—
34	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	C₂²	1	2	yes	—
38	#117×C₅₃	—	2	1	no	—
39	#117×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
42	#41×C₅₃	—	2	2	no	—
43	#41×C₅₃	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₅₃	—	2	1	no	—
46	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₅₃	—	2	1	no	—
52	#11719945	—	2	2	no	—
53	#11719945	C₂×C₄	1	4	yes	complemented
54	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂²⋊Dic₅₃	—	2	1	no	—
68	#11719925	C₂×C₄	1	4	yes	complemented
69	#11719924	C₂×C₄	1	4	yes	complemented
70	#11719923	C₂×C₄	1	4	yes	complemented
71	#11719922	C₂×C₄	1	4	yes	complemented
72	#11719921	—	2	2	no	—
73	C₄⋊Dic₅₃	C₂³	1	2	yes	—
74	#11719919	—	2	2	no	—
75	C₄×Dic₅₃	—	2	1	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	C₂×C₄	1	4	yes	complemented
79	C₂×#88322	C₂³	1	2	yes	—
80	C₂×#88322	C₂³	1	1	yes	characteristic
81	C₂×#88322	C₂³	1	1	yes	characteristic
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₅₃	—	2	1	no	—
85	Q₈×C₅₃	—	2	1	no	—
86	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
87	D₄×C₅₃	—	2	4	no	—
88	D₄×C₅₃	—	2	1	no	—
89	D₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₅₃	#41	1	2	yes	—
100	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	D₂₁₂	—	2	1	no	—
105	D₂₁₂	C₂²×C₄	1	2	yes	—
106	C₄×D₅₃	—	2	4	no	—
107	C₄×D₅₃	—	4	4	no	—
108	Dic₁₀₆	—	2	1	no	—
109	Dic₁₀₆	C₂²×C₄	1	2	yes	—
110	#88322	—	2	2	no	—
111	#88322	C₂²×C₄	1	2	yes	—
112	#88322	C₂²×C₄	1	2	yes	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	C₄×D₄	1	2	yes	—
117	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₅₃	—	2	2	no	—
119	C₂×D₅₃	—	4	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₄×C₅₃	#135	1	4	yes	complemented
125	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	Dic₅₃	—	2	2	no	—
128	Dic₅₃	—	4	2	no	—
129	C₄×#135	—	53	1	no	maximal, Sylow
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	4	2	no	—
132	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#135	1	2	yes	—
134	D₅₃	—	4	2	no	—
135	C₂×#135	—	53	1	no	—
136	C₄×#41	—	53	1	no	—
137	C₄×#41	D₅₃	1	1	yes	characteristic, complemented
138	C₄×Q₁₆	—	53	2	no	—
139	C₄×SD₁₆	—	53	2	no	—
140	C₄×SD₁₆	—	53	2	no	—
141	C₄×D₈	—	53	2	no	—
142	#427	—	53	1	no	—
143	#416	—	53	1	no	—
144	#416	—	53	1	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₅₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	53	1	no	—
148	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
149	#135	—	53	8	no	—
150	C₂×Q₁₆	—	53	1	no	—
151	C₂×SD₁₆	—	53	1	no	—
152	C₂×SD₁₆	—	53	1	no	—
153	C₂×D₈	—	53	1	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	1	no	—
156	C₄×Q₈	—	53	2	no	—
157	C₄×Q₈	C₂×D₅₃	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	53	2	no	—
160	C₄×D₄	—	106	2	no	—
161	C₄×D₄	C₂×D₅₃	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	53	1	no	—
164	#117	—	106	1	no	—
165	#117	C₂×D₅₃	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	106	1	no	—
168	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
169	#107	—	53	2	no	—
170	#106	—	53	2	no	—
171	#103	—	53	2	no	—
172	#103	—	53	2	no	—
173	#102	—	53	2	no	—
174	#102	—	53	2	no	—
175	C₄×C₈	—	53	2	no	—
176	C₄×C₈	—	53	2	no	—
177	#41	—	2	2	no	—
178	#41	—	53	4	no	—
179	#41	—	106	2	no	—
180	#41	C₄×D₅₃	1	4	yes	complemented
181	C₂×Q₈	—	53	1	no	—
182	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	53	1	no	—
185	C₂×D₄	—	106	1	no	—
186	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	106	1	no	—
190	C₂²×C₄	—	106	2	no	—
191	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
192	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
193	Q₁₆	—	53	4	no	—
194	SD₁₆	—	53	4	no	—
195	SD₁₆	—	53	4	no	—
196	D₈	—	53	4	no	—
197	C₂×C₈	—	53	4	no	—
198	C₂×C₈	—	53	1	no	—
199	C₂×C₈	—	53	1	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	106	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	53	2	no	—
204	C₄⋊C₄	—	106	2	no	—
205	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	106	1	no	—
209	C₂²⋊C₄	—	106	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	106	1	no	—
213	C₄²	—	106	2	no	—
214	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
215	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₄²	C₂²×D₅₃	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	106	1	no	—
219	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	53	2	no	—
222	Q₈	—	106	1	no	—
223	Q₈	C₂×C₄×D₅₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	53	2	no	—
227	D₄	—	106	4	no	—
228	D₄	—	106	1	no	—
229	D₄	C₂×C₄×D₅₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	106	1	no	—
236	C₂×C₄	—	106	4	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
240	C₂×C₄	#11719945	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
244	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
245	C₈	—	53	2	no	—
246	C₈	—	53	2	no	—
247	C₈	—	106	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	106	2	no	—
252	C₂²	—	212	2	no	—
253	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	106	2	no	—
259	C₄	—	212	2	no	—
260	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
261	C₄	#13133237	1	4	yes	direct product factor
262	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
263	C₄	C₂×#11719945	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	212	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂	C₂×#13133237	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	C₄×#135
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#11719945	C₂	semidirect
3. C₄×#11719925	C₂	semidirect
4. C₄×#11719924	C₂	semidirect
5. C₄×#11719923	C₂	semidirect
6. C₄×#11719922	C₂	semidirect
7. #16810659	C₂	semidirect
8. #16810619	C₂	semidirect
9. #16810617	C₂	semidirect
10. C₂×C₄×#88322	C₂	semidirect
11. C₄×#41×C₅₃	C₂	semidirect
13. #13133237	C₄	direct (N × H)
43. #41×C₅₃	C₂×C₄	semidirect
53. #11719945	C₂×C₄	semidirect
68. #11719925	C₂×C₄	semidirect
69. #11719924	C₂×C₄	semidirect
70. #11719923	C₂×C₄	semidirect
71. #11719922	C₂×C₄	semidirect
78. C₂×#88322	C₂×C₄	semidirect
124. C₄×C₅₃	#135	semidirect
137. C₄×#41	D₅₃	semidirect
146. C₅₃	C₄×#135	semidirect
180. #41	C₄×D₅₃	semidirect
261. C₄	#13133237	direct (N × H)

Subgroups — C4×#13133237 #20283485

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133237 #20283485