Subgroups — C₂²×#16810865 #28251191

Order 13568.

3006 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133197	C₂	1	2	yes	maximal
4	C₂×#16810865	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#13133140	C₂	1	2	yes	maximal, complemented
6	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Dic₁₀₆	C₂²	1	2	yes	—
8	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₁₂	—	2	8	no	—
10	C₂²×Dic₂₁₂	C₂²	1	12	yes	complemented
11	C₂³×#88322	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₅₃	—	2	8	no	—
13	C₂²×C₈×D₅₃	C₂²	1	12	yes	—
14	C₂×#13133197	—	2	12	no	—
15	C₂×#13133197	C₂²	1	24	yes	—
16	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂²×#11719921	—	2	2	no	—
18	#16810865	C₂²	1	64	yes	direct product factor
19	C₂×#13133140	C₂²	1	24	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×#107×C₅₃	C₂²	1	12	yes	complemented
23	C₂⁴×D₅₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₁₀₆	—	2	8	no	—
25	C₂²×Dic₁₀₆	—	2	8	no	—
26	C₂²×Dic₁₀₆	—	2	6	no	—
27	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
28	C₂³×Dic₅₃	—	2	2	no	—
29	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₅₃	—	2	4	no	—
31	C₂²×C₄×D₅₃	—	2	4	no	—
32	C₂²×C₄×D₅₃	D₄	1	6	yes	—
33	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
34	C₂×Dic₂₁₂	—	2	48	no	—
35	C₂×Dic₂₁₂	C₂³	1	16	yes	complemented
36	C₂²×#88322	—	2	4	no	—
37	C₂²×#88322	C₂³	1	6	yes	—
38	C₂×C₈×D₅₃	—	2	48	no	—
39	C₂×C₈×D₅₃	C₂³	1	16	yes	—
40	#13133197	—	2	48	no	—
41	#13133197	C₂³	1	32	yes	—
42	C₂×C₄⋊Dic₅₃	—	2	24	no	—
43	C₂×#11719921	—	2	24	no	—
44	#13133140	C₂³	1	32	yes	complemented
45	C₂³×C₄×C₅₃	—	2	2	no	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	8	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
49	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
51	#107×C₅₃	C₂³	1	16	yes	complemented
52	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂³×D₅₃	—	2	4	no	—
54	C₂³×D₅₃	—	2	4	no	—
55	C₂³×D₅₃	Q₁₆	1	8	yes	complemented
56	C₂³×D₅₃	C₂×D₄	1	6	yes	—
57	C₂²×C₄×C₅₃	—	2	12	no	—
58	C₂²×C₄×C₅₃	—	4	8	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₅₃	#41	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
63	C₂²×Dic₅₃	—	2	4	no	—
64	C₂²×Dic₅₃	—	2	12	no	—
65	C₂²×Dic₅₃	—	4	8	no	—
66	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
67	C₂×C₄×D₅₃	—	2	24	no	—
68	C₂×C₄×D₅₃	—	2	12	no	—
69	C₂×C₄×D₅₃	—	2	12	no	—
70	C₂×C₄×D₅₃	C₂×D₄	1	12	yes	—
71	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
72	C₂×Dic₁₀₆	—	2	24	no	—
73	C₂×Dic₁₀₆	—	2	24	no	—
74	C₂×Dic₁₀₆	—	2	12	no	—
75	C₂×Dic₁₀₆	—	4	24	no	—
76	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
77	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	—
78	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
79	C₄⋊C₄×C₅₃	—	2	12	no	—
80	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
81	#11719921	—	2	32	no	—
82	C₄⋊Dic₅₃	—	2	32	no	—
83	C₂×#88322	—	2	12	no	—
84	C₂×#88322	C₂⁴	1	4	yes	—
85	Dic₂₁₂	—	2	32	no	—
86	C₈×D₅₃	—	2	32	no	—
87	C₂³×C₅₃	#121	1	4	yes	—
88	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
89	C₂³×C₅₃	#135	1	4	yes	—
90	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
91	C₂²×D₅₃	—	2	12	no	—
92	C₂²×D₅₃	—	2	12	no	—
93	C₂²×D₅₃	—	2	32	no	—
94	C₂²×D₅₃	C₂×Q₁₆	1	24	yes	complemented
95	C₂²×D₅₃	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₅₃	—	2	8	no	—
97	C₂×C₄×C₅₃	—	4	24	no	—
98	C₂×C₄×C₅₃	#121	1	12	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
101	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
102	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	12	no	—
104	C₂×Dic₅₃	—	2	8	no	—
105	C₂×Dic₅₃	—	4	24	no	—
106	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
107	C₄×D₅₃	—	2	24	no	—
108	C₄×D₅₃	—	2	4	no	—
109	C₄×D₅₃	—	2	4	no	—
110	Dic₁₀₆	—	2	8	no	—
111	Dic₁₀₆	—	2	8	no	—
112	Dic₁₀₆	—	4	24	no	—
113	C₈×C₅₃	C₂²×D₄	1	8	yes	—
114	#88322	—	2	4	no	—
115	C₂²×#467	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	#467	1	16	yes	complemented
117	C₂²×C₅₃	C₂×#121	1	6	yes	—
118	C₂²×C₅₃	C₂²×Q₁₆	1	6	yes	—
119	C₂²×C₅₃	C₂×#135	1	6	yes	—
120	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
121	C₂×D₅₃	—	2	4	no	—
122	C₂×D₅₃	—	2	48	no	—
123	C₂×D₅₃	—	2	4	no	—
124	C₂×D₅₃	C₂²×Q₁₆	1	8	yes	complemented
125	C₄×C₅₃	—	4	8	no	—
126	C₄×C₅₃	C₂×#121	1	6	yes	—
127	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
129	Dic₅₃	—	2	4	no	—
130	Dic₅₃	—	4	8	no	—
131	C₂³×Q₁₆	—	53	1	no	—
132	C₂⁴×C₈	—	53	1	no	—
133	C₂²×C₂²⋊Q₈	—	53	2	no	—
134	C₂×#467	—	53	24	no	—
135	C₂²×#107	D₅₃	1	1	yes	characteristic, complemented
136	C₂²×#103	—	53	2	no	—
137	C₂×C₅₃	C₂×#467	1	12	yes	complemented
138	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
139	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
140	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
141	D₅₃	—	2	8	no	—
142	C₂³×Q₈	—	53	2	no	—
143	C₂⁴×C₄	—	53	1	no	—
144	C₂²×Q₁₆	—	53	12	no	—
145	C₂²×Q₁₆	—	106	8	no	—
146	C₂³×C₈	—	53	1	no	—
147	C₂³×C₈	—	53	12	no	—
148	C₂³×C₈	—	106	8	no	—
149	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
150	C₂×C₂²⋊Q₈	—	53	24	no	—
151	C₂×C₂²⋊Q₈	—	106	12	no	—
152	C₂²×C₄⋊C₄	—	106	2	no	—
153	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
154	C₂²×C₂²⋊C₄	—	106	2	no	—
155	#467	—	53	64	no	—
156	C₂×#107	C₂×D₅₃	1	12	yes	complemented
157	C₂×#103	—	53	24	no	—
158	C₅₃	C₂²×#467	1	1	yes	characteristic, complemented, Sylow
159	C₂⁵	—	53	1	no	—
160	C₂²×Q₈	—	53	12	no	—
161	C₂²×Q₈	—	106	6	no	—
162	C₂²×Q₈	—	106	8	no	—
163	C₂²×Q₈	—	106	8	no	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	—	53	1	no	—
166	C₂³×C₄	—	53	6	no	—
167	C₂³×C₄	—	53	6	no	—
168	C₂³×C₄	—	106	4	no	—
169	C₂³×C₄	—	106	4	no	—
170	C₂³×C₄	—	106	2	no	—
171	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×Q₁₆	—	53	16	no	—
173	C₂×Q₁₆	—	106	48	no	—
174	C₂²×C₈	—	53	6	no	—
175	C₂²×C₈	—	53	16	no	—
176	C₂²×C₈	—	106	4	no	—
177	C₂²×C₈	—	106	48	no	—
178	C₂²×C₈	D₂₁₂	1	8	yes	—
179	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
180	C₂²⋊Q₈	—	53	32	no	—
181	C₂²⋊Q₈	—	106	48	no	—
182	C₂×C₄⋊C₄	—	2	6	no	—
183	C₂×C₄⋊C₄	—	106	24	no	—
184	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
185	C₂×C₂²⋊C₄	—	106	24	no	—
186	#107	C₂²×D₅₃	1	16	yes	complemented
187	#103	—	53	32	no	—
188	C₂⁴	—	53	8	no	—
189	C₂⁴	—	53	6	no	—
190	C₂⁴	—	106	4	no	—
191	C₂⁴	—	106	4	no	—
192	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
193	C₂×Q₈	—	53	8	no	—
194	C₂×Q₈	—	106	12	no	—
195	C₂×Q₈	—	106	24	no	—
196	C₂×Q₈	—	106	24	no	—
197	C₂×Q₈	—	212	24	no	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	53	6	no	—
201	C₂²×C₄	—	53	12	no	—
202	C₂²×C₄	—	53	4	no	—
203	C₂²×C₄	—	106	4	no	—
204	C₂²×C₄	—	106	24	no	—
205	C₂²×C₄	—	106	12	no	—
206	C₂²×C₄	—	106	12	no	—
207	C₂²×C₄	—	106	12	no	—
208	C₂²×C₄	—	212	8	no	—
209	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
210	C₂²×C₄	D₄×D₅₃	1	3	yes	—
211	C₂²×C₄	#11719949	1	4	yes	—
212	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
213	Q₁₆	—	106	32	no	—
214	C₂×C₈	—	53	4	no	—
215	C₂×C₈	—	106	12	no	—
216	C₂×C₈	—	106	32	no	—
217	C₂×C₈	C₂×D₂₁₂	1	24	yes	—
218	C₂×C₈	C₂³×D₅₃	1	4	yes	—
219	C₄⋊C₄	—	2	12	no	—
220	C₄⋊C₄	—	106	32	no	—
221	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
222	C₂²⋊C₄	—	106	32	no	—
223	C₂³	—	53	24	no	—
224	C₂³	—	53	4	no	—
225	C₂³	—	106	12	no	—
226	C₂³	—	106	32	no	—
227	C₂³	—	106	12	no	—
228	C₂³	#13133192	1	4	yes	—
229	C₂³	Q₁₆×D₅₃	1	4	yes	—
230	C₂³	#13133236	1	4	yes	—
231	C₂³	C₂×D₄×D₅₃	1	3	yes	—
232	Q₈	—	106	8	no	—
233	Q₈	—	106	8	no	—
234	Q₈	—	212	24	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	24	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	106	12	no	—
239	C₂×C₄	—	106	24	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	—	106	8	no	—
243	C₂×C₄	—	212	24	no	—
244	C₂×C₄	#13133192	1	12	yes	—
245	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
246	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
247	C₂×C₄	C₂×#11719949	1	6	yes	—
248	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
249	C₈	—	106	4	no	—
250	C₈	C₂²×D₂₁₂	1	8	yes	—
251	C₂²	—	53	8	no	—
252	C₂²	—	106	48	no	—
253	C₂²	—	106	4	no	—
254	C₂²	—	106	4	no	—
255	C₂²	#16810865	1	16	yes	direct product factor
256	C₂²	C₂×#13133192	1	6	yes	—
257	C₂²	C₂×Q₁₆×D₅₃	1	6	yes	—
258	C₂²	C₂×#13133236	1	6	yes	—
259	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
260	C₄	—	4	8	no	—
261	C₄	—	106	4	no	—
262	C₄	—	212	8	no	—
263	C₄	C₂×#13133192	1	6	yes	—
264	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
265	C₄	C₂²×#11719949	1	1	yes	characteristic
266	C₂	—	106	8	no	—
267	C₂	C₂×#16810865	1	12	yes	direct product factor
268	C₂	C₂²×#13133192	1	1	yes	characteristic
269	C₂	C₂²×Q₁₆×D₅₃	1	1	yes	characteristic
270	C₂	C₂²×#13133236	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×D₄
prime radical	C₂⁴×D₅₃	D₄
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂²×#467
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₁₂	C₂	semidirect
4. C₂×#16810865	C₂	direct (N × H)
5. C₂²×#13133140	C₂	semidirect
6. C₂²×#107×C₅₃	C₂	semidirect
10. C₂²×Dic₂₁₂	C₂²	semidirect
18. #16810865	C₂²	direct (N × H)
19. C₂×#13133140	C₂²	semidirect
22. C₂×#107×C₅₃	C₂²	semidirect
35. C₂×Dic₂₁₂	C₂³	semidirect
44. #13133140	C₂³	semidirect
51. #107×C₅₃	C₂³	semidirect
55. C₂³×D₅₃	Q₁₆	semidirect
94. C₂²×D₅₃	C₂×Q₁₆	semidirect
116. C₂²×C₅₃	#467	semidirect
124. C₂×D₅₃	C₂²×Q₁₆	semidirect
135. C₂²×#107	D₅₃	semidirect
137. C₂×C₅₃	C₂×#467	semidirect
156. C₂×#107	C₂×D₅₃	semidirect
158. C₅₃	C₂²×#467	semidirect
186. #107	C₂²×D₅₃	semidirect
255. C₂²	#16810865	direct (N × H)
267. C₂	C₂×#16810865	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₁₂	2
C₂³×C₈×D₅₃	2
C₂²×#13133197	2
C₂×#16810865	2
C₂²×#13133140	2
C₂²×#107×C₅₃	2
C₂²×#467	53

Subgroups — C22×#16810865 #28251191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810865 #28251191