Subgroups — C₂²×C₄⋊Q₈×D₅₃ #28290780

Order 13568.

12190 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₅₃	C₂	1	2	yes	maximal
2	C₂²×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133216	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133215	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊Q₈×D₅₃	C₂	1	48	yes	maximal, direct product factor
6	C₂²×C₄⋊C₄×D₅₃	C₂	1	4	yes	maximal
7	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Dic₁₀₆	C₂²	1	2	yes	—
10	C₂³×Dic₁₀₆	C₂²	1	4	yes	—
11	C₂²×Q₈×D₅₃	—	2	32	no	—
12	C₂²×Q₈×D₅₃	C₂²	1	48	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
14	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
16	C₂×#13133217	C₂²	1	24	yes	complemented
17	C₂×#13133216	C₂²	1	96	yes	complemented
18	C₂×#13133215	C₂²	1	48	yes	complemented
19	C₄⋊Q₈×D₅₃	C₂²	1	256	yes	direct product factor
20	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
21	C₂²×#11719919	C₂²	1	8	yes	—
22	C₂×C₄⋊C₄×D₅₃	C₂²	1	96	yes	—
23	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×D₅₃	C₂²	1	24	yes	—
26	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₀₆	—	2	16	no	—
32	C₂²×Dic₁₀₆	—	2	32	no	—
33	C₂²×Dic₁₀₆	C₂³	1	24	yes	—
34	C₂²×Dic₁₀₆	C₂³	1	48	yes	—
35	C₂×Q₈×D₅₃	—	2	384	no	—
36	C₂×Q₈×D₅₃	C₂³	1	128	yes	—
37	C₂³×Dic₅₃	C₂³	1	4	yes	—
38	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂³×Dic₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×D₅₃	—	2	32	no	—
41	C₂²×C₄×D₅₃	D₄	1	16	yes	—
42	C₂²×C₄×D₅₃	Q₈	1	32	yes	complemented
43	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
44	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
45	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
46	#13133217	C₂³	1	64	yes	complemented
47	#13133216	C₂³	1	256	yes	complemented
48	#13133215	C₂³	1	128	yes	complemented
49	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	—
50	C₂×#11719919	C₂³	1	96	yes	—
51	C₄⋊C₄×D₅₃	C₂³	1	256	yes	—
52	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
53	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
54	C₄²×D₅₃	C₂³	1	64	yes	—
55	C₂²×Q₈×C₅₃	—	2	16	no	—
56	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
63	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂³×D₅₃	C₂×D₄	1	8	yes	—
65	C₂³×D₅₃	C₂×Q₈	1	16	yes	—
66	C₂³×D₅₃	C₂⁴	1	6	yes	—
67	C₂×Q₈×C₅₃	—	2	96	no	—
68	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
69	C₂²×C₄×C₅₃	—	2	16	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
71	C₂²×C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
72	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
75	C₂²×Dic₅₃	—	2	16	no	—
76	C₂²×Dic₅₃	C₂×D₄	1	8	yes	—
77	C₂²×Dic₅₃	C₂×Q₈	1	16	yes	complemented
78	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
79	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
80	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
81	Q₈×D₅₃	—	2	512	no	—
82	C₂×C₄×D₅₃	—	2	192	no	—
83	C₂×C₄×D₅₃	C₂×D₄	1	96	yes	—
84	C₂×C₄×D₅₃	C₂×Q₈	1	192	yes	complemented
85	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
86	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
87	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
88	C₂×Dic₁₀₆	—	2	96	no	—
89	C₂×Dic₁₀₆	—	2	192	no	—
90	C₂×Dic₁₀₆	C₂⁴	1	32	yes	—
91	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
92	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
93	C₄²×C₅₃	C₂⁴	1	16	yes	—
94	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
95	#11719919	C₂⁴	1	128	yes	—
96	C₄×Dic₅₃	C₂⁴	1	16	yes	—
97	C₄×Dic₅₃	C₂⁴	1	32	yes	—
98	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
99	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
100	C₂³×C₅₃	C₂⁵	1	3	yes	—
101	C₂²×D₅₃	C₄⋊Q₈	1	64	yes	direct product factor
102	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
103	C₂²×D₅₃	C₂²×Q₈	1	48	yes	—
104	C₂²×D₅₃	C₂⁵	1	4	yes	—
105	Q₈×C₅₃	—	2	64	no	—
106	C₂×C₄×C₅₃	—	2	48	no	—
107	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
108	C₂×C₄×C₅₃	C₂²×Q₈	1	48	yes	complemented
109	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
110	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
111	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
112	C₂×Dic₅₃	—	2	48	no	—
113	C₂×Dic₅₃	C₂²×D₄	1	24	yes	—
114	C₂×Dic₅₃	C₂²×Q₈	1	48	yes	complemented
115	C₂×Dic₅₃	C₂⁵	1	16	yes	—
116	C₂×Dic₅₃	C₂⁵	1	4	yes	—
117	C₂×Dic₅₃	C₂⁵	1	8	yes	—
118	C₄×D₅₃	—	2	128	no	—
119	C₄×D₅₃	C₂²×D₄	1	64	yes	—
120	C₄×D₅₃	C₂²×Q₈	1	128	yes	complemented
121	Dic₁₀₆	—	2	64	no	—
122	Dic₁₀₆	—	2	128	no	—
123	C₂³×C₄⋊Q₈	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	C₂×C₄⋊Q₈	1	16	yes	complemented
125	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
126	C₂²×C₅₃	C₂³×Q₈	1	12	yes	—
127	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
128	C₂×D₅₃	C₂×C₄⋊Q₈	1	96	yes	direct product factor
129	C₂×D₅₃	C₂³×D₄	1	8	yes	—
130	C₂×D₅₃	C₂³×Q₈	1	16	yes	—
131	C₄×C₅₃	—	2	16	no	—
132	C₄×C₅₃	C₂³×D₄	1	8	yes	—
133	C₄×C₅₃	C₂³×Q₈	1	16	yes	complemented
134	Dic₅₃	—	2	16	no	—
135	Dic₅₃	C₂³×D₄	1	8	yes	—
136	Dic₅₃	C₂³×Q₈	1	16	yes	complemented
137	C₂⁴×Q₈	—	53	2	no	—
138	C₂²×C₄⋊Q₈	—	53	1	no	—
139	C₂²×C₄⋊Q₈	—	53	2	no	—
140	C₂²×C₄⋊Q₈	—	53	48	no	—
141	C₂²×C₄⋊Q₈	—	53	4	no	—
142	C₂²×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, direct product factor
143	C₂³×C₄⋊C₄	—	53	4	no	—
144	C₂³×C₄²	—	53	1	no	—
145	C₂×C₅₃	C₂²×C₄⋊Q₈	1	12	yes	complemented
146	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
147	C₂×C₅₃	C₂⁴×Q₈	1	2	yes	—
148	D₅₃	C₂²×C₄⋊Q₈	1	16	yes	direct product factor
149	C₂³×Q₈	—	53	48	no	—
150	C₂³×Q₈	—	53	4	no	—
151	C₂³×Q₈	—	53	2	no	—
152	C₂³×Q₈	—	106	32	no	—
153	C₂³×Q₈	C₂×D₅₃	1	2	yes	—
154	C₂⁴×C₄	—	53	1	no	—
155	C₂⁴×C₄	—	53	2	no	—
156	C₂⁴×C₄	—	53	4	no	—
157	C₂×C₄⋊Q₈	—	53	48	no	—
158	C₂×C₄⋊Q₈	—	53	256	no	—
159	C₂×C₄⋊Q₈	—	53	96	no	—
160	C₂×C₄⋊Q₈	—	53	24	no	—
161	C₂×C₄⋊Q₈	C₂×D₅₃	1	24	yes	direct product factor
162	C₂²×C₄⋊C₄	—	53	8	no	—
163	C₂²×C₄⋊C₄	—	53	4	no	—
164	C₂²×C₄⋊C₄	—	53	96	no	—
165	C₂²×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
166	C₂²×C₄²	—	53	1	no	—
167	C₂²×C₄²	—	53	2	no	—
168	C₂²×C₄²	—	53	24	no	—
169	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
170	C₅₃	C₂³×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	—	53	1	no	—
172	C₂²×Q₈	—	2	16	no	—
173	C₂²×Q₈	—	53	128	no	—
174	C₂²×Q₈	—	53	48	no	—
175	C₂²×Q₈	—	53	24	no	—
176	C₂²×Q₈	—	106	32	no	—
177	C₂²×Q₈	—	106	384	no	—
178	C₂²×Q₈	—	106	16	no	—
179	C₂²×Q₈	C₂²×D₅₃	1	24	yes	—
180	C₂³×C₄	—	53	2	no	—
181	C₂³×C₄	—	53	1	no	—
182	C₂³×C₄	—	53	16	no	—
183	C₂³×C₄	—	53	32	no	—
184	C₂³×C₄	—	53	24	no	—
185	C₂³×C₄	—	53	12	no	—
186	C₂³×C₄	—	53	4	no	—
187	C₂³×C₄	—	53	48	no	—
188	C₂³×C₄	—	106	32	no	—
189	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
190	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
191	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
192	C₄⋊Q₈	—	53	128	no	—
193	C₄⋊Q₈	—	53	256	no	—
194	C₄⋊Q₈	—	53	64	no	—
195	C₄⋊Q₈	C₂²×D₅₃	1	64	yes	direct product factor
196	C₂×C₄⋊C₄	—	53	96	no	—
197	C₂×C₄⋊C₄	—	53	48	no	—
198	C₂×C₄⋊C₄	—	53	256	no	—
199	C₂×C₄⋊C₄	C₂²×D₅₃	1	48	yes	—
200	C₂×C₄²	—	53	12	no	—
201	C₂×C₄²	—	53	24	no	—
202	C₂×C₄²	—	53	64	no	—
203	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
204	C₂⁴	—	53	8	no	—
205	C₂⁴	—	53	16	no	—
206	C₂⁴	—	53	6	no	—
207	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	96	no	—
209	C₂×Q₈	—	53	64	no	—
210	C₂×Q₈	—	53	32	no	—
211	C₂×Q₈	—	106	192	no	—
212	C₂×Q₈	—	106	512	no	—
213	C₂×Q₈	—	106	96	no	—
214	C₂×Q₈	C₂³×D₅₃	1	32	yes	—
215	C₂²×C₄	—	2	16	no	—
216	C₂²×C₄	—	53	8	no	—
217	C₂²×C₄	—	53	16	no	—
218	C₂²×C₄	—	53	12	no	—
219	C₂²×C₄	—	53	6	no	—
220	C₂²×C₄	—	53	96	no	—
221	C₂²×C₄	—	53	192	no	—
222	C₂²×C₄	—	53	32	no	—
223	C₂²×C₄	—	53	16	no	—
224	C₂²×C₄	—	53	24	no	—
225	C₂²×C₄	—	53	64	no	—
226	C₂²×C₄	—	106	192	no	—
227	C₂²×C₄	—	106	16	no	—
228	C₂²×C₄	D₄×D₅₃	1	8	yes	—
229	C₂²×C₄	Q₈×D₅₃	1	16	yes	complemented
230	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
231	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
232	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
233	C₄⋊C₄	—	53	128	no	—
234	C₄⋊C₄	—	53	64	no	—
235	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
236	C₄²	—	53	16	no	—
237	C₄²	—	53	32	no	—
238	C₄²	C₂³×D₅₃	1	16	yes	—
239	C₂³	—	53	64	no	—
240	C₂³	—	53	24	no	—
241	C₂³	—	53	48	no	—
242	C₂³	—	53	4	no	—
243	C₂³	C₂×D₄×D₅₃	1	4	yes	—
244	C₂³	C₂×Q₈×D₅₃	1	8	yes	—
245	C₂³	C₂⁴×D₅₃	1	3	yes	—
246	Q₈	—	2	64	no	—
247	Q₈	—	106	128	no	—
248	Q₈	—	106	64	no	—
249	C₂×C₄	—	2	48	no	—
250	C₂×C₄	—	53	24	no	—
251	C₂×C₄	—	53	48	no	—
252	C₂×C₄	—	53	8	no	—
253	C₂×C₄	—	53	4	no	—
254	C₂×C₄	—	53	64	no	—
255	C₂×C₄	—	53	128	no	—
256	C₂×C₄	—	53	16	no	—
257	C₂×C₄	—	106	128	no	—
258	C₂×C₄	—	106	48	no	—
259	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	—
260	C₂×C₄	C₂×Q₈×D₅₃	1	48	yes	complemented
261	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
262	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
263	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
264	C₂²	—	53	8	no	—
265	C₂²	—	53	16	no	—
266	C₂²	—	53	96	no	—
267	C₂²	C₄⋊Q₈×D₅₃	1	16	yes	direct product factor
268	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
269	C₂²	C₂²×Q₈×D₅₃	1	12	yes	—
270	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
271	C₄	—	2	16	no	—
272	C₄	—	53	8	no	—
273	C₄	—	53	16	no	—
274	C₄	—	106	16	no	—
275	C₄	C₂²×D₄×D₅₃	1	8	yes	—
276	C₄	C₂²×Q₈×D₅₃	1	16	yes	complemented
277	C₂	—	53	16	no	—
278	C₂	C₂×C₄⋊Q₈×D₅₃	1	12	yes	direct product factor
279	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
280	C₂	C₂³×Q₈×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄⋊Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₅₃	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂³×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133217	C₂	semidirect
3. C₂²×#13133216	C₂	semidirect
4. C₂²×#13133215	C₂	semidirect
5. C₂×C₄⋊Q₈×D₅₃	C₂	direct (N × H)
8. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
16. C₂×#13133217	C₂²	semidirect
17. C₂×#13133216	C₂²	semidirect
18. C₂×#13133215	C₂²	semidirect
19. C₄⋊Q₈×D₅₃	C₂²	direct (N × H)
27. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
42. C₂²×C₄×D₅₃	Q₈	semidirect
46. #13133217	C₂³	semidirect
47. #13133216	C₂³	semidirect
48. #13133215	C₂³	semidirect
60. C₄⋊Q₈×C₅₃	C₂³	semidirect
71. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
77. C₂²×Dic₅₃	C₂×Q₈	semidirect
84. C₂×C₄×D₅₃	C₂×Q₈	semidirect
101. C₂²×D₅₃	C₄⋊Q₈	direct (N × H)
108. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
114. C₂×Dic₅₃	C₂²×Q₈	semidirect
120. C₄×D₅₃	C₂²×Q₈	semidirect
124. C₂²×C₅₃	C₂×C₄⋊Q₈	semidirect
128. C₂×D₅₃	C₂×C₄⋊Q₈	direct (N × H)
133. C₄×C₅₃	C₂³×Q₈	semidirect
136. Dic₅₃	C₂³×Q₈	semidirect
142. C₂²×C₄⋊Q₈	D₅₃	direct (N × H)
145. C₂×C₅₃	C₂²×C₄⋊Q₈	semidirect
148. D₅₃	C₂²×C₄⋊Q₈	direct (N × H)
161. C₂×C₄⋊Q₈	C₂×D₅₃	direct (N × H)
170. C₅₃	C₂³×C₄⋊Q₈	semidirect
195. C₄⋊Q₈	C₂²×D₅₃	direct (N × H)
229. C₂²×C₄	Q₈×D₅₃	semidirect
260. C₂×C₄	C₂×Q₈×D₅₃	semidirect
267. C₂²	C₄⋊Q₈×D₅₃	direct (N × H)
276. C₄	C₂²×Q₈×D₅₃	semidirect
278. C₂	C₂×C₄⋊Q₈×D₅₃	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×Q₈×D₅₃	2
C₂²×#13133217	2
C₂²×#13133216	2
C₂²×#13133215	2
C₂×C₄⋊Q₈×D₅₃	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂³×C₄⋊Q₈	53

Subgroups — C22×C4⋊Q8×D53 #28290780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊Q₈×D₅₃ #28290780