Subgroups — C₂×#20288117 #28290933

Order 13568.

7022 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288117	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811252	C₂	1	12	yes	maximal, complemented
3	C₂×#16811249	C₂	1	3	yes	maximal
4	C₂×#16811175	C₂	1	12	yes	maximal, complemented
5	C₂²×#13133173	C₂	1	3	yes	maximal
6	C₂×C₅₃×#543	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
8	#16811252	C₂²	1	192	yes	complemented
9	#16811249	C₂²	1	48	yes	—
10	#16811175	C₂²	1	192	yes	complemented
11	C₂×#13133216	C₂²	1	12	yes	complemented
12	C₂×#13133215	C₂²	1	12	yes	complemented
13	C₂×#13133207	C₂²	1	24	yes	complemented
14	C₂×#13133196	C₂²	1	12	yes	—
15	C₂×#13133194	C₂²	1	24	yes	—
16	C₂×Q₈×Dic₅₃	C₂²	1	4	yes	complemented
17	C₂×#13133182	C₂²	1	12	yes	complemented
18	C₂×#13133173	C₂²	1	12	yes	—
19	C₂×#13133173	C₂²	1	12	yes	—
20	C₂×#13133173	C₂²	1	48	yes	—
21	C₂²×#11719919	C₂²	1	6	yes	—
22	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
23	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
24	C₅₃×#543	C₂²	1	16	yes	—
25	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	12	yes	—
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
27	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
28	C₂³×Dic₅₃	Q₈	1	16	yes	complemented
29	C₂³×Dic₅₃	C₂³	1	6	yes	—
30	C₂³×Dic₅₃	C₂³	1	3	yes	—
31	C₂³×Dic₅₃	C₂³	1	3	yes	—
32	C₂³×Dic₅₃	C₂³	1	8	yes	—
33	#13133216	C₂³	1	96	yes	complemented
34	#13133215	C₂³	1	96	yes	complemented
35	#13133207	C₂³	1	192	yes	complemented
36	#13133196	C₂³	1	96	yes	—
37	#13133194	C₂³	1	192	yes	—
38	Q₈×Dic₅₃	C₂³	1	32	yes	complemented
39	#13133182	C₂³	1	96	yes	complemented
40	#13133173	C₂³	1	96	yes	—
41	#13133173	C₂³	1	96	yes	—
42	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
43	C₂×#11719919	C₂³	1	24	yes	—
44	C₂×#11719919	C₂³	1	48	yes	—
45	C₂×#11719919	C₂³	1	24	yes	—
46	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
47	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
48	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
49	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
51	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
54	C₂²⋊Q₈×C₅₃	C₂³	1	96	yes	—
55	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
56	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
57	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
58	C₂⁴×C₅₃	—	4	4	no	—
59	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
60	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
61	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
62	C₂×Q₈×C₅₃	—	4	8	no	—
63	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
64	C₂²×C₄×C₅₃	—	4	12	no	—
65	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
67	C₂²×Dic₅₃	—	4	12	no	—
68	C₂²×Dic₅₃	C₂×Q₈	1	24	yes	—
69	C₂²×Dic₅₃	C₂×Q₈	1	24	yes	—
70	C₂²×Dic₅₃	C₂×Q₈	1	64	yes	complemented
71	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
72	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
73	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
75	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
76	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
77	C₂×Dic₁₀₆	—	4	24	no	—
78	C₂×Dic₁₀₆	C₂⁴	1	48	yes	—
79	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
80	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
81	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
82	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
83	#11719919	C₂⁴	1	96	yes	—
84	#11719919	C₂⁴	1	96	yes	—
85	C₄×Dic₅₃	C₂⁴	1	48	yes	—
86	C₄×Dic₅₃	C₂⁴	1	48	yes	—
87	C₂³×C₅₃	—	2	12	no	—
88	C₂³×C₅₃	—	4	16	no	—
89	C₂³×C₅₃	—	4	8	no	—
90	C₂³×C₅₃	C₂²×Q₈	1	12	yes	—
91	C₂³×C₅₃	C₂²×Q₈	1	16	yes	—
92	C₂³×C₅₃	C₂⁵	1	6	yes	—
93	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
94	Q₈×C₅₃	—	4	16	no	—
95	C₂×C₄×C₅₃	—	2	24	no	—
96	C₂×C₄×C₅₃	—	4	24	no	—
97	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
98	C₂×Dic₅₃	—	2	24	no	—
99	C₂×Dic₅₃	—	4	24	no	—
100	C₂×Dic₅₃	C₂²×Q₈	1	4	yes	—
101	C₂×Dic₅₃	C₂²×Q₈	1	12	yes	—
102	C₂×Dic₅₃	C₂²×Q₈	1	48	yes	—
103	C₂×Dic₅₃	C₂²×Q₈	1	48	yes	—
104	C₂×Dic₅₃	C₂⁵	1	24	yes	—
105	C₂×Dic₅₃	C₂⁵	1	2	yes	—
106	C₂×Dic₅₃	C₂⁵	1	6	yes	—
107	Dic₁₀₆	—	4	48	no	—
108	C₂×#3458	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	12	no	—
110	C₂²×C₅₃	—	2	24	no	—
111	C₂²×C₅₃	—	4	16	no	—
112	C₂²×C₅₃	C₂³×Q₈	1	12	yes	—
113	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
114	C₂²×C₅₃	#585	1	4	yes	—
115	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
116	C₄×C₅₃	—	2	24	no	—
117	Dic₅₃	—	2	24	no	—
118	Dic₅₃	C₂³×Q₈	1	4	yes	—
119	Dic₅₃	C₂³×Q₈	1	12	yes	—
120	#3458	—	53	32	no	—
121	C₂×#544	—	53	12	no	—
122	C₂×#543	—	53	3	no	—
123	C₂×#543	D₅₃	1	1	yes	characteristic
124	C₂×#533	—	53	12	no	—
125	C₂²×#117	—	53	3	no	—
126	C₂×C₅₃	—	2	12	no	—
127	C₂×C₅₃	#3458	1	4	yes	complemented
128	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#585	1	2	yes	—
130	C₂⁴×C₄	—	53	1	no	—
131	#544	—	53	192	no	—
132	#543	—	53	48	no	—
133	#543	C₂×D₅₃	1	16	yes	—
134	#533	—	53	192	no	—
135	C₂×C₄⋊Q₈	—	53	12	no	—
136	C₂×C₄⋊Q₈	—	53	12	no	—
137	C₂×#125	—	53	24	no	—
138	C₂×C₂²⋊Q₈	—	53	12	no	—
139	C₂×C₂²⋊Q₈	—	53	24	no	—
140	C₂×C₂²⋊Q₈	C₂×D₅₃	1	12	yes	—
141	C₂×C₄×Q₈	—	53	4	no	—
142	C₂×C₄×Q₈	—	53	12	no	—
143	C₂×#117	—	53	12	no	—
144	C₂×#117	—	53	12	no	—
145	C₂×#117	—	53	48	no	—
146	C₂²×C₄⋊C₄	—	53	6	no	—
147	C₂²×C₂²⋊C₄	—	53	3	no	—
148	C₂²×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
149	C₂²×C₄²	—	53	6	no	—
150	C₅₃	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
152	C₂²×Q₈	—	53	12	no	—
153	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
154	C₂³×C₄	—	53	16	no	—
155	C₂³×C₄	—	53	3	no	—
156	C₂³×C₄	—	53	8	no	—
157	C₂³×C₄	—	53	3	no	—
158	C₂³×C₄	—	53	6	no	—
159	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
160	C₄⋊Q₈	—	53	96	no	—
161	C₄⋊Q₈	—	53	96	no	—
162	#125	—	53	192	no	—
163	C₂²⋊Q₈	—	53	96	no	—
164	C₂²⋊Q₈	—	53	192	no	—
165	C₂²⋊Q₈	C₂²×D₅₃	1	96	yes	—
166	C₄×Q₈	—	53	32	no	—
167	C₄×Q₈	—	53	96	no	—
168	#117	—	53	96	no	—
169	#117	—	53	96	no	—
170	C₂×C₄⋊C₄	—	53	24	no	—
171	C₂×C₄⋊C₄	—	53	48	no	—
172	C₂×C₄⋊C₄	—	53	24	no	—
173	C₂×C₄⋊C₄	—	53	12	no	—
174	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
175	C₂×C₂²⋊C₄	—	53	12	no	—
176	C₂×C₂²⋊C₄	—	53	24	no	—
177	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	—
178	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
179	C₂×C₄²	—	53	12	no	—
180	C₂×C₄²	—	53	12	no	—
181	C₂×C₄²	—	53	48	no	—
182	C₂⁴	—	4	4	no	—
183	C₂⁴	Q₈×D₅₃	1	8	yes	—
184	C₂⁴	C₂³×D₅₃	1	3	yes	—
185	C₂⁴	C₂³×D₅₃	1	4	yes	—
186	C₂×Q₈	—	4	8	no	—
187	C₂×Q₈	—	53	48	no	—
188	C₂×Q₈	—	212	24	no	—
189	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
190	C₂²×C₄	—	4	12	no	—
191	C₂²×C₄	—	53	24	no	—
192	C₂²×C₄	—	53	24	no	—
193	C₂²×C₄	—	53	64	no	—
194	C₂²×C₄	—	53	12	no	—
195	C₂²×C₄	—	53	12	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	3	no	—
198	C₂²×C₄	—	53	12	no	—
199	C₂²×C₄	—	53	24	no	—
200	C₂²×C₄	—	212	12	no	—
201	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
202	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
203	C₄⋊C₄	—	53	96	no	—
204	C₄⋊C₄	—	53	96	no	—
205	C₄⋊C₄	—	53	48	no	—
206	C₄⋊C₄	C₂³×D₅₃	1	48	yes	—
207	C₂²⋊C₄	—	53	48	no	—
208	C₂²⋊C₄	C₂³×D₅₃	1	48	yes	—
209	C₄²	—	53	48	no	—
210	C₄²	—	53	48	no	—
211	C₂³	—	2	12	no	—
212	C₂³	—	4	16	no	—
213	C₂³	—	4	8	no	—
214	C₂³	C₂×Q₈×D₅₃	1	12	yes	—
215	C₂³	C₂×Q₈×D₅₃	1	16	yes	—
216	C₂³	C₂⁴×D₅₃	1	6	yes	—
217	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
218	Q₈	—	4	16	no	—
219	Q₈	—	212	48	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	53	48	no	—
223	C₂×C₄	—	53	48	no	—
224	C₂×C₄	—	53	4	no	—
225	C₂×C₄	—	53	12	no	—
226	C₂×C₄	—	53	2	no	—
227	C₂×C₄	—	53	6	no	—
228	C₂×C₄	—	53	24	no	—
229	C₂×C₄	—	106	24	no	—
230	C₂×C₄	—	212	24	no	—
231	C₂×C₄	C₂⁴×D₅₃	1	24	yes	—
232	C₂²	—	2	24	no	—
233	C₂²	—	2	12	no	—
234	C₂²	—	4	16	no	—
235	C₂²	C₂²×Q₈×D₅₃	1	12	yes	—
236	C₂²	C₂²×Q₈×D₅₃	1	2	yes	—
237	C₂²	#16811458	1	4	yes	—
238	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
239	C₄	—	2	24	no	—
240	C₄	—	53	4	no	—
241	C₄	—	53	12	no	—
242	C₄	—	106	24	no	—
243	C₂	—	2	12	no	—
244	C₂	#20288117	1	4	yes	direct product factor
245	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
246	C₂	C₂×#16811458	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#543	C₂
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂×#543	D₅₃
hypocenter	C₅₃	C₂×#3458
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. #20288117	C₂	direct (N × H)
2. C₂×#16811252	C₂	semidirect
4. C₂×#16811175	C₂	semidirect
8. #16811252	C₂²	semidirect
10. #16811175	C₂²	semidirect
11. C₂×#13133216	C₂²	semidirect
12. C₂×#13133215	C₂²	semidirect
13. C₂×#13133207	C₂²	semidirect
16. C₂×Q₈×Dic₅₃	C₂²	semidirect
17. C₂×#13133182	C₂²	semidirect
28. C₂³×Dic₅₃	Q₈	semidirect
33. #13133216	C₂³	semidirect
34. #13133215	C₂³	semidirect
35. #13133207	C₂³	semidirect
38. Q₈×Dic₅₃	C₂³	semidirect
39. #13133182	C₂³	semidirect
70. C₂²×Dic₅₃	C₂×Q₈	semidirect
127. C₂×C₅₃	#3458	semidirect
150. C₅₃	C₂×#3458	semidirect
244. C₂	#20288117	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288117	2
C₂×#16811252	2
C₂×#16811249	2
C₂×#16811175	2
C₂²×#13133173	2
C₂×C₅₃×#543	2
C₂×#3458	53

Subgroups — C2×#20288117 #28290933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288117 #28290933