Subgroups — C₂³×#13133195 #28290734

Order 13568.

14078 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133195	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic
7	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₀₆	—	2	16	no	—
11	C₂³×Dic₁₀₆	C₂²	1	28	yes	complemented
12	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₃	—	2	16	no	—
16	C₂³×C₄×D₅₃	C₂²	1	28	yes	—
17	C₂×#13133195	C₂²	1	448	yes	direct product factor
18	C₂²×#11719919	C₂²	1	28	yes	complemented
19	C₂²×#11719919	C₂²	1	28	yes	complemented
20	C₂²×#11719921	C₂²	1	28	yes	—
21	C₂²×#11719921	C₂²	1	28	yes	—
22	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	complemented
26	C₂⁴×D₅₃	—	2	8	no	—
27	C₂⁴×D₅₃	—	2	8	no	—
28	C₂⁴×D₅₃	Q₈	1	16	yes	complemented
29	C₂⁴×D₅₃	C₂³	1	14	yes	—
30	C₂²×Dic₁₀₆	—	2	224	no	—
31	C₂²×Dic₁₀₆	C₂³	1	112	yes	complemented
32	C₂³×Dic₅₃	—	2	8	no	—
33	C₂³×Dic₅₃	—	2	8	no	—
34	C₂³×Dic₅₃	D₄	1	16	yes	—
35	C₂³×Dic₅₃	C₂³	1	14	yes	—
36	C₂³×Dic₅₃	C₂³	1	14	yes	—
37	C₂³×Dic₅₃	C₂³	1	14	yes	—
38	C₂²×C₄×D₅₃	—	2	224	no	—
39	C₂²×C₄×D₅₃	C₂³	1	112	yes	—
40	#13133195	C₂³	1	512	yes	direct product factor
41	C₂×#11719919	C₂³	1	112	yes	complemented
42	C₂×#11719919	C₂³	1	112	yes	complemented
43	C₂×#11719921	C₂³	1	112	yes	—
44	C₂×#11719921	C₂³	1	112	yes	—
45	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	—	2	8	no	—
47	C₂³×C₄×C₅₃	—	2	8	no	—
48	C₂³×C₄×C₅₃	—	2	8	no	—
49	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	complemented
53	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
54	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
55	C₂⁴×C₅₃	#41	1	8	yes	—
56	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
57	C₂³×D₅₃	—	2	128	no	—
58	C₂³×D₅₃	—	2	56	no	—
59	C₂³×D₅₃	—	2	56	no	—
60	C₂³×D₅₃	C₂×Q₈	1	112	yes	complemented
61	C₂³×D₅₃	C₂⁴	1	28	yes	—
62	C₂²×C₄×C₅₃	—	2	56	no	—
63	C₂²×C₄×C₅₃	—	2	56	no	—
64	C₂²×C₄×C₅₃	—	2	56	no	—
65	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
68	C₂²×Dic₅₃	—	2	56	no	—
69	C₂²×Dic₅₃	—	2	56	no	—
70	C₂²×Dic₅₃	C₂×D₄	1	112	yes	—
71	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
72	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
73	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
74	C₂×C₄×D₅₃	—	2	448	no	—
75	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
76	C₂×Dic₁₀₆	—	2	448	no	—
77	C₂×Dic₁₀₆	C₂⁴	1	64	yes	complemented
78	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
79	#11719921	C₂⁴	1	64	yes	—
80	#11719921	C₂⁴	1	64	yes	—
81	#11719919	C₂⁴	1	64	yes	complemented
82	#11719919	C₂⁴	1	64	yes	complemented
83	C₂³×C₅₃	C₂²⋊Q₈	1	64	yes	complemented
84	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
85	C₂³×C₅₃	C₂²×Q₈	1	28	yes	—
86	C₂³×C₅₃	C₂×#41	1	28	yes	—
87	C₂³×C₅₃	C₂⁵	1	7	yes	—
88	C₂²×D₅₃	—	2	448	no	—
89	C₂²×D₅₃	—	2	56	no	—
90	C₂²×D₅₃	—	2	56	no	—
91	C₂²×D₅₃	C₂²×Q₈	1	112	yes	complemented
92	C₂²×D₅₃	C₂⁵	1	8	yes	—
93	C₂×C₄×C₅₃	—	2	56	no	—
94	C₂×C₄×C₅₃	—	2	56	no	—
95	C₂×C₄×C₅₃	—	2	56	no	—
96	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
97	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
98	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
99	C₂×Dic₅₃	—	2	56	no	—
100	C₂×Dic₅₃	—	2	56	no	—
101	C₂×Dic₅₃	C₂²×D₄	1	112	yes	—
102	C₂×Dic₅₃	C₂⁵	1	8	yes	—
103	C₂×Dic₅₃	C₂⁵	1	8	yes	—
104	C₂×Dic₅₃	C₂⁵	1	8	yes	—
105	C₄×D₅₃	—	2	128	no	—
106	Dic₁₀₆	—	2	128	no	—
107	C₂³×C₂²⋊Q₈	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	112	yes	complemented
109	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
110	C₂²×C₅₃	C₂³×Q₈	1	14	yes	—
111	C₂²×C₅₃	C₂²×#41	1	14	yes	—
112	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
113	C₂×D₅₃	—	2	8	no	—
114	C₂×D₅₃	—	2	8	no	—
115	C₂×D₅₃	—	2	224	no	—
116	C₂×D₅₃	C₂³×Q₈	1	16	yes	complemented
117	C₄×C₅₃	—	2	8	no	—
118	C₄×C₅₃	—	2	8	no	—
119	C₄×C₅₃	—	2	8	no	—
120	Dic₅₃	—	2	8	no	—
121	Dic₅₃	—	2	8	no	—
122	Dic₅₃	C₂³×D₄	1	16	yes	—
123	C₂⁴×Q₈	—	53	1	no	—
124	C₂⁵×C₄	—	53	1	no	—
125	C₂²×C₂²⋊Q₈	—	53	56	no	—
126	C₂³×C₄⋊C₄	—	53	1	no	—
127	C₂³×C₄⋊C₄	—	53	1	no	—
128	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
129	C₂³×C₂²⋊C₄	—	53	1	no	—
130	C₂³×C₂²⋊C₄	—	53	1	no	—
131	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	28	yes	complemented
132	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
133	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
134	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
135	D₅₃	—	2	16	no	—
136	C₂⁶	—	53	1	no	—
137	C₂³×Q₈	—	53	28	no	—
138	C₂³×Q₈	—	106	16	no	—
139	C₂⁴×C₄	—	53	1	no	—
140	C₂⁴×C₄	—	53	1	no	—
141	C₂⁴×C₄	—	53	1	no	—
142	C₂⁴×C₄	—	53	28	no	—
143	C₂⁴×C₄	—	106	16	no	—
144	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
145	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
146	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
147	C₂×C₂²⋊Q₈	—	53	448	no	—
148	C₂²×C₄⋊C₄	—	53	28	no	—
149	C₂²×C₄⋊C₄	—	53	28	no	—
150	C₂²×C₄⋊C₄	C₂×D₅₃	1	28	yes	complemented
151	C₂²×C₂²⋊C₄	—	53	28	no	—
152	C₂²×C₂²⋊C₄	—	53	28	no	—
153	C₅₃	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
154	C₂⁵	—	53	16	no	—
155	C₂⁵	—	53	14	no	—
156	C₂⁵	—	106	8	no	—
157	C₂⁵	—	106	8	no	—
158	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
159	C₂²×Q₈	—	53	112	no	—
160	C₂²×Q₈	—	106	224	no	—
161	C₂³×C₄	—	2	8	no	—
162	C₂³×C₄	—	2	8	no	—
163	C₂³×C₄	—	2	8	no	—
164	C₂³×C₄	—	53	16	no	—
165	C₂³×C₄	—	53	14	no	—
166	C₂³×C₄	—	53	14	no	—
167	C₂³×C₄	—	53	14	no	—
168	C₂³×C₄	—	53	112	no	—
169	C₂³×C₄	—	106	224	no	—
170	C₂³×C₄	—	106	8	no	—
171	C₂³×C₄	—	106	8	no	—
172	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
173	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
174	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
175	C₂²⋊Q₈	—	53	512	no	—
176	C₂×C₄⋊C₄	—	53	112	no	—
177	C₂×C₄⋊C₄	—	53	112	no	—
178	C₂×C₄⋊C₄	C₂²×D₅₃	1	112	yes	complemented
179	C₂×C₂²⋊C₄	—	53	112	no	—
180	C₂×C₂²⋊C₄	—	53	112	no	—
181	C₂⁴	—	53	112	no	—
182	C₂⁴	—	53	28	no	—
183	C₂⁴	—	106	128	no	—
184	C₂⁴	—	106	56	no	—
185	C₂⁴	—	106	56	no	—
186	C₂⁴	#11719945	1	8	yes	—
187	C₂⁴	D₄×D₅₃	1	8	yes	—
188	C₂⁴	Q₈×D₅₃	1	8	yes	—
189	C₂⁴	C₂³×D₅₃	1	7	yes	—
190	C₂×Q₈	—	53	64	no	—
191	C₂×Q₈	—	106	448	no	—
192	C₂²×C₄	—	2	56	no	—
193	C₂²×C₄	—	2	56	no	—
194	C₂²×C₄	—	2	56	no	—
195	C₂²×C₄	—	53	112	no	—
196	C₂²×C₄	—	53	28	no	—
197	C₂²×C₄	—	53	28	no	—
198	C₂²×C₄	—	53	28	no	—
199	C₂²×C₄	—	53	64	no	—
200	C₂²×C₄	—	106	448	no	—
201	C₂²×C₄	—	106	56	no	—
202	C₂²×C₄	—	106	56	no	—
203	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
204	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
205	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
206	C₄⋊C₄	—	53	64	no	—
207	C₄⋊C₄	—	53	64	no	—
208	C₄⋊C₄	C₂³×D₅₃	1	64	yes	complemented
209	C₂²⋊C₄	—	53	64	no	—
210	C₂²⋊C₄	—	53	64	no	—
211	C₂³	—	53	112	no	—
212	C₂³	—	53	8	no	—
213	C₂³	—	106	448	no	—
214	C₂³	—	106	56	no	—
215	C₂³	—	106	56	no	—
216	C₂³	#13133195	1	64	yes	direct product factor
217	C₂³	C₂×D₄×D₅₃	1	28	yes	—
218	C₂³	C₂×Q₈×D₅₃	1	28	yes	—
219	C₂³	C₂×#11719945	1	28	yes	—
220	C₂³	C₂⁴×D₅₃	1	7	yes	—
221	Q₈	—	106	128	no	—
222	C₂×C₄	—	2	56	no	—
223	C₂×C₄	—	2	56	no	—
224	C₂×C₄	—	2	56	no	—
225	C₂×C₄	—	53	112	no	—
226	C₂×C₄	—	53	8	no	—
227	C₂×C₄	—	53	8	no	—
228	C₂×C₄	—	53	8	no	—
229	C₂×C₄	—	106	128	no	—
230	C₂×C₄	—	106	56	no	—
231	C₂×C₄	—	106	56	no	—
232	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
233	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
234	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
235	C₂²	—	53	16	no	—
236	C₂²	—	106	224	no	—
237	C₂²	—	106	8	no	—
238	C₂²	—	106	8	no	—
239	C₂²	C₂×#13133195	1	112	yes	direct product factor
240	C₂²	C₂²×D₄×D₅₃	1	14	yes	—
241	C₂²	C₂²×Q₈×D₅₃	1	14	yes	—
242	C₂²	C₂²×#11719945	1	14	yes	—
243	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	2	8	no	—
247	C₄	—	53	16	no	—
248	C₄	—	106	8	no	—
249	C₄	—	106	8	no	—
250	C₂	—	106	16	no	—
251	C₂	C₂²×#13133195	1	28	yes	direct product factor
252	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
253	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
254	C₂	C₂³×#11719945	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₀₆	C₂	semidirect
3. C₂²×#13133195	C₂	direct (N × H)
4. C₂³×#11719919	C₂	semidirect
5. C₂³×#11719919	C₂	semidirect
8. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
11. C₂³×Dic₁₀₆	C₂²	semidirect
17. C₂×#13133195	C₂²	direct (N × H)
18. C₂²×#11719919	C₂²	semidirect
19. C₂²×#11719919	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
28. C₂⁴×D₅₃	Q₈	semidirect
31. C₂²×Dic₁₀₆	C₂³	semidirect
40. #13133195	C₂³	direct (N × H)
41. C₂×#11719919	C₂³	semidirect
42. C₂×#11719919	C₂³	semidirect
52. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
60. C₂³×D₅₃	C₂×Q₈	semidirect
77. C₂×Dic₁₀₆	C₂⁴	semidirect
78. C₄⋊C₄×C₅₃	C₂⁴	semidirect
81. #11719919	C₂⁴	semidirect
82. #11719919	C₂⁴	semidirect
83. C₂³×C₅₃	C₂²⋊Q₈	semidirect
91. C₂²×D₅₃	C₂²×Q₈	semidirect
108. C₂²×C₅₃	C₂×C₂²⋊Q₈	semidirect
116. C₂×D₅₃	C₂³×Q₈	semidirect
128. C₂³×C₄⋊C₄	D₅₃	semidirect
131. C₂×C₅₃	C₂²×C₂²⋊Q₈	semidirect
150. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
153. C₅₃	C₂³×C₂²⋊Q₈	semidirect
178. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
208. C₄⋊C₄	C₂³×D₅₃	semidirect
216. C₂³	#13133195	direct (N × H)
239. C₂²	C₂×#13133195	direct (N × H)
251. C₂	C₂²×#13133195	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₀₆	2
C₂⁴×C₄×D₅₃	2
C₂²×#13133195	2
C₂³×#11719919	2
C₂³×#11719921	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×C₂²⋊Q₈	53

Subgroups — C23×#13133195 #28290734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133195 #28290734