Subgroups — C₂³×#13133248 #28297890

Order 13568.

11166 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133248	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11719925	C₂	1	2	yes	maximal, complemented
4	C₂³×#11719924	C₂	1	2	yes	maximal, complemented
5	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719945	—	2	14	no	—
8	C₂²×#11719945	C₂²	1	28	yes	—
9	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
11	C₂³×C₂²⋊D₅₃	—	2	1	no	—
12	C₂³×C₄×D₅₃	—	2	1	no	—
13	C₂×#13133248	C₂²	1	448	yes	direct product factor
14	C₂²×#11719925	C₂²	1	56	yes	complemented
15	C₂²×#11719924	C₂²	1	56	yes	complemented
16	C₂²×#11719917	C₂²	1	28	yes	complemented
17	C₂³×#88322	C₂²	1	2	yes	—
18	C₂³×Q₈×C₅₃	—	2	1	no	—
19	C₂³×Q₈×C₅₃	—	2	14	no	—
20	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₂³×Q₈×C₅₃	C₂²	1	28	yes	complemented
22	C₂⁴×C₄×C₅₃	—	2	1	no	—
23	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂⁴×D₅₃	—	2	1	no	—
25	C₂×#11719945	—	2	168	no	—
26	C₂×#11719945	C₂³	1	112	yes	—
27	C₂²×Dic₁₀₆	—	2	7	no	—
28	C₂²×Dic₁₀₆	C₂³	1	14	yes	—
29	C₂²×D₂₁₂	—	2	7	no	—
30	C₂²×D₂₁₂	C₂³	1	14	yes	—
31	C₂²×C₂²⋊D₅₃	—	2	28	no	—
32	C₂³×Dic₅₃	—	2	1	no	—
33	C₂²×C₄×D₅₃	—	2	28	no	—
34	#13133248	C₂³	1	512	yes	direct product factor
35	C₂×#11719925	C₂³	1	224	yes	complemented
36	C₂×#11719924	C₂³	1	224	yes	complemented
37	C₂×#11719917	C₂³	1	112	yes	complemented
38	C₂²×#88322	C₂³	1	28	yes	—
39	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×Q₈×C₅₃	—	2	14	no	—
41	C₂²×Q₈×C₅₃	—	2	28	no	—
42	C₂²×Q₈×C₅₃	—	2	168	no	—
43	C₂²×Q₈×C₅₃	C₂³	1	28	yes	—
44	C₂²×Q₈×C₅₃	C₂³	1	112	yes	complemented
45	C₂³×C₄×C₅₃	—	2	2	no	—
46	C₂³×C₄×C₅₃	—	2	28	no	—
47	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	D₄	1	14	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
51	C₂⁴×C₅₃	—	2	8	no	—
52	C₂⁴×C₅₃	C₂×D₄	1	14	yes	—
53	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂³×D₅₃	—	2	14	no	—
55	C₂³×D₅₃	—	4	8	no	—
56	C₂×Q₈×C₅₃	—	2	84	no	—
57	C₂×Q₈×C₅₃	—	2	112	no	—
58	C₂×Q₈×C₅₃	—	2	224	no	—
59	C₂×Q₈×C₅₃	C₂⁴	1	56	yes	—
60	C₂×Q₈×C₅₃	C₂⁴	1	64	yes	complemented
61	C₂²×C₄×C₅₃	—	2	28	no	—
62	C₂²×C₄×C₅₃	—	2	112	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	14	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	84	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
68	C₂×C₂²⋊D₅₃	—	2	112	no	—
69	C₂²×Dic₅₃	—	2	14	no	—
70	#11719945	—	2	224	no	—
71	#11719945	C₂⁴	1	64	yes	—
72	C₂×D₂₁₂	—	2	42	no	—
73	C₂×D₂₁₂	C₂⁴	1	28	yes	—
74	C₂×C₄×D₅₃	—	2	112	no	—
75	C₂×Dic₁₀₆	—	2	42	no	—
76	C₂×Dic₁₀₆	C₂⁴	1	28	yes	—
77	#11719925	C₂⁴	1	128	yes	complemented
78	#11719924	C₂⁴	1	128	yes	complemented
79	#11719917	C₂⁴	1	64	yes	complemented
80	C₂×#88322	C₂⁴	1	56	yes	—
81	C₂³×C₅₃	—	2	56	no	—
82	C₂³×C₅₃	#137	1	8	yes	complemented
83	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
84	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
85	C₂²×D₅₃	—	2	28	no	—
86	C₂²×D₅₃	—	4	56	no	—
87	Q₈×C₅₃	—	2	56	no	—
88	Q₈×C₅₃	—	2	64	no	—
89	Q₈×C₅₃	C₂⁵	1	16	yes	—
90	C₂×C₄×C₅₃	—	2	56	no	—
91	C₂×C₄×C₅₃	—	2	64	no	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	21	yes	—
93	C₂×C₄×C₅₃	C₂²×D₄	1	28	yes	—
94	C₂×C₄×C₅₃	C₂²×D₄	1	56	yes	—
95	C₂×C₄×C₅₃	C₂⁵	1	7	yes	—
96	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
97	C₂²⋊D₅₃	—	2	64	no	—
98	C₂×Dic₅₃	—	2	28	no	—
99	D₂₁₂	—	2	28	no	—
100	D₂₁₂	C₂⁵	1	8	yes	—
101	C₄×D₅₃	—	2	64	no	—
102	Dic₁₀₆	—	2	28	no	—
103	Dic₁₀₆	C₂⁵	1	8	yes	—
104	#88322	C₂⁵	1	16	yes	—
105	C₂³×#137	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	56	no	—
107	C₂²×C₅₃	C₂×#137	1	28	yes	complemented
108	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
109	C₂²×C₅₃	C₂³×D₄	1	7	yes	—
110	C₂×D₅₃	—	2	8	no	—
111	C₂×D₅₃	—	4	56	no	—
112	C₄×C₅₃	—	2	16	no	—
113	C₄×C₅₃	C₂³×D₄	1	7	yes	—
114	C₄×C₅₃	C₂³×D₄	1	8	yes	—
115	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
116	Dic₅₃	—	2	8	no	—
117	C₂³×#41	—	53	1	no	—
118	C₂⁴×Q₈	D₅₃	1	1	yes	characteristic, complemented
119	C₂²×#137	—	53	56	no	—
120	C₂³×Q₁₆	—	53	2	no	—
121	C₂³×SD₁₆	—	53	2	no	—
122	C₂³×M₁₆	—	53	1	no	—
123	C₂×C₅₃	—	2	8	no	—
124	C₂×C₅₃	C₂²×#137	1	14	yes	complemented
125	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
126	D₅₃	—	4	8	no	—
127	C₂²×#41	—	53	28	no	—
128	C₂²×#41	—	106	14	no	—
129	C₂³×Q₈	—	2	14	no	—
130	C₂³×Q₈	—	2	1	no	—
131	C₂³×Q₈	—	53	1	no	—
132	C₂³×Q₈	C₂×D₅₃	1	2	yes	—
133	C₂³×Q₈	C₂×D₅₃	1	28	yes	complemented
134	C₂³×D₄	—	53	1	no	—
135	C₂³×D₄	—	106	1	no	—
136	C₂⁴×C₄	—	2	1	no	—
137	C₂⁴×C₄	—	106	1	no	—
138	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
139	C₂×#137	—	53	448	no	—
140	C₂²×Q₁₆	—	53	56	no	—
141	C₂²×SD₁₆	—	53	56	no	—
142	C₂²×M₁₆	—	53	28	no	—
143	C₂³×C₈	—	53	2	no	—
144	C₅₃	C₂³×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂⁵	—	106	1	no	—
146	C₂⁵	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×#41	—	53	112	no	—
148	C₂×#41	—	106	168	no	—
149	C₂²×Q₈	—	2	14	no	—
150	C₂²×Q₈	—	2	28	no	—
151	C₂²×Q₈	—	2	168	no	—
152	C₂²×Q₈	—	53	14	no	—
153	C₂²×Q₈	—	106	7	no	—
154	C₂²×Q₈	C₂²×D₅₃	1	28	yes	—
155	C₂²×Q₈	C₂²×D₅₃	1	112	yes	complemented
156	C₂²×D₄	—	53	14	no	—
157	C₂²×D₄	—	106	28	no	—
158	C₂²×D₄	—	106	7	no	—
159	C₂³×C₄	—	2	28	no	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	—	106	1	no	—
162	C₂³×C₄	—	106	28	no	—
163	C₂³×C₄	C₂²⋊D₅₃	1	14	yes	—
164	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
166	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
167	#137	—	53	512	no	—
168	C₂×Q₁₆	—	53	224	no	—
169	C₂×SD₁₆	—	53	224	no	—
170	C₂×M₁₆	—	53	112	no	—
171	C₂²×C₈	—	53	28	no	—
172	C₂⁴	—	2	8	no	—
173	C₂⁴	—	106	14	no	—
174	C₂⁴	—	212	8	no	—
175	C₂⁴	C₂×C₂²⋊D₅₃	1	14	yes	—
176	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	#41	—	53	64	no	—
178	#41	—	106	224	no	—
179	C₂×Q₈	—	2	84	no	—
180	C₂×Q₈	—	2	112	no	—
181	C₂×Q₈	—	2	224	no	—
182	C₂×Q₈	—	53	28	no	—
183	C₂×Q₈	—	106	42	no	—
184	C₂×Q₈	C₂³×D₅₃	1	56	yes	—
185	C₂×Q₈	C₂³×D₅₃	1	64	yes	complemented
186	C₂×D₄	—	53	28	no	—
187	C₂×D₄	—	106	112	no	—
188	C₂×D₄	—	106	42	no	—
189	C₂²×C₄	—	2	28	no	—
190	C₂²×C₄	—	2	112	no	—
191	C₂²×C₄	—	106	14	no	—
192	C₂²×C₄	—	106	112	no	—
193	C₂²×C₄	C₂×C₂²⋊D₅₃	1	14	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	7	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₅₃	1	84	yes	—
196	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
197	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
198	Q₁₆	—	53	128	no	—
199	SD₁₆	—	53	128	no	—
200	M₁₆	—	53	64	no	—
201	C₂×C₈	—	53	56	no	—
202	C₂³	—	2	56	no	—
203	C₂³	—	106	28	no	—
204	C₂³	—	212	56	no	—
205	C₂³	#13133248	1	8	yes	direct product factor
206	C₂³	C₂²×C₂²⋊D₅₃	1	28	yes	—
207	C₂³	C₂²×C₂²⋊D₅₃	1	7	yes	—
208	Q₈	—	2	56	no	—
209	Q₈	—	2	64	no	—
210	Q₈	—	53	8	no	—
211	Q₈	—	106	28	no	—
212	Q₈	C₂⁴×D₅₃	1	16	yes	—
213	D₄	—	53	8	no	—
214	D₄	—	106	64	no	—
215	D₄	—	106	28	no	—
216	C₂×C₄	—	2	56	no	—
217	C₂×C₄	—	2	64	no	—
218	C₂×C₄	—	106	28	no	—
219	C₂×C₄	—	106	64	no	—
220	C₂×C₄	C₂²×C₂²⋊D₅₃	1	28	yes	—
221	C₂×C₄	C₂²×C₂²⋊D₅₃	1	21	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₅₃	1	56	yes	—
223	C₂×C₄	C₂⁴×D₅₃	1	7	yes	—
224	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
225	C₈	—	53	16	no	—
226	C₂²	—	2	56	no	—
227	C₂²	—	106	8	no	—
228	C₂²	—	212	56	no	—
229	C₂²	C₂×#13133248	1	28	yes	direct product factor
230	C₂²	C₂³×C₂²⋊D₅₃	1	8	yes	—
231	C₂²	C₂³×C₂²⋊D₅₃	1	7	yes	—
232	C₄	—	2	16	no	—
233	C₄	—	106	8	no	—
234	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
235	C₄	C₂³×C₂²⋊D₅₃	1	7	yes	—
236	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
237	C₂	—	2	8	no	—
238	C₂	—	212	8	no	—
239	C₂	C₂²×#13133248	1	14	yes	direct product factor
240	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂⁴×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×D₄
prime radical	C₂³×#11719945	C₂
hypercenter	C₂⁴×Q₈	D₅₃
hypocenter	C₅₃	C₂³×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133248	C₂	direct (N × H)
3. C₂³×#11719925	C₂	semidirect
4. C₂³×#11719924	C₂	semidirect
5. C₂³×#11719917	C₂	semidirect
6. C₂⁴×Q₈×C₅₃	C₂	semidirect
13. C₂×#13133248	C₂²	direct (N × H)
14. C₂²×#11719925	C₂²	semidirect
15. C₂²×#11719924	C₂²	semidirect
16. C₂²×#11719917	C₂²	semidirect
21. C₂³×Q₈×C₅₃	C₂²	semidirect
34. #13133248	C₂³	direct (N × H)
35. C₂×#11719925	C₂³	semidirect
36. C₂×#11719924	C₂³	semidirect
37. C₂×#11719917	C₂³	semidirect
44. C₂²×Q₈×C₅₃	C₂³	semidirect
60. C₂×Q₈×C₅₃	C₂⁴	semidirect
77. #11719925	C₂⁴	semidirect
78. #11719924	C₂⁴	semidirect
79. #11719917	C₂⁴	semidirect
82. C₂³×C₅₃	#137	semidirect
107. C₂²×C₅₃	C₂×#137	semidirect
118. C₂⁴×Q₈	D₅₃	semidirect
124. C₂×C₅₃	C₂²×#137	semidirect
133. C₂³×Q₈	C₂×D₅₃	semidirect
144. C₅₃	C₂³×#137	semidirect
155. C₂²×Q₈	C₂²×D₅₃	semidirect
185. C₂×Q₈	C₂³×D₅₃	semidirect
205. C₂³	#13133248	direct (N × H)
229. C₂²	C₂×#13133248	direct (N × H)
239. C₂	C₂²×#13133248	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×#11719945	2
C₂²×#13133248	2
C₂³×#11719925	2
C₂³×#11719924	2
C₂³×#11719917	2
C₂⁴×Q₈×C₅₃	2
C₂³×#137	53

Subgroups — C23×#13133248 #28297890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133248 #28297890