Subgroups — C₂²×D₅₃⋊M₃₂ #28330786

Order 13568.

2334 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133292	C₂	1	2	yes	maximal, complemented
2	C₂×D₅₃⋊M₃₂	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133291	C₂	1	2	yes	maximal, complemented
4	C₂²×D₅₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	C₂²	1	24	yes	complemented
7	D₅₃⋊M₃₂	C₂²	1	64	yes	direct product factor
8	C₂×#13133291	C₂²	1	24	yes	complemented
9	C₂×D₅₃⋊C₁₆	C₂²	1	24	yes	complemented
10	C₂²×#11719910	C₂²	1	4	yes	complemented
11	C₂³×C₄×D₅₃	C₄	1	1	yes	characteristic
12	C₂³×#88322	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₅₃	C₄	1	12	yes	—
14	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
15	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	12	yes	—
18	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
20	#13133292	C₂³	1	32	yes	complemented
21	#13133291	C₂³	1	32	yes	complemented
22	D₅₃⋊C₁₆	C₂³	1	32	yes	complemented
23	C₂×#11719910	C₂³	1	24	yes	complemented
24	C₂⁴×D₅₃	C₈	1	1	yes	characteristic
25	C₂³×Dic₅₃	C₈	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₈	1	12	yes	—
28	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
30	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	6	yes	—
33	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
34	C₂²×#88322	C₂³	1	1	yes	characteristic
35	C₂²×#88322	C₂³	1	6	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
37	C₂×C₈×D₅₃	C₂×C₄	1	48	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	12	yes	—
39	C₂×C₈×D₅₃	C₂×C₄	1	12	yes	—
40	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
41	C₂×C₈×D₅₃	C₂³	1	6	yes	—
42	C₂×C₈×D₅₃	C₂³	1	16	yes	—
43	C₂×C₈×D₅₃	C₂³	1	6	yes	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂³×D₅₃	—	2	8	no	—
50	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂³×D₅₃	C₂×C₈	1	12	yes	—
52	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₂×C₈	1	12	yes	—
58	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₅₃	C₂×C₈	1	12	yes	—
61	C₂×C₄×D₅₃	C₂×C₈	1	6	yes	—
62	C₂×C₄×D₅₃	C₂×C₈	1	48	yes	—
63	C₂×C₄×D₅₃	C₂×C₈	1	6	yes	—
64	C₂×C₄×D₅₃	C₂×C₈	1	12	yes	—
65	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
66	C₂×C₄×D₅₃	C₂²×C₄	1	6	yes	—
67	C₂×C₄×D₅₃	C₂²×C₄	1	6	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	16	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
71	C₂×#88322	C₂²×C₄	1	3	yes	—
72	C₂×#88322	C₂²×C₄	1	12	yes	—
73	C₂×#88322	C₂²×C₄	1	6	yes	—
74	C₂×#88322	C₂⁴	1	3	yes	—
75	C₂×#88322	C₂⁴	1	4	yes	—
76	C₈×D₅₃	C₂²×C₄	1	12	yes	—
77	C₈×D₅₃	C₂²×C₄	1	16	yes	—
78	C₈×D₅₃	C₂²×C₄	1	16	yes	—
79	C₈×D₅₃	C₂²×C₄	1	12	yes	—
80	C₈×D₅₃	C₂⁴	1	4	yes	—
81	C₈×D₅₃	C₂⁴	1	4	yes	—
82	#11719910	C₂⁴	1	16	yes	complemented
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₂²×C₈	1	6	yes	—
85	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	4	no	—
87	C₂²×D₅₃	—	2	48	no	—
88	C₂²×D₅₃	M₃₂	1	8	yes	complemented
89	C₂²×D₅₃	C₂²×C₈	1	6	yes	—
90	C₂²×D₅₃	C₂²×C₈	1	16	yes	—
91	C₂²×D₅₃	C₂²×C₈	1	6	yes	—
92	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
97	C₂×Dic₅₃	C₂²×C₈	1	16	yes	—
98	C₂×Dic₅₃	C₂²×C₈	1	6	yes	—
99	C₂×Dic₅₃	C₂²×C₈	1	6	yes	—
100	C₄×D₅₃	C₂²×C₈	1	16	yes	—
101	C₄×D₅₃	C₂²×C₈	1	12	yes	—
102	C₄×D₅₃	C₂²×C₈	1	12	yes	—
103	C₄×D₅₃	C₂²×C₈	1	16	yes	—
104	C₄×D₅₃	C₂³×C₄	1	4	yes	—
105	C₄×D₅₃	C₂³×C₄	1	4	yes	—
106	C₈×C₅₃	C₂³×C₄	1	4	yes	—
107	C₈×C₅₃	C₂³×C₄	1	4	yes	—
108	#88322	C₂³×C₄	1	3	yes	—
109	#88322	C₂³×C₄	1	4	yes	—
110	#88322	C₂⁵	1	1	yes	characteristic
111	C₂³×M₃₂	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	12	no	—
113	C₂²×C₅₃	C₂×M₃₂	1	4	yes	complemented
114	C₂²×C₅₃	C₂³×C₈	1	4	yes	—
115	C₂²×C₅₃	C₂³×C₈	1	3	yes	—
116	C₂×D₅₃	—	2	12	no	—
117	C₂×D₅₃	—	2	32	no	—
118	C₂×D₅₃	C₂×M₃₂	1	24	yes	complemented
119	C₂×D₅₃	C₂³×C₈	1	4	yes	—
120	C₂×D₅₃	C₂³×C₈	1	4	yes	—
121	C₄×C₅₃	C₂³×C₈	1	3	yes	—
122	C₄×C₅₃	C₂³×C₈	1	4	yes	—
123	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
124	Dic₅₃	C₂³×C₈	1	4	yes	—
125	Dic₅₃	C₂³×C₈	1	4	yes	—
126	C₂⁴×C₈	—	53	1	no	—
127	C₂²×M₃₂	—	53	2	no	—
128	C₂²×M₃₂	—	53	24	no	—
129	C₂²×M₃₂	—	53	2	no	—
130	C₂³×C₁₆	—	53	2	no	—
131	C₂×C₅₃	—	2	4	no	—
132	C₂×C₅₃	C₂²×M₃₂	1	6	yes	complemented
133	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
134	D₅₃	—	2	4	no	—
135	D₅₃	C₂²×M₃₂	1	8	yes	complemented
136	C₂⁴×C₄	—	53	1	no	—
137	C₂³×C₈	—	53	1	no	—
138	C₂³×C₈	—	53	1	no	—
139	C₂³×C₈	—	53	12	no	—
140	C₂³×C₈	—	53	12	no	—
141	C₂³×C₈	—	53	1	no	—
142	C₂³×C₈	—	53	1	no	—
143	C₂³×C₈	—	53	1	no	—
144	C₂³×C₈	#6157	1	1	yes	characteristic
145	C₂×M₃₂	—	53	24	no	—
146	C₂×M₃₂	—	53	64	no	—
147	C₂×M₃₂	—	53	24	no	—
148	C₂²×C₁₆	—	53	4	no	—
149	C₂²×C₁₆	—	53	24	no	—
150	C₅₃	C₂³×M₃₂	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	—	53	1	no	—
152	C₂³×C₄	—	53	1	no	—
153	C₂³×C₄	—	53	12	no	—
154	C₂³×C₄	—	53	1	no	—
155	C₂³×C₄	—	53	1	no	—
156	C₂³×C₄	—	53	1	no	—
157	C₂³×C₄	—	53	12	no	—
158	C₂³×C₄	—	53	1	no	—
159	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
160	C₂²×C₈	—	53	6	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	6	no	—
164	C₂²×C₈	—	53	6	no	—
165	C₂²×C₈	—	53	48	no	—
166	C₂²×C₈	—	53	12	no	—
167	C₂²×C₈	—	53	12	no	—
168	C₂²×C₈	—	53	6	no	—
169	C₂²×C₈	—	53	6	no	—
170	C₂²×C₈	—	53	6	no	—
171	C₂²×C₈	—	53	16	no	—
172	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
173	C₂²×C₈	C₂×#6157	1	12	yes	—
174	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
175	M₃₂	—	53	32	no	—
176	M₃₂	—	53	32	no	—
177	C₂×C₁₆	—	53	24	no	—
178	C₂×C₁₆	—	53	32	no	—
179	C₂⁴	—	53	1	no	—
180	C₂⁴	—	53	12	no	—
181	C₂⁴	—	53	1	no	—
182	C₂⁴	—	106	8	no	—
183	C₂⁴	D₅₃⋊C₈	1	1	yes	characteristic
184	C₂²×C₄	—	53	12	no	—
185	C₂²×C₄	—	53	1	no	—
186	C₂²×C₄	—	53	1	no	—
187	C₂²×C₄	—	53	12	no	—
188	C₂²×C₄	—	53	6	no	—
189	C₂²×C₄	—	53	48	no	—
190	C₂²×C₄	—	53	12	no	—
191	C₂²×C₄	—	53	6	no	—
192	C₂²×C₄	—	53	6	no	—
193	C₂²×C₄	—	53	16	no	—
194	C₂²×C₄	—	53	6	no	—
195	C₂²×C₄	D₅₃⋊C₈	1	6	yes	—
196	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²×#6157	1	6	yes	—
198	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
199	C₂×C₈	—	53	3	no	—
200	C₂×C₈	—	53	4	no	—
201	C₂×C₈	—	53	3	no	—
202	C₂×C₈	—	53	12	no	—
203	C₂×C₈	—	53	6	no	—
204	C₂×C₈	—	53	4	no	—
205	C₂×C₈	—	53	4	no	—
206	C₂×C₈	—	53	12	no	—
207	C₂×C₈	—	53	12	no	—
208	C₂×C₈	—	53	16	no	—
209	C₂×C₈	—	53	16	no	—
210	C₂×C₈	C₂²×#6157	1	16	yes	—
211	C₂×C₈	C₂²×#6157	1	6	yes	—
212	C₂×C₈	C₂²×#6157	1	6	yes	—
213	C₁₆	—	53	16	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	53	8	no	—
216	C₂³	—	53	6	no	—
217	C₂³	—	53	16	no	—
218	C₂³	—	53	6	no	—
219	C₂³	—	106	48	no	—
220	C₂³	—	106	4	no	—
221	C₂³	C₂×D₅₃⋊C₈	1	6	yes	—
222	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	—	53	16	no	—
224	C₂×C₄	—	53	6	no	—
225	C₂×C₄	—	53	6	no	—
226	C₂×C₄	—	53	16	no	—
227	C₂×C₄	—	53	12	no	—
228	C₂×C₄	—	53	12	no	—
229	C₂×C₄	—	53	16	no	—
230	C₂×C₄	—	53	4	no	—
231	C₂×C₄	—	53	4	no	—
232	C₂×C₄	C₂×D₅₃⋊C₈	1	12	yes	—
233	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
234	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
235	C₂×C₄	C₂³×#6157	1	4	yes	—
236	C₂×C₄	C₂³×#6157	1	3	yes	—
237	C₈	—	53	3	no	—
238	C₈	—	53	4	no	—
239	C₈	—	53	1	no	—
240	C₈	C₂³×#6157	1	4	yes	—
241	C₈	C₂³×#6157	1	4	yes	—
242	C₂²	—	2	12	no	—
243	C₂²	—	53	24	no	—
244	C₂²	—	53	4	no	—
245	C₂²	—	53	4	no	—
246	C₂²	—	106	32	no	—
247	C₂²	—	106	12	no	—
248	C₂²	D₅₃⋊M₃₂	1	4	yes	direct product factor
249	C₂²	C₂²×D₅₃⋊C₈	1	4	yes	—
250	C₂²	C₂²×D₅₃⋊C₈	1	3	yes	—
251	C₄	—	53	4	no	—
252	C₄	—	53	4	no	—
253	C₄	C₂²×D₅₃⋊C₈	1	3	yes	—
254	C₄	C₂²×D₅₃⋊C₈	1	4	yes	—
255	C₄	C₂⁴×#6157	1	1	yes	characteristic
256	C₂	—	2	4	no	—
257	C₂	—	53	8	no	—
258	C₂	—	106	4	no	—
259	C₂	C₂×D₅₃⋊M₃₂	1	6	yes	direct product factor
260	C₂	C₂³×D₅₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×D₅₃	C₈
cosocle	#88322	C₂⁵
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂³×M₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133292	C₂	semidirect
2. C₂×D₅₃⋊M₃₂	C₂	direct (N × H)
3. C₂²×#13133291	C₂	semidirect
4. C₂²×D₅₃⋊C₁₆	C₂	semidirect
6. C₂×#13133292	C₂²	semidirect
7. D₅₃⋊M₃₂	C₂²	direct (N × H)
8. C₂×#13133291	C₂²	semidirect
9. C₂×D₅₃⋊C₁₆	C₂²	semidirect
10. C₂²×#11719910	C₂²	semidirect
20. #13133292	C₂³	semidirect
21. #13133291	C₂³	semidirect
22. D₅₃⋊C₁₆	C₂³	semidirect
23. C₂×#11719910	C₂³	semidirect
82. #11719910	C₂⁴	semidirect
88. C₂²×D₅₃	M₃₂	semidirect
113. C₂²×C₅₃	C₂×M₃₂	semidirect
118. C₂×D₅₃	C₂×M₃₂	semidirect
132. C₂×C₅₃	C₂²×M₃₂	semidirect
135. D₅₃	C₂²×M₃₂	semidirect
150. C₅₃	C₂³×M₃₂	semidirect
248. C₂²	D₅₃⋊M₃₂	direct (N × H)
259. C₂	C₂×D₅₃⋊M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133292	2
C₂×D₅₃⋊M₃₂	2
C₂²×#13133291	2
C₂²×D₅₃⋊C₁₆	2
C₂³×C₈×D₅₃	2
C₂³×M₃₂	53

Subgroups — C22×D53⋊M32 #28330786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₅₃⋊M₃₂ #28330786