Subgroups — C₂²×#16810646 #28247703

Order 13568.

3438 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810646	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133146	C₂	1	4	yes	maximal, complemented
5	C₂³×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₅₃	C₄	1	2	yes	—
7	C₂²×#11719917	—	2	8	no	—
8	C₂²×#11719917	C₂²	1	12	yes	—
9	C₂³×#88322	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₅₃	C₄	1	12	yes	—
11	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
14	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
15	#16810646	C₂²	1	64	yes	direct product factor
16	C₂×#13133146	C₂²	1	48	yes	complemented
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₅₃	—	2	8	no	—
19	C₂²×M₁₆×C₅₃	C₂²	1	12	yes	—
20	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂³×Dic₅₃	—	2	16	no	—
22	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂³×Dic₅₃	C₂×C₄	1	24	yes	—
24	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×#11719917	—	2	48	no	—
26	C₂×#11719917	C₂³	1	16	yes	—
27	C₂²×#88322	—	2	8	no	—
28	C₂²×#88322	C₂³	1	12	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
35	#13133146	C₂³	1	64	yes	complemented
36	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
38	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	D₄	1	8	yes	—
40	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₅₃	—	2	48	no	—
44	C₂×M₁₆×C₅₃	C₂³	1	16	yes	—
45	C₂²×C₈×C₅₃	—	2	8	no	—
46	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
47	C₂⁴×C₅₃	—	2	8	no	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
49	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
50	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	48	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
58	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
59	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
62	C₂²×Dic₅₃	—	2	8	no	—
63	C₂²×Dic₅₃	—	2	96	no	—
64	C₂²×Dic₅₃	M₁₆	1	16	yes	complemented
65	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
66	C₂²×Dic₅₃	C₂²×C₄	1	32	yes	—
67	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
68	M₁₆×C₅₃	—	2	32	no	—
69	C₂×C₈×C₅₃	—	2	24	no	—
70	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
71	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
72	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
73	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
74	C₄×Dic₅₃	C₂⁴	1	8	yes	—
75	#11719917	—	2	32	no	—
76	C₂×#88322	—	2	24	no	—
77	C₂×#88322	C₂⁴	1	8	yes	—
78	C₂³×C₅₃	—	2	24	no	—
79	C₂³×C₅₃	—	2	32	no	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
82	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
83	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
84	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
90	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
92	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
93	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
94	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
95	C₂×Dic₅₃	—	2	24	no	—
96	C₂×Dic₅₃	—	2	64	no	—
97	C₂×Dic₅₃	C₂×M₁₆	1	48	yes	complemented
98	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
99	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
100	C₈×C₅₃	—	2	8	no	—
101	#88322	—	2	8	no	—
102	C₂²×#423	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	48	no	—
104	C₂²×C₅₃	—	2	8	no	—
105	C₂²×C₅₃	#423	1	16	yes	complemented
106	C₂²×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	6	yes	—
108	C₂²×C₅₃	C₂²×M₁₆	1	12	yes	—
109	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
110	C₄×C₅₃	C₂²×C₄⋊C₄	1	6	yes	—
111	C₄×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
112	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
113	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
114	Dic₅₃	—	2	8	no	—
115	Dic₅₃	C₂²×M₁₆	1	16	yes	complemented
116	C₂³×M₁₆	—	53	1	no	—
117	C₂³×M₁₆	D₅₃	1	1	yes	characteristic
118	C₂³×C₄²	—	53	1	no	—
119	C₂×#423	—	53	24	no	—
120	C₂²×C₄⋊C₈	—	53	4	no	—
121	C₂×C₅₃	—	2	8	no	—
122	C₂×C₅₃	C₂×#423	1	12	yes	complemented
123	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂³×M₁₆	1	2	yes	—
125	C₂⁴×C₄	—	53	2	no	—
126	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
127	C₂²×M₁₆	—	2	8	no	—
128	C₂²×M₁₆	—	53	12	no	—
129	C₂²×M₁₆	—	106	8	no	—
130	C₂²×M₁₆	C₂×D₅₃	1	12	yes	—
131	C₂³×C₈	—	53	2	no	—
132	C₂³×C₈	C₂×D₅₃	1	2	yes	—
133	C₂²×C₄²	—	53	1	no	—
134	C₂²×C₄²	—	53	12	no	—
135	C₂²×C₄²	—	53	1	no	—
136	C₂²×C₄²	—	53	12	no	—
137	C₂²×C₄²	—	53	2	no	—
138	#423	—	53	64	no	—
139	C₂×C₄⋊C₈	—	53	48	no	—
140	C₅₃	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
142	C₂³×C₄	—	53	2	no	—
143	C₂³×C₄	—	53	24	no	—
144	C₂³×C₄	—	53	2	no	—
145	C₂³×C₄	—	106	16	no	—
146	C₂³×C₄	Dic₁₀₆	1	8	yes	—
147	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
148	C₂³×C₄	C₄×D₅₃	1	6	yes	—
149	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
151	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	48	no	—
153	C₂×M₁₆	—	53	16	no	—
154	C₂×M₁₆	—	106	48	no	—
155	C₂×M₁₆	C₂²×D₅₃	1	16	yes	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	53	12	no	—
158	C₂²×C₈	—	106	8	no	—
159	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
160	C₂×C₄²	—	53	12	no	—
161	C₂×C₄²	—	53	12	no	—
162	C₂×C₄²	—	53	48	no	—
163	C₂×C₄²	—	53	12	no	—
164	C₂×C₄²	—	53	12	no	—
165	C₂×C₄²	—	53	16	no	—
166	C₄⋊C₈	—	53	64	no	—
167	C₂⁴	—	2	8	no	—
168	C₂⁴	#11719919	1	8	yes	—
169	C₂⁴	C₂×C₄×D₅₃	1	6	yes	—
170	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	—	53	16	no	—
172	C₂²×C₄	—	53	12	no	—
173	C₂²×C₄	—	53	32	no	—
174	C₂²×C₄	—	53	12	no	—
175	C₂²×C₄	—	106	96	no	—
176	C₂²×C₄	—	106	8	no	—
177	C₂²×C₄	#11719919	1	48	yes	—
178	C₂²×C₄	#11719919	1	8	yes	—
179	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
180	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
181	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
182	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
183	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
185	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
186	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
187	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
188	M₁₆	—	2	32	no	—
189	M₁₆	—	106	32	no	—
190	C₂×C₈	—	2	24	no	—
191	C₂×C₈	—	53	8	no	—
192	C₂×C₈	—	106	24	no	—
193	C₂×C₈	C₂³×D₅₃	1	8	yes	—
194	C₄²	—	53	16	no	—
195	C₄²	—	53	24	no	—
196	C₄²	—	53	16	no	—
197	C₄²	—	53	8	no	—
198	C₂³	—	2	24	no	—
199	C₂³	—	2	32	no	—
200	C₂³	C₂×#11719919	1	24	yes	—
201	C₂³	C₂×#11719919	1	4	yes	—
202	C₂³	M₁₆×D₅₃	1	8	yes	—
203	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
204	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
205	C₂×C₄	—	53	48	no	—
206	C₂×C₄	—	53	8	no	—
207	C₂×C₄	—	53	8	no	—
208	C₂×C₄	—	106	64	no	—
209	C₂×C₄	—	106	24	no	—
210	C₂×C₄	C₂×#11719919	1	48	yes	—
211	C₂×C₄	C₂×#11719919	1	24	yes	—
212	C₂×C₄	C₂×#11719919	1	12	yes	—
213	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
214	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
216	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
217	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
218	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
219	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
220	C₈	—	2	8	no	—
221	C₈	—	106	8	no	—
222	C₂²	—	2	48	no	—
223	C₂²	—	2	8	no	—
224	C₂²	#16810646	1	16	yes	direct product factor
225	C₂²	C₂²×#11719919	1	8	yes	—
226	C₂²	C₂²×#11719919	1	6	yes	—
227	C₂²	C₂×M₁₆×D₅₃	1	12	yes	—
228	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
229	C₄	—	53	16	no	—
230	C₄	—	106	8	no	—
231	C₄	C₂²×#11719919	1	6	yes	—
232	C₄	C₂²×#11719919	1	8	yes	—
233	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
234	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
235	C₂	—	2	8	no	—
236	C₂	C₂×#16810646	1	12	yes	direct product factor
237	C₂	C₂³×#11719919	1	1	yes	characteristic
238	C₂	C₂²×M₁₆×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×M₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×M₁₆	D₅₃
hypocenter	C₅₃	C₂²×#423
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16810646	C₂	direct (N × H)
4. C₂²×#13133146	C₂	semidirect
15. #16810646	C₂²	direct (N × H)
16. C₂×#13133146	C₂²	semidirect
35. #13133146	C₂³	semidirect
64. C₂²×Dic₅₃	M₁₆	semidirect
97. C₂×Dic₅₃	C₂×M₁₆	semidirect
105. C₂²×C₅₃	#423	semidirect
115. Dic₅₃	C₂²×M₁₆	semidirect
122. C₂×C₅₃	C₂×#423	semidirect
140. C₅₃	C₂²×#423	semidirect
224. C₂²	#16810646	direct (N × H)
236. C₂	C₂×#16810646	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×#11719917	2
C₂³×C₄×Dic₅₃	2
C₂×#16810646	2
C₂²×#13133146	2
C₂³×M₁₆×C₅₃	2
C₂²×#423	53

Subgroups — C22×#16810646 #28247703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810646 #28247703