Subgroups — C₂²×#16810388 #27381509

Order 13568.

2238 conjugacy classes of subgroups.

The subgroup lattice has 236 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	2	yes	maximal
2	C₂×#16810388	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	6	no	—
5	C₂²×#11719917	—	2	8	no	—
6	C₂²×#11719917	C₄	1	16	yes	—
7	C₂²×#11719917	C₂²	1	12	yes	—
8	C₂³×#88322	—	2	2	no	—
9	#16810388	C₂²	1	16	yes	direct product factor
10	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	—	2	8	no	—
12	C₂³×C₈×C₅₃	C₄	1	6	yes	—
13	C₂³×C₈×C₅₃	C₄	1	2	yes	—
14	C₂³×C₈×C₅₃	C₂²	1	6	yes	—
15	C₂×#11719917	—	2	48	no	—
16	C₂×#11719917	—	2	12	no	—
17	C₂×#11719917	—	2	24	no	—
18	C₂×#11719917	C₂×C₄	1	48	yes	—
19	C₂×#11719917	C₂³	1	8	yes	—
20	C₂²×#88322	—	2	16	no	—
21	C₂²×#88322	—	2	12	no	—
22	C₂⁵×C₅₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	D₄	1	3	yes	—
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	8	yes	—
29	C₂³×C₄×C₅₃	Q₈	1	3	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
31	C₂²×C₈×C₅₃	—	2	24	no	—
32	C₂²×C₈×C₅₃	—	2	8	no	—
33	C₂²×C₈×C₅₃	—	2	24	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
37	C₂⁴×C₅₃	—	2	8	no	—
38	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
39	C₂⁴×C₅₃	C₄⋊C₄	1	3	yes	—
40	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
42	C₂²×C₄×C₅₃	C₄²	1	12	yes	—
43	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	24	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	24	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	24	no	—
61	C₂×C₈×C₅₃	—	2	24	no	—
62	C₂×C₈×C₅₃	—	2	8	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
64	#11719917	—	2	48	no	—
65	#11719917	—	2	8	no	—
66	#11719917	C₂²×C₄	1	16	yes	—
67	C₂×#88322	—	2	48	no	—
68	C₂×#88322	—	2	8	no	—
69	C₂³×C₅₃	—	2	32	no	—
70	C₂³×C₅₃	—	2	12	no	—
71	C₂³×C₅₃	—	2	12	no	—
72	C₂³×C₅₃	#95	1	12	yes	—
73	C₂³×C₅₃	#95	1	4	yes	—
74	C₂³×C₅₃	#108	1	8	yes	—
75	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
76	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#95	1	24	yes	—
80	C₂×C₄×C₅₃	#95	1	12	yes	—
81	C₂×C₄×C₅₃	#95	1	12	yes	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₅₃	—	2	8	no	—
96	#88322	—	2	16	no	—
97	C₂²×#341	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	6	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	2	24	no	—
101	C₂²×C₅₃	—	2	24	no	—
102	C₂²×C₅₃	#341	1	16	yes	complemented
103	C₂²×C₅₃	C₂×#95	1	6	yes	—
104	C₂²×C₅₃	C₂×#95	1	6	yes	—
105	C₂²×C₅₃	C₂×#108	1	12	yes	—
106	C₂²×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₅₃	C₂×#95	1	6	yes	—
109	C₄×C₅₃	C₂×#95	1	6	yes	—
110	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
112	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×M₁₆	—	53	2	no	—
114	C₂⁴×C₈	D₅₃	1	1	yes	characteristic
115	C₂×#341	—	53	12	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	—	2	6	no	—
118	C₂×C₅₃	C₂×#341	1	12	yes	complemented
119	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
120	C₂×C₅₃	C₂²×#108	1	2	yes	—
121	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
122	C₂²×M₁₆	—	53	16	no	—
123	C₂²×M₁₆	—	53	12	no	—
124	C₂²×M₁₆	—	106	6	no	—
125	C₂²×M₁₆	—	106	8	no	—
126	C₂³×C₈	—	2	8	no	—
127	C₂³×C₈	—	106	2	no	—
128	C₂³×C₈	Dic₅₃	1	2	yes	—
129	C₂³×C₈	Dic₅₃	1	6	yes	—
130	C₂³×C₈	C₂×D₅₃	1	6	yes	—
131	#341	—	53	16	no	—
132	C₅₃	C₂²×#341	1	1	yes	characteristic, complemented, Sylow
133	C₂⁵	Dic₁₀₆	1	1	yes	characteristic
134	C₂³×C₄	Dic₁₀₆	1	3	yes	—
135	C₂³×C₄	C₄×D₅₃	1	8	yes	—
136	C₂³×C₄	D₂₁₂	1	3	yes	—
137	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
138	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
140	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
141	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
142	C₂×M₁₆	—	53	48	no	—
143	C₂×M₁₆	—	53	8	no	—
144	C₂×M₁₆	—	106	12	no	—
145	C₂×M₁₆	—	106	24	no	—
146	C₂×M₁₆	—	106	48	no	—
147	C₂²×C₈	—	2	24	no	—
148	C₂²×C₈	—	2	24	no	—
149	C₂²×C₈	—	2	8	no	—
150	C₂²×C₈	—	106	12	no	—
151	C₂²×C₈	—	106	16	no	—
152	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
153	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
154	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
155	C₂⁴	—	2	8	no	—
156	C₂⁴	#11719919	1	8	yes	—
157	C₂⁴	C₄⋊Dic₅₃	1	3	yes	—
158	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₂⁴	C₂×Dic₁₀₆	1	3	yes	—
160	C₂²×C₄	C₄×Dic₅₃	1	12	yes	—
161	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
162	C₂²×C₄	#11719919	1	24	yes	—
163	C₂²×C₄	C₄⋊Dic₅₃	1	6	yes	—
164	C₂²×C₄	C₄⋊Dic₅₃	1	3	yes	—
165	C₂²×C₄	C₄⋊Dic₅₃	1	3	yes	—
166	C₂²×C₄	#11719921	1	8	yes	—
167	C₂²×C₄	#11719921	1	24	yes	—
168	C₂²×C₄	C₂²⋊Dic₅₃	1	12	yes	—
169	C₂²×C₄	C₂²⋊Dic₅₃	1	4	yes	—
170	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
171	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
172	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
173	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
174	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
175	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
176	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
177	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
178	M₁₆	—	53	16	no	—
179	M₁₆	—	106	48	no	—
180	M₁₆	—	106	8	no	—
181	C₂×C₈	—	2	24	no	—
182	C₂×C₈	—	2	24	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	106	48	no	—
185	C₂×C₈	—	106	8	no	—
186	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
187	C₂³	—	2	32	no	—
188	C₂³	—	2	12	no	—
189	C₂³	—	2	12	no	—
190	C₂³	#13133124	1	12	yes	—
191	C₂³	#13133124	1	4	yes	—
192	C₂³	#13133153	1	8	yes	—
193	C₂³	C₂×#11719919	1	12	yes	—
194	C₂³	C₂×C₄⋊Dic₅₃	1	3	yes	—
195	C₂³	C₂×C₄⋊Dic₅₃	1	3	yes	—
196	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
197	C₂×C₄	#13133124	1	24	yes	—
198	C₂×C₄	#13133124	1	12	yes	—
199	C₂×C₄	#13133124	1	12	yes	—
200	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
201	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
202	C₂×C₄	C₂×#11719921	1	12	yes	—
203	C₂×C₄	C₂×#11719921	1	12	yes	—
204	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	6	yes	—
205	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	6	yes	—
206	C₂×C₄	C₂×#11719919	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
209	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
210	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
212	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
213	C₈	—	2	8	no	—
214	C₈	—	106	16	no	—
215	C₂²	—	2	24	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	6	no	—
218	C₂²	—	2	2	no	—
219	C₂²	#16810388	1	16	yes	direct product factor
220	C₂²	C₂×#13133124	1	6	yes	—
221	C₂²	C₂×#13133124	1	6	yes	—
222	C₂²	C₂×#13133153	1	12	yes	—
223	C₂²	C₂²×#11719919	1	2	yes	—
224	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₄	C₂×#13133124	1	6	yes	—
226	C₄	C₂×#13133124	1	6	yes	—
227	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
228	C₄	C₂²×#11719921	1	2	yes	—
229	C₄	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	—	2	6	no	—
232	C₂	C₂×#16810388	1	12	yes	direct product factor
233	C₂	C₂²×#13133124	1	1	yes	characteristic
234	C₂	C₂²×#13133153	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂⁴×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₄⋊C₄
prime radical	C₂⁵×C₅₃	Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂⁴×C₈	D₅₃
hypocenter	C₅₃	C₂²×#341
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810388	C₂	direct (N × H)
9. #16810388	C₂²	direct (N × H)
102. C₂²×C₅₃	#341	semidirect
118. C₂×C₅₃	C₂×#341	semidirect
132. C₅₃	C₂²×#341	semidirect
219. C₂²	#16810388	direct (N × H)
232. C₂	C₂×#16810388	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
C₂×#16810388	2
C₂⁴×C₈×C₅₃	2
C₂²×#341	53

Subgroups — C22×#16810388 #27381509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810388 #27381509