Subgroups — C₂²×#16810672 #28247785

Order 13568.

2718 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×#11719917	C₂	1	2	yes	maximal
2	C₂×#16810672	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133153	C₂	1	4	yes	maximal, complemented
4	C₂³×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	2	no	—
6	C₂²×#11719917	—	2	12	no	—
7	C₂²×#11719917	C₂²	1	4	yes	—
8	C₂²×#11719917	C₂²	1	24	yes	—
9	C₂³×#88322	—	2	2	no	—
10	#16810672	C₂²	1	64	yes	direct product factor
11	C₂×#13133153	C₂²	1	48	yes	complemented
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	C₄	1	4	yes	—
14	C₂²×M₁₆×C₅₃	C₄	1	12	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	12	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#11719917	—	2	24	no	—
18	C₂×#11719917	—	2	12	no	—
19	C₂×#11719917	—	2	48	no	—
20	C₂×#11719917	C₂³	1	24	yes	—
21	C₂×#11719917	C₂³	1	32	yes	—
22	C₂²×#88322	—	2	24	no	—
23	C₂²×#88322	—	2	4	no	—
24	#13133153	C₂³	1	64	yes	complemented
25	C₂⁵×C₅₃	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
28	C₂³×C₄×C₅₃	D₄	1	2	yes	—
29	C₂³×C₄×C₅₃	D₄	1	6	yes	—
30	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	Q₈	1	6	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	48	yes	—
35	C₂×M₁₆×C₅₃	C₂×C₄	1	48	yes	—
36	C₂×M₁₆×C₅₃	C₂³	1	16	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
40	C₂⁴×C₅₃	—	4	4	no	—
41	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂⁴×C₅₃	C₄⋊C₄	1	6	yes	—
43	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂⁴×C₅₃	C₂×Q₈	1	6	yes	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	24	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
60	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
63	M₁₆×C₅₃	C₂²×C₄	1	64	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	—
66	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
67	#11719917	—	2	32	no	—
68	#11719917	—	2	24	no	—
69	#11719917	C₂⁴	1	16	yes	—
70	C₂×#88322	—	2	24	no	—
71	C₂×#88322	—	2	32	no	—
72	C₂³×C₅₃	—	2	8	no	—
73	C₂³×C₅₃	—	2	4	no	—
74	C₂³×C₅₃	—	4	12	no	—
75	C₂³×C₅₃	—	4	12	no	—
76	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
79	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₅₃	C₂²×Q₈	1	3	yes	—
81	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
93	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
94	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
96	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
97	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
98	C₈×C₅₃	C₂³×C₄	1	16	yes	—
99	#88322	—	2	16	no	—
100	C₂²×#430	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	24	no	—
102	C₂²×C₅₃	—	2	6	no	—
103	C₂²×C₅₃	—	2	6	no	—
104	C₂²×C₅₃	—	4	12	no	—
105	C₂²×C₅₃	—	4	4	no	—
106	C₂²×C₅₃	#430	1	4	yes	complemented
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
108	C₂²×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
109	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
110	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
112	C₄×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
113	C₄×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
114	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
115	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
116	C₂³×M₁₆	—	53	2	no	—
117	C₂³×M₁₆	D₅₃	1	1	yes	characteristic
118	C₂×#430	—	53	24	no	—
119	C₂²×#108	—	53	4	no	—
120	C₂×C₅₃	—	2	8	no	—
121	C₂×C₅₃	—	2	3	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	C₂×#430	1	6	yes	complemented
124	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
125	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
126	C₂²×M₁₆	—	53	4	no	—
127	C₂²×M₁₆	—	53	24	no	—
128	C₂²×M₁₆	—	106	12	no	—
129	C₂²×M₁₆	—	106	2	no	—
130	C₂²×M₁₆	Dic₅₃	1	4	yes	—
131	C₂²×M₁₆	Dic₅₃	1	12	yes	—
132	C₂²×M₁₆	C₂×D₅₃	1	12	yes	—
133	C₂³×C₈	—	106	2	no	—
134	C₂³×C₈	C₂×D₅₃	1	2	yes	—
135	#430	—	53	64	no	—
136	C₂×#108	—	53	48	no	—
137	C₅₃	C₂²×#430	1	1	yes	characteristic, complemented, Sylow
138	C₂⁵	Dic₁₀₆	1	1	yes	characteristic
139	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
140	C₂³×C₄	Dic₁₀₆	1	6	yes	—
141	C₂³×C₄	D₂₁₂	1	6	yes	—
142	C₂³×C₄	D₂₁₂	1	2	yes	—
143	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
144	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
145	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
146	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×M₁₆	—	53	24	no	—
148	C₂×M₁₆	—	53	32	no	—
149	C₂×M₁₆	—	106	24	no	—
150	C₂×M₁₆	—	106	12	no	—
151	C₂×M₁₆	—	106	48	no	—
152	C₂×M₁₆	C₂×Dic₅₃	1	48	yes	—
153	C₂×M₁₆	C₂×Dic₅₃	1	48	yes	—
154	C₂×M₁₆	C₂²×D₅₃	1	16	yes	—
155	C₂²×C₈	—	106	24	no	—
156	C₂²×C₈	—	106	4	no	—
157	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
158	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
159	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
160	#108	—	53	64	no	—
161	C₂⁴	—	4	4	no	—
162	C₂⁴	C₄⋊Dic₅₃	1	2	yes	—
163	C₂⁴	C₄⋊Dic₅₃	1	6	yes	—
164	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
165	C₂⁴	C₂×Dic₁₀₆	1	6	yes	—
166	C₂²×C₄	C₄⋊Dic₅₃	1	6	yes	—
167	C₂²×C₄	C₄⋊Dic₅₃	1	12	yes	—
168	C₂²×C₄	C₄⋊Dic₅₃	1	24	yes	—
169	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
170	C₂²×C₄	C₄⋊Dic₅₃	1	12	yes	—
171	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
172	C₂²×C₄	C₂×Dic₁₀₆	1	12	yes	—
173	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
174	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
175	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
176	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
177	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
180	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
181	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
182	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
183	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
184	M₁₆	—	53	16	no	—
185	M₁₆	—	106	32	no	—
186	M₁₆	—	106	24	no	—
187	M₁₆	C₂²×Dic₅₃	1	64	yes	—
188	C₂×C₈	—	106	24	no	—
189	C₂×C₈	—	106	32	no	—
190	C₂×C₈	C₂²×Dic₅₃	1	24	yes	—
191	C₂×C₈	C₂²×Dic₅₃	1	24	yes	—
192	C₂×C₈	C₂³×D₅₃	1	8	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	8	no	—
195	C₂³	—	4	12	no	—
196	C₂³	—	4	12	no	—
197	C₂³	C₂×C₄⋊Dic₅₃	1	12	yes	—
198	C₂³	C₂×C₄⋊Dic₅₃	1	3	yes	—
199	C₂³	C₂×C₄⋊Dic₅₃	1	12	yes	—
200	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₂³	C₂²×Dic₁₀₆	1	3	yes	—
202	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₅₃	1	24	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₅₃	1	24	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
210	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
211	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
212	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
213	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
214	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
215	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
216	C₂×C₄	C₂²×Dic₁₀₆	1	3	yes	—
217	C₂×C₄	C₂²×Dic₁₀₆	1	4	yes	—
218	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
219	C₈	—	106	16	no	—
220	C₈	C₂³×Dic₅₃	1	16	yes	—
221	C₂²	—	2	6	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	4	12	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#16810672	1	4	yes	direct product factor
227	C₂²	C₂²×C₄⋊Dic₅₃	1	3	yes	—
228	C₂²	C₂²×C₄⋊Dic₅₃	1	8	yes	—
229	C₂²	C₂²×C₄⋊Dic₅₃	1	3	yes	—
230	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊Dic₅₃	1	3	yes	—
232	C₄	C₂²×C₄⋊Dic₅₃	1	8	yes	—
233	C₄	C₂²×C₄⋊Dic₅₃	1	3	yes	—
234	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
235	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
236	C₂	—	2	8	no	—
237	C₂	—	2	3	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×#16810672	1	6	yes	direct product factor
240	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₅₃	Q₈
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×M₁₆	D₅₃
hypocenter	C₅₃	C₂²×#430
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810672	C₂	direct (N × H)
3. C₂²×#13133153	C₂	semidirect
10. #16810672	C₂²	direct (N × H)
11. C₂×#13133153	C₂²	semidirect
24. #13133153	C₂³	semidirect
106. C₂²×C₅₃	#430	semidirect
123. C₂×C₅₃	C₂×#430	semidirect
137. C₅₃	C₂²×#430	semidirect
226. C₂²	#16810672	direct (N × H)
239. C₂	C₂×#16810672	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×#16810672	2
C₂²×#13133153	2
C₂³×M₁₆×C₅₃	2
C₂²×#430	53

Subgroups — C22×#16810672 #28247785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810672 #28247785