Subgroups — C₂²×#16811295 #28291044

Order 13568.

12918 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719949	C₂	1	2	yes	maximal, complemented
2	C₂×#16811295	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133214	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133192	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×D₂₁₂	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719949	—	2	32	no	—
8	C₂²×#11719949	C₂²	1	48	yes	complemented
9	C₂³×D₂₁₂	C₂²	1	3	yes	—
10	C₂³×D₂₁₂	C₂²	1	6	yes	—
11	C₂³×D₂₁₂	C₂²	1	6	yes	—
12	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
13	#16811295	C₂²	1	256	yes	direct product factor
14	C₂×#13133214	C₂²	1	72	yes	complemented
15	C₂×#13133192	C₂²	1	144	yes	complemented
16	C₂×C₄×D₂₁₂	C₂²	1	72	yes	complemented
17	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂²×#11719921	C₂²	1	6	yes	—
19	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	complemented
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
23	C₂⁴×D₅₃	C₂³	1	6	yes	—
24	C₂×#11719949	—	2	384	no	—
25	C₂×#11719949	C₂³	1	128	yes	complemented
26	C₂²×D₂₁₂	—	2	48	no	—
27	C₂²×D₂₁₂	—	2	24	no	—
28	C₂²×D₂₁₂	—	4	24	no	—
29	C₂²×D₂₁₂	C₂³	1	36	yes	—
30	C₂²×D₂₁₂	C₂³	1	72	yes	—
31	C₂²×D₂₁₂	C₂³	1	72	yes	—
32	C₂³×Dic₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×D₅₃	—	2	48	no	—
34	C₂²×C₄×D₅₃	C₂³	1	72	yes	—
35	#13133214	C₂³	1	192	yes	complemented
36	#13133192	C₂³	1	384	yes	complemented
37	C₄×D₂₁₂	C₂³	1	192	yes	complemented
38	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
39	C₂×#11719921	C₂³	1	72	yes	—
40	C₂²×Q₈×C₅₃	D₄	1	16	yes	complemented
41	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	C₂³	1	64	yes	complemented
46	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	36	yes	—
48	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂³×D₅₃	—	2	24	no	—
50	C₂³×D₅₃	—	2	24	no	—
51	C₂³×D₅₃	—	2	24	no	—
52	C₂³×D₅₃	C₂⁴	1	36	yes	—
53	C₂×Q₈×C₅₃	C₂×D₄	1	96	yes	complemented
54	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
55	C₂²×C₄×C₅₃	—	2	12	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₅₃	#41	1	8	yes	complemented
58	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
61	C₂²×Dic₅₃	—	2	8	no	—
62	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
63	#11719949	—	2	512	no	—
64	C₂×D₂₁₂	—	2	288	no	—
65	C₂×D₂₁₂	—	2	144	no	—
66	C₂×D₂₁₂	—	4	144	no	—
67	C₂×D₂₁₂	C₂⁴	1	48	yes	—
68	C₂×D₂₁₂	C₂⁴	1	96	yes	—
69	C₂×D₂₁₂	C₂⁴	1	96	yes	—
70	C₂×C₄×D₅₃	—	2	288	no	—
71	C₂×C₄×D₅₃	C₂⁴	1	96	yes	—
72	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
73	C₄²×C₅₃	C₂⁴	1	48	yes	—
74	#11719921	C₂⁴	1	96	yes	—
75	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
76	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
77	C₂³×C₅₃	C₂×#41	1	4	yes	—
78	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
79	C₂³×C₅₃	C₂⁵	1	3	yes	—
80	C₂²×D₅₃	—	2	72	no	—
81	C₂²×D₅₃	—	2	72	no	—
82	C₂²×D₅₃	—	2	72	no	—
83	C₂²×D₅₃	—	4	96	no	—
84	C₂²×D₅₃	C₂⁵	1	24	yes	—
85	Q₈×C₅₃	C₂²×D₄	1	64	yes	complemented
86	C₂×C₄×C₅₃	—	2	36	no	—
87	C₂×C₄×C₅₃	C₂²×D₄	1	72	yes	—
88	C₂×C₄×C₅₃	C₂×#41	1	24	yes	complemented
89	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
90	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
91	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
92	C₂×Dic₅₃	—	2	24	no	—
93	C₂×Dic₅₃	C₂⁵	1	8	yes	—
94	D₂₁₂	—	2	192	no	—
95	D₂₁₂	—	2	96	no	—
96	D₂₁₂	—	4	96	no	—
97	C₄×D₅₃	—	2	192	no	—
98	C₂²×#550	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#550	1	16	yes	complemented
100	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
101	C₂²×C₅₃	C₂²×#41	1	6	yes	—
102	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
103	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
104	C₂×D₅₃	—	2	24	no	—
105	C₂×D₅₃	—	2	24	no	—
106	C₂×D₅₃	—	2	24	no	—
107	C₂×D₅₃	—	4	144	no	—
108	C₄×C₅₃	—	2	12	no	—
109	C₄×C₅₃	C₂³×D₄	1	24	yes	—
110	C₄×C₅₃	C₂²×#41	1	8	yes	complemented
111	Dic₅₃	—	2	8	no	—
112	C₂³×#41	—	53	2	no	—
113	C₂×#550	—	53	48	no	—
114	C₂²×#127	—	53	3	no	—
115	C₂²×#121	—	53	6	no	—
116	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
117	C₂²×C₄×D₄	—	53	3	no	—
118	C₂×C₅₃	C₂×#550	1	12	yes	complemented
119	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
120	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
121	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
122	D₅₃	—	4	24	no	—
123	C₂²×#41	—	53	48	no	—
124	C₂²×#41	—	106	32	no	—
125	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
126	C₂³×D₄	—	53	6	no	—
127	C₂³×D₄	—	53	3	no	—
128	C₂³×D₄	—	53	6	no	—
129	C₂⁴×C₄	—	53	6	no	—
130	#550	—	53	256	no	—
131	C₂×#127	—	53	72	no	—
132	C₂×#121	—	53	144	no	—
133	C₂×C₄×Q₈	C₂×D₅₃	1	24	yes	complemented
134	C₂×C₄×D₄	—	53	72	no	—
135	C₂²×C₄⋊C₄	—	53	1	no	—
136	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
137	C₂²×C₂²⋊C₄	—	53	6	no	—
138	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
139	C₅₃	C₂²×#550	1	1	yes	characteristic, complemented, Sylow
140	C₂⁵	—	53	6	no	—
141	C₂×#41	—	53	128	no	—
142	C₂×#41	—	106	384	no	—
143	C₂²×Q₈	D₂₁₂	1	16	yes	complemented
144	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
145	C₂²×D₄	—	53	36	no	—
146	C₂²×D₄	—	53	72	no	—
147	C₂²×D₄	—	53	72	no	—
148	C₂²×D₄	—	106	48	no	—
149	C₂²×D₄	—	106	24	no	—
150	C₂²×D₄	—	212	24	no	—
151	C₂³×C₄	—	53	2	no	—
152	C₂³×C₄	—	53	72	no	—
153	C₂³×C₄	—	106	48	no	—
154	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
155	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
156	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
157	#127	—	53	192	no	—
158	#121	—	53	384	no	—
159	C₄×Q₈	C₂²×D₅₃	1	64	yes	complemented
160	C₄×D₄	—	53	192	no	—
161	C₂×C₄⋊C₄	—	53	12	no	—
162	C₂×C₄⋊C₄	C₂²×D₅₃	1	36	yes	—
163	C₂×C₂²⋊C₄	—	53	72	no	—
164	C₂×C₄²	C₂²×D₅₃	1	36	yes	—
165	C₂⁴	—	53	36	no	—
166	C₂⁴	—	106	24	no	—
167	C₂⁴	—	106	24	no	—
168	C₂⁴	—	106	24	no	—
169	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
170	#41	—	106	512	no	—
171	C₂×Q₈	C₂×D₂₁₂	1	96	yes	complemented
172	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
173	C₂×D₄	—	53	48	no	—
174	C₂×D₄	—	53	96	no	—
175	C₂×D₄	—	53	96	no	—
176	C₂×D₄	—	106	288	no	—
177	C₂×D₄	—	106	144	no	—
178	C₂×D₄	—	212	144	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	—	53	12	no	—
181	C₂²×C₄	—	53	96	no	—
182	C₂²×C₄	—	106	8	no	—
183	C₂²×C₄	—	106	288	no	—
184	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
185	C₂²×C₄	#11719949	1	8	yes	complemented
186	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
187	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
188	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
189	C₄⋊C₄	—	53	16	no	—
190	C₄⋊C₄	C₂³×D₅₃	1	48	yes	—
191	C₂²⋊C₄	—	53	96	no	—
192	C₄²	C₂³×D₅₃	1	48	yes	—
193	C₂³	—	53	24	no	—
194	C₂³	—	106	72	no	—
195	C₂³	—	106	72	no	—
196	C₂³	—	106	72	no	—
197	C₂³	—	212	96	no	—
198	C₂³	C₂²×D₂₁₂	1	4	yes	—
199	C₂³	C₂×#11719949	1	4	yes	—
200	C₂³	#13133263	1	4	yes	—
201	C₂³	C₂⁴×D₅₃	1	3	yes	—
202	Q₈	C₂²×D₂₁₂	1	64	yes	complemented
203	D₄	—	106	192	no	—
204	D₄	—	106	96	no	—
205	D₄	—	212	96	no	—
206	C₂×C₄	—	2	36	no	—
207	C₂×C₄	—	53	8	no	—
208	C₂×C₄	—	106	24	no	—
209	C₂×C₄	—	106	192	no	—
210	C₂×C₄	C₂²×D₂₁₂	1	72	yes	—
211	C₂×C₄	C₂×#11719949	1	24	yes	complemented
212	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
213	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
214	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
215	C₂²	—	106	24	no	—
216	C₂²	—	106	24	no	—
217	C₂²	—	106	24	no	—
218	C₂²	—	212	144	no	—
219	C₂²	#16811295	1	16	yes	direct product factor
220	C₂²	C₂³×D₂₁₂	1	6	yes	—
221	C₂²	C₂²×#11719949	1	6	yes	—
222	C₂²	C₂×#13133263	1	6	yes	—
223	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
224	C₄	—	2	12	no	—
225	C₄	—	106	8	no	—
226	C₄	C₂³×D₂₁₂	1	24	yes	—
227	C₄	C₂²×#11719949	1	8	yes	complemented
228	C₂	—	212	24	no	—
229	C₂	C₂×#16811295	1	12	yes	direct product factor
230	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
231	C₂	C₂³×#11719949	1	1	yes	characteristic
232	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×#550
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719949	C₂	semidirect
2. C₂×#16811295	C₂	direct (N × H)
3. C₂²×#13133214	C₂	semidirect
4. C₂²×#13133192	C₂	semidirect
5. C₂²×C₄×D₂₁₂	C₂	semidirect
6. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
8. C₂²×#11719949	C₂²	semidirect
13. #16811295	C₂²	direct (N × H)
14. C₂×#13133214	C₂²	semidirect
15. C₂×#13133192	C₂²	semidirect
16. C₂×C₄×D₂₁₂	C₂²	semidirect
20. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
25. C₂×#11719949	C₂³	semidirect
35. #13133214	C₂³	semidirect
36. #13133192	C₂³	semidirect
37. C₄×D₂₁₂	C₂³	semidirect
40. C₂²×Q₈×C₅₃	D₄	semidirect
45. C₄×Q₈×C₅₃	C₂³	semidirect
53. C₂×Q₈×C₅₃	C₂×D₄	semidirect
57. C₂²×C₄×C₅₃	#41	semidirect
85. Q₈×C₅₃	C₂²×D₄	semidirect
88. C₂×C₄×C₅₃	C₂×#41	semidirect
99. C₂²×C₅₃	#550	semidirect
110. C₄×C₅₃	C₂²×#41	semidirect
116. C₂²×C₄×Q₈	D₅₃	semidirect
118. C₂×C₅₃	C₂×#550	semidirect
133. C₂×C₄×Q₈	C₂×D₅₃	semidirect
139. C₅₃	C₂²×#550	semidirect
143. C₂²×Q₈	D₂₁₂	semidirect
159. C₄×Q₈	C₂²×D₅₃	semidirect
171. C₂×Q₈	C₂×D₂₁₂	semidirect
185. C₂²×C₄	#11719949	semidirect
202. Q₈	C₂²×D₂₁₂	semidirect
211. C₂×C₄	C₂×#11719949	semidirect
219. C₂²	#16811295	direct (N × H)
227. C₄	C₂²×#11719949	semidirect
229. C₂	C₂×#16811295	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×#11719949	2
C₂×#16811295	2
C₂²×#13133214	2
C₂²×#13133192	2
C₂²×C₄×D₂₁₂	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×#550	53

Subgroups — C22×#16811295 #28291044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811295 #28291044