Subgroups — C₂²×#16811173 #28290790

Order 13568.

12878 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	2	yes	maximal, complemented
2	C₂×#16811173	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133205	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133191	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
8	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719945	—	2	32	no	—
10	C₂²×#11719945	C₂²	1	48	yes	complemented
11	C₂³×Dic₁₀₆	C₂²	1	2	yes	—
12	C₂³×D₂₁₂	C₂²	1	4	yes	—
13	C₂³×D₂₁₂	C₂²	1	2	yes	—
14	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	—
15	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
16	#16811173	C₂²	1	256	yes	direct product factor
17	C₂×#13133217	C₂²	1	24	yes	complemented
18	C₂×#13133214	C₂²	1	24	yes	complemented
19	C₂×#13133205	C₂²	1	48	yes	complemented
20	C₂×#13133191	C₂²	1	96	yes	complemented
21	C₂×C₄×D₂₁₂	C₂²	1	96	yes	complemented
22	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
23	C₂²×#11719921	C₂²	1	8	yes	—
24	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
25	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₅₃	C₂²	1	24	yes	complemented
30	C₂⁴×D₅₃	C₂³	1	4	yes	—
31	C₂×#11719945	—	2	384	no	—
32	C₂×#11719945	C₂³	1	128	yes	complemented
33	C₂²×Dic₁₀₆	—	2	16	no	—
34	C₂²×Dic₁₀₆	C₂³	1	24	yes	—
35	C₂²×D₂₁₂	—	2	16	no	—
36	C₂²×D₂₁₂	—	2	32	no	—
37	C₂²×D₂₁₂	C₂³	1	48	yes	—
38	C₂²×D₂₁₂	C₂³	1	24	yes	—
39	C₂²×C₂²⋊D₅₃	—	2	32	no	—
40	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	—
41	C₂³×Dic₅₃	C₂³	1	4	yes	—
42	C₂²×C₄×D₅₃	—	2	32	no	—
43	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
44	#13133217	C₂³	1	64	yes	complemented
45	#13133214	C₂³	1	64	yes	complemented
46	#13133205	C₂³	1	128	yes	complemented
47	#13133191	C₂³	1	256	yes	complemented
48	C₄×D₂₁₂	C₂³	1	256	yes	complemented
49	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	—
50	C₂×#11719921	C₂³	1	96	yes	—
51	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	—	2	16	no	—
53	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
54	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
61	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	64	yes	complemented
64	C₂⁴×C₅₃	—	2	8	no	—
65	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
66	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
67	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂³×D₅₃	—	2	16	no	—
69	C₂³×D₅₃	—	2	32	no	—
70	C₂³×D₅₃	C₂⁴	1	24	yes	—
71	C₂²×C₄×C₅₃	—	2	8	no	—
72	C₂²×C₄×C₅₃	—	2	96	no	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	96	yes	complemented
76	C₂²×C₄×C₅₃	#41	1	16	yes	complemented
77	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
79	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
83	C₂×C₂²⋊D₅₃	—	2	192	no	—
84	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
85	C₂²×Dic₅₃	—	2	16	no	—
86	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
87	#11719945	—	2	512	no	—
88	C₂×D₂₁₂	—	2	96	no	—
89	C₂×D₂₁₂	—	2	192	no	—
90	C₂×D₂₁₂	C₂⁴	1	64	yes	—
91	C₂×D₂₁₂	C₂⁴	1	32	yes	—
92	C₂×C₄×D₅₃	—	2	192	no	—
93	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
94	C₂×Dic₁₀₆	—	2	96	no	—
95	C₂×Dic₁₀₆	C₂⁴	1	32	yes	—
96	C₄²×C₅₃	C₂⁴	1	32	yes	—
97	C₄²×C₅₃	C₂⁴	1	16	yes	—
98	C₄²×C₅₃	C₂⁴	1	16	yes	—
99	#11719921	C₂⁴	1	128	yes	—
100	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
101	C₂³×C₅₃	—	2	32	no	—
102	C₂³×C₅₃	—	2	24	no	—
103	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
104	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
105	C₂³×C₅₃	C₂×#41	1	8	yes	—
106	C₂³×C₅₃	C₂⁵	1	4	yes	—
107	C₂³×C₅₃	C₂⁵	1	3	yes	—
108	C₂²×D₅₃	—	2	48	no	—
109	C₂²×D₅₃	—	2	96	no	—
110	C₂²×D₅₃	—	4	64	no	—
111	C₂²×D₅₃	C₂⁵	1	16	yes	—
112	C₂×C₄×C₅₃	—	2	24	no	—
113	C₂×C₄×C₅₃	—	2	64	no	—
114	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
115	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
116	C₂×C₄×C₅₃	C₂²×D₄	1	64	yes	complemented
117	C₂×C₄×C₅₃	C₂×#41	1	48	yes	complemented
118	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
119	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
120	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
121	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
122	C₂²⋊D₅₃	—	2	128	no	—
123	C₂×Dic₅₃	—	2	48	no	—
124	C₂×Dic₅₃	C₂⁵	1	16	yes	—
125	D₂₁₂	—	2	64	no	—
126	D₂₁₂	—	2	128	no	—
127	C₄×D₅₃	—	2	128	no	—
128	Dic₁₀₆	—	2	64	no	—
129	C₂²×#532	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	8	no	—
131	C₂²×C₅₃	—	2	48	no	—
132	C₂²×C₅₃	#532	1	16	yes	complemented
133	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
134	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
135	C₂²×C₅₃	C₂²×#41	1	12	yes	—
136	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
137	C₂×D₅₃	—	2	16	no	—
138	C₂×D₅₃	—	2	32	no	—
139	C₂×D₅₃	—	4	96	no	—
140	C₄×C₅₃	—	2	8	no	—
141	C₄×C₅₃	C₂³×D₄	1	8	yes	—
142	C₄×C₅₃	C₂³×D₄	1	8	yes	—
143	C₄×C₅₃	C₂²×#41	1	16	yes	complemented
144	Dic₅₃	—	2	16	no	—
145	C₂³×#41	—	53	2	no	—
146	C₂×#532	—	53	48	no	—
147	C₂²×C₄⋊Q₈	—	53	1	no	—
148	C₂²×#127	—	53	1	no	—
149	C₂²×#124	—	53	2	no	—
150	C₂²×#121	—	53	4	no	—
151	C₂²×C₄×D₄	—	53	4	no	—
152	C₂³×C₄²	D₅₃	1	1	yes	characteristic, complemented
153	C₂×C₅₃	—	2	8	no	—
154	C₂×C₅₃	C₂×#532	1	12	yes	complemented
155	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
156	C₂×C₅₃	C₂³×#41	1	2	yes	—
157	D₅₃	—	4	16	no	—
158	C₂²×#41	—	53	48	no	—
159	C₂²×#41	—	106	32	no	—
160	C₂³×Q₈	—	53	2	no	—
161	C₂³×D₄	—	53	4	no	—
162	C₂³×D₄	—	53	2	no	—
163	C₂³×D₄	—	53	4	no	—
164	C₂⁴×C₄	—	53	4	no	—
165	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
166	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
167	#532	—	53	256	no	—
168	C₂×C₄⋊Q₈	—	53	24	no	—
169	C₂×#127	—	53	24	no	—
170	C₂×#124	—	53	48	no	—
171	C₂×#121	—	53	96	no	—
172	C₂×C₄×D₄	—	53	96	no	—
173	C₂²×C₄⋊C₄	—	53	4	no	—
174	C₂²×C₂²⋊C₄	—	53	8	no	—
175	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
176	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
177	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
178	C₂²×C₄²	C₂×D₅₃	1	24	yes	complemented
179	C₅₃	C₂²×#532	1	1	yes	characteristic, complemented, Sylow
180	C₂⁵	—	53	4	no	—
181	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
182	C₂×#41	—	53	128	no	—
183	C₂×#41	—	106	384	no	—
184	C₂²×Q₈	—	53	24	no	—
185	C₂²×Q₈	—	106	16	no	—
186	C₂²×D₄	—	53	48	no	—
187	C₂²×D₄	—	53	48	no	—
188	C₂²×D₄	—	53	24	no	—
189	C₂²×D₄	—	106	32	no	—
190	C₂²×D₄	—	106	16	no	—
191	C₂²×D₄	—	106	32	no	—
192	C₂³×C₄	—	2	16	no	—
193	C₂³×C₄	—	53	4	no	—
194	C₂³×C₄	—	53	48	no	—
195	C₂³×C₄	—	106	32	no	—
196	C₂³×C₄	D₂₁₂	1	16	yes	complemented
197	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
198	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
199	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
201	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
202	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	53	64	no	—
204	#127	—	53	64	no	—
205	#124	—	53	128	no	—
206	#121	—	53	256	no	—
207	C₄×D₄	—	53	256	no	—
208	C₂×C₄⋊C₄	—	53	48	no	—
209	C₂×C₂²⋊C₄	—	53	96	no	—
210	C₂×C₄²	C₂²×D₅₃	1	24	yes	—
211	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
212	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
213	C₂×C₄²	C₂²×D₅₃	1	64	yes	complemented
214	C₂⁴	—	2	8	no	—
215	C₂⁴	—	53	24	no	—
216	C₂⁴	—	106	16	no	—
217	C₂⁴	—	106	32	no	—
218	C₂⁴	C₂×D₂₁₂	1	8	yes	—
219	C₂⁴	C₂³×D₅₃	1	6	yes	—
220	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
221	#41	—	106	512	no	—
222	C₂×Q₈	—	53	32	no	—
223	C₂×Q₈	—	106	96	no	—
224	C₂×D₄	—	53	64	no	—
225	C₂×D₄	—	53	64	no	—
226	C₂×D₄	—	53	32	no	—
227	C₂×D₄	—	106	192	no	—
228	C₂×D₄	—	106	96	no	—
229	C₂×D₄	—	106	192	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	96	no	—
232	C₂²×C₄	—	53	24	no	—
233	C₂²×C₄	—	53	64	no	—
234	C₂²×C₄	—	106	16	no	—
235	C₂²×C₄	—	106	192	no	—
236	C₂²×C₄	C₂×D₂₁₂	1	8	yes	—
237	C₂²×C₄	C₂×D₂₁₂	1	8	yes	—
238	C₂²×C₄	C₂×D₂₁₂	1	96	yes	complemented
239	C₂²×C₄	#11719945	1	16	yes	complemented
240	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
241	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
242	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
243	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
244	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
245	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
246	C₄⋊C₄	—	53	64	no	—
247	C₂²⋊C₄	—	53	128	no	—
248	C₄²	C₂³×D₅₃	1	32	yes	—
249	C₄²	C₂³×D₅₃	1	16	yes	—
250	C₄²	C₂³×D₅₃	1	16	yes	—
251	C₂³	—	2	32	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	53	16	no	—
254	C₂³	—	106	48	no	—
255	C₂³	—	106	96	no	—
256	C₂³	—	212	64	no	—
257	C₂³	C₂²×D₂₁₂	1	24	yes	—
258	C₂³	C₂²×D₂₁₂	1	4	yes	—
259	C₂³	C₂×#11719945	1	8	yes	—
260	C₂³	C₂⁴×D₅₃	1	4	yes	—
261	C₂³	C₂⁴×D₅₃	1	3	yes	—
262	Q₈	—	106	64	no	—
263	D₄	—	106	128	no	—
264	D₄	—	106	64	no	—
265	D₄	—	106	128	no	—
266	C₂×C₄	—	2	24	no	—
267	C₂×C₄	—	2	64	no	—
268	C₂×C₄	—	53	16	no	—
269	C₂×C₄	—	106	48	no	—
270	C₂×C₄	—	106	128	no	—
271	C₂×C₄	C₂²×D₂₁₂	1	24	yes	—
272	C₂×C₄	C₂²×D₂₁₂	1	24	yes	—
273	C₂×C₄	C₂²×D₂₁₂	1	64	yes	complemented
274	C₂×C₄	C₂×#11719945	1	48	yes	complemented
275	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
276	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
277	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
278	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	48	no	—
281	C₂²	—	106	16	no	—
282	C₂²	—	106	32	no	—
283	C₂²	—	212	96	no	—
284	C₂²	#16811173	1	16	yes	direct product factor
285	C₂²	C₂³×D₂₁₂	1	8	yes	—
286	C₂²	C₂³×D₂₁₂	1	6	yes	—
287	C₂²	C₂²×#11719945	1	12	yes	—
288	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
289	C₄	—	2	8	no	—
290	C₄	—	106	16	no	—
291	C₄	C₂³×D₂₁₂	1	8	yes	—
292	C₄	C₂³×D₂₁₂	1	8	yes	—
293	C₄	C₂²×#11719945	1	16	yes	complemented
294	C₂	—	2	8	no	—
295	C₂	—	212	16	no	—
296	C₂	C₂×#16811173	1	12	yes	direct product factor
297	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
298	C₂	C₂³×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#532
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719945	C₂	semidirect
2. C₂×#16811173	C₂	direct (N × H)
3. C₂²×#13133217	C₂	semidirect
4. C₂²×#13133214	C₂	semidirect
5. C₂²×#13133205	C₂	semidirect
6. C₂²×#13133191	C₂	semidirect
7. C₂²×C₄×D₂₁₂	C₂	semidirect
8. C₂³×C₄²×C₅₃	C₂	semidirect
10. C₂²×#11719945	C₂²	semidirect
16. #16811173	C₂²	direct (N × H)
17. C₂×#13133217	C₂²	semidirect
18. C₂×#13133214	C₂²	semidirect
19. C₂×#13133205	C₂²	semidirect
20. C₂×#13133191	C₂²	semidirect
21. C₂×C₄×D₂₁₂	C₂²	semidirect
29. C₂²×C₄²×C₅₃	C₂²	semidirect
32. C₂×#11719945	C₂³	semidirect
44. #13133217	C₂³	semidirect
45. #13133214	C₂³	semidirect
46. #13133205	C₂³	semidirect
47. #13133191	C₂³	semidirect
48. C₄×D₂₁₂	C₂³	semidirect
53. C₂³×C₄×C₅₃	D₄	semidirect
63. C₂×C₄²×C₅₃	C₂³	semidirect
75. C₂²×C₄×C₅₃	C₂×D₄	semidirect
76. C₂²×C₄×C₅₃	#41	semidirect
116. C₂×C₄×C₅₃	C₂²×D₄	semidirect
117. C₂×C₄×C₅₃	C₂×#41	semidirect
132. C₂²×C₅₃	#532	semidirect
143. C₄×C₅₃	C₂²×#41	semidirect
152. C₂³×C₄²	D₅₃	semidirect
154. C₂×C₅₃	C₂×#532	semidirect
178. C₂²×C₄²	C₂×D₅₃	semidirect
179. C₅₃	C₂²×#532	semidirect
196. C₂³×C₄	D₂₁₂	semidirect
213. C₂×C₄²	C₂²×D₅₃	semidirect
238. C₂²×C₄	C₂×D₂₁₂	semidirect
239. C₂²×C₄	#11719945	semidirect
273. C₂×C₄	C₂²×D₂₁₂	semidirect
274. C₂×C₄	C₂×#11719945	semidirect
284. C₂²	#16811173	direct (N × H)
293. C₄	C₂²×#11719945	semidirect
296. C₂	C₂×#16811173	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×#11719945	2
C₂×#16811173	2
C₂²×#13133217	2
C₂²×#13133214	2
C₂²×#13133205	2
C₂²×#13133191	2
C₂²×C₄×D₂₁₂	2
C₂³×C₄²×C₅₃	2
C₂²×#532	53

Subgroups — C22×#16811173 #28290790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811173 #28290790