Subgroups — C₂²×#16811292 #28291041

Order 13568.

12918 conjugacy classes of subgroups.

The subgroup lattice has 324 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	1	yes	maximal, characteristic, complemented
2	C₂×#16811292	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133213	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133191	C₂	1	3	yes	maximal, complemented
5	C₂²×#13133188	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×C₂²⋊D₅₃	C₂	1	3	yes	maximal, complemented
8	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719949	—	2	16	no	—
10	C₂²×#11719949	C₂²	1	24	yes	complemented
11	C₂³×D₂₁₂	C₂²	1	3	yes	—
12	C₂³×C₂²⋊D₅₃	C₂²	1	3	yes	—
13	C₂³×C₂²⋊D₅₃	C₂²	1	6	yes	—
14	C₂³×C₄×D₅₃	C₂²	1	3	yes	—
15	#16811292	C₂²	1	256	yes	direct product factor
16	C₂×#13133213	C₂²	1	72	yes	complemented
17	C₂×#13133191	C₂²	1	72	yes	complemented
18	C₂×#13133188	C₂²	1	72	yes	complemented
19	C₂×Q₈×Dic₅₃	C₂²	1	24	yes	complemented
20	C₂×C₄×C₂²⋊D₅₃	C₂²	1	72	yes	complemented
21	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	—
22	C₂²×#11719919	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
24	C₂²×#11719921	C₂²	1	3	yes	—
25	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
26	C₂²×#41×C₅₃	—	2	16	no	—
27	C₂²×#41×C₅₃	C₂²	1	24	yes	complemented
28	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
30	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
31	C₂⁴×D₅₃	C₂³	1	3	yes	—
32	C₂×#11719949	—	2	192	no	—
33	C₂×#11719949	C₂³	1	64	yes	complemented
34	C₂²×D₂₁₂	—	2	24	no	—
35	C₂²×D₂₁₂	C₂³	1	36	yes	—
36	C₂²×C₂²⋊D₅₃	—	2	24	no	—
37	C₂²×C₂²⋊D₅₃	—	4	24	no	—
38	C₂²×C₂²⋊D₅₃	C₂³	1	36	yes	—
39	C₂²×C₂²⋊D₅₃	C₂³	1	72	yes	—
40	C₂³×Dic₅₃	C₂³	1	3	yes	—
41	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
42	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×D₅₃	—	2	24	no	—
44	C₂²×C₄×D₅₃	C₂³	1	36	yes	—
45	#13133213	C₂³	1	192	yes	complemented
46	#13133191	C₂³	1	192	yes	complemented
47	#13133188	C₂³	1	192	yes	complemented
48	Q₈×Dic₅₃	C₂³	1	64	yes	complemented
49	C₄×C₂²⋊D₅₃	C₂³	1	192	yes	complemented
50	C₂×C₄⋊Dic₅₃	C₂³	1	36	yes	—
51	C₂×#11719919	C₂³	1	12	yes	—
52	C₂×C₂²⋊Dic₅₃	C₂³	1	36	yes	—
53	C₂×#11719921	C₂³	1	36	yes	—
54	C₂×C₄×Dic₅₃	C₂³	1	36	yes	—
55	C₂⁵×C₅₃	C₂³	1	3	yes	—
56	C₂×#41×C₅₃	—	2	192	no	—
57	C₂×#41×C₅₃	C₂³	1	64	yes	complemented
58	C₂²×Q₈×C₅₃	D₄	1	16	yes	complemented
59	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
60	C₂²×D₄×C₅₃	—	2	24	no	—
61	C₂²×D₄×C₅₃	C₂³	1	36	yes	—
62	C₂³×C₄×C₅₃	—	2	24	no	—
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂³	1	36	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
66	C₂⁴×C₅₃	—	2	12	no	—
67	C₂⁴×C₅₃	—	2	12	no	—
68	C₂⁴×C₅₃	—	2	12	no	—
69	C₂⁴×C₅₃	C₂⁴	1	18	yes	—
70	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂³×D₅₃	—	2	12	no	—
72	C₂³×D₅₃	—	2	12	no	—
73	C₂³×D₅₃	—	2	12	no	—
74	C₂³×D₅₃	C₂⁴	1	18	yes	—
75	#41×C₅₃	—	2	256	no	—
76	C₂×Q₈×C₅₃	C₂×D₄	1	96	yes	complemented
77	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
78	C₂×D₄×C₅₃	—	2	144	no	—
79	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
80	C₂²×C₄×C₅₃	—	2	4	no	—
81	C₂²×C₄×C₅₃	—	2	144	no	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
83	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₅₃	C₂⁴	1	48	yes	—
85	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
86	C₂×C₂²⋊D₅₃	—	2	144	no	—
87	C₂×C₂²⋊D₅₃	—	4	144	no	—
88	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
89	C₂×C₂²⋊D₅₃	C₂⁴	1	96	yes	—
90	C₂²×Dic₅₃	—	2	4	no	—
91	C₂²×Dic₅₃	—	2	12	no	—
92	C₂²×Dic₅₃	#41	1	8	yes	complemented
93	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
94	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
95	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
96	#11719949	—	2	256	no	—
97	C₂×D₂₁₂	—	2	144	no	—
98	C₂×D₂₁₂	C₂⁴	1	48	yes	—
99	C₂×C₄×D₅₃	—	2	144	no	—
100	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
101	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
102	#11719921	C₂⁴	1	48	yes	—
103	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
104	#11719919	C₂⁴	1	16	yes	—
105	C₄×Dic₅₃	C₂⁴	1	48	yes	—
106	C₂³×C₅₃	—	2	36	no	—
107	C₂³×C₅₃	—	2	36	no	—
108	C₂³×C₅₃	—	2	36	no	—
109	C₂³×C₅₃	—	4	48	no	—
110	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
111	C₂³×C₅₃	C₂×#41	1	4	yes	—
112	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
113	C₂³×C₅₃	C₂⁵	1	12	yes	—
114	C₂³×C₅₃	C₂⁵	1	3	yes	—
115	C₂²×D₅₃	—	2	36	no	—
116	C₂²×D₅₃	—	2	36	no	—
117	C₂²×D₅₃	—	2	36	no	—
118	C₂²×D₅₃	—	4	48	no	—
119	C₂²×D₅₃	C₂⁵	1	12	yes	—
120	Q₈×C₅₃	C₂²×D₄	1	64	yes	complemented
121	D₄×C₅₃	—	2	96	no	—
122	C₂×C₄×C₅₃	—	2	12	no	—
123	C₂×C₄×C₅₃	—	2	96	no	—
124	C₂×C₄×C₅₃	C₂²×D₄	1	72	yes	—
125	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
126	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
127	C₂²⋊D₅₃	—	2	96	no	—
128	C₂²⋊D₅₃	—	4	96	no	—
129	C₂×Dic₅₃	—	2	12	no	—
130	C₂×Dic₅₃	—	2	36	no	—
131	C₂×Dic₅₃	C₂×#41	1	24	yes	complemented
132	C₂×Dic₅₃	C₂⁵	1	4	yes	—
133	C₂×Dic₅₃	C₂⁵	1	12	yes	—
134	C₂×Dic₅₃	C₂⁵	1	4	yes	—
135	D₂₁₂	—	2	96	no	—
136	C₄×D₅₃	—	2	96	no	—
137	C₂²×#550	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	12	no	—
139	C₂²×C₅₃	—	2	12	no	—
140	C₂²×C₅₃	—	2	12	no	—
141	C₂²×C₅₃	—	4	72	no	—
142	C₂²×C₅₃	#550	1	16	yes	complemented
143	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
144	C₂²×C₅₃	C₂²×#41	1	6	yes	—
145	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
146	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
147	C₂×D₅₃	—	2	12	no	—
148	C₂×D₅₃	—	2	12	no	—
149	C₂×D₅₃	—	2	12	no	—
150	C₂×D₅₃	—	4	72	no	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	C₂³×D₄	1	24	yes	—
153	Dic₅₃	—	2	4	no	—
154	Dic₅₃	—	2	12	no	—
155	Dic₅₃	C₂²×#41	1	8	yes	complemented
156	C₂³×#41	—	53	1	no	—
157	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
158	C₂×#550	—	53	48	no	—
159	C₂²×#127	—	53	3	no	—
160	C₂²×#121	—	53	3	no	—
161	C₂²×#121	—	53	3	no	—
162	C₂²×C₄×Q₈	—	53	1	no	—
163	C₂²×C₄×D₄	—	53	3	no	—
164	C₂×C₅₃	—	4	12	no	—
165	C₂×C₅₃	C₂×#550	1	12	yes	complemented
166	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
167	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
168	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
169	D₅₃	—	4	12	no	—
170	C₂²×#41	—	2	16	no	—
171	C₂²×#41	—	53	24	no	—
172	C₂²×#41	—	106	16	no	—
173	C₂²×#41	C₂×D₅₃	1	24	yes	complemented
174	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
175	C₂³×D₄	—	53	3	no	—
176	C₂³×D₄	—	53	6	no	—
177	C₂³×D₄	—	53	3	no	—
178	C₂³×D₄	C₂×D₅₃	1	3	yes	—
179	C₂⁴×C₄	—	53	3	no	—
180	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
181	#550	—	53	256	no	—
182	C₂×#127	—	53	72	no	—
183	C₂×#121	—	53	72	no	—
184	C₂×#121	—	53	72	no	—
185	C₂×C₄×Q₈	—	53	24	no	—
186	C₂×C₄×D₄	—	53	72	no	—
187	C₂²×C₄⋊C₄	—	53	1	no	—
188	C₂²×C₄⋊C₄	—	53	3	no	—
189	C₂²×C₂²⋊C₄	—	53	3	no	—
190	C₂²×C₂²⋊C₄	—	53	3	no	—
191	C₂²×C₄²	—	53	3	no	—
192	C₅₃	C₂²×#550	1	1	yes	characteristic, complemented, Sylow
193	C₂⁵	—	53	3	no	—
194	C₂⁵	C₂²×D₅₃	1	3	yes	—
195	C₂×#41	—	2	192	no	—
196	C₂×#41	—	53	64	no	—
197	C₂×#41	—	106	192	no	—
198	C₂×#41	C₂²×D₅₃	1	64	yes	complemented
199	C₂²×Q₈	C₂²⋊D₅₃	1	16	yes	complemented
200	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
201	C₂²×D₄	—	2	24	no	—
202	C₂²×D₄	—	53	36	no	—
203	C₂²×D₄	—	53	72	no	—
204	C₂²×D₄	—	53	36	no	—
205	C₂²×D₄	—	106	24	no	—
206	C₂²×D₄	—	106	24	no	—
207	C₂²×D₄	—	212	24	no	—
208	C₂²×D₄	C₂²×D₅₃	1	36	yes	—
209	C₂³×C₄	—	2	24	no	—
210	C₂³×C₄	—	53	1	no	—
211	C₂³×C₄	—	53	1	no	—
212	C₂³×C₄	—	53	3	no	—
213	C₂³×C₄	—	53	36	no	—
214	C₂³×C₄	—	106	24	no	—
215	C₂³×C₄	C₂²×D₅₃	1	36	yes	—
216	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
217	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
218	#127	—	53	192	no	—
219	#121	—	53	192	no	—
220	#121	—	53	192	no	—
221	C₄×Q₈	—	53	64	no	—
222	C₄×D₄	—	53	192	no	—
223	C₂×C₄⋊C₄	—	53	12	no	—
224	C₂×C₄⋊C₄	—	53	36	no	—
225	C₂×C₂²⋊C₄	—	53	36	no	—
226	C₂×C₂²⋊C₄	—	53	36	no	—
227	C₂×C₄²	—	53	36	no	—
228	C₂⁴	—	2	12	no	—
229	C₂⁴	—	2	12	no	—
230	C₂⁴	—	2	12	no	—
231	C₂⁴	—	53	18	no	—
232	C₂⁴	—	106	12	no	—
233	C₂⁴	—	106	12	no	—
234	C₂⁴	—	106	12	no	—
235	C₂⁴	C₂³×D₅₃	1	18	yes	—
236	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
237	#41	—	2	256	no	—
238	#41	—	106	256	no	—
239	C₂×Q₈	C₂×C₂²⋊D₅₃	1	96	yes	complemented
240	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
241	C₂×D₄	—	2	144	no	—
242	C₂×D₄	—	53	48	no	—
243	C₂×D₄	—	53	96	no	—
244	C₂×D₄	—	53	48	no	—
245	C₂×D₄	—	106	144	no	—
246	C₂×D₄	—	106	144	no	—
247	C₂×D₄	—	212	144	no	—
248	C₂×D₄	C₂³×D₅₃	1	48	yes	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	144	no	—
251	C₂²×C₄	—	53	8	no	—
252	C₂²×C₄	—	53	6	no	—
253	C₂²×C₄	—	53	18	no	—
254	C₂²×C₄	—	53	6	no	—
255	C₂²×C₄	—	53	48	no	—
256	C₂²×C₄	—	106	4	no	—
257	C₂²×C₄	—	106	12	no	—
258	C₂²×C₄	—	106	144	no	—
259	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
260	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
261	C₂²×C₄	C₂³×D₅₃	1	48	yes	—
262	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
263	C₄⋊C₄	—	53	16	no	—
264	C₄⋊C₄	—	53	48	no	—
265	C₂²⋊C₄	—	53	48	no	—
266	C₂²⋊C₄	—	53	48	no	—
267	C₄²	—	53	48	no	—
268	C₂³	—	2	36	no	—
269	C₂³	—	2	36	no	—
270	C₂³	—	2	36	no	—
271	C₂³	—	4	48	no	—
272	C₂³	—	53	12	no	—
273	C₂³	—	106	36	no	—
274	C₂³	—	106	36	no	—
275	C₂³	—	106	36	no	—
276	C₂³	—	212	48	no	—
277	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
278	C₂³	#41×D₅₃	1	4	yes	—
279	C₂³	#13133263	1	4	yes	—
280	C₂³	C₂⁴×D₅₃	1	12	yes	—
281	C₂³	C₂⁴×D₅₃	1	3	yes	—
282	Q₈	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
283	D₄	—	2	96	no	—
284	D₄	—	106	96	no	—
285	D₄	—	106	96	no	—
286	D₄	—	212	96	no	—
287	C₂×C₄	—	2	12	no	—
288	C₂×C₄	—	2	96	no	—
289	C₂×C₄	—	53	24	no	—
290	C₂×C₄	—	53	4	no	—
291	C₂×C₄	—	53	12	no	—
292	C₂×C₄	—	53	4	no	—
293	C₂×C₄	—	106	12	no	—
294	C₂×C₄	—	106	36	no	—
295	C₂×C₄	—	106	96	no	—
296	C₂×C₄	C₂²×C₂²⋊D₅₃	1	72	yes	—
297	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
298	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
299	C₂²	—	2	12	no	—
300	C₂²	—	2	12	no	—
301	C₂²	—	2	12	no	—
302	C₂²	—	4	72	no	—
303	C₂²	—	106	12	no	—
304	C₂²	—	106	12	no	—
305	C₂²	—	106	12	no	—
306	C₂²	—	212	72	no	—
307	C₂²	#16811292	1	16	yes	direct product factor
308	C₂²	C₂³×C₂²⋊D₅₃	1	6	yes	—
309	C₂²	C₂×#41×D₅₃	1	6	yes	—
310	C₂²	C₂×#13133263	1	6	yes	—
311	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	53	8	no	—
314	C₄	—	106	4	no	—
315	C₄	—	106	12	no	—
316	C₄	C₂³×C₂²⋊D₅₃	1	24	yes	—
317	C₂	—	4	12	no	—
318	C₂	—	212	12	no	—
319	C₂	C₂×#16811292	1	12	yes	direct product factor
320	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
321	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic
322	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂²×#550
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719949	C₂	semidirect
2. C₂×#16811292	C₂	direct (N × H)
3. C₂²×#13133213	C₂	semidirect
4. C₂²×#13133191	C₂	semidirect
5. C₂²×#13133188	C₂	semidirect
6. C₂²×Q₈×Dic₅₃	C₂	semidirect
7. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
8. C₂³×#41×C₅₃	C₂	semidirect
10. C₂²×#11719949	C₂²	semidirect
15. #16811292	C₂²	direct (N × H)
16. C₂×#13133213	C₂²	semidirect
17. C₂×#13133191	C₂²	semidirect
18. C₂×#13133188	C₂²	semidirect
19. C₂×Q₈×Dic₅₃	C₂²	semidirect
20. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
27. C₂²×#41×C₅₃	C₂²	semidirect
33. C₂×#11719949	C₂³	semidirect
45. #13133213	C₂³	semidirect
46. #13133191	C₂³	semidirect
47. #13133188	C₂³	semidirect
48. Q₈×Dic₅₃	C₂³	semidirect
49. C₄×C₂²⋊D₅₃	C₂³	semidirect
57. C₂×#41×C₅₃	C₂³	semidirect
58. C₂²×Q₈×C₅₃	D₄	semidirect
76. C₂×Q₈×C₅₃	C₂×D₄	semidirect
92. C₂²×Dic₅₃	#41	semidirect
120. Q₈×C₅₃	C₂²×D₄	semidirect
131. C₂×Dic₅₃	C₂×#41	semidirect
142. C₂²×C₅₃	#550	semidirect
155. Dic₅₃	C₂²×#41	semidirect
157. C₂³×#41	D₅₃	semidirect
165. C₂×C₅₃	C₂×#550	semidirect
173. C₂²×#41	C₂×D₅₃	semidirect
192. C₅₃	C₂²×#550	semidirect
198. C₂×#41	C₂²×D₅₃	semidirect
199. C₂²×Q₈	C₂²⋊D₅₃	semidirect
239. C₂×Q₈	C₂×C₂²⋊D₅₃	semidirect
282. Q₈	C₂²×C₂²⋊D₅₃	semidirect
307. C₂²	#16811292	direct (N × H)
319. C₂	C₂×#16811292	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×#16811292	2
C₂²×#13133213	2
C₂²×#13133191	2
C₂²×#13133188	2
C₂²×Q₈×Dic₅₃	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂³×#41×C₅₃	2
C₂²×#550	53

Subgroups — C22×#16811292 #28291041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811292 #28291041