Subgroups — C₂×C₈×#11719949 #28248525

Order 13568.

2566 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	C₂×C₄×#11719949	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719949	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₂₁₂	C₂	1	3	yes	maximal, complemented
5	C₂×#16810688	C₂	1	3	yes	maximal, complemented
6	C₂×#16810675	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×C₈×D₅₃	C₂	1	3	yes	maximal, complemented
8	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719949	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅₃	C₂²	1	3	yes	—
11	C₂²×C₈×D₅₃	C₂²	1	3	yes	—
12	C₄×#11719949	C₄	1	8	yes	—
13	C₄×#11719949	C₂²	1	8	yes	—
14	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₂₁₂	C₂²	1	3	yes	—
16	C₂×#13133179	C₄	1	3	yes	—
17	C₂×#13133175	C₄	1	3	yes	—
18	C₂×C₄²×D₅₃	C₂²	1	3	yes	—
19	Q₈×#88322	C₂²	1	8	yes	complemented
20	C₈×D₂₁₂	C₂²	1	24	yes	complemented
21	#16810688	C₂²	1	24	yes	complemented
22	#16810675	C₂²	1	24	yes	complemented
23	C₂×C₄⋊#88322	C₂²	1	3	yes	—
24	C₂×#13133129	C₂²	1	6	yes	—
25	C₂×C₄×#88322	C₂²	1	3	yes	—
26	C₂×C₈×Dic₅₃	C₂²	1	3	yes	—
27	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₅₃	C₂²	1	24	yes	complemented
29	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₅₃	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
33	C₂×#11719949	C₈	1	16	yes	direct product factor
34	C₂×#11719949	C₂×C₄	1	8	yes	—
35	C₂²×D₂₁₂	C₂×C₄	1	3	yes	—
36	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
37	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
38	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
39	C₂²×#88322	C₂³	1	3	yes	—
40	C₂²×#88322	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₅₃	—	2	12	no	—
42	C₂×C₈×D₅₃	—	2	12	no	—
43	C₂×C₈×D₅₃	C₂³	1	12	yes	—
44	C₂×C₈×D₅₃	C₂³	1	12	yes	—
45	Q₈×Dic₅₃	C₂×C₄	1	8	yes	—
46	C₄×D₂₁₂	C₂×C₄	1	12	yes	—
47	C₄×D₂₁₂	C₂³	1	12	yes	—
48	#13133179	C₂×C₄	1	24	yes	—
49	#13133175	C₂×C₄	1	24	yes	—
50	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
51	C₂×#11719921	C₂×C₄	1	6	yes	—
52	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
54	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
55	C₄²×D₅₃	C₂×C₄	1	12	yes	—
56	C₄²×D₅₃	C₂³	1	12	yes	—
57	C₄⋊#88322	C₂³	1	12	yes	—
58	#13133129	C₂³	1	24	yes	—
59	C₄×#88322	C₂³	1	12	yes	—
60	C₈×Dic₅₃	C₂³	1	4	yes	—
61	C₈×Dic₅₃	C₂³	1	12	yes	—
62	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
66	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
69	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
70	C₄×C₈×C₅₃	C₂³	1	12	yes	—
71	C₂³×D₅₃	C₂²×C₄	1	3	yes	—
72	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	complemented
73	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
78	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
80	#11719949	C₂×C₈	1	64	yes	direct product factor
81	C₂×D₂₁₂	C₂×C₈	1	24	yes	complemented
82	C₂×D₂₁₂	C₂²×C₄	1	12	yes	—
83	C₂×C₄×D₅₃	—	2	6	no	—
84	C₂×C₄×D₅₃	—	2	6	no	—
85	C₂×C₄×D₅₃	—	2	12	no	—
86	C₂×C₄×D₅₃	C₂×C₈	1	24	yes	complemented
87	C₂×C₄×D₅₃	C₂²×C₄	1	6	yes	—
88	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
89	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
90	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
91	C₂×C₈×C₅₃	—	2	6	no	—
92	C₂×C₈×C₅₃	#41	1	4	yes	complemented
93	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
94	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
95	C₄⋊C₄×C₅₃	C₂²×C₄	1	12	yes	—
96	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
97	C₄²×C₅₃	C₂⁴	1	6	yes	—
98	#11719921	C₂²×C₄	1	24	yes	—
99	C₄⋊Dic₅₃	C₂²×C₄	1	12	yes	—
100	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
101	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
102	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
103	C₄×Dic₅₃	C₂⁴	1	2	yes	—
104	C₂×#88322	—	2	6	no	—
105	C₂×#88322	#41	1	4	yes	complemented
106	C₂×#88322	C₂⁴	1	6	yes	—
107	C₂×#88322	C₂⁴	1	2	yes	—
108	C₈×D₅₃	—	2	24	no	—
109	C₈×D₅₃	—	2	24	no	—
110	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²×D₅₃	—	2	6	no	—
112	C₂²×D₅₃	—	2	6	no	—
113	C₂²×D₅₃	C₂²×C₈	1	12	yes	—
114	C₂²×D₅₃	C₂³×C₄	1	6	yes	—
115	Q₈×C₅₃	C₂²×C₈	1	16	yes	complemented
116	C₂×C₄×C₅₃	—	2	6	no	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
121	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	6	no	—
125	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
126	C₂×Dic₅₃	#131	1	4	yes	—
127	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
128	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
129	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
130	D₂₁₂	C₂²×C₈	1	48	yes	complemented
131	C₄×D₅₃	—	2	12	no	—
132	C₄×D₅₃	—	2	6	no	—
133	C₄×D₅₃	—	2	6	no	—
134	C₄×D₅₃	—	2	24	no	—
135	C₄×D₅₃	C₂²×C₈	1	48	yes	complemented
136	C₈×C₅₃	—	2	6	no	—
137	C₈×C₅₃	C₂×#41	1	4	yes	complemented
138	#88322	—	2	6	no	—
139	#88322	C₂×#41	1	4	yes	complemented
140	C₂×C₈×#41	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	C₄×#41	1	2	yes	—
142	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
143	C₂²×C₅₃	C₂×#131	1	2	yes	—
144	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
145	C₂×D₅₃	—	2	6	no	—
146	C₂×D₅₃	—	2	24	no	—
147	C₂×D₅₃	—	2	6	no	—
148	C₂×D₅₃	C₂³×C₈	1	12	yes	—
149	C₄×C₅₃	—	2	6	no	—
150	C₄×C₅₃	C₄×#41	1	2	yes	—
151	C₄×C₅₃	C₂³×C₈	1	12	yes	—
152	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
154	Dic₅₃	—	2	6	no	—
155	Dic₅₃	C₂³×C₈	1	4	yes	—
156	Dic₅₃	C₂×#131	1	4	yes	—
157	C₂×C₄×#41	—	53	1	no	—
158	C₈×#41	—	53	16	no	—
159	C₂×C₈×Q₈	—	53	1	no	—
160	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
161	C₂×C₈×D₄	—	53	3	no	—
162	C₂×C₈×D₄	—	53	3	no	—
163	C₂×#431	—	53	3	no	—
164	C₂²×C₄×C₈	—	53	3	no	—
165	C₂×C₅₃	C₈×#41	1	4	yes	complemented
166	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
167	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
168	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
169	D₅₃	—	2	12	no	—
170	C₂²×#41	—	53	1	no	—
171	C₂³×C₈	—	53	3	no	—
172	C₂³×C₈	—	53	3	no	—
173	C₄×#41	—	53	8	no	—
174	C₄×#41	—	53	8	no	—
175	C₂×C₄×Q₈	—	53	1	no	—
176	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
177	C₂×C₄×D₄	—	53	3	no	—
178	C₂×C₄×D₄	—	53	3	no	—
179	C₂×#117	—	53	3	no	—
180	C₂²×C₄²	—	53	3	no	—
181	C₈×Q₈	—	53	8	no	—
182	C₈×Q₈	C₂×D₅₃	1	8	yes	complemented
183	C₈×D₄	—	53	24	no	—
184	C₈×D₄	—	53	24	no	—
185	#431	—	53	24	no	—
186	C₂×C₄⋊C₈	—	53	3	no	—
187	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
188	C₂×C₂²⋊C₈	—	53	6	no	—
189	C₂×C₄×C₈	—	53	3	no	—
190	C₂×C₄×C₈	—	53	1	no	—
191	C₂×C₄×C₈	—	53	3	no	—
192	C₂×C₄×C₈	—	53	24	no	—
193	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
194	C₅₃	C₂×C₈×#41	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	—	53	8	no	—
196	C₂×#41	—	53	16	no	—
197	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×D₄	—	53	3	no	—
199	C₂³×C₄	—	53	3	no	—
200	C₂³×C₄	—	53	3	no	—
201	C₂³×C₄	—	53	3	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	—	53	3	no	—
204	C₂²×C₈	—	53	12	no	—
205	C₂²×C₈	—	53	12	no	—
206	C₂²×C₈	—	106	12	no	—
207	C₂²×C₈	—	106	12	no	—
208	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₄×Q₈	—	53	8	no	—
211	C₄×Q₈	C₄×D₅₃	1	4	yes	—
212	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
213	C₄×D₄	—	53	24	no	—
214	C₄×D₄	—	53	12	no	—
215	C₄×D₄	—	53	12	no	—
216	#117	—	53	24	no	—
217	C₂×C₄⋊C₄	—	53	3	no	—
218	C₂×C₄⋊C₄	C₄×D₅₃	1	3	yes	—
219	C₂×C₂²⋊C₄	—	53	6	no	—
220	C₂×C₄²	—	53	3	no	—
221	C₂×C₄²	—	53	3	no	—
222	C₂×C₄²	—	53	1	no	—
223	C₂×C₄²	—	53	12	no	—
224	C₂×C₄²	—	53	12	no	—
225	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
226	C₄⋊C₈	—	53	12	no	—
227	C₄⋊C₈	C₂²×D₅₃	1	12	yes	—
228	C₂²⋊C₈	—	53	24	no	—
229	C₄×C₈	—	53	4	no	—
230	C₄×C₈	—	53	12	no	—
231	C₄×C₈	—	53	12	no	—
232	C₄×C₈	C₂²×D₅₃	1	12	yes	—
233	C₂⁴	—	53	3	no	—
234	#41	—	53	64	no	—
235	C₂×Q₈	C₈×D₅₃	1	8	yes	complemented
236	C₂×Q₈	C₂×C₄×D₅₃	1	4	yes	—
237	C₂×D₄	—	53	24	no	—
238	C₂×D₄	—	53	12	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	24	no	—
242	C₂²×C₄	—	53	12	no	—
243	C₂²×C₄	—	53	3	no	—
244	C₂²×C₄	—	53	12	no	—
245	C₂²×C₄	—	53	6	no	—
246	C₂²×C₄	—	53	6	no	—
247	C₂²×C₄	—	106	12	no	—
248	C₂²×C₄	—	106	6	no	—
249	C₂²×C₄	—	106	6	no	—
250	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
251	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
252	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
253	C₂×C₈	—	2	6	no	—
254	C₂×C₈	—	53	4	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	6	no	—
257	C₂×C₈	—	106	24	no	—
258	C₂×C₈	—	106	6	no	—
259	C₂×C₈	—	106	24	no	—
260	C₂×C₈	#11719949	1	4	yes	direct product factor
261	C₂×C₈	C₂³×D₅₃	1	6	yes	—
262	C₂×C₈	C₂³×D₅₃	1	2	yes	—
263	C₄⋊C₄	—	53	12	no	—
264	C₄⋊C₄	C₂×C₄×D₅₃	1	12	yes	—
265	C₂²⋊C₄	—	53	24	no	—
266	C₄²	—	53	2	no	—
267	C₄²	—	53	12	no	—
268	C₄²	—	53	12	no	—
269	C₄²	—	53	2	no	—
270	C₄²	C₂×C₄×D₅₃	1	6	yes	—
271	C₄²	C₂³×D₅₃	1	6	yes	—
272	C₂³	—	53	12	no	—
273	C₂³	—	53	6	no	—
274	C₂³	—	106	6	no	—
275	C₂³	—	106	6	no	—
276	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
277	Q₈	C₂×C₈×D₅₃	1	16	yes	complemented
278	D₄	—	53	48	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	53	4	no	—
281	C₂×C₄	—	53	4	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	53	48	no	—
285	C₂×C₄	—	53	6	no	—
286	C₂×C₄	—	106	24	no	—
287	C₂×C₄	—	106	6	no	—
288	C₂×C₄	—	106	12	no	—
289	C₂×C₄	—	106	6	no	—
290	C₂×C₄	—	106	6	no	—
291	C₂×C₄	C₂×C₈×D₅₃	1	12	yes	—
292	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
293	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
295	C₂×C₄	#41×D₅₃	1	2	yes	—
296	C₂×C₄	C₂×#11719949	1	2	yes	—
297	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
298	C₈	—	2	6	no	—
299	C₈	—	53	4	no	—
300	C₈	—	106	6	no	—
301	C₈	C₂×#11719949	1	4	yes	direct product factor
302	C₂²	—	53	12	no	—
303	C₂²	—	106	24	no	—
304	C₂²	—	106	6	no	—
305	C₂²	—	106	6	no	—
306	C₂²	C₄×#11719949	1	2	yes	—
307	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
308	C₂²	#131×D₅₃	1	2	yes	—
309	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
310	C₄	—	2	6	no	—
311	C₄	—	53	4	no	—
312	C₄	—	53	4	no	—
313	C₄	—	106	6	no	—
314	C₄	C₄×#11719949	1	2	yes	—
315	C₄	C₂²×C₈×D₅₃	1	12	yes	—
316	C₄	C₂×#41×D₅₃	1	1	yes	characteristic
317	C₄	C₂²×#11719949	1	1	yes	characteristic
318	C₂	—	106	12	no	—
319	C₂	C₈×#11719949	1	4	yes	direct product factor
320	C₂	C₂×C₄×#11719949	1	1	yes	characteristic
321	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
322	C₂	C₂×#131×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×#11719949	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	C₂×C₈×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₈×#11719949	C₂	direct (N × H)
3. C₂×Q₈×#88322	C₂	semidirect
4. C₂×C₈×D₂₁₂	C₂	semidirect
5. C₂×#16810688	C₂	semidirect
6. C₂×#16810675	C₂	semidirect
7. C₂×C₄×C₈×D₅₃	C₂	semidirect
8. C₂×C₈×Q₈×C₅₃	C₂	semidirect
19. Q₈×#88322	C₂²	semidirect
20. C₈×D₂₁₂	C₂²	semidirect
21. #16810688	C₂²	semidirect
22. #16810675	C₂²	semidirect
28. C₄×C₈×D₅₃	C₂²	semidirect
30. C₈×Q₈×C₅₃	C₂²	semidirect
33. C₂×#11719949	C₈	direct (N × H)
72. C₂×Q₈×C₅₃	C₂×C₈	semidirect
80. #11719949	C₂×C₈	direct (N × H)
81. C₂×D₂₁₂	C₂×C₈	semidirect
86. C₂×C₄×D₅₃	C₂×C₈	semidirect
92. C₂×C₈×C₅₃	#41	semidirect
105. C₂×#88322	#41	semidirect
115. Q₈×C₅₃	C₂²×C₈	semidirect
130. D₂₁₂	C₂²×C₈	semidirect
135. C₄×D₅₃	C₂²×C₈	semidirect
137. C₈×C₅₃	C₂×#41	semidirect
139. #88322	C₂×#41	semidirect
160. C₂×C₈×Q₈	D₅₃	semidirect
165. C₂×C₅₃	C₈×#41	semidirect
182. C₈×Q₈	C₂×D₅₃	semidirect
194. C₅₃	C₂×C₈×#41	semidirect
235. C₂×Q₈	C₈×D₅₃	semidirect
260. C₂×C₈	#11719949	direct (N × H)
277. Q₈	C₂×C₈×D₅₃	semidirect
301. C₈	C₂×#11719949	direct (N × H)
319. C₂	C₈×#11719949	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#11719949	2
C₈×#11719949	2
C₂×Q₈×#88322	2
C₂×C₈×D₂₁₂	2
C₂×#16810688	2
C₂×#16810675	2
C₂×C₄×C₈×D₅₃	2
C₂×C₈×Q₈×C₅₃	2
C₂×C₈×#41	53

Subgroups — C2×C8×#11719949 #28248525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#11719949 #28248525