Subgroups — C₄×C₈×D₂₁₂ #27382407

Order 13568.

1422 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×C₈×D₂₁₂	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₄²×D₂₁₂	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal, complemented
5	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11719921	C₂	1	4	yes	maximal, complemented
7	C₄×#13133129	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₃	C₂²	1	4	yes	—
10	C₂×C₄×D₂₁₂	C₄	1	2	yes	—
11	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₅₃	C₂²	1	2	yes	—
13	C₈×D₂₁₂	C₄	1	16	yes	direct product factor
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×#13133129	C₂²	1	4	yes	—
16	C₂×C₄×#88322	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
18	C₄×C₈×D₅₃	—	2	8	no	—
19	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
21	C₄×#11719921	C₄	1	4	yes	—
22	C₄×#11719921	C₂²	1	2	yes	—
23	#16810378	C₂²	1	2	yes	—
24	#16810377	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄³×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
33	C₂²×#88322	C₂³	1	4	yes	—
34	C₂×C₈×D₅₃	—	2	8	no	—
35	C₂×C₈×D₅₃	—	2	8	no	—
36	C₂×C₈×D₅₃	C₂×C₄	1	16	yes	complemented
37	C₄×D₂₁₂	C₈	1	16	yes	direct product factor
38	C₄×D₂₁₂	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
41	C₂×#11719921	C₂×C₄	1	2	yes	—
42	C₂×#11719921	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
45	C₄²×D₅₃	—	2	4	no	—
46	C₄²×D₅₃	—	2	4	no	—
47	C₄⋊#88322	C₂×C₄	1	8	yes	complemented
48	#13133129	C₂×C₄	1	16	yes	complemented
49	C₄×#88322	—	2	4	no	—
50	C₈×Dic₅₃	—	2	8	no	—
51	#13133124	C₂×C₄	1	2	yes	—
52	#13133124	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₃	—	2	4	no	—
61	C₄×C₈×C₅₃	—	2	2	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
63	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
64	C₂³×D₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
71	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
72	C₂×D₂₁₂	C₄²	1	4	yes	—
73	C₂×D₂₁₂	C₂×C₈	1	8	yes	—
74	C₂×C₄×D₅₃	—	2	8	no	—
75	C₂×C₄×D₅₃	—	2	8	no	—
76	C₂×C₄×D₅₃	—	2	2	no	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	—	2	4	no	—
79	C₂×C₄×D₅₃	C₂×C₈	1	16	yes	complemented
80	C₂×C₄×D₅₃	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₅₃	#41	1	4	yes	—
87	C₄²×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	C₂×C₈	1	8	yes	complemented
90	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
91	C₄²×C₅₃	C₂×D₄	1	2	yes	—
92	C₄²×C₅₃	#41	1	2	yes	—
93	#11719921	C₄²	1	8	yes	—
94	#11719921	C₂×C₈	1	16	yes	complemented
95	C₄⋊Dic₅₃	C₄²	1	4	yes	—
96	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	complemented
97	C₄×Dic₅₃	—	2	8	no	—
98	C₄×Dic₅₃	—	2	2	no	—
99	C₄×Dic₅₃	—	2	2	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	C₂²×C₄	1	8	yes	—
103	C₈×D₅₃	—	2	32	no	—
104	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
105	C₂²×D₅₃	—	2	2	no	—
106	C₂²×D₅₃	—	2	2	no	—
107	C₂²×D₅₃	—	2	4	no	—
108	C₂²×D₅₃	C₂×C₄²	1	4	yes	—
109	C₂²×D₅₃	C₂²×C₈	1	8	yes	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
118	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
119	C₂×C₄×C₅₃	#131	1	4	yes	—
120	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	2	4	no	—
128	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
129	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
130	D₂₁₂	C₄×C₈	1	16	yes	direct product factor
131	C₄×D₅₃	—	2	8	no	—
132	C₄×D₅₃	—	2	8	no	—
133	C₄×D₅₃	—	2	32	no	—
134	C₈×C₅₃	—	2	4	no	—
135	C₈×C₅₃	C₄×D₄	1	8	yes	complemented
136	#88322	—	2	8	no	—
137	C₄×C₈×D₄	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
141	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
142	C₂²×C₅₃	C₂×#131	1	2	yes	—
143	C₂×D₅₃	—	2	4	no	—
144	C₂×D₅₃	—	2	8	no	—
145	C₂×D₅₃	—	2	8	no	—
146	C₂×D₅₃	—	2	4	no	—
147	C₂×D₅₃	C₂×C₄×C₈	1	8	yes	complemented
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	4	no	—
150	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	complemented
151	C₄×C₅₃	C₈×D₄	1	8	yes	complemented
152	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
153	C₄×C₅₃	C₄×#41	1	2	yes	—
154	Dic₅₃	—	2	8	no	—
155	Dic₅₃	—	2	4	no	—
156	C₂×C₈×D₄	—	53	2	no	—
157	C₂²×C₄×C₈	—	53	2	no	—
158	C₄²×D₄	—	53	1	no	—
159	C₈×C₄⋊C₄	—	53	2	no	—
160	C₄×C₄⋊C₈	—	53	1	no	—
161	C₈×C₂²⋊C₄	—	53	4	no	—
162	C₄×C₂²⋊C₈	—	53	2	no	—
163	C₄²×C₈	D₅₃	1	1	yes	characteristic, complemented
164	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
165	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
166	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
167	C₂×C₅₃	C₂×C₈×D₄	1	2	yes	—
168	C₂×C₅₃	C₈×#41	1	2	yes	—
169	D₅₃	—	2	8	no	—
170	C₂³×C₈	—	53	4	no	—
171	C₂×C₄×D₄	—	53	2	no	—
172	C₂×C₄×D₄	—	53	1	no	—
173	C₂²×C₄²	—	53	2	no	—
174	C₈×D₄	—	53	16	no	—
175	C₂×C₄⋊C₈	—	53	2	no	—
176	C₂×C₂²⋊C₈	—	53	4	no	—
177	C₂×C₄×C₈	—	53	4	no	—
178	C₂×C₄×C₈	—	53	2	no	—
179	C₂×C₄×C₈	—	106	8	no	—
180	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
182	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
183	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	53	2	no	—
185	C₄×C₄⋊C₄	—	53	1	no	—
186	C₄×C₂²⋊C₄	—	53	4	no	—
187	C₄×C₂²⋊C₄	—	53	2	no	—
188	C₄³	C₂×D₅₃	1	1	yes	characteristic
189	#336	—	53	4	no	—
190	#336	—	53	2	no	—
191	C₅₃	C₄×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	53	1	no	—
193	C₂³×C₄	—	53	4	no	—
194	C₂³×C₄	—	53	2	no	—
195	C₂²×C₈	—	53	4	no	—
196	C₂²×C₈	—	53	16	no	—
197	C₂²×C₈	—	106	8	no	—
198	C₂²×C₈	—	106	8	no	—
199	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
200	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
201	C₄×D₄	—	53	8	no	—
202	C₄×D₄	—	53	16	no	—
203	C₂×C₄⋊C₄	—	53	1	no	—
204	C₂×C₄⋊C₄	—	53	2	no	—
205	C₂×C₂²⋊C₄	—	53	2	no	—
206	C₂×C₂²⋊C₄	—	53	4	no	—
207	C₂×C₄²	—	53	4	no	—
208	C₂×C₄²	—	53	2	no	—
209	C₂×C₄²	—	106	4	no	—
210	C₂×C₄²	—	106	4	no	—
211	C₂×C₄²	C₄×D₅₃	1	2	yes	—
212	C₂×C₄²	C₄×D₅₃	1	2	yes	—
213	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	53	8	no	—
217	C₂²⋊C₈	—	53	16	no	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	106	8	no	—
221	C₄×C₈	—	106	4	no	—
222	C₄×C₈	C₄×D₅₃	1	8	yes	complemented
223	C₄×C₈	D₂₁₂	1	4	yes	direct product factor
224	#95	—	53	2	no	—
225	#95	—	53	4	no	—
226	C₂⁴	—	53	2	no	—
227	C₂×D₄	—	53	4	no	—
228	C₂×D₄	—	53	8	no	—
229	C₂²×C₄	—	53	2	no	—
230	C₂²×C₄	—	53	4	no	—
231	C₂²×C₄	—	53	16	no	—
232	C₂²×C₄	—	53	8	no	—
233	C₂²×C₄	—	106	8	no	—
234	C₂²×C₄	—	106	8	no	—
235	C₂²×C₄	—	106	2	no	—
236	C₂²×C₄	—	106	2	no	—
237	C₂²×C₄	—	106	4	no	—
238	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
242	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	53	8	no	—
246	C₂×C₈	—	106	4	no	—
247	C₂×C₈	—	106	4	no	—
248	C₂×C₈	—	106	32	no	—
249	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
250	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
251	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
252	C₂×C₈	#11719945	1	4	yes	—
253	C₄⋊C₄	—	53	4	no	—
254	C₄⋊C₄	—	53	8	no	—
255	C₂²⋊C₄	—	53	8	no	—
256	C₂²⋊C₄	—	53	16	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	106	8	no	—
260	C₄²	—	106	2	no	—
261	C₄²	—	106	2	no	—
262	C₄²	C₈×D₅₃	1	8	yes	complemented
263	C₄²	C₂×C₄×D₅₃	1	4	yes	—
264	C₄²	C₂×D₂₁₂	1	2	yes	—
265	C₄²	#11719945	1	2	yes	—
266	C₂³	—	53	4	no	—
267	C₂³	—	53	8	no	—
268	C₂³	—	106	2	no	—
269	C₂³	—	106	2	no	—
270	C₂³	—	106	4	no	—
271	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
272	D₄	—	53	16	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	53	4	no	—
277	C₂×C₄	—	53	8	no	—
278	C₂×C₄	—	106	2	no	—
279	C₂×C₄	—	106	4	no	—
280	C₂×C₄	—	106	2	no	—
281	C₂×C₄	—	106	4	no	—
282	C₂×C₄	—	106	8	no	—
283	C₂×C₄	—	106	8	no	—
284	C₂×C₄	—	106	32	no	—
285	C₂×C₄	C₄²×D₅₃	1	2	yes	—
286	C₂×C₄	C₄²×D₅₃	1	2	yes	—
287	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
288	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
289	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
290	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
291	C₂×C₄	#13133223	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
293	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719945	1	2	yes	—
295	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
296	C₈	—	2	4	no	—
297	C₈	—	106	8	no	—
298	C₈	C₄×D₂₁₂	1	8	yes	direct product factor
299	C₂²	—	53	8	no	—
300	C₂²	—	106	8	no	—
301	C₂²	—	106	4	no	—
302	C₂²	—	106	8	no	—
303	C₂²	—	106	4	no	—
304	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
305	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
306	C₂²	C₄×#11719945	1	1	yes	characteristic
307	C₂²	C₂²×C₈×D₅₃	1	2	yes	—
308	C₂²	C₂×#13133223	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	106	4	no	—
312	C₄	—	106	8	no	—
313	C₄	C₄×C₈×D₅₃	1	4	yes	complemented
314	C₄	C₈×D₂₁₂	1	8	yes	direct product factor
315	C₄	C₂×C₄×D₂₁₂	1	2	yes	—
316	C₄	C₄×#11719945	1	2	yes	—
317	C₂	—	106	8	no	—
318	C₂	C₄²×D₂₁₂	1	1	yes	characteristic
319	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
320	C₂	C₄×#13133223	1	1	yes	characteristic
321	C₂	C₂×C₈×D₂₁₂	1	2	yes	—
322	C₂	C₈×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	C₄×C₈×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₈×D₅₃	C₂	semidirect
4. C₈×C₄⋊Dic₅₃	C₂	semidirect
5. C₄×C₄⋊#88322	C₂	semidirect
6. C₈×#11719921	C₂	semidirect
7. C₄×#13133129	C₂	semidirect
8. C₄²×C₈×C₅₃	C₂	semidirect
13. C₈×D₂₁₂	C₄	direct (N × H)
36. C₂×C₈×D₅₃	C₂×C₄	semidirect
37. C₄×D₂₁₂	C₈	direct (N × H)
47. C₄⋊#88322	C₂×C₄	semidirect
48. #13133129	C₂×C₄	semidirect
62. C₄×C₈×C₅₃	C₂×C₄	semidirect
63. C₄×C₈×C₅₃	D₄	semidirect
79. C₂×C₄×D₅₃	C₂×C₈	semidirect
89. C₄²×C₅₃	C₂×C₈	semidirect
94. #11719921	C₂×C₈	semidirect
96. C₄⋊Dic₅₃	C₂×C₈	semidirect
130. D₂₁₂	C₄×C₈	direct (N × H)
135. C₈×C₅₃	C₄×D₄	semidirect
147. C₂×D₅₃	C₂×C₄×C₈	semidirect
150. C₄×C₅₃	C₂×C₄×C₈	semidirect
151. C₄×C₅₃	C₈×D₄	semidirect
163. C₄²×C₈	D₅₃	semidirect
191. C₅₃	C₄×C₈×D₄	semidirect
222. C₄×C₈	C₄×D₅₃	semidirect
223. C₄×C₈	D₂₁₂	direct (N × H)
262. C₄²	C₈×D₅₃	semidirect
298. C₈	C₄×D₂₁₂	direct (N × H)
313. C₄	C₄×C₈×D₅₃	semidirect
314. C₄	C₈×D₂₁₂	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×C₈×D₂₁₂	2
C₂×C₄×C₈×D₅₃	2
C₄²×D₂₁₂	2
C₈×C₄⋊Dic₅₃	2
C₄×C₄⋊#88322	2
C₈×#11719921	2
C₄×#13133129	2
C₄²×C₈×C₅₃	2
C₄×C₈×D₄	53

Subgroups — C4×C8×D212 #27382407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×D₂₁₂ #27382407