Subgroups — C₄⋊C₄×Dic₂₁₂ #27402525

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₀₆	C₂	1	2	yes	maximal
4	#20275448	C₂	1	2	yes	maximal
5	#20274972	C₂	1	4	yes	maximal
6	#20274875	C₂	1	1	yes	maximal, characteristic
7	#20274721	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
12	C₄×Dic₂₁₂	—	2	8	no	—
13	C₄×Dic₂₁₂	C₄	1	8	yes	complemented
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133151	C₂²	1	2	yes	—
16	C₂×#13133151	C₂²	1	1	yes	characteristic
17	C₂×#13133142	C₂²	1	4	yes	—
18	C₂×#13133142	C₂²	1	2	yes	—
19	#16810521	C₂²	1	4	yes	—
20	#16810520	—	2	4	no	—
21	#16810493	C₂²	1	2	yes	—
22	#16810492	—	2	2	no	—
23	C₄⋊C₄×Dic₅₃	—	2	2	no	—
24	#16810386	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	1	yes	characteristic
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
31	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₂₁₂	D₄	1	4	yes	—
33	C₂×Dic₂₁₂	Q₈	1	4	yes	—
34	C₄×Dic₁₀₆	—	2	4	no	—
35	C₄×Dic₁₀₆	—	2	8	no	—
36	C₄×Dic₁₀₆	—	4	4	no	—
37	C₄×Dic₁₀₆	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	—	2	2	no	—
40	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
42	C₂×#11719919	—	2	2	no	—
43	C₂×#11719919	—	2	4	no	—
44	C₂×#11719919	—	2	4	no	—
45	C₂×C₄×Dic₅₃	—	2	2	no	—
46	C₂×C₄×Dic₅₃	—	2	4	no	—
47	#13133151	—	2	4	no	—
48	#13133151	—	4	2	no	—
49	#13133151	C₂×C₄	1	4	yes	—
50	#13133142	—	2	8	no	—
51	#13133142	C₂×C₄	1	8	yes	—
52	#13133124	—	2	4	no	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	2	no	—
61	C₄×C₈×C₅₃	—	2	4	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
63	#95×C₅₃	D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	2	2	no	—
70	C₂²×Dic₅₃	—	2	2	no	—
71	C₂²×Dic₅₃	—	2	4	no	—
72	C₂×Dic₁₀₆	—	2	2	no	—
73	C₂×Dic₁₀₆	—	2	2	no	—
74	C₂×Dic₁₀₆	—	2	2	no	—
75	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
76	C₂×Dic₁₀₆	C₂×D₄	1	4	yes	—
77	C₂×Dic₁₀₆	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	Q₁₆	1	4	yes	complemented
89	C₄²×C₅₃	—	2	2	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₅₃	#41	1	2	yes	—
93	C₄⋊Dic₅₃	—	2	2	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	—	4	4	no	—
97	C₄⋊Dic₅₃	—	8	2	no	—
98	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
99	#11719919	—	2	8	no	—
100	#11719919	—	4	8	no	—
101	#11719919	—	8	4	no	—
102	C₄×Dic₅₃	—	2	8	no	—
103	C₄×Dic₅₃	—	4	8	no	—
104	Dic₂₁₂	C₄⋊C₄	1	16	yes	direct product factor
105	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
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	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
116	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	#135	1	2	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	2	4	no	—
126	C₂×Dic₅₃	—	2	4	no	—
127	C₂×Dic₅₃	—	2	4	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	—	4	2	no	—
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	—	4	4	no	—
132	C₂×Dic₅₃	—	4	4	no	—
133	Dic₁₀₆	—	2	4	no	—
134	Dic₁₀₆	—	2	4	no	—
135	Dic₁₀₆	—	2	4	no	—
136	Dic₁₀₆	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₄⋊C₄×Q₁₆	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₅₃	D₄²	1	1	yes	characteristic
143	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
144	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
146	C₂²×C₅₃	#576	1	1	yes	characteristic
147	C₂²×C₅₃	#578	1	1	yes	characteristic
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	4	no	—
150	C₄×C₅₃	—	4	2	no	—
151	C₄×C₅₃	C₄×Q₁₆	1	4	yes	complemented
152	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₅₃	#520	1	1	yes	characteristic
154	C₄×C₅₃	#547	1	1	yes	characteristic
155	C₄×C₅₃	#554	1	1	yes	characteristic
156	Dic₅₃	—	2	8	no	—
157	Dic₅₃	—	4	4	no	—
158	Dic₅₃	—	8	4	no	—
159	C₂×C₄×Q₁₆	—	53	1	no	—
160	C₂×C₄×Q₁₆	—	53	2	no	—
161	Q₈×C₄⋊C₄	—	53	2	no	—
162	#1942	—	53	2	no	—
163	#1863	—	53	4	no	—
164	#1845	—	53	1	no	—
165	#1803	—	53	2	no	—
166	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
167	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
169	C₂×C₅₃	#2951	1	1	yes	characteristic
170	C₂×C₅₃	D₄×Q₁₆	1	1	yes	characteristic
171	C₂×C₅₃	#3287	1	1	yes	characteristic
172	C₂×C₅₃	Q₈×Q₁₆	1	1	yes	characteristic
173	C₂×C₅₃	#3383	1	1	yes	characteristic
174	C₂²×Q₁₆	—	53	1	no	—
175	C₂×C₄×Q₈	—	53	2	no	—
176	C₂×C₄×Q₈	—	53	4	no	—
177	C₄×Q₁₆	—	53	8	no	—
178	C₄×Q₁₆	—	106	8	no	—
179	C₂×#107	—	53	1	no	—
180	C₂×#107	—	53	2	no	—
181	C₂×#107	—	53	2	no	—
182	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#103	—	53	2	no	—
184	C₂×#103	—	53	4	no	—
185	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
186	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
187	#389	—	53	4	no	—
188	#389	—	106	4	no	—
189	#384	—	53	2	no	—
190	#384	—	106	2	no	—
191	C₄×C₄⋊C₄	—	106	2	no	—
192	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
193	#340	—	53	4	no	—
194	#336	C₂×D₅₃	1	2	yes	—
195	C₅₃	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	53	2	no	—
197	C₂×Q₁₆	—	53	4	no	—
198	C₂×Q₁₆	—	53	4	no	—
199	C₂×Q₁₆	—	53	4	no	—
200	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
201	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₄×Q₈	—	53	8	no	—
204	C₄×Q₈	—	106	8	no	—
205	C₄×Q₈	—	106	4	no	—
206	C₄×Q₈	—	212	4	no	—
207	C₂×C₄⋊C₄	—	53	2	no	—
208	C₂×C₄⋊C₄	—	53	4	no	—
209	C₂×C₄⋊C₄	—	106	2	no	—
210	C₂×C₄⋊C₄	—	106	4	no	—
211	C₂×C₄⋊C₄	—	106	2	no	—
212	C₂×C₄⋊C₄	—	106	4	no	—
213	C₂×C₄⋊C₄	—	106	2	no	—
214	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
215	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
216	C₂×C₄²	—	106	2	no	—
217	C₂×C₄²	—	106	4	no	—
218	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
219	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
220	#107	—	53	4	no	—
221	#107	—	106	4	no	—
222	#107	—	212	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	#103	—	53	8	no	—
225	#103	—	106	8	no	—
226	C₄×C₈	—	2	4	no	—
227	C₄×C₈	C₄×D₅₃	1	4	yes	—
228	#95	—	106	4	no	—
229	#95	D₂₁₂	1	2	yes	—
230	C₂×Q₈	—	53	4	no	—
231	C₂×Q₈	—	53	4	no	—
232	C₂×Q₈	—	53	4	no	—
233	C₂×Q₈	—	106	2	no	—
234	C₂×Q₈	—	106	2	no	—
235	C₂×Q₈	—	106	2	no	—
236	C₂²×C₄	—	106	2	no	—
237	C₂²×C₄	—	106	2	no	—
238	C₂²×C₄	—	106	4	no	—
239	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
240	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
242	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
244	Q₁₆	—	53	16	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
252	C₂×C₈	D₄×D₅₃	1	2	yes	—
253	C₂×C₈	Q₈×D₅₃	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	53	4	no	—
256	C₄⋊C₄	—	106	8	no	—
257	C₄⋊C₄	—	106	2	no	—
258	C₄⋊C₄	—	106	4	no	—
259	C₄⋊C₄	—	106	4	no	—
260	C₄⋊C₄	—	424	4	no	—
261	C₄⋊C₄	—	424	2	no	—
262	C₄⋊C₄	—	212	8	no	—
263	C₄⋊C₄	—	212	4	no	—
264	C₄⋊C₄	Dic₂₁₂	1	4	yes	direct product factor
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	106	8	no	—
268	C₄²	—	212	8	no	—
269	C₄²	C₂×C₄×D₅₃	1	2	yes	—
270	C₄²	#11719945	1	2	yes	—
271	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
272	Q₈	—	53	8	no	—
273	Q₈	—	106	4	no	—
274	Q₈	—	106	4	no	—
275	Q₈	—	106	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	106	4	no	—
284	C₂×C₄	—	106	4	no	—
285	C₂×C₄	—	106	4	no	—
286	C₂×C₄	—	106	4	no	—
287	C₂×C₄	—	212	4	no	—
288	C₂×C₄	—	212	4	no	—
289	C₂×C₄	—	212	4	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
293	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
294	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
295	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
296	C₂×C₄	#13133239	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
301	C₂×C₄	#13133263	1	1	yes	characteristic
302	C₂×C₄	#13133264	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	4	2	no	—
305	C₈	C₄⋊C₄×D₅₃	1	4	yes	—
306	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
307	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
308	C₂²	Q₈×D₂₁₂	1	1	yes	characteristic
309	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
310	C₂²	C₂×#13133239	1	1	yes	characteristic
311	C₂²	#16811428	1	1	yes	characteristic
312	C₂²	#16811440	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	106	8	no	—
317	C₄	—	424	4	no	—
318	C₄	—	212	4	no	—
319	C₄	C₄×Dic₂₁₂	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
321	C₄	#16811103	1	1	yes	characteristic
322	C₄	#16811278	1	1	yes	characteristic
323	C₄	#16811333	1	1	yes	characteristic
324	C₂	C₄⋊C₄×D₂₁₂	1	1	yes	characteristic
325	C₂	C₂×C₄×Dic₂₁₂	1	1	yes	characteristic
326	C₂	#20283592	1	1	yes	characteristic
327	C₂	D₄×Dic₂₁₂	1	1	yes	characteristic
328	C₂	#20286475	1	1	yes	characteristic
329	C₂	Q₈×Dic₂₁₂	1	1	yes	characteristic
330	C₂	#20287607	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₂₁₂	C₄	semidirect
88. C₄⋊C₄×C₅₃	Q₁₆	semidirect
104. Dic₂₁₂	C₄⋊C₄	direct (N × H)
151. C₄×C₅₃	C₄×Q₁₆	semidirect
195. C₅₃	C₄⋊C₄×Q₁₆	semidirect
264. C₄⋊C₄	Dic₂₁₂	direct (N × H)
319. C₄	C₄×Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
C₄⋊C₄×Dic₁₀₆	2
#20275448	2
#20274972	2
#20274875	2
#20274721	2
C₈×C₄⋊C₄×C₅₃	2
C₄⋊C₄×Q₁₆	53

Subgroups — C4⋊C4×Dic212 #27402525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×Dic₂₁₂ #27402525