Subgroups — C₄⋊C₄×Dic₅₂ #15881352

Order 3328.

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×Dic₅₂	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₂₆	C₂	1	2	yes	maximal
4	#13115569	C₂	1	2	yes	maximal
5	#13115093	C₂	1	4	yes	maximal
6	#13114996	C₂	1	1	yes	maximal, characteristic
7	#13114842	C₂	1	2	yes	maximal
8	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₂²	1	4	yes	—
11	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	C₂×#88056	C₂²	1	4	yes	—
20	C₂×#88056	C₂²	1	2	yes	—
21	#11718637	—	2	2	no	—
22	C₄⋊C₄×Dic₁₃	—	2	2	no	—
23	#11718596	C₂²	1	2	yes	—
24	#11718589	C₂²	1	4	yes	—
25	C₄×Dic₅₂	—	2	8	no	—
26	C₄×Dic₅₂	C₄	1	8	yes	complemented
27	#11718308	—	2	4	no	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	#11718143	C₂²	1	4	yes	—
30	C₂²×Dic₂₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
42	C₂×#6101	—	2	2	no	—
43	C₂×#6101	—	2	4	no	—
44	C₂×#6101	—	2	4	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	—	2	4	no	—
47	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
48	C₂×Dic₅₂	D₄	1	4	yes	—
49	C₂×Dic₅₂	Q₈	1	4	yes	—
50	C₄×Dic₂₆	—	2	4	no	—
51	C₄×Dic₂₆	—	2	8	no	—
52	C₄×Dic₂₆	—	4	4	no	—
53	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₄×C₈×C₁₃	—	2	4	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
57	C₁₃×#95	D₄	1	2	yes	—
58	#88071	—	2	4	no	—
59	#88058	—	2	4	no	—
60	#88058	—	4	2	no	—
61	#88058	C₂×C₄	1	4	yes	—
62	#88056	—	2	8	no	—
63	#88056	C₂×C₄	1	8	yes	—
64	C₄⋊C₄×Q₁₆	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	—	2	2	no	—
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	2	no	—
75	C₂×Dic₂₆	—	2	2	no	—
76	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
77	C₂×Dic₂₆	C₂×D₄	1	4	yes	—
78	C₂×Dic₂₆	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	Q₁₆	1	4	yes	complemented
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	C₄⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	4	no	—
98	C₄⋊Dic₁₃	—	8	2	no	—
99	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
100	#6101	—	2	8	no	—
101	#6101	—	4	8	no	—
102	#6101	—	8	4	no	—
103	C₄×Dic₁₃	—	2	8	no	—
104	C₄×Dic₁₃	—	4	8	no	—
105	Dic₅₂	C₄⋊C₄	1	16	yes	direct product factor
106	C₂×C₄×Q₁₆	—	13	1	no	—
107	C₂×C₄×Q₁₆	—	13	2	no	—
108	Q₈×C₄⋊C₄	—	13	2	no	—
109	#1942	—	13	2	no	—
110	#1863	—	13	4	no	—
111	#1845	—	13	1	no	—
112	#1803	—	13	2	no	—
113	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
114	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
125	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	#135	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	4	no	—
134	C₂×Dic₁₃	—	2	4	no	—
135	C₂×Dic₁₃	—	2	4	no	—
136	C₂×Dic₁₃	—	2	4	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	4	4	no	—
141	C₂×Dic₁₃	—	4	4	no	—
142	Dic₂₆	—	2	4	no	—
143	Dic₂₆	—	2	4	no	—
144	Dic₂₆	—	2	4	no	—
145	Dic₂₆	C₂×C₄⋊C₄	1	8	yes	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	4	2	no	—
148	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
149	C₂²×Q₁₆	—	13	1	no	—
150	C₂×C₄×Q₈	—	13	2	no	—
151	C₂×C₄×Q₈	—	13	4	no	—
152	C₄×Q₁₆	—	13	8	no	—
153	C₄×Q₁₆	—	26	8	no	—
154	C₂×#107	—	13	1	no	—
155	C₂×#107	—	13	2	no	—
156	C₂×#107	—	13	2	no	—
157	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
158	C₂×#103	—	13	2	no	—
159	C₂×#103	—	13	4	no	—
160	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	#389	—	13	4	no	—
163	#389	—	26	4	no	—
164	#384	—	13	2	no	—
165	#384	—	26	2	no	—
166	C₄×C₄⋊C₄	—	26	2	no	—
167	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
168	#340	—	13	4	no	—
169	#336	C₂×D₁₃	1	2	yes	—
170	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₁₃	D₄²	1	1	yes	characteristic
172	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
173	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
175	C₂²×C₁₃	#576	1	1	yes	characteristic
176	C₂²×C₁₃	#578	1	1	yes	characteristic
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	—	4	2	no	—
180	C₄×C₁₃	C₄×Q₁₆	1	4	yes	complemented
181	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
182	C₄×C₁₃	#520	1	1	yes	characteristic
183	C₄×C₁₃	#547	1	1	yes	characteristic
184	C₄×C₁₃	#554	1	1	yes	characteristic
185	Dic₁₃	—	2	8	no	—
186	Dic₁₃	—	4	4	no	—
187	Dic₁₃	—	8	4	no	—
188	C₂²×Q₈	—	13	2	no	—
189	C₂×Q₁₆	—	13	4	no	—
190	C₂×Q₁₆	—	13	4	no	—
191	C₂×Q₁₆	—	13	4	no	—
192	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×Q₈	—	13	8	no	—
196	C₄×Q₈	—	26	8	no	—
197	C₄×Q₈	—	26	4	no	—
198	C₄×Q₈	—	52	4	no	—
199	C₂×C₄⋊C₄	—	13	2	no	—
200	C₂×C₄⋊C₄	—	13	4	no	—
201	C₂×C₄⋊C₄	—	26	2	no	—
202	C₂×C₄⋊C₄	—	26	4	no	—
203	C₂×C₄⋊C₄	—	26	2	no	—
204	C₂×C₄⋊C₄	—	26	4	no	—
205	C₂×C₄⋊C₄	—	26	2	no	—
206	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	—	26	4	no	—
210	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	#107	—	13	4	no	—
213	#107	—	26	4	no	—
214	#107	—	52	2	no	—
215	C₄⋊C₈	—	2	2	no	—
216	#103	—	13	8	no	—
217	#103	—	26	8	no	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	C₄×D₁₃	1	4	yes	—
220	#95	—	26	4	no	—
221	#95	D₅₂	1	2	yes	—
222	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
224	C₂×C₁₃	#2951	1	1	yes	characteristic
225	C₂×C₁₃	D₄×Q₁₆	1	1	yes	characteristic
226	C₂×C₁₃	#3287	1	1	yes	characteristic
227	C₂×C₁₃	Q₈×Q₁₆	1	1	yes	characteristic
228	C₂×C₁₃	#3383	1	1	yes	characteristic
229	C₂×Q₈	—	13	4	no	—
230	C₂×Q₈	—	13	4	no	—
231	C₂×Q₈	—	13	4	no	—
232	C₂×Q₈	—	26	2	no	—
233	C₂×Q₈	—	26	2	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂²×C₄	—	26	2	no	—
236	C₂²×C₄	—	26	2	no	—
237	C₂²×C₄	—	26	4	no	—
238	C₂²×C₄	C₂×D₅₂	1	2	yes	—
239	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₅₂	1	2	yes	—
241	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	Q₁₆	—	13	16	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
251	C₂×C₈	D₄×D₁₃	1	2	yes	—
252	C₂×C₈	Q₈×D₁₃	1	2	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	13	4	no	—
255	C₄⋊C₄	—	26	8	no	—
256	C₄⋊C₄	—	26	2	no	—
257	C₄⋊C₄	—	26	4	no	—
258	C₄⋊C₄	—	26	4	no	—
259	C₄⋊C₄	—	52	8	no	—
260	C₄⋊C₄	—	52	4	no	—
261	C₄⋊C₄	—	104	4	no	—
262	C₄⋊C₄	—	104	2	no	—
263	C₄⋊C₄	Dic₅₂	1	4	yes	direct product factor
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	26	8	no	—
267	C₄²	—	52	8	no	—
268	C₄²	C₂×C₄×D₁₃	1	2	yes	—
269	C₄²	#6127	1	2	yes	—
270	C₁₃	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂²×D₅₂	1	1	yes	characteristic
272	Q₈	—	13	8	no	—
273	Q₈	—	26	4	no	—
274	Q₈	—	26	4	no	—
275	Q₈	—	26	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	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	—	26	4	no	—
284	C₂×C₄	—	26	4	no	—
285	C₂×C₄	—	26	4	no	—
286	C₂×C₄	—	26	4	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	—	52	2	no	—
289	C₂×C₄	—	52	4	no	—
290	C₂×C₄	—	52	4	no	—
291	C₂×C₄	—	52	4	no	—
292	C₂×C₄	C₄×D₅₂	1	2	yes	—
293	C₂×C₄	C₄×D₅₂	1	2	yes	—
294	C₂×C₄	#88158	1	2	yes	—
295	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
296	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂×#6127	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
301	C₂×C₄	#88256	1	1	yes	characteristic
302	C₂×C₄	#88257	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	4	2	no	—
305	C₈	C₄⋊C₄×D₁₃	1	4	yes	—
306	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
307	C₂²	D₄×D₅₂	1	1	yes	characteristic
308	C₂²	Q₈×D₅₂	1	1	yes	characteristic
309	C₂²	C₂×#88158	1	1	yes	characteristic
310	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
311	C₂²	#11719457	1	1	yes	characteristic
312	C₂²	#11719458	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	26	8	no	—
317	C₄	—	52	4	no	—
318	C₄	—	104	4	no	—
319	C₄	C₄×Dic₅₂	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
321	C₄	#11719228	1	1	yes	characteristic
322	C₄	#11719256	1	1	yes	characteristic
323	C₄	#11719284	1	1	yes	characteristic
324	C₂	C₄⋊C₄×D₅₂	1	1	yes	characteristic
325	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
326	C₂	#13123713	1	1	yes	characteristic
327	C₂	D₄×Dic₅₂	1	1	yes	characteristic
328	C₂	#13126596	1	1	yes	characteristic
329	C₂	Q₈×Dic₅₂	1	1	yes	characteristic
330	C₂	#13127728	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
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
26. C₄×Dic₅₂	C₄	semidirect
89. C₁₃×C₄⋊C₄	Q₁₆	semidirect
105. Dic₅₂	C₄⋊C₄	direct (N × H)
180. C₄×C₁₃	C₄×Q₁₆	semidirect
263. C₄⋊C₄	Dic₅₂	direct (N × H)
270. C₁₃	C₄⋊C₄×Q₁₆	semidirect
319. C₄	C₄×Dic₅₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
C₄⋊C₄×Dic₂₆	2
#13115569	2
#13115093	2
#13114996	2
#13114842	2
C₈×C₁₃×C₄⋊C₄	2
C₄⋊C₄×Q₁₆	13

Subgroups — C4⋊C4×Dic52 #15881352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×Dic₅₂ #15881352