Subgroups — D₄×C₄⋊#261 #13131348

Order 1664.

776 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#261	C₂	1	2	yes	maximal
2	C₂²×C₄⋊#261	C₂	1	2	yes	maximal, complemented
3	C₄×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
4	#11719136	C₂	1	2	yes	maximal, complemented
5	#11719083	C₂	1	2	yes	maximal, complemented
6	#11719070	C₂	1	4	yes	maximal, complemented
7	#11719059	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊#261	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×#261	C₂²	1	4	yes	—
10	C₂×D₄×D₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊#261	C₂²	1	4	yes	—
13	D₄×#261	—	2	8	no	—
14	C₂×C₄⋊#261	—	2	8	no	—
15	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#261	C₂²	1	2	yes	—
17	C₂×C₄⋊#261	C₂²	1	2	yes	—
18	C₂×C₄⋊#261	C₂²	1	4	yes	—
19	C₂×C₄⋊#261	C₂²	1	2	yes	—
20	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
21	C₂×C₄×#261	C₂²	1	2	yes	—
22	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
23	D₄×Dic₁₃	C₄	1	1	yes	characteristic, complemented
24	C₄×C₂²⋊D₁₃	C₄	1	2	yes	complemented
25	#88147	C₄	1	1	yes	characteristic, complemented
26	C₄⋊C₄×D₁₃	C₂²	1	1	yes	characteristic
27	#88135	C₄	1	2	yes	complemented
28	C₂²⋊C₄×D₁₃	C₂²	1	2	yes	—
29	C₄×D₅₂	C₄	1	1	yes	characteristic, complemented
30	C₄²×D₁₃	C₂²	1	1	yes	characteristic
31	(C₂×#261)⋊C₄	C₂²	1	2	yes	—
32	(C₂×#261)⋊C₄	C₂²	1	4	yes	—
33	C₂³×D₁₃	C₂³	1	2	yes	—
34	C₂²×#261	—	2	8	no	—
35	C₂²×#261	—	2	8	no	—
36	C₂²×#261	—	4	4	no	—
37	C₂²×#261	C₂³	1	2	yes	—
38	C₂²×#261	C₂³	1	4	yes	—
39	C₂²×#261	C₂³	1	2	yes	—
40	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	complemented
42	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
43	C₂²×Dic₁₃	C₂×C₄	1	2	yes	complemented
44	D₄×D₁₃	D₄	1	4	yes	—
45	D₄×D₁₃	Q₈	1	4	yes	—
46	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	—	2	4	no	—
48	C₂×C₄×D₁₃	—	2	4	no	—
49	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	complemented
50	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	complemented
51	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₂³	1	2	yes	—
53	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
55	C₂²⋊#261	—	2	8	no	—
56	C₄⋊#261	—	2	8	no	—
57	C₄⋊#261	—	2	4	no	—
58	C₄⋊#261	—	2	4	no	—
59	C₄⋊#261	—	2	2	no	—
60	C₄⋊#261	—	4	2	no	—
61	C₄⋊#261	D₄	1	4	yes	direct product factor
62	C₄×#261	—	2	4	no	—
63	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic, complemented
64	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	complemented
65	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
66	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	complemented
67	#6103	C₂×C₄	1	2	yes	complemented
68	#6103	C₂×C₄	1	2	yes	complemented
69	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic, complemented
70	#6101	C₂×C₄	1	2	yes	complemented
71	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic, complemented
72	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic, complemented
73	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic, complemented
74	D₄×C₄⋊C₄	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	—	2	2	no	—
78	C₂²×D₁₃	—	2	2	no	—
79	C₂²×D₁₃	—	2	2	no	—
80	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₃	C₂×D₄	1	4	yes	—
82	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
83	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×#261	—	2	2	no	—
85	C₂×#261	—	2	2	no	—
86	C₂×#261	—	2	2	no	—
87	C₂×#261	—	2	4	no	—
88	C₂×#261	—	2	4	no	—
89	C₂×#261	—	2	2	no	—
90	C₂×#261	—	2	4	no	—
91	C₂×#261	—	2	2	no	—
92	C₂×#261	—	4	16	no	—
93	C₂×#261	C₂×D₄	1	4	yes	—
94	D₄×C₁₃	C₄⋊C₄	1	4	yes	complemented
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₁₃	C₄⋊C₄	1	8	yes	complemented
101	C₂×Dic₁₃	—	2	4	no	—
102	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
104	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
106	D₅₂	C₄⋊C₄	1	4	yes	complemented
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	—	2	2	no	—
109	C₄×D₁₃	—	2	1	no	—
110	C₄×D₁₃	—	2	1	no	—
111	C₄×D₁₃	—	2	4	no	—
112	C₄×D₁₃	—	2	4	no	—
113	C₄×D₁₃	C₂×D₄	1	2	yes	—
114	C₄×D₁₃	C₂×D₄	1	2	yes	—
115	C₄×D₁₃	C₂×Q₈	1	2	yes	—
116	C₄×D₁₃	#41	1	2	yes	—
117	C₂×C₄×D₄	—	13	1	no	—
118	C₂×C₄×D₄	—	13	2	no	—
119	C₂²×C₄⋊C₄	—	13	2	no	—
120	#389	—	13	2	no	—
121	#385	—	13	4	no	—
122	#384	—	13	1	no	—
123	#380	—	13	2	no	—
124	C₄×C₄⋊C₄	—	13	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	C₂×C₄⋊C₄	1	4	yes	complemented
128	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×D₁₃	—	2	2	no	—
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	—	2	4	no	—
132	C₂×D₁₃	—	2	4	no	—
133	C₂×D₁₃	—	2	4	no	—
134	C₂×D₁₃	—	2	4	no	—
135	C₂×D₁₃	C₂×C₄⋊C₄	1	4	yes	complemented
136	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×D₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×D₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	#261	—	2	4	no	—
143	#261	—	4	4	no	—
144	#261	—	4	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	complemented
148	C₄×C₁₃	C₄×D₄	1	2	yes	complemented
149	Dic₁₃	—	2	2	no	—
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	C₂×C₄⋊C₄	1	2	yes	complemented
152	Dic₁₃	C₄×D₄	1	2	yes	complemented
153	C₂²×D₄	—	13	1	no	—
154	C₂³×C₄	—	13	2	no	—
155	C₂³×C₄	—	13	4	no	—
156	C₄×D₄	—	13	1	no	—
157	C₄×D₄	—	13	2	no	—
158	C₄×D₄	—	13	1	no	—
159	C₄×D₄	—	13	2	no	—
160	C₄×D₄	—	13	1	no	—
161	C₄×D₄	—	26	8	no	—
162	C₄×D₄	#261	1	1	yes	characteristic, complemented
163	C₂×C₄⋊C₄	—	13	1	no	—
164	C₂×C₄⋊C₄	—	13	1	no	—
165	C₂×C₄⋊C₄	—	13	2	no	—
166	C₂×C₄⋊C₄	—	13	4	no	—
167	C₂×C₄⋊C₄	—	13	2	no	—
168	C₂×C₄⋊C₄	—	13	1	no	—
169	C₂×C₄⋊C₄	—	13	2	no	—
170	C₂×C₄⋊C₄	—	26	8	no	—
171	C₂×C₂²⋊C₄	—	13	2	no	—
172	C₂×C₂²⋊C₄	—	13	4	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	13	2	no	—
175	#95	—	13	2	no	—
176	#95	—	13	4	no	—
177	C₂×C₁₃	—	2	4	no	—
178	C₂×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×C₁₃	#520	1	1	yes	characteristic
181	D₁₃	—	2	4	no	—
182	D₁₃	D₄²	1	1	yes	characteristic
183	D₁₃	#547	1	1	yes	characteristic
184	D₁₃	D₄×Q₈	1	1	yes	characteristic
185	D₁₃	#554	1	1	yes	characteristic
186	C₂⁴	—	13	2	no	—
187	C₂×D₄	—	13	2	no	—
188	C₂×D₄	—	13	4	no	—
189	C₂×D₄	—	13	4	no	—
190	C₂×D₄	—	13	1	no	—
191	C₂×D₄	C₂×#261	1	1	yes	characteristic
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	2	no	—
201	C₂²×C₄	—	13	4	no	—
202	C₂²×C₄	—	26	4	no	—
203	C₂²×C₄	—	26	4	no	—
204	C₂²×C₄	—	26	8	no	—
205	C₂²×C₄	—	26	8	no	—
206	C₂²×C₄	—	52	4	no	—
207	C₂²×C₄	C₂×#261	1	2	yes	complemented
208	C₄⋊C₄	—	13	4	no	—
209	C₄⋊C₄	—	13	2	no	—
210	C₄⋊C₄	—	13	1	no	—
211	C₄⋊C₄	—	26	4	no	—
212	C₄⋊C₄	—	26	8	no	—
213	C₄⋊C₄	—	26	4	no	—
214	C₄⋊C₄	—	26	2	no	—
215	C₄⋊C₄	—	52	2	no	—
216	C₄⋊C₄	C₂×#261	1	1	yes	characteristic, complemented
217	C₂²⋊C₄	—	13	2	no	—
218	C₂²⋊C₄	—	13	2	no	—
219	C₂²⋊C₄	—	13	2	no	—
220	C₂²⋊C₄	—	26	8	no	—
221	C₂²⋊C₄	C₂×#261	1	2	yes	complemented
222	C₄²	—	13	1	no	—
223	C₄²	—	13	1	no	—
224	C₄²	—	13	1	no	—
225	C₄²	—	26	4	no	—
226	C₄²	C₂×#261	1	1	yes	characteristic, complemented
227	C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	13	1	no	—
229	C₂³	—	13	4	no	—
230	C₂³	—	13	4	no	—
231	C₂³	—	13	2	no	—
232	C₂³	—	26	2	no	—
233	C₂³	—	26	2	no	—
234	C₂³	—	26	2	no	—
235	C₂³	—	26	2	no	—
236	C₂³	C₂²×#261	1	2	yes	—
237	D₄	—	13	8	no	—
238	D₄	—	13	4	no	—
239	D₄	C₄⋊#261	1	4	yes	direct product factor
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	13	2	no	—
247	C₂×C₄	—	13	4	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	26	4	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	4	no	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	—	26	4	no	—
264	C₂×C₄	—	26	2	no	—
265	C₂×C₄	—	52	16	no	—
266	C₂×C₄	C₂²×#261	1	1	yes	characteristic
267	C₂×C₄	C₂²×#261	1	2	yes	—
268	C₂×C₄	C₂²×#261	1	1	yes	characteristic
269	C₂×C₄	C₂²×#261	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	13	1	no	—
273	C₂²	—	13	1	no	—
274	C₂²	—	13	1	no	—
275	C₂²	—	13	1	no	—
276	C₂²	—	13	1	no	—
277	C₂²	—	13	1	no	—
278	C₂²	—	13	4	no	—
279	C₂²	—	26	4	no	—
280	C₂²	—	26	4	no	—
281	C₂²	—	26	4	no	—
282	C₂²	—	26	4	no	—
283	C₂²	—	26	2	no	—
284	C₂²	—	26	2	no	—
285	C₂²	C₂×C₄⋊#261	1	4	yes	complemented
286	C₂²	C₂³×#261	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	13	2	no	—
290	C₄	—	13	2	no	—
291	C₄	—	26	4	no	—
292	C₄	—	26	2	no	—
293	C₄	—	26	1	no	—
294	C₄	—	52	2	no	—
295	C₄	—	52	4	no	—
296	C₄	C₂×C₄⋊#261	1	2	yes	complemented
297	C₄	D₄×#261	1	2	yes	complemented
298	C₂	—	2	4	no	—
299	C₂	—	13	1	no	—
300	C₂	—	13	1	no	—
301	C₂	—	13	1	no	—
302	C₂	—	13	1	no	—
303	C₂	—	26	4	no	—
304	C₂	C₂²×C₄⋊#261	1	1	yes	characteristic
305	C₂	C₂×D₄×#261	1	1	yes	characteristic
306	C₂	#11719631	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×D₄×C₁₃	C₄
prime radical	C₂×D₄×D₁₃	C₂²
cosocle	C₂²×D₁₃	C₂⁴
hypercenter	C₄×D₄	#261
hypocenter	C₁₃	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊#261	C₂	semidirect
4. #11719136	C₂	semidirect
5. #11719083	C₂	semidirect
6. #11719070	C₂	semidirect
7. #11719059	C₂	semidirect
8. C₄×C₄⋊#261	C₂	semidirect
22. C₄×D₄×C₁₃	C₄	semidirect
23. D₄×Dic₁₃	C₄	semidirect
24. C₄×C₂²⋊D₁₃	C₄	semidirect
25. #88147	C₄	semidirect
27. #88135	C₄	semidirect
29. C₄×D₅₂	C₄	semidirect
41. C₂²×C₄×C₁₃	C₂×C₄	semidirect
43. C₂²×Dic₁₃	C₂×C₄	semidirect
49. C₂×C₄×D₁₃	C₂×C₄	semidirect
50. C₂×C₄×D₁₃	C₂×C₄	semidirect
61. C₄⋊#261	D₄	direct (N × H)
63. C₁₃×C₄⋊C₄	C₂×C₄	semidirect
64. C₁₃×C₂²⋊C₄	C₂×C₄	semidirect
65. C₄²×C₁₃	C₂×C₄	semidirect
66. C₂²⋊Dic₁₃	C₂×C₄	semidirect
67. #6103	C₂×C₄	semidirect
68. #6103	C₂×C₄	semidirect
69. C₄⋊Dic₁₃	C₂×C₄	semidirect
70. #6101	C₂×C₄	semidirect
71. C₄×Dic₁₃	C₂×C₄	semidirect
72. C₄×Dic₁₃	C₂×C₄	semidirect
73. C₄×Dic₁₃	C₂×C₄	semidirect
94. D₄×C₁₃	C₄⋊C₄	semidirect
100. C₂²⋊D₁₃	C₄⋊C₄	semidirect
106. D₅₂	C₄⋊C₄	semidirect
127. C₂²×C₁₃	C₂×C₄⋊C₄	semidirect
135. C₂×D₁₃	C₂×C₄⋊C₄	semidirect
147. C₄×C₁₃	C₂×C₄⋊C₄	semidirect
148. C₄×C₁₃	C₄×D₄	semidirect
151. Dic₁₃	C₂×C₄⋊C₄	semidirect
152. Dic₁₃	C₄×D₄	semidirect
162. C₄×D₄	#261	semidirect
207. C₂²×C₄	C₂×#261	semidirect
216. C₄⋊C₄	C₂×#261	semidirect
221. C₂²⋊C₄	C₂×#261	semidirect
226. C₄²	C₂×#261	semidirect
227. C₁₃	D₄×C₄⋊C₄	semidirect
239. D₄	C₄⋊#261	direct (N × H)
285. C₂²	C₂×C₄⋊#261	semidirect
296. C₄	C₂×C₄⋊#261	semidirect
297. C₄	D₄×#261	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×#261	2
C₂²×C₄⋊#261	2
C₄×D₄×D₁₃	2
#11719136	2
#11719083	2
#11719070	2
#11719059	2
C₄×C₄⋊#261	2
D₄×C₄⋊C₄	13

Subgroups — D4×C4⋊#261 #13131348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₄⋊#261 #13131348