Subgroups — D₄×Q₈×D₁₃ #13128306

Order 1664.

1568 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₃	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
4	#11719432	C₂	1	3	yes	maximal, complemented
5	D₁₃×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
6	#11719337	C₂	1	6	yes	maximal, complemented
7	D₁₃×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
8	Q₈×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₃	C₂	1	3	yes	maximal
11	D₄×Dic₂₆	C₂	1	3	yes	maximal, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	complemented
13	C₂×Q₈×D₁₃	—	2	16	no	—
14	C₂×Q₈×D₁₃	C₂²	1	6	yes	—
15	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×D₁₃	C₂²	1	6	yes	complemented
17	C₂×Q₈×D₁₃	C₂²	1	2	yes	complemented
18	C₂×D₄×D₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₃	C₂²	1	6	yes	—
20	C₂²×Dic₂₆	C₂²	1	6	yes	complemented
21	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
22	C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	complemented
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
24	C₄×D₄×C₁₃	C₂²	1	3	yes	—
25	#88200	C₂²	1	6	yes	complemented
26	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
27	#88198	C₂²	1	6	yes	complemented
28	D₄×Dic₁₃	C₂²	1	3	yes	—
29	C₄×C₂²⋊D₁₃	C₂²	1	6	yes	—
30	#88178	C₂²	1	6	yes	complemented
31	#88151	C₂²	1	6	yes	complemented
32	#88150	C₂²	1	12	yes	complemented
33	#88147	C₂²	1	3	yes	—
34	C₄⋊C₄×D₁₃	C₂²	1	3	yes	—
35	C₄⋊C₄×D₁₃	C₂²	1	6	yes	—
36	C₄⋊C₄×D₁₃	C₂²	1	3	yes	—
37	#88142	C₂²	1	6	yes	complemented
38	#88142	C₂²	1	6	yes	complemented
39	#88141	C₂²	1	3	yes	complemented
40	#88135	C₂²	1	6	yes	—
41	C₂²⋊C₄×D₁₃	C₂²	1	6	yes	—
42	#88132	C₂²	1	12	yes	complemented
43	C₄×D₅₂	C₂²	1	3	yes	—
44	C₄²×D₁₃	C₂²	1	3	yes	—
45	#88123	C₂²	1	3	yes	complemented
46	C₄×Dic₂₆	C₂²	1	3	yes	complemented
47	C₂³×D₁₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₁₃	—	2	8	no	—
49	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
50	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
53	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
54	C₂²×Dic₁₃	C₂³	1	6	yes	—
55	Q₈×D₁₃	—	2	8	no	—
56	Q₈×D₁₃	—	2	24	no	—
57	Q₈×D₁₃	—	2	24	no	—
58	Q₈×D₁₃	D₄	1	8	yes	direct product factor
59	D₄×D₁₃	Q₈	1	8	yes	direct product factor
60	C₂×D₅₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₃	—	2	24	no	—
62	C₂×C₄×D₁₃	C₂³	1	3	yes	—
63	C₂×C₄×D₁₃	C₂³	1	6	yes	—
64	C₂×C₄×D₁₃	C₂³	1	6	yes	—
65	C₂×C₄×D₁₃	C₂³	1	6	yes	—
66	C₂×C₄×D₁₃	C₂³	1	3	yes	—
67	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₂₆	—	2	24	no	—
69	C₂×Dic₂₆	C₂³	1	6	yes	—
70	C₂×Dic₂₆	C₂³	1	12	yes	—
71	C₂×Dic₂₆	C₂³	1	3	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
74	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
76	C₄²×C₁₃	C₂³	1	3	yes	—
77	C₂²⋊Dic₁₃	C₂³	1	6	yes	—
78	#6103	C₂³	1	6	yes	—
79	#6103	C₂³	1	6	yes	—
80	C₄⋊Dic₁₃	C₂³	1	3	yes	—
81	C₄⋊Dic₁₃	C₂³	1	3	yes	—
82	C₄⋊Dic₁₃	C₂³	1	6	yes	—
83	#6101	C₂³	1	6	yes	—
84	#6101	C₂³	1	12	yes	—
85	#6101	C₂³	1	6	yes	—
86	C₄×Dic₁₃	C₂³	1	3	yes	—
87	C₄×Dic₁₃	C₂³	1	3	yes	—
88	C₄×Dic₁₃	C₂³	1	3	yes	—
89	C₂×D₄×Q₈	—	13	1	no	maximal, Sylow
90	C₂³×C₁₃	C₂⁴	1	2	yes	—
91	C₂²×D₁₃	—	2	4	no	—
92	C₂²×D₁₃	—	2	4	no	—
93	C₂²×D₁₃	C₂×Q₈	1	8	yes	complemented
94	C₂²×D₁₃	C₂⁴	1	2	yes	—
95	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
96	Q₈×C₁₃	—	2	12	no	—
97	Q₈×C₁₃	C₂×D₄	1	4	yes	complemented
98	D₄×C₁₃	C₂×Q₈	1	4	yes	complemented
99	C₂×C₄×C₁₃	—	2	12	no	—
100	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₃	C₂×Q₈	1	8	yes	complemented
105	C₂×Dic₁₃	—	2	12	no	—
106	C₂×Dic₁₃	C₂⁴	1	3	yes	—
107	C₂×Dic₁₃	C₂⁴	1	6	yes	—
108	C₂×Dic₁₃	C₂⁴	1	3	yes	—
109	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
110	D₅₂	C₂×Q₈	1	4	yes	complemented
111	C₄×D₁₃	—	2	12	no	—
112	C₄×D₁₃	—	2	6	no	—
113	C₄×D₁₃	—	2	12	no	—
114	C₄×D₁₃	—	2	12	no	—
115	C₄×D₁₃	C₂×D₄	1	12	yes	—
116	C₄×D₁₃	C₂×Q₈	1	4	yes	complemented
117	Dic₂₆	—	2	12	no	—
118	Dic₂₆	—	2	24	no	—
119	Dic₂₆	C₂×D₄	1	12	yes	complemented
120	C₂³×Q₈	—	13	2	no	—
121	D₄×Q₈	—	13	2	no	—
122	D₄×Q₈	—	13	3	no	—
123	D₄×Q₈	—	13	6	no	—
124	D₄×Q₈	—	13	1	no	—
125	D₄×Q₈	—	13	3	no	—
126	D₄×Q₈	D₁₃	1	1	yes	characteristic, direct product factor
127	C₂×C₄⋊Q₈	—	13	3	no	—
128	C₂×C₂²⋊Q₈	—	13	6	no	—
129	C₂×C₄×Q₈	—	13	1	no	—
130	C₂×C₄×D₄	—	13	3	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	C₂²×Q₈	1	4	yes	complemented
134	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	2	2	no	—
137	C₂×D₁₃	—	2	16	no	—
138	C₂×D₁₃	C₂²×D₄	1	2	yes	—
139	C₂×D₁₃	C₂²×Q₈	1	2	yes	—
140	C₂×D₁₃	C₂²×Q₈	1	4	yes	complemented
141	C₂×D₁₃	2₋¹⁺⁴	1	2	yes	—
142	C₄×C₁₃	—	2	6	no	—
143	C₄×C₁₃	—	2	3	no	—
144	C₄×C₁₃	C₂²×D₄	1	6	yes	—
145	C₄×C₁₃	C₂²×Q₈	1	2	yes	complemented
146	Dic₁₃	—	2	6	no	—
147	Dic₁₃	—	2	3	no	—
148	Dic₁₃	C₂²×D₄	1	6	yes	—
149	Dic₁₃	C₂²×Q₈	1	2	yes	complemented
150	C₂²×Q₈	—	13	1	no	—
151	C₂²×Q₈	—	13	6	no	—
152	C₂²×Q₈	—	13	6	no	—
153	C₂²×Q₈	—	13	2	no	—
154	C₂²×Q₈	—	13	6	no	—
155	C₂²×Q₈	—	26	16	no	—
156	C₂²×Q₈	C₂×D₁₃	1	2	yes	complemented
157	C₂²×D₄	—	13	1	no	—
158	C₂³×C₄	—	13	6	no	—
159	C₄⋊Q₈	—	13	6	no	—
160	C₄⋊Q₈	—	13	6	no	—
161	C₄⋊Q₈	—	13	6	no	—
162	C₄⋊Q₈	—	13	3	no	—
163	C₄⋊Q₈	C₂×D₁₃	1	3	yes	complemented
164	C₂²⋊Q₈	—	13	6	no	—
165	C₂²⋊Q₈	—	13	6	no	—
166	C₂²⋊Q₈	—	13	12	no	—
167	C₂²⋊Q₈	—	13	6	no	—
168	C₂²⋊Q₈	—	13	12	no	—
169	C₂²⋊Q₈	C₂×D₁₃	1	6	yes	complemented
170	C₄×Q₈	—	13	1	no	—
171	C₄×Q₈	—	13	3	no	—
172	C₄×Q₈	—	13	3	no	—
173	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
174	C₄×D₄	—	13	3	no	—
175	C₄×D₄	—	13	6	no	—
176	C₄×D₄	—	13	3	no	—
177	C₄×D₄	—	13	6	no	—
178	C₄×D₄	—	13	3	no	—
179	C₄×D₄	C₂×D₁₃	1	3	yes	—
180	C₂×C₄⋊C₄	—	13	3	no	—
181	C₂×C₄⋊C₄	—	13	3	no	—
182	C₂×C₄⋊C₄	—	13	6	no	—
183	C₂×C₂²⋊C₄	—	13	6	no	—
184	C₂×C₄²	—	13	3	no	—
185	C₂×C₁₃	—	2	4	no	—
186	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
187	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
188	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
189	D₁₃	—	2	4	no	—
190	D₁₃	D₄×Q₈	1	4	yes	direct product factor
191	C₂⁴	—	13	2	no	—
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	13	8	no	—
194	C₂×Q₈	—	13	3	no	—
195	C₂×Q₈	—	13	12	no	—
196	C₂×Q₈	—	13	6	no	—
197	C₂×Q₈	—	26	24	no	—
198	C₂×Q₈	—	26	24	no	—
199	C₂×Q₈	—	26	24	no	—
200	C₂×Q₈	—	26	8	no	—
201	C₂×Q₈	C₂²×D₁₃	1	6	yes	—
202	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₂×D₄	—	13	2	no	—
204	C₂×D₄	—	13	8	no	—
205	C₂×D₄	—	13	1	no	—
206	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	13	3	no	—
209	C₂²×C₄	—	13	6	no	—
210	C₂²×C₄	—	13	3	no	—
211	C₂²×C₄	—	13	6	no	—
212	C₂²×C₄	—	13	6	no	—
213	C₂²×C₄	—	13	6	no	—
214	C₂²×C₄	—	26	24	no	—
215	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
216	C₄⋊C₄	—	13	6	no	—
217	C₄⋊C₄	—	13	3	no	—
218	C₄⋊C₄	—	13	6	no	—
219	C₄⋊C₄	—	13	3	no	—
220	C₄⋊C₄	—	13	12	no	—
221	C₄⋊C₄	—	13	6	no	—
222	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
223	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
224	C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
225	C₂²⋊C₄	—	13	6	no	—
226	C₂²⋊C₄	—	13	6	no	—
227	C₂²⋊C₄	—	13	6	no	—
228	C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
229	C₄²	—	13	3	no	—
230	C₄²	—	13	3	no	—
231	C₄²	—	13	3	no	—
232	C₄²	C₂²×D₁₃	1	3	yes	—
233	C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	13	1	no	—
235	C₂³	—	13	8	no	—
236	C₂³	—	13	2	no	—
237	C₂³	—	26	4	no	—
238	C₂³	—	26	4	no	—
239	C₂³	C₂³×D₁₃	1	2	yes	—
240	Q₈	—	2	12	no	—
241	Q₈	—	13	12	no	—
242	Q₈	—	26	24	no	—
243	Q₈	—	26	12	no	—
244	Q₈	D₄×D₁₃	1	4	yes	direct product factor
245	D₄	—	13	8	no	—
246	D₄	—	13	4	no	—
247	D₄	Q₈×D₁₃	1	4	yes	direct product factor
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	3	no	—
251	C₂×C₄	—	13	12	no	—
252	C₂×C₄	—	13	4	no	—
253	C₂×C₄	—	13	3	no	—
254	C₂×C₄	—	13	6	no	—
255	C₂×C₄	—	26	12	no	—
256	C₂×C₄	—	26	6	no	—
257	C₂×C₄	—	26	12	no	—
258	C₂×C₄	—	26	12	no	—
259	C₂×C₄	—	26	12	no	—
260	C₂×C₄	C₂³×D₁₃	1	3	yes	—
261	C₂×C₄	C₂³×D₁₃	1	6	yes	—
262	C₂×C₄	C₂³×D₁₃	1	3	yes	—
263	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	13	2	no	—
267	C₂²	—	13	2	no	—
268	C₂²	—	13	2	no	—
269	C₂²	—	13	4	no	—
270	C₂²	—	26	16	no	—
271	C₂²	—	26	2	no	—
272	C₂²	—	26	2	no	—
273	C₂²	C₂×Q₈×D₁₃	1	4	yes	complemented
274	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
275	C₄	—	2	6	no	—
276	C₄	—	2	3	no	—
277	C₄	—	13	6	no	—
278	C₄	—	13	2	no	—
279	C₄	—	26	6	no	—
280	C₄	—	26	3	no	—
281	C₄	C₂×D₄×D₁₃	1	6	yes	—
282	C₄	C₂×Q₈×D₁₃	1	2	yes	complemented
283	C₂	—	2	4	no	—
284	C₂	—	13	4	no	—
285	C₂	—	26	4	no	—
286	C₂	C₂²×D₄×D₁₃	1	1	yes	characteristic
287	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
288	C₂	D₁₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	D₄×Q₈×C₁₃	C₂
prime radical	C₂×D₄×D₁₃	C₂²
hypercenter	D₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₃	C₂	semidirect
2. D₄×Q₈×C₁₃	C₂	semidirect
3. Q₈×C₂²⋊D₁₃	C₂	semidirect
4. #11719432	C₂	semidirect
5. D₁₃×C₄⋊Q₈	C₂	semidirect
6. #11719337	C₂	semidirect
7. D₁₃×C₂²⋊Q₈	C₂	semidirect
8. Q₈×D₅₂	C₂	semidirect
9. C₄×Q₈×D₁₃	C₂	semidirect
11. D₄×Dic₂₆	C₂	semidirect
12. C₂²×Q₈×C₁₃	C₂²	semidirect
16. C₂×Q₈×D₁₃	C₂²	semidirect
17. C₂×Q₈×D₁₃	C₂²	semidirect
20. C₂²×Dic₂₆	C₂²	semidirect
21. C₁₃×C₄⋊Q₈	C₂²	semidirect
22. C₁₃×C₂²⋊Q₈	C₂²	semidirect
23. C₄×Q₈×C₁₃	C₂²	semidirect
25. #88200	C₂²	semidirect
26. Q₈×Dic₁₃	C₂²	semidirect
27. #88198	C₂²	semidirect
30. #88178	C₂²	semidirect
31. #88151	C₂²	semidirect
32. #88150	C₂²	semidirect
37. #88142	C₂²	semidirect
38. #88142	C₂²	semidirect
39. #88141	C₂²	semidirect
42. #88132	C₂²	semidirect
45. #88123	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
58. Q₈×D₁₃	D₄	direct (N × H)
59. D₄×D₁₃	Q₈	direct (N × H)
93. C₂²×D₁₃	C₂×Q₈	semidirect
97. Q₈×C₁₃	C₂×D₄	semidirect
98. D₄×C₁₃	C₂×Q₈	semidirect
104. C₂²⋊D₁₃	C₂×Q₈	semidirect
110. D₅₂	C₂×Q₈	semidirect
116. C₄×D₁₃	C₂×Q₈	semidirect
119. Dic₂₆	C₂×D₄	semidirect
126. D₄×Q₈	D₁₃	direct (N × H)
133. C₂²×C₁₃	C₂²×Q₈	semidirect
140. C₂×D₁₃	C₂²×Q₈	semidirect
145. C₄×C₁₃	C₂²×Q₈	semidirect
149. Dic₁₃	C₂²×Q₈	semidirect
156. C₂²×Q₈	C₂×D₁₃	semidirect
163. C₄⋊Q₈	C₂×D₁₃	semidirect
169. C₂²⋊Q₈	C₂×D₁₃	semidirect
173. C₄×Q₈	C₂×D₁₃	semidirect
190. D₁₃	D₄×Q₈	direct (N × H)
233. C₁₃	C₂×D₄×Q₈	semidirect
244. Q₈	D₄×D₁₃	direct (N × H)
247. D₄	Q₈×D₁₃	direct (N × H)
273. C₂²	C₂×Q₈×D₁₃	semidirect
282. C₄	C₂×Q₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₃	2
D₄×Q₈×C₁₃	2
Q₈×C₂²⋊D₁₃	2
#11719432	2
D₁₃×C₄⋊Q₈	2
#11719337	2
D₁₃×C₂²⋊Q₈	2
Q₈×D₅₂	2
C₄×Q₈×D₁₃	2
C₄×D₄×D₁₃	2
D₄×Dic₂₆	2
C₂×D₄×Q₈	13

Subgroups — D4×Q8×D13 #13128306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₈×D₁₃ #13128306