Subgroups — D₁₃×#2176 #16016212

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₁₃	C₂	1	2	yes	maximal
3	#13116708	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116707	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116706	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116705	C₂	1	2	yes	maximal, complemented
7	#13116704	C₂	1	2	yes	maximal, complemented
8	D₁₃×#367	C₂	1	1	yes	maximal, characteristic
9	D₁₃×#358	C₂	1	2	yes	maximal
10	D₁₃×#346	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	2	yes	—
13	C₄×Q₈×D₁₃	—	2	2	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
17	C₂×#88069	C₂²	1	2	yes	—
18	C₄×#6107	—	2	2	no	—
19	D₁₃×Q₃₂	C₄	1	8	yes	complemented
20	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	D₁₃×#107	C₂²	1	2	yes	—
24	#11718534	C₂²	1	2	yes	—
25	D₁₃×#103	—	2	2	no	—
26	#11718451	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#346	C₂²	1	1	yes	characteristic
32	#11718156	C₂²	1	2	yes	—
33	#11718097	C₂²	1	1	yes	characteristic
34	#11718095	C₂²	1	2	yes	—
35	#11718094	C₂²	1	1	yes	characteristic
36	#11718094	C₂²	1	1	yes	characteristic
37	#11718084	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	#11718053	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₁₃	—	2	2	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	Q₈×Dic₁₃	—	2	2	no	—
44	C₂×#6107	—	2	2	no	—
45	Q₁₆×D₁₃	—	2	4	no	—
46	Q₁₆×D₁₃	C₂×C₄	1	8	yes	—
47	C₂×Dic₅₂	C₂³	1	2	yes	—
48	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×D₁₃	—	2	2	no	—
51	C₄⋊C₄×D₁₃	—	4	2	no	—
52	#88141	—	2	2	no	—
53	#88141	—	4	2	no	—
54	C₄²×D₁₃	—	4	2	no	—
55	C₄²×D₁₃	D₄	1	1	yes	characteristic
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	—	4	2	no	—
58	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#103	—	2	2	no	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88075	—	2	2	no	—
65	#88069	C₂×C₄	1	8	yes	complemented
66	#88058	C₂³	1	2	yes	—
67	#88056	—	2	2	no	—
68	C₈×Dic₁₃	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88050	—	2	2	no	—
73	#88047	—	2	2	no	—
74	C₄×#1055	D₄	1	1	yes	characteristic
75	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
76	C₁₆×D₁₃	—	2	2	no	—
77	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
78	C₂×#2176	—	13	1	no	maximal, Sylow
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	Q₈×D₁₃	—	2	8	no	—
81	Q₈×D₁₃	—	4	4	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×C₄×D₁₃	—	4	2	no	—
84	C₂×C₄×D₁₃	D₈	1	2	yes	—
85	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₂₆	—	2	2	no	—
87	C₂×Dic₂₆	—	4	2	no	—
88	C₁₃×Q₁₆	—	2	2	no	—
89	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₄²×C₁₃	—	4	2	no	—
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	#6107	—	2	8	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	#6101	—	4	2	no	—
100	#6101	—	4	2	no	—
101	#6101	—	4	2	no	—
102	C₄×Dic₁₃	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₄×Dic₁₃	—	4	2	no	—
105	C₄×Dic₁₃	D₈	1	2	yes	—
106	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	Dic₅₂	—	2	2	no	—
110	Dic₅₂	C₂²×C₄	1	4	yes	—
111	C₈×D₁₃	—	2	2	no	—
112	C₈×D₁₃	C₂²×C₄	1	2	yes	—
113	C₈×D₁₃	#41	1	2	yes	—
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
116	#6090	—	2	1	no	—
117	#6090	C₂²×C₄	1	2	yes	—
118	C₂²×Q₃₂	—	13	1	no	—
119	C₂×C₄×Q₁₆	—	13	2	no	—
120	#2176	—	13	1	no	—
121	#2176	—	13	2	no	—
122	#2176	—	13	1	no	—
123	#2176	—	13	2	no	—
124	#2176	—	13	1	no	—
125	#2176	D₁₃	1	1	yes	characteristic, direct product factor
126	C₂×#367	—	13	1	no	—
127	C₂×#358	—	13	2	no	—
128	C₂×#346	—	13	1	no	—
129	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₁₃	—	2	4	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
139	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
140	C₄×D₁₃	—	2	4	no	—
141	C₄×D₁₃	—	4	8	no	—
142	C₄×D₁₃	—	8	4	no	—
143	C₄×D₁₃	C₄×D₄	1	2	yes	—
144	C₄×D₁₃	#135	1	2	yes	—
145	Dic₂₆	—	2	4	no	—
146	Dic₂₆	—	4	2	no	—
147	Dic₂₆	—	4	8	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
151	#1055	—	2	1	no	—
152	#1055	C₄×D₄	1	2	yes	—
153	C₂²×Q₁₆	—	13	2	no	—
154	C₂×C₄×Q₈	—	26	2	no	—
155	C₂×Q₃₂	—	13	2	no	—
156	C₂×Q₃₂	—	13	8	no	—
157	C₂×Q₃₂	—	13	1	no	—
158	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
159	C₂²×C₁₆	—	13	1	no	—
160	C₂²×C₁₆	—	13	1	no	—
161	C₄×Q₁₆	—	13	2	no	—
162	C₄×Q₁₆	—	13	2	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	26	2	no	—
165	C₄×Q₁₆	—	26	2	no	—
166	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
167	C₂×#107	—	13	2	no	—
168	C₂×#103	—	26	2	no	—
169	C₂×C₄×C₈	—	13	1	no	—
170	#367	—	13	1	no	—
171	#367	—	13	2	no	—
172	#367	C₂×D₁₃	1	1	yes	characteristic
173	#358	—	13	2	no	—
174	#358	—	13	2	no	—
175	#358	—	13	2	no	—
176	#358	C₂×D₁₃	1	2	yes	—
177	#346	—	13	1	no	—
178	#346	—	13	1	no	—
179	#346	—	13	1	no	—
180	#346	C₂×D₁₃	1	1	yes	characteristic
181	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
182	C₂×D₁₃	C₄×D₈	1	2	yes	—
183	C₂×D₁₃	#510	1	4	yes	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	4	4	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	4	4	no	—
191	Dic₁₃	—	8	2	no	—
192	Dic₁₃	C₄×D₈	1	2	yes	—
193	C₂²×Q₈	—	26	2	no	—
194	C₂×Q₁₆	—	13	8	no	—
195	C₂×Q₁₆	—	13	2	no	—
196	C₂×Q₁₆	—	26	4	no	—
197	C₂×Q₁₆	—	26	2	no	—
198	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	26	2	no	—
203	C₄×Q₈	—	26	2	no	—
204	C₄×Q₈	—	26	2	no	—
205	C₄×Q₈	—	52	2	no	—
206	C₄×Q₈	—	52	2	no	—
207	C₂×C₄⋊C₄	—	26	2	no	—
208	C₂×C₄⋊C₄	—	52	2	no	—
209	C₂×C₄²	—	13	1	no	—
210	C₂×C₄²	—	52	2	no	—
211	Q₃₂	—	13	8	no	—
212	Q₃₂	—	13	4	no	—
213	Q₃₂	C₄×D₁₃	1	4	yes	complemented
214	C₂×C₁₆	—	13	1	no	—
215	C₂×C₁₆	—	13	4	no	—
216	C₂×C₁₆	—	13	1	no	—
217	C₂×C₁₆	—	26	2	no	—
218	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
219	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	#107	—	13	2	no	—
221	#107	—	26	2	no	—
222	#107	C₂²×D₁₃	1	2	yes	—
223	#103	—	2	2	no	—
224	#103	—	26	2	no	—
225	#103	—	26	2	no	—
226	#103	—	26	2	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
231	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#510	1	2	yes	—
233	D₁₃	#2176	1	4	yes	direct product factor
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	26	8	no	—
236	C₂×Q₈	—	26	2	no	—
237	C₂×Q₈	—	52	2	no	—
238	C₂×Q₈	—	52	4	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	2	no	—
241	C₂²×C₄	—	52	2	no	—
242	C₂²×C₄	—	52	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	13	4	no	—
245	Q₁₆	—	26	8	no	—
246	Q₁₆	—	26	2	no	—
247	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	26	2	no	—
253	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₄⋊C₄	—	52	2	no	—
261	C₄⋊C₄	—	52	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	13	2	no	—
264	C₄²	—	13	1	no	—
265	C₄²	—	52	2	no	—
266	C₄²	—	52	2	no	—
267	C₄²	—	52	2	no	—
268	C₄²	D₄×D₁₃	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	13	2	no	—
271	C₁₆	—	26	1	no	—
272	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
273	C₁₃	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	13	1	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	26	4	no	—
278	Q₈	—	52	8	no	—
279	Q₈	—	52	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	13	1	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	26	4	no	—
287	C₂×C₄	—	52	8	no	—
288	C₂×C₄	—	52	2	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	104	4	no	—
291	C₂×C₄	D₈×D₁₃	1	2	yes	—
292	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	13	2	no	—
295	C₈	—	26	1	no	—
296	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
297	C₈	#41×D₁₃	1	1	yes	characteristic
298	C₂²	—	13	4	no	—
299	C₂²	—	13	2	no	—
300	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	13	2	no	—
305	C₄	—	26	2	no	—
306	C₄	—	52	4	no	—
307	C₄	—	104	2	no	—
308	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
309	C₄	D₁₃×#135	1	1	yes	characteristic
310	C₂	—	13	4	no	—
311	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
312	C₂	D₁₃×#510	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2176	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2176	D₁₃
hypocenter	C₁₃	C₂×#2176
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #13116708	C₂	semidirect
4. #13116707	C₂	semidirect
5. #13116706	C₂	semidirect
6. #13116705	C₂	semidirect
7. #13116704	C₂	semidirect
11. C₁₃×#2176	C₂	semidirect
19. D₁₃×Q₃₂	C₄	semidirect
58. C₁₃×Q₃₂	C₂×C₄	semidirect
65. #88069	C₂×C₄	semidirect
75. Dic₁₀₄	C₂×C₄	semidirect
125. #2176	D₁₃	direct (N × H)
213. Q₃₂	C₄×D₁₃	semidirect
233. D₁₃	#2176	direct (N × H)
273. C₁₃	C₂×#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×Q₃₂	2
C₄×Q₁₆×D₁₃	2
#13116708	2
#13116707	2
#13116706	2
#13116705	2
#13116704	2
D₁₃×#367	2
D₁₃×#358	2
D₁₃×#346	2
C₁₃×#2176	2
C₂×#2176	13

Subgroups — D13×#2176 #16016212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#2176 #16016212