Subgroups — Dic₁₃×#427 #15860971

Order 3328.

1182 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#117×Dic₁₃	C₂	1	2	yes	maximal
4	#107×Dic₁₃	C₂	1	4	yes	maximal, complemented
5	#106×Dic₁₃	C₂	1	4	yes	maximal, complemented
6	#13114552	C₂	1	2	yes	maximal
7	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#427	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	#11718767	C₄	1	8	yes	complemented
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88057	C₂²	1	2	yes	—
19	C₂×C₈×Dic₁₃	C₄	1	8	yes	complemented
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₁₃	C₂²	1	8	yes	—
23	C₂²⋊C₄×Dic₁₃	—	2	4	no	—
24	#11718593	—	2	2	no	—
25	C₄²×Dic₁₃	—	2	2	no	—
26	#11718121	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×Dic₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	—	2	4	no	—
35	C₁₃×#117	C₂×C₄	1	8	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₁₃	—	2	2	no	—
39	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
40	#88180	—	2	4	no	—
41	#88180	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
43	C₂×C₄×Dic₁₃	—	2	2	no	—
44	C₂×C₄×Dic₁₃	—	2	8	no	—
45	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₁₃	Q₈	1	4	yes	—
49	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×#1055	—	2	2	no	—
51	C₂²×#1055	—	2	2	no	—
52	C₂²×#1055	C₂×C₄	1	4	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₁₃×#107	C₂×C₄	1	8	yes	complemented
55	C₁₃×#106	C₂×C₄	1	8	yes	complemented
56	#88071	—	2	8	no	—
57	#88058	C₂×C₄	1	8	yes	complemented
58	#88057	C₂×C₄	1	8	yes	complemented
59	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
60	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
61	C₄×#427	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	8	no	—
73	C₂²×Dic₁₃	—	2	4	no	—
74	C₂²×Dic₁₃	C₄⋊C₄	1	8	yes	—
75	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
76	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
86	C₁₃×C₂²⋊C₄	—	2	8	no	—
87	C₄²×C₁₃	—	2	8	no	—
88	C₂²⋊Dic₁₃	—	2	8	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
91	C₄×Dic₁₃	—	2	8	no	—
92	C₄×Dic₁₃	—	4	16	no	—
93	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
94	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
95	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
96	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
97	C₂×#1055	—	2	8	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	C₂²×C₄	1	4	yes	—
101	C₂×#427	—	13	1	no	—
102	C₂×#427	D₁₃	1	1	yes	characteristic
103	C₂²×C₄×C₈	—	13	1	no	—
104	C₄×#117	—	13	2	no	—
105	C₄×#107	—	13	4	no	—
106	C₄×#106	—	13	4	no	—
107	#1735	—	13	2	no	—
108	C₂³×C₁₃	—	2	4	no	—
109	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₁₃	C₄×D₄	1	2	yes	—
111	C₂³×C₁₃	C₄×D₄	1	2	yes	—
112	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	8	no	—
114	C₂×C₄×C₁₃	—	4	8	no	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	#117	1	4	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	4	no	—
129	C₂×Dic₁₃	—	2	16	no	—
130	C₂×Dic₁₃	—	2	8	no	—
131	C₂×Dic₁₃	—	4	8	no	—
132	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×Dic₁₃	#135	1	8	yes	—
135	C₈×C₁₃	C₂×C₄²	1	4	yes	—
136	C₈×C₁₃	C₂×C₄²	1	4	yes	—
137	#1055	—	2	4	no	—
138	C₂³×C₈	—	13	1	no	—
139	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#117	—	13	2	no	—
141	C₂×#117	C₂×D₁₃	1	2	yes	—
142	C₂²×C₄²	—	13	1	no	—
143	#427	—	13	8	no	—
144	#427	Dic₁₃	1	8	yes	direct product factor
145	C₂×#107	—	13	2	no	—
146	C₂×#107	C₂×D₁₃	1	2	yes	—
147	C₂×#106	—	13	2	no	—
148	C₂×#106	C₂×D₁₃	1	2	yes	—
149	C₂×C₄×C₈	—	13	8	no	—
150	C₂×C₄×C₈	—	13	2	no	—
151	C₂×C₄×C₈	—	13	2	no	—
152	C₄×C₄⋊C₄	—	13	8	no	—
153	C₄×C₂²⋊C₄	—	26	4	no	—
154	#376	—	26	2	no	—
155	C₄³	—	26	2	no	—
156	#340	—	13	8	no	—
157	C₂²×C₁₃	—	2	4	no	—
158	C₂²×C₁₃	—	2	4	no	—
159	C₂²×C₁₃	—	2	4	no	—
160	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
161	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×#135	1	4	yes	—
165	C₄×C₁₃	—	4	8	no	—
166	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
167	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
168	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
169	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	Dic₁₃	—	2	4	no	—
172	Dic₁₃	—	4	8	no	—
173	Dic₁₃	#427	1	8	yes	direct product factor
174	C₂³×C₄	—	13	2	no	—
175	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	—	13	4	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	26	2	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	C₄×D₁₃	1	4	yes	—
181	C₂²×C₈	C₄×D₁₃	1	4	yes	—
182	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	#117	—	2	4	no	—
186	#117	—	13	8	no	—
187	#117	—	26	4	no	—
188	#117	C₂×Dic₁₃	1	8	yes	—
189	C₂×C₄⋊C₄	—	13	4	no	—
190	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	26	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	13	4	no	—
195	C₂×C₄²	—	13	1	no	—
196	C₂×C₄²	—	13	1	no	—
197	C₂×C₄²	—	13	4	no	—
198	C₂×C₄²	—	13	2	no	—
199	C₂×C₄²	—	26	2	no	—
200	C₂×C₄²	—	26	8	no	—
201	#107	—	13	8	no	—
202	#107	C₂×Dic₁₃	1	8	yes	complemented
203	#106	—	13	8	no	—
204	#106	C₂×Dic₁₃	1	8	yes	complemented
205	C₄×C₈	—	13	8	no	—
206	C₄×C₈	—	13	8	no	—
207	#95	—	26	8	no	—
208	C₂×C₁₃	—	2	4	no	—
209	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₁₃	C₂×#427	1	2	yes	—
211	C₂×C₁₃	C₄×#135	1	2	yes	—
212	C₂×C₁₃	C₄×#135	1	2	yes	—
213	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	13	2	no	—
216	C₂²×C₄	—	13	8	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	26	8	no	—
219	C₂²×C₄	—	26	4	no	—
220	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
222	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	#6129	1	2	yes	—
224	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
225	C₂²×C₄	#6131	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
227	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
228	C₂×C₈	—	13	4	no	—
229	C₂×C₈	—	26	8	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
233	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
237	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
238	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	13	8	no	—
241	C₄⋊C₄	—	26	4	no	—
242	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₂²⋊C₄	—	26	8	no	—
245	C₄²	—	2	8	no	—
246	C₄²	—	13	4	no	—
247	C₄²	—	13	4	no	—
248	C₄²	—	13	4	no	—
249	C₄²	—	13	4	no	—
250	C₄²	—	26	8	no	—
251	C₄²	—	52	16	no	—
252	C₁₃	C₄×#427	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	2	4	no	—
254	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
255	C₂³	#88147	1	2	yes	—
256	C₂³	D₄×Dic₁₃	1	2	yes	—
257	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	13	4	no	—
261	C₂×C₄	—	13	8	no	—
262	C₂×C₄	—	13	4	no	—
263	C₂×C₄	—	26	16	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	26	8	no	—
266	C₂×C₄	—	52	8	no	—
267	C₂×C₄	#88141	1	4	yes	—
268	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
269	C₂×C₄	#88146	1	4	yes	—
270	C₂×C₄	#88147	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
272	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
273	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
274	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#6129	1	1	yes	characteristic
277	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#6131	1	1	yes	characteristic
279	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
280	C₈	—	26	4	no	—
281	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
282	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	C₄⋊C₄×Dic₁₃	1	4	yes	—
287	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
288	C₂²	C₂×#88147	1	1	yes	characteristic
289	C₂²	D₁₃×#135	1	4	yes	—
290	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
291	C₄	—	4	8	no	—
292	C₄	—	13	8	no	—
293	C₄	—	26	4	no	—
294	C₄	—	52	8	no	—
295	C₄	C₄⋊C₄×Dic₁₃	1	4	yes	—
296	C₄	C₂×#88141	1	1	yes	characteristic
297	C₄	C₂×#88146	1	1	yes	characteristic
298	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
299	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
302	C₂	D₁₃×#427	1	2	yes	—
303	C₂	#13123607	1	2	yes	—
304	C₂	#135×Dic₁₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	C₄×#427
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #107×Dic₁₃	C₂	semidirect
5. #106×Dic₁₃	C₂	semidirect
13. C₁₃×#427	C₄	semidirect
16. #11718767	C₄	semidirect
19. C₂×C₈×Dic₁₃	C₄	semidirect
54. C₁₃×#107	C₂×C₄	semidirect
55. C₁₃×#106	C₂×C₄	semidirect
57. #88058	C₂×C₄	semidirect
58. #88057	C₂×C₄	semidirect
77. C₂×C₈×C₁₃	C₄²	semidirect
144. #427	Dic₁₃	direct (N × H)
173. Dic₁₃	#427	direct (N × H)
202. #107	C₂×Dic₁₃	semidirect
204. #106	C₂×Dic₁₃	semidirect
232. C₂×C₈	C₄×Dic₁₃	semidirect
252. C₁₃	C₄×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
C₂²×C₈×Dic₁₃	2
#117×Dic₁₃	2
#107×Dic₁₃	2
#106×Dic₁₃	2
#13114552	2
C₂×C₁₃×#427	2
C₄×#427	13

Subgroups — Dic13×#427 #15860971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃×#427 #15860971