Subgroups — C₄×D₁₃×#108 #15861013

Order 3328.

1086 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×#108	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₁₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
4	#108×Dic₁₃	C₂	1	2	yes	maximal, complemented
5	C₄×#88062	C₂	1	2	yes	maximal, complemented
6	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114737	C₂	1	2	yes	maximal, complemented
8	D₁₃×#341	C₂	1	2	yes	maximal
9	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₁₃	C₂²	1	4	yes	—
11	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88062	C₂²	1	4	yes	—
18	M₁₆×Dic₁₃	C₂²	1	4	yes	—
19	C₂×#88059	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₄×#6099	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	D₁₃×#108	C₄	1	16	yes	direct product factor
25	#11718368	—	2	2	no	—
26	D₁₃×#97	—	2	2	no	—
27	C₄×#6094	—	2	2	no	—
28	C₄×C₈×D₁₃	—	2	2	no	—
29	C₄×C₈×D₁₃	C₄	1	4	yes	—
30	C₄×C₈×D₁₃	C₄	1	4	yes	—
31	C₁₃×#341	C₂²	1	2	yes	—
32	#11718150	C₂²	1	4	yes	—
33	#11718144	C₂²	1	2	yes	—
34	C₂²×C₄×D₁₃	Q₈	1	2	yes	—
35	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6099	C₂³	1	4	yes	—
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	C₂²×#1055	C₂³	1	1	yes	characteristic
45	M₁₆×D₁₃	—	2	8	no	—
46	M₁₆×D₁₃	C₂×C₄	1	16	yes	—
47	C₂×#6094	—	2	4	no	—
48	C₂×C₈×D₁₃	—	2	4	no	—
49	C₂×C₈×D₁₃	—	2	4	no	—
50	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
53	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
55	C₄²×D₁₃	C₂×C₄	1	2	yes	—
56	C₄²×D₁₃	C₂×C₄	1	2	yes	—
57	C₄²×D₁₃	D₄	1	1	yes	characteristic
58	C₄²×D₁₃	D₄	1	1	yes	characteristic
59	C₄²×D₁₃	D₄	1	1	yes	characteristic
60	C₄²×D₁₃	D₄	1	1	yes	characteristic
61	C₁₃×#108	C₂×C₄	1	8	yes	complemented
62	C₁₃×#97	—	2	2	no	—
63	C₄×C₈×C₁₃	—	2	2	no	—
64	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
65	#88062	C₂×C₄	1	16	yes	complemented
66	#88059	C₂×C₄	1	8	yes	complemented
67	#88055	—	2	4	no	—
68	C₈×Dic₁₃	—	2	4	no	—
69	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
70	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
71	#88043	—	2	2	no	—
72	C₄×#1055	—	2	2	no	—
73	C₄×#1055	C₂×C₄	1	4	yes	—
74	C₂×C₄×#108	—	13	1	no	maximal, Sylow
75	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
79	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	—	2	8	no	—
81	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	#41	1	4	yes	—
91	C₁₃×M₁₆	—	2	4	no	—
92	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
103	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
106	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
108	#6099	—	2	4	no	—
109	#6099	C₂²×C₄	1	8	yes	—
110	C₂×#1055	—	2	4	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	C₂²×C₄	1	2	yes	—
113	C₂×#1055	C₂²×C₄	1	2	yes	—
114	C₂×#1055	C₂²×C₄	1	4	yes	—
115	#6094	—	2	16	no	—
116	C₈×D₁₃	—	2	4	no	—
117	C₈×D₁₃	—	2	4	no	—
118	C₈×D₁₃	—	2	16	no	—
119	C₈×D₁₃	C₄²	1	8	yes	—
120	C₈×D₁₃	C₄²	1	8	yes	—
121	C₂²×#108	—	13	2	no	—
122	C₂×C₄×M₁₆	—	13	2	no	—
123	C₂²×C₄×C₈	—	13	1	no	—
124	C₄×#108	—	13	2	no	—
125	C₄×#108	—	13	2	no	—
126	C₄×#108	—	13	1	no	—
127	C₄×#108	—	13	2	no	—
128	C₄×#108	D₁₃	1	1	yes	characteristic, direct product factor
129	C₂×#341	—	13	2	no	—
130	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂²×D₁₃	—	2	4	no	—
132	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×D₁₃	C₄×Q₈	1	4	yes	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×Dic₁₃	—	2	4	no	—
143	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
148	C₄×D₁₃	—	2	4	no	—
149	C₄×D₁₃	—	2	4	no	—
150	C₄×D₁₃	—	2	4	no	—
151	C₄×D₁₃	#108	1	8	yes	direct product factor
152	C₄×D₁₃	C₂×C₄²	1	2	yes	—
153	C₄×D₁₃	C₂×C₄²	1	2	yes	—
154	C₄×D₁₃	#117	1	2	yes	—
155	C₄×D₁₃	#117	1	2	yes	—
156	C₄×D₁₃	C₄×D₄	1	4	yes	—
157	C₄×D₁₃	C₄×D₄	1	4	yes	—
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	—	2	8	no	—
160	C₈×C₁₃	C₂×C₄²	1	4	yes	—
161	#1055	—	2	2	no	—
162	#1055	—	2	8	no	—
163	#1055	C₂×C₄²	1	4	yes	—
164	C₂²×M₁₆	—	13	4	no	—
165	C₂³×C₈	—	13	1	no	—
166	C₂³×C₈	—	13	1	no	—
167	C₂²×C₄²	—	13	1	no	—
168	C₂×#108	—	13	4	no	—
169	C₂×#108	—	13	2	no	—
170	C₂×#108	—	13	16	no	—
171	C₂×#108	C₂×D₁₃	1	2	yes	—
172	C₄×M₁₆	—	13	4	no	—
173	C₄×M₁₆	—	13	2	no	—
174	C₄×M₁₆	—	26	2	no	—
175	C₄×M₁₆	—	26	2	no	—
176	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
177	C₂×#97	—	26	2	no	—
178	C₂×C₄×C₈	—	13	1	no	—
179	C₂×C₄×C₈	—	13	1	no	—
180	C₂×C₄×C₈	—	13	1	no	—
181	C₂×C₄×C₈	—	13	4	no	—
182	C₂×C₄×C₈	—	13	4	no	—
183	C₂×C₄×C₈	—	26	2	no	—
184	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
185	#341	—	13	4	no	—
186	#341	—	13	2	no	—
187	#341	C₂×D₁₃	1	2	yes	—
188	C₂²×C₁₃	—	2	2	no	—
189	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
191	C₂×D₁₃	—	2	2	no	—
192	C₂×D₁₃	—	2	8	no	—
193	C₂×D₁₃	C₄×C₄⋊C₄	1	2	yes	—
194	C₂×D₁₃	C₄×C₄⋊C₄	1	2	yes	—
195	C₂×D₁₃	C₂×#108	1	4	yes	—
196	C₄×C₁₃	—	2	2	no	—
197	C₄×C₁₃	C₂×#108	1	4	yes	complemented
198	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
199	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
200	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
201	Dic₁₃	—	2	2	no	—
202	Dic₁₃	C₄×C₄⋊C₄	1	2	yes	—
203	Dic₁₃	C₄×C₄⋊C₄	1	2	yes	—
204	Dic₁₃	C₂×#108	1	4	yes	complemented
205	C₂³×C₄	—	13	1	no	—
206	C₂³×C₄	—	13	2	no	—
207	C₂×M₁₆	—	13	4	no	—
208	C₂×M₁₆	—	13	16	no	—
209	C₂×M₁₆	—	26	4	no	—
210	C₂×M₁₆	—	26	8	no	—
211	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	2	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	8	no	—
216	C₂²×C₈	—	13	1	no	—
217	C₂²×C₈	—	13	2	no	—
218	C₂²×C₈	—	13	2	no	—
219	C₂²×C₈	—	26	4	no	—
220	C₂²×C₈	—	26	4	no	—
221	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×C₄²	—	13	2	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	2	no	—
226	C₂×C₄²	—	13	2	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	13	1	no	—
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	#108	—	13	16	no	—
233	#108	—	13	8	no	—
234	#108	C₄×D₁₃	1	8	yes	direct product factor
235	#97	—	2	2	no	—
236	#97	—	26	4	no	—
237	#97	—	26	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	—	13	4	no	—
240	C₄×C₈	—	13	4	no	—
241	C₄×C₈	—	13	4	no	—
242	C₄×C₈	—	26	4	no	—
243	C₄×C₈	—	26	2	no	—
244	C₄×C₈	C₄×D₁₃	1	4	yes	—
245	C₂×C₁₃	—	2	2	no	—
246	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₁₃	C₂²×#108	1	2	yes	—
248	D₁₃	—	2	2	no	—
249	D₁₃	C₄×#108	1	4	yes	direct product factor
250	C₂⁴	—	13	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	4	no	—
256	C₂²×C₄	—	13	1	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	4	no	—
259	C₂²×C₄	—	13	2	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	2	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂²×C₄	—	26	8	no	—
264	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
265	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
266	M₁₆	—	2	4	no	—
267	M₁₆	—	13	8	no	—
268	M₁₆	—	26	16	no	—
269	M₁₆	—	26	4	no	—
270	M₁₆	C₂×C₄×D₁₃	1	8	yes	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	13	4	no	—
275	C₂×C₈	—	13	8	no	—
276	C₂×C₈	—	13	8	no	—
277	C₂×C₈	—	13	2	no	—
278	C₂×C₈	—	26	4	no	—
279	C₂×C₈	—	26	4	no	—
280	C₂×C₈	—	26	16	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	26	2	no	—
283	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
284	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
285	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
286	C₄²	—	13	2	no	—
287	C₄²	—	13	2	no	—
288	C₄²	—	13	2	no	—
289	C₄²	—	13	2	no	—
290	C₄²	—	13	2	no	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	1	no	—
293	C₄²	C₂×C₄×D₁₃	1	2	yes	—
294	C₄²	D₄×D₁₃	1	1	yes	characteristic
295	C₄²	D₄×D₁₃	1	1	yes	characteristic
296	C₁₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
297	C₂³	—	13	1	no	—
298	C₂³	—	13	1	no	—
299	C₂³	—	13	4	no	—
300	C₂³	—	26	4	no	—
301	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	13	4	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	13	2	no	—
311	C₂×C₄	—	13	4	no	—
312	C₂×C₄	—	13	4	no	—
313	C₂×C₄	—	13	8	no	—
314	C₂×C₄	—	13	2	no	—
315	C₂×C₄	—	26	4	no	—
316	C₂×C₄	—	26	4	no	—
317	C₂×C₄	—	26	4	no	—
318	C₂×C₄	—	26	4	no	—
319	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
320	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
323	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
324	C₂×C₄	#41×D₁₃	1	2	yes	—
325	C₈	—	2	2	no	—
326	C₈	—	2	8	no	—
327	C₈	—	13	4	no	—
328	C₈	—	26	2	no	—
329	C₈	—	26	8	no	—
330	C₈	C₄²×D₁₃	1	4	yes	—
331	C₂²	—	2	2	no	—
332	C₂²	—	13	4	no	—
333	C₂²	—	13	2	no	—
334	C₂²	—	13	2	no	—
335	C₂²	—	26	8	no	—
336	C₂²	—	26	2	no	—
337	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
338	C₂²	C₄×Q₈×D₁₃	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	13	4	no	—
341	C₄	—	13	2	no	—
342	C₄	—	13	2	no	—
343	C₄	—	26	2	no	—
344	C₄	D₁₃×#108	1	4	yes	direct product factor
345	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
346	C₄	D₁₃×#117	1	1	yes	characteristic
347	C₄	C₄×D₄×D₁₃	1	2	yes	—
348	C₂	—	2	2	no	—
349	C₂	—	13	4	no	—
350	C₂	—	26	2	no	—
351	C₂	C₄×C₄⋊C₄×D₁₃	1	1	yes	characteristic
352	C₂	C₂×D₁₃×#108	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	C₂×C₄×#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #108×Dic₁₃	C₂	semidirect
5. C₄×#88062	C₂	semidirect
6. C₄×#88059	C₂	semidirect
7. #13114737	C₂	semidirect
9. C₄×C₁₃×#108	C₂	semidirect
24. D₁₃×#108	C₄	direct (N × H)
61. C₁₃×#108	C₂×C₄	semidirect
65. #88062	C₂×C₄	semidirect
66. #88059	C₂×C₄	semidirect
128. C₄×#108	D₁₃	direct (N × H)
151. C₄×D₁₃	#108	direct (N × H)
197. C₄×C₁₃	C₂×#108	semidirect
204. Dic₁₃	C₂×#108	semidirect
234. #108	C₄×D₁₃	direct (N × H)
249. D₁₃	C₄×#108	direct (N × H)
296. C₁₃	C₂×C₄×#108	semidirect
344. C₄	D₁₃×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₁₃×#108	2
C₄×M₁₆×D₁₃	2
C₂×C₄×C₈×D₁₃	2
#108×Dic₁₃	2
C₄×#88062	2
C₄×#88059	2
#13114737	2
D₁₃×#341	2
C₄×C₁₃×#108	2
C₂×C₄×#108	13

Subgroups — C4×D13×#108 #15861013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₁₃×#108 #15861013