Subgroups — #15863233

Order 3328.

758 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114979	C₂	1	2	yes	maximal, complemented
2	#13114956	C₂	1	2	yes	maximal, complemented
3	#13114904	C₂	1	2	yes	maximal
4	#13114820	C₂	1	2	yes	maximal
5	#13114689	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114602	C₂	1	2	yes	maximal
8	#13114588	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
15	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
16	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
17	C₁₃×#376	C₂²	1	1	yes	characteristic
18	C₁₃×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₁₃	C₂²	1	1	yes	characteristic
20	C₄³×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄⋊#1055	C₂²	1	2	yes	—
24	C₂×#88043	C₂²	1	2	yes	—
25	C₂×C₄×#1055	C₂²	1	2	yes	—
26	#11718116	C₂²	1	2	yes	—
27	#11718116	C₂²	1	2	yes	—
28	#11718116	C₂²	1	4	yes	—
29	#11718116	C₂²	1	4	yes	—
30	#11718116	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#117	—	2	2	no	—
34	C₁₃×#117	—	2	2	no	—
35	C₁₃×#117	—	2	4	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	—	2	4	no	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×#1055	C₂³	1	4	yes	—
55	C₂²×#1055	C₂³	1	4	yes	—
56	C₁₃×#95	C₂×C₄	1	2	yes	—
57	C₁₃×#95	C₂×C₄	1	2	yes	—
58	C₁₃×#95	C₂×C₄	1	2	yes	—
59	C₁₃×#95	C₂×C₄	1	2	yes	—
60	C₂²⋊#1055	—	2	8	no	—
61	C₄⋊#1055	—	2	4	no	—
62	C₄⋊#1055	—	2	4	no	—
63	#88043	—	2	4	no	—
64	C₄×#1055	—	2	4	no	—
65	#20781	—	13	1	no	maximal, Sylow
66	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	#41	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₂²⋊C₄	—	2	2	no	—
88	C₁₃×C₂²⋊C₄	—	2	2	no	—
89	C₁₃×C₂²⋊C₄	—	2	4	no	—
90	C₄²×C₁₃	—	2	4	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	4	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	—	2	1	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	#41	1	2	yes	—
101	C₄²×C₁₃	#41	1	2	yes	—
102	C₄²×C₁₃	#41	1	4	yes	—
103	C₂×#1055	—	2	4	no	—
104	C₂×#1055	—	2	4	no	—
105	C₂×#1055	—	2	4	no	—
106	C₂×#1055	—	2	4	no	—
107	C₂×#1055	—	2	4	no	—
108	C₂×#1055	—	2	4	no	—
109	C₄×#117	D₁₃	1	1	yes	characteristic
110	#1839	—	13	2	no	—
111	#1832	—	13	2	no	—
112	#1817	—	13	2	no	—
113	#1798	—	13	2	no	—
114	#1767	—	13	1	no	—
115	C₄×C₄⋊C₈	—	13	1	no	—
116	#1750	—	13	2	no	—
117	#1746	—	13	2	no	—
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	—	2	1	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	—	2	2	no	—
122	C₂³×C₁₃	#117	1	2	yes	—
123	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	4	8	no	—
145	C₂×C₄×C₁₃	#117	1	2	yes	—
146	C₂×C₄×C₁₃	#117	1	2	yes	—
147	C₂×C₄×C₁₃	#117	1	2	yes	—
148	C₂×C₄×C₁₃	#117	1	4	yes	—
149	C₂×C₄×C₁₃	#131	1	4	yes	—
150	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	#1055	—	4	4	no	—
155	#1055	—	4	4	no	—
156	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	13	2	no	—
160	C₂×C₄⋊C₈	—	13	2	no	—
161	C₂×C₂²⋊C₈	—	13	4	no	—
162	C₂×#97	—	13	2	no	—
163	C₂×C₄×C₈	—	13	2	no	—
164	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
165	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
166	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
167	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
168	#376	C₂×D₁₃	1	1	yes	characteristic
169	#376	C₂×D₁₃	1	1	yes	characteristic
170	C₄³	C₂×D₁₃	1	1	yes	characteristic
171	C₄³	C₂×D₁₃	1	1	yes	characteristic
172	#336	—	13	4	no	—
173	#336	—	13	4	no	—
174	#336	—	13	2	no	—
175	#336	—	13	2	no	—
176	#336	—	13	4	no	—
177	C₂²×C₁₃	—	2	4	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	4	1	no	—
180	C₂²×C₁₃	—	4	1	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
183	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
184	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#131	1	2	yes	—
186	C₂²×C₁₃	#568	1	2	yes	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	4	no	—
190	C₄×C₁₃	—	2	4	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	4	1	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	—	4	2	no	—
195	C₄×C₁₃	—	4	2	no	—
196	C₄×C₁₃	#529	1	2	yes	—
197	C₄×C₁₃	#529	1	2	yes	—
198	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
199	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
200	C₂²×C₈	—	13	4	no	—
201	C₂²×C₈	—	13	4	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	4	no	—
205	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
208	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
213	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
214	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
215	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
216	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
217	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
223	C₄⋊C₈	—	26	4	no	—
224	C₄⋊C₈	—	26	4	no	—
225	C₂²⋊C₈	—	26	8	no	—
226	#97	—	26	4	no	—
227	C₄×C₈	—	26	4	no	—
228	#95	C₂×Dic₁₃	1	2	yes	—
229	#95	C₂×Dic₁₃	1	2	yes	—
230	#95	C₂×Dic₁₃	1	2	yes	—
231	#95	C₂×Dic₁₃	1	2	yes	—
232	C₂×C₁₃	—	4	2	no	—
233	C₂×C₁₃	#2283	1	1	yes	characteristic
234	C₂×C₁₃	#2922	1	1	yes	characteristic
235	C₂×C₁₃	#2923	1	1	yes	characteristic
236	C₂×C₁₃	#2924	1	2	yes	—
237	C₂×C₁₃	#2925	1	2	yes	—
238	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	#6127	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
255	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
256	C₂×C₈	—	26	4	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	26	4	no	—
259	C₂×C₈	—	26	4	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	26	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	#6127	1	2	yes	—
279	C₄²	#6127	1	2	yes	—
280	C₄²	#6127	1	4	yes	—
281	C₁₃	#20781	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	2	2	no	—
286	C₂³	#88180	1	2	yes	—
287	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	8	no	—
309	C₂×C₄	#88180	1	2	yes	—
310	C₂×C₄	#88180	1	2	yes	—
311	C₂×C₄	#88180	1	2	yes	—
312	C₂×C₄	#88180	1	4	yes	—
313	C₂×C₄	#88202	1	4	yes	—
314	C₂×C₄	C₂×#6127	1	2	yes	—
315	C₂×C₄	C₂×#6127	1	2	yes	—
316	C₂×C₄	C₂×#6127	1	1	yes	characteristic
317	C₂×C₄	C₂×#6127	1	1	yes	characteristic
318	C₈	—	52	4	no	—
319	C₈	—	52	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	2	no	—
325	C₂²	C₂×#88180	1	1	yes	characteristic
326	C₂²	C₂×#88180	1	1	yes	characteristic
327	C₂²	C₂×#88180	1	1	yes	characteristic
328	C₂²	C₂×#88202	1	2	yes	—
329	C₂²	#11719524	1	2	yes	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	#11719184	1	2	yes	—
340	C₄	#11719184	1	2	yes	—
341	C₂	—	4	2	no	—
342	C₂	#13117234	1	1	yes	characteristic
343	C₂	#13123480	1	1	yes	characteristic
344	C₂	#13123487	1	1	yes	characteristic
345	C₂	#13123494	1	2	yes	—
346	C₂	#13123502	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#117	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#20781
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #13114979	C₂	semidirect
2. #13114956	C₂	semidirect
5. #13114689	C₂	semidirect
6. C₄×C₄⋊#1055	C₂	semidirect
8. #13114588	C₂	semidirect
281. C₁₃	#20781	semidirect

Maximal subgroups

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

subgroup	index
#13114979	2
#13114956	2
#13114904	2
#13114820	2
#13114689	2
C₄×C₄⋊#1055	2
#13114602	2
#13114588	2
C₄×C₁₃×#117	2
#20781	13