Subgroups — #15768133

Order 3328.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718762	C₂	1	2	yes	maximal
2	C₂×#11718148	C₂	1	2	yes	maximal
3	#13114575	C₂	1	1	yes	maximal, characteristic
4	#13114569	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₁₃×#405	—	2	2	no	—
12	C₁₃×#405	—	2	2	no	—
13	C₂×C₁₃×#97	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
15	#11718762	—	2	4	no	—
16	#11718762	C₄	1	8	yes	complemented
17	C₂×#88055	—	2	2	no	—
18	C₂×C₈×Dic₁₃	—	2	2	no	—
19	#11718148	C₄	1	8	yes	complemented
20	#11718122	C₄	1	4	yes	complemented
21	#11718118	C₄	1	4	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	1	no	—
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
35	C₁₃×#117	—	2	2	no	—
36	C₁₃×#117	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	C₂×C₄	1	4	yes	—
44	#88180	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	C₂×C₄	1	4	yes	—
51	C₂×#6099	C₂×C₄	1	4	yes	—
52	C₂×#6099	C₂×C₄	1	4	yes	—
53	C₂²×#1055	—	2	2	no	—
54	C₁₃×#97	—	2	4	no	—
55	C₄×C₈×C₁₃	—	2	2	no	—
56	C₄×C₈×C₁₃	—	2	2	no	—
57	#88055	—	2	8	no	—
58	C₈×Dic₁₃	—	2	8	no	—
59	#10711	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₄²	1	4	yes	—
78	C₂×C₈×C₁₃	C₄²	1	2	yes	—
79	C₂×C₈×C₁₃	C₄²	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₂²⋊C₄	—	2	4	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₂²⋊Dic₁₃	—	2	4	no	—
94	C₂²⋊Dic₁₃	C₄²	1	8	yes	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄×Dic₁₃	—	2	4	no	—
98	C₄×Dic₁₃	—	2	4	no	—
99	#6099	—	2	4	no	—
100	#6099	—	2	4	no	—
101	#6099	—	2	4	no	—
102	#6099	C₄²	1	8	yes	—
103	C₂×#1055	—	2	4	no	—
104	C₂×#1055	—	2	4	no	—
105	C₂×#1055	—	2	4	no	—
106	C₂×#405	—	13	2	no	—
107	C₂×#405	D₁₃	1	1	yes	characteristic
108	C₂×#344	—	13	2	no	—
109	#1742	—	13	1	no	—
110	#1740	—	13	1	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂³×C₁₃	C₄×D₄	1	2	yes	—
115	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#117	1	2	yes	—
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#117	1	2	yes	—
129	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
137	C₂×Dic₁₃	—	2	8	no	—
138	C₂×Dic₁₃	—	4	8	no	—
139	C₈×C₁₃	—	2	4	no	—
140	C₈×C₁₃	#95	1	4	yes	—
141	C₈×C₁₃	#95	1	2	yes	—
142	C₈×C₁₃	#95	1	2	yes	—
143	#1055	—	2	8	no	—
144	C₂²×M₁₆	—	13	2	no	—
145	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
146	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₂×#117	—	13	2	no	—
148	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
149	#405	—	2	2	no	—
150	#405	—	2	2	no	—
151	#405	—	13	8	no	—
152	#405	—	26	4	no	—
153	C₂×#97	—	26	2	no	—
154	C₂×#97	Dic₁₃	1	2	yes	complemented
155	C₂×C₄×C₈	—	26	2	no	—
156	C₂×C₄×C₈	Dic₁₃	1	2	yes	complemented
157	#344	—	13	8	no	—
158	#343	—	13	4	no	—
159	#342	—	13	4	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	4	2	no	—
163	C₂²×C₁₃	—	4	1	no	—
164	C₂²×C₁₃	—	4	1	no	—
165	C₂²×C₁₃	C₄³	1	1	yes	characteristic
166	C₂²×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
167	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
169	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₁₃	—	4	4	no	—
171	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
172	C₄×C₁₃	#376	1	1	yes	characteristic
173	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
174	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
175	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
176	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
177	Dic₁₃	—	4	8	no	—
178	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	13	4	no	—
183	C₂×M₁₆	—	13	4	no	—
184	C₂×M₁₆	—	13	4	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	Dic₂₆	1	2	yes	—
191	C₂²×C₈	C₄×D₁₃	1	4	yes	—
192	C₂²×C₈	D₅₂	1	2	yes	—
193	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
194	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
197	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	13	4	no	—
201	#117	—	13	4	no	—
202	#117	—	26	2	no	—
203	#117	—	26	2	no	—
204	C₂×C₄⋊C₄	—	26	2	no	—
205	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
206	C₂×C₂²⋊C₄	—	13	4	no	—
207	C₂×C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
210	#97	—	2	4	no	—
211	#97	—	26	8	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	—	26	8	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	—	2	1	no	—
217	C₂×C₁₃	—	2	2	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	C₄×#95	1	1	yes	characteristic
220	C₂×C₁₃	#1757	1	2	yes	—
221	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂²×C₄	—	26	4	no	—
223	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
224	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
228	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
229	C₂²×C₄	#6127	1	2	yes	—
230	C₂²×C₄	#6127	1	1	yes	characteristic
231	C₂²×C₄	#6127	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	13	8	no	—
237	M₁₆	—	26	4	no	—
238	M₁₆	—	26	4	no	—
239	M₁₆	—	26	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	26	4	no	—
243	C₂×C₈	—	26	4	no	—
244	C₂×C₈	—	26	4	no	—
245	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
246	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
247	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
248	C₂×C₈	#6101	1	4	yes	—
249	C₂×C₈	#6101	1	4	yes	—
250	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
251	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
252	C₂×C₈	#6103	1	4	yes	—
253	C₂×C₈	#6103	1	4	yes	—
254	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
256	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	26	4	no	—
259	C₄⋊C₄	—	26	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	13	8	no	—
262	C₂²⋊C₄	—	26	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	26	4	no	—
266	C₄²	—	26	4	no	—
267	C₁₃	#10711	1	1	yes	characteristic, complemented, Sylow
268	C₂³	—	4	2	no	—
269	C₂³	C₄×Dic₂₆	1	1	yes	characteristic
270	C₂³	C₄²×D₁₃	1	2	yes	—
271	C₂³	C₄×D₅₂	1	1	yes	characteristic
272	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
273	C₂³	C₄×C₂²⋊D₁₃	1	2	yes	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	26	8	no	—
277	C₂×C₄	—	52	8	no	—
278	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
279	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
280	C₂×C₄	C₄²×D₁₃	1	2	yes	—
281	C₂×C₄	#88126	1	2	yes	—
282	C₂×C₄	#88126	1	2	yes	—
283	C₂×C₄	C₄×D₅₂	1	2	yes	—
284	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂×#6101	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
289	C₂×C₄	#88180	1	1	yes	characteristic
290	C₂×C₄	#88180	1	1	yes	characteristic
291	C₂×C₄	C₂×#6103	1	2	yes	—
292	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
293	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
294	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
296	C₈	—	2	4	no	—
297	C₈	—	26	8	no	—
298	C₈	#88071	1	4	yes	—
299	C₈	#88071	1	2	yes	—
300	C₈	#88071	1	2	yes	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	C₄²×Dic₁₃	1	1	yes	characteristic
307	C₂²	C₄×#6101	1	2	yes	—
308	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
309	C₂²	C₄×#6103	1	2	yes	—
310	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
311	C₄	—	4	4	no	—
312	C₄	—	52	8	no	—
313	C₄	C₄×#6101	1	2	yes	—
314	C₄	#11718593	1	1	yes	characteristic
315	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
316	C₄	C₄×#6103	1	2	yes	—
317	C₄	C₂×#88071	1	1	yes	characteristic
318	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	C₄×#88071	1	1	yes	characteristic
324	C₂	#13114636	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₁₃
commutator	C₂²×C₁₃	C₄³
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#10711
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
13. C₂×C₁₃×#97	C₄	semidirect
14. C₂×C₄×C₈×C₁₃	C₄	semidirect
16. #11718762	C₄	semidirect
19. #11718148	C₄	semidirect
20. #11718122	C₄	semidirect
154. C₂×#97	Dic₁₃	semidirect
156. C₂×C₄×C₈	Dic₁₃	semidirect
267. C₁₃	#10711	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718762	2
C₂×#11718148	2
#13114575	2
#13114569	2
C₂×C₁₃×#405	2
#10711	13