Subgroups — #15861582

Order 3328.

866 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718492	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718491	C₂	1	1	yes	maximal, characteristic
3	#13117313	C₂	1	1	yes	maximal, characteristic
4	#13114709	C₂	1	2	yes	maximal
5	#13114706	C₂	1	1	yes	maximal, characteristic
6	#13114702	C₂	1	2	yes	maximal
7	#13114701	C₂	1	1	yes	maximal, characteristic
8	#13114694	C₂	1	2	yes	maximal
9	#13114522	C₂	1	2	yes	maximal
10	C₄×#88043	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88055	C₂²	1	2	yes	—
18	C₂×#88055	C₂²	1	2	yes	—
19	C₂×#88054	C₂²	1	2	yes	—
20	C₂×#88054	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	#11718594	C₄	1	4	yes	—
25	#11718593	C₂²	1	2	yes	—
26	C₄×#6101	C₂²	1	2	yes	—
27	#11718590	C₄	1	2	yes	—
28	#11718589	C₄	1	2	yes	—
29	C₂×#88043	C₂²	1	1	yes	characteristic
30	C₂×#88043	C₂²	1	1	yes	characteristic
31	C₂×#88043	C₂²	1	1	yes	characteristic
32	C₂×#88043	C₂²	1	1	yes	characteristic
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
35	#11718492	C₄	1	8	yes	complemented
36	#11718491	C₄	1	8	yes	complemented
37	C₁₃×#336	C₂²	1	2	yes	—
38	#11718142	C₂²	1	2	yes	—
39	#11718142	C₂²	1	2	yes	—
40	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#6101	C₂×C₄	1	4	yes	—
51	C₂×#6101	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	C₂²×#1055	C₂³	1	1	yes	characteristic
57	C₂²×#1055	C₂³	1	1	yes	characteristic
58	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
59	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₁₃	—	2	4	no	—
63	#88071	C₂×C₄	1	4	yes	—
64	#88071	C₂×C₄	1	4	yes	—
65	#88071	C₂×C₄	1	4	yes	—
66	#88071	C₂×C₄	1	2	yes	—
67	#88055	—	2	4	no	—
68	#88055	C₂×C₄	1	8	yes	—
69	#88054	C₂×C₄	1	8	yes	—
70	#88054	C₂×C₄	1	8	yes	—
71	C₈×Dic₁₃	—	2	4	no	—
72	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
73	#88043	—	2	2	no	—
74	#88043	—	2	2	no	—
75	#88043	—	2	2	no	—
76	#88043	Q₈	1	4	yes	complemented
77	C₄×#1055	C₂×C₄	1	4	yes	—
78	C₄×#1055	C₂×C₄	1	4	yes	—
79	#20521	—	13	1	no	maximal, Sylow
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
86	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₂₆	—	2	4	no	—
89	C₂×Dic₂₆	C₄²	1	4	yes	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₃	—	2	4	no	—
97	C₄²×C₁₃	—	2	1	no	—
98	C₄²×C₁₃	—	2	1	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃	#41	1	2	yes	—
104	C₄⋊Dic₁₃	—	2	4	no	—
105	C₄⋊Dic₁₃	C₄²	1	4	yes	—
106	#6101	—	4	4	no	—
107	#6101	C₄²	1	8	yes	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
111	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
112	C₂×#1055	—	2	1	no	—
113	C₂×#1055	—	2	1	no	—
114	C₂×#1055	—	2	1	no	—
115	C₂×#1055	—	2	1	no	—
116	C₂×#1055	C₂²×C₄	1	2	yes	—
117	C₂×#1055	C₂²×C₄	1	2	yes	—
118	C₂×#1055	C₂²×C₄	1	2	yes	—
119	C₂×#1055	C₂²×C₄	1	2	yes	—
120	C₂×#1055	C₂×Q₈	1	2	yes	—
121	C₂×#1055	C₂×Q₈	1	2	yes	—
122	C₂×#1055	#41	1	2	yes	—
123	C₂×#1055	#41	1	2	yes	—
124	C₂×#446	—	13	1	no	—
125	C₂×C₈×Q₈	—	13	1	no	—
126	#2295	—	13	1	no	—
127	#1771	—	13	2	no	—
128	#1770	—	13	1	no	—
129	#1770	—	13	2	no	—
130	C₈×C₄⋊C₄	—	13	1	no	—
131	#1768	—	13	2	no	—
132	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
133	#1726	—	13	2	no	—
134	C₄×#97	—	13	1	no	—
135	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	—	2	2	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	C₂×C₄²	1	2	yes	—
143	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
145	C₂×C₄×C₁₃	#131	1	4	yes	—
146	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₁₃	—	2	4	no	—
153	C₂×Dic₁₃	—	2	4	no	—
154	C₂×Dic₁₃	—	2	4	no	—
155	C₂×Dic₁₃	—	2	4	no	—
156	C₂×Dic₁₃	—	2	4	no	—
157	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
158	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
159	Dic₂₆	—	2	8	no	—
160	C₈×C₁₃	—	2	4	no	—
161	C₈×C₁₃	—	2	4	no	—
162	#1055	—	2	2	no	—
163	#1055	—	2	2	no	—
164	#1055	C₄×Q₈	1	4	yes	—
165	#1055	C₄×Q₈	1	4	yes	—
166	C₂×C₄×Q₈	—	13	1	no	—
167	#446	—	13	8	no	—
168	C₈×Q₈	—	13	8	no	—
169	C₂×C₄⋊C₈	—	13	2	no	—
170	C₂×C₄⋊C₈	—	13	2	no	—
171	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×#97	—	13	2	no	—
174	C₂×#97	—	13	1	no	—
175	C₂×#97	—	13	2	no	—
176	C₂×#97	—	13	1	no	—
177	C₂×#97	—	13	1	no	—
178	C₂×#97	—	13	1	no	—
179	C₂×C₄×C₈	—	13	2	no	—
180	C₂×C₄×C₈	—	13	1	no	—
181	C₂×C₄×C₈	—	13	1	no	—
182	C₂×C₄×C₈	—	13	2	no	—
183	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
184	#391	—	13	2	no	—
185	#389	—	13	2	no	—
186	#387	—	13	4	no	—
187	C₄×C₄⋊C₄	—	13	2	no	—
188	C₄×C₄⋊C₄	—	13	1	no	—
189	#376	—	13	2	no	—
190	C₄³	C₂×D₁₃	1	1	yes	characteristic
191	#336	—	13	2	no	—
192	#336	—	13	2	no	—
193	#336	C₂×D₁₃	1	2	yes	—
194	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
197	C₂²×C₁₃	C₂×#131	1	2	yes	—
198	C₂²×C₁₃	#568	1	2	yes	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	4	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	#405	1	4	yes	—
203	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
204	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
205	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
206	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
207	Dic₁₃	—	2	4	no	—
208	Dic₁₃	—	2	4	no	—
209	Dic₁₃	—	4	4	no	—
210	C₂²×Q₈	—	13	1	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	1	no	—
214	C₂²×C₈	—	13	1	no	—
215	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
216	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
217	C₄×Q₈	—	13	4	no	—
218	C₄×Q₈	—	13	4	no	—
219	C₂×C₄⋊C₄	—	13	2	no	—
220	C₂×C₄⋊C₄	—	13	1	no	—
221	C₂×C₄⋊C₄	—	13	2	no	—
222	C₂×C₄⋊C₄	—	13	4	no	—
223	C₂×C₄²	—	13	2	no	—
224	C₂×C₄²	—	13	2	no	—
225	C₂×C₄²	C₄×D₁₃	1	2	yes	—
226	C₂×C₄²	C₄×D₁₃	1	2	yes	—
227	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
230	C₄⋊C₈	—	13	8	no	—
231	C₄⋊C₈	—	13	8	no	—
232	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
233	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
234	#97	—	13	8	no	—
235	#97	—	13	4	no	—
236	#97	—	26	2	no	—
237	#97	—	26	2	no	—
238	#97	—	26	4	no	—
239	#97	—	26	2	no	—
240	C₄×C₈	—	2	4	no	—
241	C₄×C₈	—	13	8	no	—
242	C₄×C₈	—	13	4	no	—
243	C₄×C₈	—	13	4	no	—
244	C₄×C₈	—	26	4	no	—
245	#95	—	13	4	no	—
246	#95	—	13	4	no	—
247	#95	—	13	2	no	—
248	#95	—	13	4	no	—
249	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
251	C₂×C₁₃	#2875	1	1	yes	characteristic
252	C₂×C₁₃	#2962	1	2	yes	—
253	C₂×C₁₃	#2969	1	2	yes	—
254	C₂×Q₈	—	13	4	no	—
255	C₂×Q₈	—	26	4	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	4	no	—
259	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
260	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
262	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	—	13	2	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	13	2	no	—
275	C₂×C₈	—	13	2	no	—
276	C₂×C₈	—	26	1	no	—
277	C₂×C₈	—	26	1	no	—
278	C₂×C₈	—	26	1	no	—
279	C₂×C₈	—	26	1	no	—
280	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
281	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
282	C₄⋊C₄	—	13	8	no	—
283	C₄⋊C₄	—	13	4	no	—
284	C₄⋊C₄	—	26	4	no	—
285	C₄⋊C₄	—	52	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	4	2	no	—
290	C₄²	—	13	4	no	—
291	C₄²	—	13	4	no	—
292	C₄²	—	26	2	no	—
293	C₄²	—	26	2	no	—
294	C₄²	C₂×C₄×D₁₃	1	2	yes	—
295	C₄²	C₂×C₄×D₁₃	1	2	yes	—
296	C₄²	#6129	1	2	yes	—
297	C₄²	Q₈×D₁₃	1	2	yes	—
298	C₁₃	#20521	1	1	yes	characteristic, complemented, Sylow
299	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
300	Q₈	—	26	8	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	—	13	4	no	—
308	C₂×C₄	—	13	4	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	26	4	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	—	26	4	no	—
313	C₂×C₄	—	26	4	no	—
314	C₂×C₄	C₄²×D₁₃	1	2	yes	—
315	C₂×C₄	C₄²×D₁₃	1	2	yes	—
316	C₂×C₄	#88141	1	4	yes	—
317	C₂×C₄	#88155	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
320	C₂×C₄	C₂×#6129	1	1	yes	characteristic
321	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
322	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
323	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	13	4	no	—
327	C₈	—	13	4	no	—
328	C₈	—	26	2	no	—
329	C₈	—	26	2	no	—
330	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
331	C₂²	C₂×#88141	1	1	yes	characteristic
332	C₂²	#11719227	1	1	yes	characteristic
333	C₂²	C₂×#88155	1	2	yes	—
334	C₂²	#11719450	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	26	4	no	—
339	C₄	—	26	4	no	—
340	C₄	—	52	4	no	—
341	C₄	#11718350	1	4	yes	—
342	C₄	C₄×#6129	1	1	yes	characteristic
343	C₄	C₄×#6129	1	1	yes	characteristic
344	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
345	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
346	C₂	C₄×#88141	1	1	yes	characteristic
347	C₂	C₂×#11718350	1	1	yes	characteristic
348	C₂	#13123265	1	1	yes	characteristic
349	C₂	#13123779	1	2	yes	—
350	C₂	#13123834	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#20521
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
35. #11718492	C₄	semidirect
36. #11718491	C₄	semidirect
76. #88043	Q₈	semidirect
298. C₁₃	#20521	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718492	2
C₂×#11718491	2
#13117313	2
#13114709	2
#13114706	2
#13114702	2
#13114701	2
#13114694	2
#13114522	2
C₄×#88043	2
C₄×C₁₃×C₄⋊C₈	2
#20521	13