Subgroups — #15881851

Order 3328.

712 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718685	C₂	1	2	yes	maximal
2	#13115560	C₂	1	2	yes	maximal
3	#13115560	C₂	1	2	yes	maximal
4	#13115555	C₂	1	2	yes	maximal
5	#13114956	C₂	1	2	yes	maximal
6	#13114953	C₂	1	1	yes	maximal, characteristic
7	#13114693	C₂	1	2	yes	maximal
8	#13114689	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2295	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#391	C₄	1	1	yes	characteristic
12	C₁₃×#391	C₄	1	1	yes	characteristic
13	C₁₃×#389	C₄	1	1	yes	characteristic
14	C₁₃×#389	C₄	1	1	yes	characteristic
15	C₁₃×#387	C₄	1	2	yes	—
16	C₁₃×#387	C₄	1	2	yes	—
17	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₁₃×#376	C₂²	1	2	yes	—
20	C₄³×C₁₃	C₂²	1	1	yes	characteristic
21	#11718685	—	2	8	no	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄⋊#1055	C₂²	1	2	yes	—
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₂×C₄⋊#1055	C₂²	1	2	yes	—
26	C₂×C₄⋊#1055	C₂²	1	2	yes	—
27	C₂×C₄⋊#1055	C₂²	1	2	yes	—
28	C₂×#88043	C₂²	1	2	yes	—
29	C₂×#88043	C₂²	1	2	yes	—
30	C₂×C₄×#1055	C₂²	1	2	yes	—
31	#11718116	C₂²	1	1	yes	characteristic
32	#11718116	C₂²	1	1	yes	characteristic
33	#11718116	C₂²	1	2	yes	—
34	#11718116	C₂²	1	1	yes	characteristic
35	#11718116	C₂²	1	1	yes	characteristic
36	#11718116	C₂²	1	2	yes	—
37	#11718116	C₂²	1	2	yes	—
38	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	D₄	1	4	yes	—
40	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×#1055	C₂³	1	2	yes	—
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₂²×#1055	C₂³	1	2	yes	—
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₁₃×#95	C₂×C₄	1	2	yes	—
61	C₁₃×#95	C₂×C₄	1	2	yes	—
62	C₁₃×#95	C₂×C₄	1	2	yes	—
63	C₁₃×#95	C₂×C₄	1	2	yes	—
64	C₁₃×#95	C₂×C₄	1	2	yes	—
65	C₁₃×#95	C₂×C₄	1	2	yes	—
66	C₁₃×#95	C₂×C₄	1	2	yes	—
67	C₄⋊#1055	—	2	4	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	—	2	4	no	—
70	C₄⋊#1055	—	2	4	no	—
71	C₄⋊#1055	—	2	4	no	—
72	C₄⋊#1055	—	2	4	no	—
73	#88043	—	2	4	no	—
74	#88043	—	2	4	no	—
75	C₄×#1055	—	2	4	no	—
76	#22737	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₄²×C₁₃	—	4	1	no	—
104	C₄²×C₁₃	—	4	1	no	—
105	C₄²×C₁₃	C₂×D₄	1	4	yes	—
106	C₄²×C₁₃	C₂×D₄	1	2	yes	—
107	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
108	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
109	C₂×#1055	—	2	2	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	2	no	—
120	C₂×#1055	#41	1	4	yes	—
121	C₂×#446	—	13	2	no	—
122	#2295	D₁₃	1	1	yes	characteristic
123	#1940	—	13	2	no	—
124	#1940	—	13	2	no	—
125	#1939	—	13	2	no	—
126	#1832	—	13	2	no	—
127	#1831	—	13	1	no	—
128	#1768	—	13	2	no	—
129	#1767	—	13	1	no	—
130	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	Q₈×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	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	1	no	—
138	C₂×C₄×C₁₃	—	2	1	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	1	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	—	2	1	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	1	no	—
150	C₂×C₄×C₁₃	—	2	1	no	—
151	C₂×C₄×C₁₃	—	2	1	no	—
152	C₂×C₄×C₁₃	—	2	1	no	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
158	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
160	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₁₃	#131	1	2	yes	—
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
165	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
166	#1055	—	2	4	no	—
167	#1055	—	4	2	no	—
168	#1055	—	4	2	no	—
169	#1055	—	4	2	no	—
170	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
171	#446	—	26	8	no	—
172	C₂×C₄⋊C₈	—	13	2	no	—
173	C₂×C₄⋊C₈	—	13	2	no	—
174	C₂×C₄⋊C₈	—	13	2	no	—
175	C₂×C₄⋊C₈	—	13	2	no	—
176	C₂×C₄⋊C₈	—	13	2	no	—
177	C₂×C₄⋊C₈	—	13	2	no	—
178	C₂×#97	—	13	2	no	—
179	C₂×#97	—	13	2	no	—
180	C₂×C₄×C₈	—	13	2	no	—
181	#391	Dic₁₃	1	1	yes	characteristic
182	#391	Dic₁₃	1	1	yes	characteristic
183	#389	Dic₁₃	1	1	yes	characteristic
184	#389	Dic₁₃	1	1	yes	characteristic
185	#387	Dic₁₃	1	2	yes	—
186	#387	Dic₁₃	1	2	yes	—
187	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
188	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
189	#376	C₂×D₁₃	1	2	yes	—
190	C₄³	C₂×D₁₃	1	1	yes	characteristic
191	#336	—	13	2	no	—
192	#336	—	13	1	no	—
193	#336	—	13	1	no	—
194	#336	—	13	1	no	—
195	#336	—	13	2	no	—
196	#336	—	13	2	no	—
197	#336	—	13	1	no	—
198	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
200	C₂²×C₁₃	#519	1	1	yes	characteristic
201	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
202	C₂²×C₁₃	#568	1	1	yes	characteristic
203	C₂²×C₁₃	#568	1	1	yes	characteristic
204	C₂²×C₁₃	#568	1	1	yes	characteristic
205	C₄×C₁₃	—	2	2	no	—
206	C₄×C₁₃	—	2	4	no	—
207	C₄×C₁₃	—	2	2	no	—
208	C₄×C₁₃	—	2	4	no	—
209	C₄×C₁₃	—	2	2	no	—
210	C₄×C₁₃	—	4	1	no	—
211	C₄×C₁₃	—	4	2	no	—
212	C₄×C₁₃	—	4	2	no	—
213	C₄×C₁₃	—	4	1	no	—
214	C₄×C₁₃	—	4	1	no	—
215	C₄×C₁₃	#548	1	2	yes	—
216	C₄×C₁₃	#557	1	2	yes	—
217	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₈	—	13	2	no	—
219	C₂²×C₈	—	13	2	no	—
220	C₂²×C₈	—	13	2	no	—
221	C₂²×C₈	—	13	2	no	—
222	C₄×Q₈	Dic₂₆	1	4	yes	—
223	C₄×Q₈	D₅₂	1	4	yes	—
224	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
226	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
230	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
237	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
239	C₄⋊C₈	—	26	4	no	—
240	C₄⋊C₈	—	26	4	no	—
241	C₄⋊C₈	—	26	4	no	—
242	C₄⋊C₈	—	26	4	no	—
243	C₄⋊C₈	—	26	4	no	—
244	C₄⋊C₈	—	26	4	no	—
245	#97	—	26	4	no	—
246	#97	—	26	4	no	—
247	C₄×C₈	—	26	4	no	—
248	#95	C₂×Dic₁₃	1	2	yes	—
249	#95	C₂×Dic₁₃	1	2	yes	—
250	#95	C₂×Dic₁₃	1	2	yes	—
251	#95	C₂×Dic₁₃	1	2	yes	—
252	#95	C₂×Dic₁₃	1	2	yes	—
253	#95	C₂×Dic₁₃	1	2	yes	—
254	#95	C₂×Dic₁₃	1	2	yes	—
255	C₂×C₁₃	#2351	1	1	yes	characteristic
256	C₂×C₁₃	#2905	1	1	yes	characteristic
257	C₂×C₁₃	#2906	1	1	yes	characteristic
258	C₂×C₁₃	#2967	1	1	yes	characteristic
259	C₂×C₁₃	#2968	1	1	yes	characteristic
260	C₂×C₁₃	#2985	1	1	yes	characteristic
261	C₂×C₁₃	#2987	1	1	yes	characteristic
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
265	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
268	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
269	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
271	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
273	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
275	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
276	C₂×C₈	—	13	4	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	—	26	2	no	—
279	C₂×C₈	—	26	2	no	—
280	C₂×C₈	—	26	2	no	—
281	C₂×C₈	—	26	2	no	—
282	C₂×C₈	—	26	2	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	—	26	2	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	—	26	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
293	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×Dic₂₆	1	2	yes	—
303	C₄²	C₂×Dic₂₆	1	4	yes	—
304	C₄²	C₂×D₅₂	1	4	yes	—
305	C₄²	C₂×D₅₂	1	2	yes	—
306	C₁₃	#22737	1	1	yes	characteristic, complemented, Sylow
307	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
335	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
336	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
337	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
338	C₂×C₄	#88202	1	2	yes	—
339	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
341	C₂×C₄	#88254	1	2	yes	—
342	C₂×C₄	#41×D₁₃	1	2	yes	—
343	C₈	—	26	4	no	—
344	C₈	—	52	2	no	—
345	C₈	—	52	2	no	—
346	C₈	—	52	2	no	—
347	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
348	C₂²	#11719517	1	1	yes	characteristic
349	C₂²	C₂×#88202	1	1	yes	characteristic
350	C₂²	#11719524	1	1	yes	characteristic
351	C₂²	#11719524	1	1	yes	characteristic
352	C₂²	#11719524	1	1	yes	characteristic
353	C₂²	#41×Dic₁₃	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	4	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	4	no	—
358	C₄	—	2	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	#11719274	1	2	yes	—
365	C₄	#11719281	1	2	yes	—
366	C₂	#13117757	1	1	yes	characteristic
367	C₂	#13123400	1	1	yes	characteristic
368	C₂	#13123404	1	1	yes	characteristic
369	C₂	#13123823	1	1	yes	characteristic
370	C₂	#13123832	1	1	yes	characteristic
371	C₂	#13123931	1	1	yes	characteristic
372	C₂	#13123942	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₁₃×#2295	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2295	D₁₃
hypocenter	C₁₃	#22737
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
306. C₁₃	#22737	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718685	2
#13115560	2
#13115555	2
#13114956	2
#13114953	2
#13114693	2
#13114689	2
C₁₃×#2295	2
#22737	13