Subgroups — #15861678

Order 3328.

922 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	#13117243	C₂	1	1	yes	maximal, characteristic
3	#13114706	C₂	1	2	yes	maximal, complemented
4	#13114703	C₂	1	2	yes	maximal, complemented
5	#13114701	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
7	#13114604	C₂	1	4	yes	maximal
8	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₁₃×#431	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	#11718873	C₄	1	2	yes	—
17	#11718873	C₄	1	2	yes	—
18	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
19	C₂×#88055	C₂²	1	4	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
21	#11718597	C₄	1	2	yes	—
22	#11718596	C₄	1	2	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
25	#11718586	C₄	1	8	yes	complemented
26	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₂×C₄×#1055	C₂²	1	2	yes	—
29	#11718142	C₂²	1	4	yes	—
30	#11718142	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
43	#88180	—	4	2	no	—
44	#88180	—	4	2	no	—
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
55	#88071	C₂×C₄	1	4	yes	—
56	#88071	C₂×C₄	1	4	yes	—
57	C₂²⋊#1055	C₂×C₄	1	8	yes	—
58	#88055	—	4	4	no	—
59	C₈×Dic₁₃	—	4	4	no	—
60	C₄⋊#1055	C₂×C₄	1	4	yes	—
61	C₄⋊#1055	C₂×C₄	1	4	yes	—
62	C₄×#1055	C₂×C₄	1	8	yes	—
63	#20538	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
72	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	C₄²	1	4	yes	—
89	C₄²×C₁₃	C₄²	1	4	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂²⋊Dic₁₃	—	4	4	no	—
95	C₄⋊Dic₁₃	—	2	8	no	—
96	C₄⋊Dic₁₃	—	2	8	no	—
97	C₄⋊Dic₁₃	—	4	4	no	—
98	C₄×Dic₁₃	—	2	4	no	—
99	C₄×Dic₁₃	—	2	4	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	C₂²×C₄	1	4	yes	—
105	C₂×#1055	C₂²×C₄	1	4	yes	—
106	C₂×#431	—	13	1	no	—
107	C₂×#431	D₁₃	1	1	yes	characteristic
108	#2285	—	13	1	no	—
109	#1770	—	13	2	no	—
110	#1770	—	13	2	no	—
111	C₈×C₄⋊C₄	—	13	2	no	—
112	C₈×C₄⋊C₄	—	13	2	no	—
113	#1750	—	13	4	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	—	2	1	no	—
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	8	no	—
129	C₂×C₄×C₁₃	—	2	8	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₁₃	#131	1	4	yes	—
136	C₂×C₄×C₁₃	#131	1	4	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	8	no	—
144	C₂×Dic₁₃	—	2	8	no	—
145	C₂×Dic₁₃	—	2	8	no	—
146	C₈×C₁₃	—	2	4	no	—
147	C₈×C₁₃	—	2	4	no	—
148	#1055	—	2	4	no	—
149	#1055	—	2	4	no	—
150	C₂×#117	—	13	2	no	—
151	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
152	#431	—	13	8	no	—
153	#431	Dic₁₃	1	8	yes	complemented
154	C₂×C₄⋊C₈	—	13	1	no	—
155	C₂×C₄⋊C₈	—	13	1	no	—
156	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	—	13	2	no	—
159	C₂×C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
160	C₂×#97	—	13	4	no	—
161	C₂×C₄×C₈	—	13	2	no	—
162	C₂×C₄×C₈	—	13	4	no	—
163	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
164	#384	—	13	2	no	—
165	#382	—	13	2	no	—
166	#380	—	13	2	no	—
167	#380	—	13	2	no	—
168	C₄×C₄⋊C₄	—	13	2	no	—
169	C₄×C₄⋊C₄	—	13	2	no	—
170	#336	—	13	4	no	—
171	#336	—	13	4	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	4	2	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
178	C₂²×C₁₃	#518	1	1	yes	characteristic
179	C₂²×C₁₃	#519	1	1	yes	characteristic
180	C₂²×C₁₃	C₂×#131	1	2	yes	—
181	C₂²×C₁₃	C₂×#131	1	2	yes	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	#405	1	4	yes	—
186	C₄×C₁₃	#405	1	4	yes	—
187	C₄×C₁₃	#520	1	1	yes	characteristic
188	C₄×C₁₃	#520	1	1	yes	characteristic
189	C₄×C₁₃	#520	1	1	yes	characteristic
190	C₄×C₁₃	#520	1	1	yes	characteristic
191	Dic₁₃	—	4	8	no	—
192	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
198	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
199	#117	—	52	2	no	—
200	#117	—	52	2	no	—
201	C₂×C₄⋊C₄	—	13	4	no	—
202	C₂×C₄⋊C₄	—	13	4	no	—
203	C₂×C₄⋊C₄	—	13	4	no	—
204	C₂×C₂²⋊C₄	—	13	4	no	—
205	C₂×C₄²	—	13	4	no	—
206	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₄×D₁₃	1	4	yes	—
209	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
210	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	C₄⋊C₈	—	13	4	no	—
213	C₄⋊C₈	—	13	4	no	—
214	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
215	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
216	C₂²⋊C₈	—	13	8	no	—
217	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
218	#97	—	52	4	no	—
219	C₄×C₈	—	13	8	no	—
220	C₄×C₈	—	52	4	no	—
221	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
222	#95	—	13	4	no	—
223	#95	—	13	4	no	—
224	C₂×C₁₃	—	4	2	no	—
225	C₂×C₁₃	#2274	1	1	yes	characteristic
226	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
227	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
228	C₂×C₁₃	#2979	1	2	yes	—
229	C₂×C₁₃	#2986	1	2	yes	—
230	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	13	8	no	—
238	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
239	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
244	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
247	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	13	4	no	—
253	C₂×C₈	—	13	4	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
259	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
260	C₄⋊C₄	—	26	8	no	—
261	C₄⋊C₄	—	26	8	no	—
262	C₄⋊C₄	—	52	4	no	—
263	C₂²⋊C₄	—	52	4	no	—
264	C₄²	—	26	4	no	—
265	C₄²	—	26	4	no	—
266	C₄²	C₄×Dic₁₃	1	4	yes	—
267	C₄²	C₄×Dic₁₃	1	4	yes	—
268	C₄²	C₂×C₄×D₁₃	1	2	yes	—
269	C₄²	C₂×C₄×D₁₃	1	2	yes	—
270	C₄²	C₂²×Dic₁₃	1	2	yes	—
271	C₄²	C₂²×Dic₁₃	1	2	yes	—
272	C₁₃	#20538	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
278	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	26	8	no	—
290	C₂×C₄	—	26	8	no	—
291	C₂×C₄	—	26	8	no	—
292	C₂×C₄	#88155	1	4	yes	—
293	C₂×C₄	#88161	1	4	yes	—
294	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
295	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
296	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
297	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
298	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	#88256	1	1	yes	characteristic
301	C₂×C₄	#88256	1	1	yes	characteristic
302	C₂×C₄	#88257	1	1	yes	characteristic
303	C₂×C₄	#88257	1	1	yes	characteristic
304	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
305	C₈	—	2	4	no	—
306	C₈	—	2	4	no	—
307	C₈	—	26	4	no	—
308	C₈	—	26	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	#11719221	1	1	yes	characteristic
315	C₂²	#11719226	1	1	yes	characteristic
316	C₂²	C₂×#88155	1	2	yes	—
317	C₂²	C₂×#88161	1	2	yes	—
318	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	52	8	no	—
323	C₄	#11718762	1	4	yes	—
324	C₄	#11718783	1	4	yes	—
325	C₄	#11719228	1	1	yes	characteristic
326	C₄	#11719228	1	1	yes	characteristic
327	C₄	#11719532	1	1	yes	characteristic
328	C₄	#11719532	1	1	yes	characteristic
329	C₂	—	4	2	no	—
330	C₂	#13117192	1	1	yes	characteristic
331	C₂	C₂×#11718762	1	1	yes	characteristic
332	C₂	C₂×#11718783	1	1	yes	characteristic
333	C₂	#13123902	1	2	yes	—
334	C₂	#13123937	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₂×C₁₃×#431	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#20538
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13114706	C₂	semidirect
4. #13114703	C₂	semidirect
5. #13114701	C₂	semidirect
6. C₈×C₄⋊Dic₁₃	C₂	semidirect
11. C₁₃×#431	C₄	semidirect
25. #11718586	C₄	semidirect
153. #431	Dic₁₃	semidirect
272. C₁₃	#20538	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
#13117243	2
#13114706	2
#13114703	2
#13114701	2
C₈×C₄⋊Dic₁₃	2
#13114604	2
C₂×C₁₃×#431	2
#20538	13