Subgroups — #15863491

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117445	C₂	1	2	yes	maximal
2	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114996	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114970	C₂	1	2	yes	maximal, complemented
5	#13114869	C₂	1	2	yes	maximal
6	#13114865	C₂	1	2	yes	maximal
7	#13114733	C₂	1	2	yes	maximal, complemented
8	C₄×#88058	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
13	C₁₃×#405	C₄	1	8	yes	complemented
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#11718873	C₂²	1	4	yes	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88058	C₂²	1	2	yes	—
22	C₂×#88057	C₂²	1	2	yes	—
23	#11718642	—	2	4	no	—
24	#11718612	—	2	4	no	—
25	#11718597	C₂²	1	2	yes	—
26	#11718596	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
28	#11718143	C₂²	1	4	yes	—
29	#11718143	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×#117	C₂×C₄	1	4	yes	—
42	C₁₃×#117	Q₈	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
44	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
48	C₂×C₄⋊Dic₁₃	—	2	4	no	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
51	C₂×C₄×Dic₁₃	—	2	4	no	—
52	C₁₃×#97	C₂×C₄	1	4	yes	—
53	C₁₃×#97	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
56	#88071	—	2	4	no	—
57	#88071	—	2	4	no	—
58	#88071	—	2	4	no	—
59	#88058	—	2	4	no	—
60	#88058	—	2	4	no	—
61	#88058	—	2	4	no	—
62	#88057	—	2	4	no	—
63	#20821	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	2	4	no	—
76	C₂²×Dic₁₃	—	2	4	no	—
77	C₁₃×M₁₆	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	8	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
92	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
96	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂²⋊Dic₁₃	—	8	4	no	—
98	C₄⋊Dic₁₃	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	2	4	no	—
101	C₄⋊Dic₁₃	—	2	4	no	—
102	C₄⋊Dic₁₃	—	8	4	no	—
103	C₄×Dic₁₃	—	8	4	no	—
104	C₂×#405	D₁₃	1	1	yes	characteristic
105	#2314	—	13	2	no	—
106	#1846	—	13	1	no	—
107	#1845	—	13	1	no	—
108	#1836	—	13	2	no	—
109	#1809	—	13	2	no	—
110	#1808	—	13	2	no	—
111	#1776	—	13	2	no	—
112	C₄×#107	—	13	2	no	—
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	—	2	2	no	—
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	2	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	2	no	—
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	4	4	no	—
141	C₂×Dic₁₃	—	4	4	no	—
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	4	4	no	—
144	C₈×C₁₃	—	2	2	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	#117	1	4	yes	—
147	C₈×C₁₃	#117	1	4	yes	—
148	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
149	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
150	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
151	C₂×#107	—	13	2	no	—
152	C₂×#107	—	13	2	no	—
153	C₂×#107	—	13	2	no	—
154	C₂×#106	—	13	2	no	—
155	#405	Dic₁₃	1	8	yes	complemented
156	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
160	#388	—	26	4	no	—
161	#387	—	26	4	no	—
162	#384	—	13	2	no	—
163	#382	—	13	2	no	—
164	#380	—	13	4	no	—
165	C₄×C₄⋊C₄	—	13	4	no	—
166	#340	—	13	4	no	—
167	#340	—	13	4	no	—
168	C₂²×C₁₃	—	2	4	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	4	2	no	—
171	C₂²×C₁₃	—	4	1	no	—
172	C₂²×C₁₃	—	4	1	no	—
173	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₁₃	#532	1	1	yes	characteristic
175	C₂²×C₁₃	#532	1	1	yes	characteristic
176	C₂²×C₁₃	#576	1	2	yes	—
177	C₂²×C₁₃	#578	1	2	yes	—
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	—	2	4	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
182	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
183	C₄×C₁₃	#533	1	1	yes	characteristic
184	C₄×C₁₃	#533	1	1	yes	characteristic
185	Dic₁₃	—	8	4	no	—
186	Dic₁₃	—	8	4	no	—
187	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	#117	Dic₂₆	1	4	yes	—
199	#117	C₂×Dic₁₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	13	4	no	—
201	C₂×C₄⋊C₄	—	13	4	no	—
202	C₂×C₄⋊C₄	—	26	4	no	—
203	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
204	C₂×C₂²⋊C₄	—	26	4	no	—
205	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
206	C₂×C₄²	—	26	4	no	—
207	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	#107	—	26	4	no	—
210	#107	—	26	4	no	—
211	#107	—	26	4	no	—
212	#106	—	26	4	no	—
213	#97	C₂×Dic₁₃	1	4	yes	—
214	#97	C₂×Dic₁₃	1	4	yes	—
215	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
216	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
217	#95	—	26	4	no	—
218	#95	—	26	4	no	—
219	#95	—	26	4	no	—
220	C₂×C₁₃	—	4	2	no	—
221	C₂×C₁₃	#2285	1	1	yes	characteristic
222	C₂×C₁₃	#2913	1	2	yes	—
223	C₂×C₁₃	#3175	1	2	yes	—
224	C₂×C₁₃	#3176	1	1	yes	characteristic
225	C₂×C₁₃	#3177	1	1	yes	characteristic
226	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	26	4	no	—
232	C₂²×C₄	—	26	4	no	—
233	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
234	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₅₂	1	2	yes	—
236	C₂²×C₄	C₂×D₅₂	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
238	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
239	M₁₆	—	2	8	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	#6127	1	2	yes	—
246	C₂×C₈	#6127	1	2	yes	—
247	C₂×C₈	#6127	1	2	yes	—
248	C₂×C₈	#6127	1	2	yes	—
249	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
250	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
251	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
252	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
253	C₄⋊C₄	—	26	4	no	—
254	C₄⋊C₄	—	26	4	no	—
255	C₄⋊C₄	—	26	4	no	—
256	C₄⋊C₄	—	26	4	no	—
257	C₄⋊C₄	—	104	4	no	—
258	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
259	C₂²⋊C₄	—	104	4	no	—
260	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
261	C₄²	—	104	4	no	—
262	C₄²	C₂×Dic₂₆	1	2	yes	—
263	C₄²	C₂×Dic₂₆	1	2	yes	—
264	C₄²	C₂²×Dic₁₃	1	2	yes	—
265	C₄²	C₂²×Dic₁₃	1	2	yes	—
266	C₁₃	#20821	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
272	C₂³	C₂²×D₅₂	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	52	4	no	—
284	C₂×C₄	—	52	4	no	—
285	C₂×C₄	—	52	4	no	—
286	C₂×C₄	—	52	4	no	—
287	C₂×C₄	—	52	4	no	—
288	C₂×C₄	—	52	4	no	—
289	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
292	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
293	C₂×C₄	C₂×#6127	1	1	yes	characteristic
294	C₂×C₄	C₂×#6127	1	1	yes	characteristic
295	C₂×C₄	C₂×#6127	1	1	yes	characteristic
296	C₂×C₄	C₂×#6127	1	1	yes	characteristic
297	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	#88180	1	4	yes	—
301	C₈	#88180	1	4	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	#11719182	1	1	yes	characteristic
308	C₂²	#11719182	1	1	yes	characteristic
309	C₂²	#11719457	1	2	yes	—
310	C₂²	#11719458	1	2	yes	—
311	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	104	4	no	—
316	C₄	—	104	4	no	—
317	C₄	#11719175	1	1	yes	characteristic
318	C₄	#11719175	1	1	yes	characteristic
319	C₄	C₂×#88180	1	1	yes	characteristic
320	C₄	C₂×#88180	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	#13117243	1	1	yes	characteristic
323	C₂	#13123436	1	2	yes	—
324	C₂	#13125475	1	2	yes	—
325	C₂	#13125486	1	1	yes	characteristic
326	C₂	#13125495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#20821
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13115000	C₂	semidirect
3. #13114996	C₂	semidirect
4. #13114970	C₂	semidirect
7. #13114733	C₂	semidirect
8. C₄×#88058	C₂	semidirect
13. C₁₃×#405	C₄	semidirect
155. #405	Dic₁₃	semidirect
266. C₁₃	#20821	semidirect

Maximal subgroups

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

subgroup	index
#13117445	2
#13115000	2
#13114996	2
#13114970	2
#13114869	2
#13114865	2
#13114733	2
C₄×#88058	2
C₂×C₁₃×#405	2
#20821	13