Subgroups — #15880522

Order 3328.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123494	C₂	1	2	yes	maximal
2	#13115590	C₂	1	2	yes	maximal, complemented
3	#13115437	C₂	1	4	yes	maximal, complemented
4	#13114935	C₂	1	2	yes	maximal
5	#13114874	C₂	1	1	yes	maximal, characteristic
6	#13114722	C₂	1	2	yes	maximal, complemented
7	#13114612	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	2	yes	—
11	C₁₃×#427	C₄	1	1	yes	characteristic, complemented
12	C₁₃×#427	C₄	1	1	yes	characteristic, complemented
13	C₁₃×#427	C₄	1	2	yes	complemented
14	C₁₃×#427	C₂²	1	2	yes	—
15	C₁₃×#427	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₃×#107	C₄	1	1	yes	characteristic, complemented
19	C₂×C₁₃×#106	C₄	1	2	yes	complemented
20	C₄×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	#11718868	C₂²	1	2	yes	—
22	C₂×C₂²⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×#88059	C₂²	1	2	yes	—
24	C₂×#88055	C₂²	1	2	yes	—
25	#11718661	—	2	4	no	—
26	#11718660	C₂²	1	4	yes	—
27	#11718658	C₂²	1	4	yes	—
28	#11718587	—	2	2	no	—
29	#11718586	—	2	2	no	—
30	#11718144	C₂²	1	2	yes	—
31	#11718142	C₂²	1	2	yes	—
32	#11718124	C₂²	1	4	yes	—
33	#11718124	C₂²	1	4	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	—	2	4	no	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	—	2	2	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₁₃×#117	C₂×C₄	1	2	yes	—
45	C₁₃×#117	C₂×C₄	1	2	yes	—
46	C₁₃×#117	C₂³	1	2	yes	—
47	C₁₃×#117	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
53	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
54	C₂×#6099	C₂³	1	4	yes	—
55	C₂²×#1055	C₂³	1	2	yes	—
56	C₁₃×#107	C₂×C₄	1	2	yes	—
57	C₁₃×#107	C₂×C₄	1	2	yes	—
58	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
59	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
60	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
61	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
62	C₁₃×#106	C₂×C₄	1	2	yes	—
63	C₁₃×#106	C₂×C₄	1	2	yes	—
64	C₁₃×#106	C₂×C₄	1	2	yes	—
65	C₁₃×#106	C₂×C₄	1	2	yes	—
66	#88071	—	2	1	no	—
67	C₂²⋊#1055	—	2	2	no	—
68	C₂²⋊#1055	—	2	2	no	—
69	C₂²⋊#1055	D₄	1	2	yes	—
70	C₂²⋊#1055	D₄	1	2	yes	—
71	#88059	—	2	4	no	—
72	#88055	—	2	4	no	—
73	C₄⋊#1055	—	2	4	no	—
74	C₄⋊#1055	—	2	4	no	—
75	#88043	—	2	4	no	—
76	C₄×#1055	—	2	4	no	—
77	#22607	—	13	1	no	maximal, Sylow
78	C₂⁴×C₁₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₁₃	—	2	2	no	—
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	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	4	yes	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
103	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
104	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
105	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
106	C₁₃×C₂²⋊C₄	—	2	2	no	—
107	C₁₃×C₂²⋊C₄	—	2	2	no	—
108	C₁₃×C₂²⋊C₄	—	2	2	no	—
109	C₁₃×C₂²⋊C₄	—	2	2	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	—	2	2	no	—
114	C₂²⋊Dic₁₃	—	4	2	no	—
115	C₄×Dic₁₃	—	2	2	no	—
116	C₄×Dic₁₃	—	2	2	no	—
117	#6099	—	2	4	no	—
118	#6099	—	2	4	no	—
119	C₂×#1055	—	2	2	no	—
120	C₂×#1055	—	2	2	no	—
121	C₂×#1055	—	2	4	no	—
122	C₂×#1055	C₂×D₄	1	4	yes	—
123	#2924	—	13	2	no	—
124	C₂×#427	D₁₃	1	1	yes	characteristic
125	#1946	—	13	2	no	—
126	#1922	—	13	4	no	—
127	#1826	—	13	2	no	—
128	#1810	—	13	1	no	—
129	#1773	—	13	2	no	—
130	#1752	—	13	1	no	—
131	C₂³×C₁₃	—	2	2	no	—
132	C₂³×C₁₃	—	2	2	no	—
133	C₂³×C₁₃	—	4	1	no	—
134	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	4	no	—
149	C₂×C₄×C₁₃	—	8	2	no	—
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
155	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
156	C₂×Dic₁₃	—	4	2	no	—
157	C₂×Dic₁₃	—	4	2	no	—
158	C₂×Dic₁₃	—	4	2	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	—	2	1	no	—
161	C₈×C₁₃	#117	1	2	yes	—
162	C₈×C₁₃	#117	1	2	yes	—
163	#1055	—	2	4	no	—
164	#1055	—	4	4	no	—
165	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₂×#117	C₂×D₁₃	1	2	yes	—
167	#433	—	26	4	no	—
168	#432	—	26	2	no	—
169	#431	—	26	2	no	—
170	C₂×#108	—	13	2	no	—
171	#427	Dic₁₃	1	1	yes	characteristic, complemented
172	#427	Dic₁₃	1	1	yes	characteristic, complemented
173	#427	Dic₁₃	1	2	yes	complemented
174	#427	C₂×D₁₃	1	2	yes	—
175	#427	C₂×D₁₃	1	1	yes	characteristic
176	#427	C₂×D₁₃	1	1	yes	characteristic
177	C₂×#107	Dic₁₃	1	1	yes	characteristic, complemented
178	C₂×#107	Dic₁₃	1	1	yes	characteristic, complemented
179	C₂×#106	Dic₁₃	1	2	yes	complemented
180	#424	—	13	4	no	—
181	#407	—	13	2	no	—
182	C₂×C₂²⋊C₈	—	13	1	no	—
183	#405	—	13	4	no	—
184	C₂×#97	—	13	2	no	—
185	C₄×C₂²⋊C₄	—	13	1	no	—
186	#341	—	13	2	no	—
187	#339	—	13	4	no	—
188	#339	—	13	4	no	—
189	#336	—	13	2	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	2	1	no	—
195	C₂²×C₁₃	—	2	1	no	—
196	C₂²×C₁₃	—	2	1	no	—
197	C₂²×C₁₃	—	4	4	no	—
198	C₂²×C₁₃	#444	1	2	yes	—
199	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₁₃	#552	1	2	yes	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	4	2	no	—
204	C₄×C₁₃	—	4	2	no	—
205	C₄×C₁₃	—	4	2	no	—
206	C₄×C₁₃	—	4	2	no	—
207	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
208	C₄×C₁₃	#519	1	1	yes	characteristic
209	C₄×C₁₃	#547	1	2	yes	—
210	Dic₁₃	—	8	2	no	—
211	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
212	C₂×M₁₆	—	13	4	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
217	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	2	2	no	—
223	#117	C₂×Dic₁₃	1	2	yes	—
224	#117	C₂×Dic₁₃	1	2	yes	—
225	#117	C₂²×D₁₃	1	2	yes	—
226	#117	C₂²×D₁₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	26	1	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	13	2	no	—
233	#108	—	26	4	no	—
234	#107	C₂×Dic₁₃	1	2	yes	—
235	#107	C₂×Dic₁₃	1	2	yes	—
236	#107	C₂×Dic₁₃	1	1	yes	characteristic
237	#107	C₂×Dic₁₃	1	1	yes	characteristic
238	#107	C₂×Dic₁₃	1	1	yes	characteristic
239	#107	C₂×Dic₁₃	1	1	yes	characteristic
240	#106	C₂×Dic₁₃	1	2	yes	—
241	#106	C₂×Dic₁₃	1	2	yes	—
242	#106	C₂×Dic₁₃	1	2	yes	—
243	#106	C₂×Dic₁₃	1	2	yes	—
244	C₄⋊C₈	—	26	4	no	—
245	C₄⋊C₈	—	26	4	no	—
246	C₂²⋊C₈	—	13	2	no	—
247	C₂²⋊C₈	—	13	2	no	—
248	C₂²⋊C₈	—	26	2	no	—
249	C₂²⋊C₈	—	26	2	no	—
250	#97	—	26	4	no	—
251	#97	—	26	4	no	—
252	C₄×C₈	—	26	4	no	—
253	#95	—	26	1	no	—
254	C₂×C₁₃	—	2	2	no	—
255	C₂×C₁₃	—	2	2	no	—
256	C₂×C₁₃	—	4	1	no	—
257	C₂×C₁₃	#2344	1	1	yes	characteristic
258	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
259	C₂×C₁₃	#2957	1	1	yes	characteristic
260	C₂⁴	#6129	1	1	yes	characteristic
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	26	2	no	—
266	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
268	C₂²×C₄	#6129	1	1	yes	characteristic
269	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
272	M₁₆	—	26	4	no	—
273	M₁₆	—	26	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	13	4	no	—
280	C₂×C₈	—	26	2	no	—
281	C₂×C₈	—	26	2	no	—
282	C₂×C₈	—	26	4	no	—
283	C₂×C₈	#6127	1	4	yes	—
284	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
285	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
286	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
291	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
292	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
293	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	52	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	26	2	no	—
304	C₄²	—	26	2	no	—
305	C₁₃	#22607	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
310	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
311	C₂³	C₂×#6129	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#6127	1	2	yes	—
331	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
332	C₂×C₄	#88254	1	2	yes	—
333	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	26	4	no	—
337	C₈	—	52	4	no	—
338	C₈	#88180	1	2	yes	—
339	C₈	#88180	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#11718858	1	2	yes	—
349	C₂²	#11719218	1	2	yes	—
350	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	104	2	no	—
358	C₄	#11719212	1	2	yes	—
359	C₄	C₂×#88180	1	1	yes	characteristic
360	C₄	#11719517	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#13117705	1	1	yes	characteristic
365	C₂	C₂×#11718858	1	1	yes	characteristic
366	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#22607
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #13115590	C₂	semidirect
3. #13115437	C₂	semidirect
6. #13114722	C₂	semidirect
11. C₁₃×#427	C₄	semidirect
12. C₁₃×#427	C₄	semidirect
13. C₁₃×#427	C₄	semidirect
17. C₂×C₁₃×#107	C₄	semidirect
18. C₂×C₁₃×#107	C₄	semidirect
19. C₂×C₁₃×#106	C₄	semidirect
171. #427	Dic₁₃	semidirect
172. #427	Dic₁₃	semidirect
173. #427	Dic₁₃	semidirect
177. C₂×#107	Dic₁₃	semidirect
178. C₂×#107	Dic₁₃	semidirect
179. C₂×#106	Dic₁₃	semidirect
305. C₁₃	#22607	semidirect

Maximal subgroups

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

subgroup	index
#13123494	2
#13115590	2
#13115437	2
#13114935	2
#13114874	2
#13114722	2
#13114612	2
C₂×C₁₃×#427	2
#22607	13