Subgroups — #15752770

Order 3328.

648 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	#13123469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13112700	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2283	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×#6099	—	2	1	no	—
7	C₁₃×#383	C₂²	1	2	yes	—
8	C₁₃×#382	C₄	1	2	yes	—
9	C₁₃×#381	—	2	1	no	—
10	C₁₃×#380	—	2	1	no	—
11	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
12	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
13	C₁₃×#376	—	2	1	no	—
14	#11718868	—	2	1	no	—
15	#11718587	C₄	1	2	yes	complemented
16	#11718587	C₂²	1	2	yes	—
17	#11718586	—	2	1	no	—
18	#11718586	—	2	1	no	—
19	#11718585	—	2	1	no	—
20	C₄×#6099	—	2	1	no	—
21	C₄×#6099	—	2	1	no	—
22	C₄×#6099	C₄	1	2	yes	complemented
23	C₄×#6099	C₄	1	2	yes	complemented
24	C₄×#6099	C₂²	1	2	yes	—
25	C₂×#88043	—	2	1	no	—
26	C₂×C₄×#1055	C₄	1	2	yes	complemented
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
32	C₂×C₄²×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×#6099	—	2	8	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂²×#1055	—	2	2	no	—
50	C₁₃×#95	—	2	2	no	—
51	C₁₃×#95	—	2	2	no	—
52	C₁₃×#95	—	2	2	no	—
53	C₁₃×#95	C₂×C₄	1	4	yes	—
54	C₂²⋊#1055	—	2	2	no	—
55	C₂²⋊#1055	—	2	2	no	—
56	C₄⋊#1055	—	2	2	no	—
57	C₄⋊#1055	—	2	2	no	—
58	#88043	—	2	2	no	—
59	#88043	—	2	2	no	—
60	#88043	C₂×C₄	1	4	yes	—
61	C₄×#1055	—	2	2	no	—
62	C₄×#1055	—	2	2	no	—
63	C₄×#1055	C₂×C₄	1	4	yes	—
64	#7974	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₂²⋊C₄	—	4	4	no	—
90	C₄²×C₁₃	—	4	4	no	—
91	C₄²×C₁₃	C₄²	1	1	yes	characteristic
92	C₄²×C₁₃	C₄²	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
107	#6099	—	2	8	no	—
108	#6099	—	2	8	no	—
109	#6099	—	4	4	no	—
110	C₂×#1055	—	2	8	no	—
111	C₂×#1055	—	2	4	no	—
112	C₂×#1055	—	2	4	no	—
113	#2920	—	13	1	no	—
114	C₂×C₄×M₁₆	—	13	1	no	—
115	#2283	D₁₃	1	1	yes	characteristic
116	#1288	—	13	4	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	—	4	1	no	—
120	C₂³×C₁₃	#95	1	1	yes	characteristic
121	C₂³×C₁₃	#95	1	1	yes	characteristic
122	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	4	no	—
143	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#104	1	4	yes	—
150	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	#1055	—	2	8	no	—
157	#1055	—	4	4	no	—
158	C₂²×M₁₆	—	26	1	no	—
159	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
160	#432	—	13	2	no	—
161	#432	—	13	2	no	—
162	#431	—	26	1	no	—
163	#431	—	26	1	no	—
164	#423	—	26	1	no	—
165	#407	—	26	1	no	—
166	C₄×M₁₆	—	13	2	no	—
167	C₄×M₁₆	—	13	2	no	—
168	C₄×M₁₆	—	13	2	no	—
169	C₄×M₁₆	—	26	1	no	—
170	C₄×M₁₆	—	26	1	no	—
171	C₂×#97	—	26	1	no	—
172	C₂×C₄×C₈	—	13	2	no	—
173	#383	C₂×D₁₃	1	2	yes	—
174	#382	Dic₁₃	1	2	yes	—
175	#381	—	2	1	no	—
176	#380	—	2	1	no	—
177	C₄×C₄⋊C₄	—	2	1	no	—
178	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
179	#376	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	4	4	no	—
188	C₂²×C₁₃	H(3,ℤ/4)	1	2	yes	—
189	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
190	C₂²×C₁₃	C₂×#104	1	2	yes	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	8	4	no	—
201	C₄×C₁₃	#339	1	2	yes	—
202	C₄×C₁₃	#339	1	2	yes	—
203	C₄×C₁₃	#344	1	2	yes	—
204	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
206	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
207	C₂×M₁₆	—	26	8	no	—
208	C₂×M₁₆	—	26	2	no	—
209	C₂×M₁₆	—	26	2	no	—
210	C₂×M₁₆	—	26	2	no	—
211	C₂²×C₈	—	26	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
216	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
221	C₂×C₄²	D₅₂	1	1	yes	characteristic
222	C₂×C₄²	D₅₂	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
227	C₄⋊C₈	—	26	2	no	—
228	C₄⋊C₈	—	26	2	no	—
229	C₂²⋊C₈	—	26	2	no	—
230	C₂²⋊C₈	—	26	2	no	—
231	#97	—	13	4	no	—
232	#97	—	26	2	no	—
233	#97	—	26	2	no	—
234	C₄×C₈	—	13	4	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	—	26	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	C₂×Dic₁₃	1	4	yes	—
241	C₂×C₁₃	—	2	2	no	—
242	C₂×C₁₃	—	2	2	no	—
243	C₂×C₁₃	—	4	1	no	—
244	C₂×C₁₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#339	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#344	1	1	yes	characteristic
247	C₂⁴	C₂²⋊Dic₁₃	1	1	yes	characteristic
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
254	C₂²×C₄	#6101	1	1	yes	characteristic
255	C₂²×C₄	#6101	1	1	yes	characteristic
256	C₂²×C₄	#6101	1	1	yes	characteristic
257	C₂²×C₄	#6101	1	1	yes	characteristic
258	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
260	C₂²×C₄	#6103	1	1	yes	characteristic
261	C₂²×C₄	#6103	1	1	yes	characteristic
262	C₂²×C₄	#6103	1	1	yes	characteristic
263	C₂²×C₄	#6103	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
270	M₁₆	—	26	8	no	—
271	M₁₆	—	26	8	no	—
272	M₁₆	—	52	4	no	—
273	C₂×C₈	—	26	8	no	—
274	C₂×C₈	—	26	4	no	—
275	C₂×C₈	—	26	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₄²	—	4	4	no	—
279	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
280	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
281	C₄²	#6101	1	1	yes	characteristic
282	C₄²	#6101	1	1	yes	characteristic
283	C₄²	#6101	1	1	yes	characteristic
284	C₄²	#6101	1	1	yes	characteristic
285	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
286	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
287	C₄²	#6103	1	1	yes	characteristic
288	C₄²	#6103	1	1	yes	characteristic
289	C₄²	#6103	1	1	yes	characteristic
290	C₄²	#6103	1	1	yes	characteristic
291	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
292	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
293	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
294	C₄²	C₂×D₅₂	1	1	yes	characteristic
295	C₁₃	#7974	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	4	1	no	—
299	C₂³	#88071	1	1	yes	characteristic
300	C₂³	#88071	1	1	yes	characteristic
301	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
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	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	#88071	1	1	yes	characteristic
323	C₂×C₄	#88071	1	1	yes	characteristic
324	C₂×C₄	#88071	1	1	yes	characteristic
325	C₂×C₄	#88071	1	1	yes	characteristic
326	C₂×C₄	#88071	1	1	yes	characteristic
327	C₂×C₄	#88071	1	1	yes	characteristic
328	C₂×C₄	#88077	1	4	yes	—
329	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#6101	1	1	yes	characteristic
331	C₂×C₄	C₂×#6101	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#6103	1	1	yes	characteristic
334	C₂×C₄	C₂×#6103	1	1	yes	characteristic
335	C₈	—	26	8	no	—
336	C₈	—	52	4	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	4	no	—
345	C₂²	#11718123	1	2	yes	—
346	C₂²	C₂×#88071	1	1	yes	characteristic
347	C₂²	C₂×#88077	1	2	yes	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	8	4	no	—
358	C₄	#11718124	1	2	yes	—
359	C₄	#11718124	1	2	yes	—
360	C₄	#11718125	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	C₂×#11718123	1	1	yes	characteristic
365	C₂	C₂×#11718124	1	1	yes	characteristic
366	C₂	C₂×#11718125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88071
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2283	C₂
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2283	D₁₃
hypocenter	C₁₃	#7974
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #13112700	C₂	semidirect
15. #11718587	C₄	semidirect
22. C₄×#6099	C₄	semidirect
23. C₄×#6099	C₄	semidirect
26. C₂×C₄×#1055	C₄	semidirect
295. C₁₃	#7974	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
C₂×C₄×#6099	2
#13112700	2
C₁₃×#2283	2
#7974	13