Subgroups — #15833057

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124840	C₂	1	1	yes	maximal, characteristic
2	#13123859	C₂	1	1	yes	maximal, characteristic
3	#13113691	C₂	1	2	yes	maximal, complemented
4	#13113670	C₂	1	2	yes	maximal, complemented
5	#13113646	C₂	1	2	yes	maximal, complemented
6	C₈×#6106	C₂	1	2	yes	maximal, complemented
7	#13113617	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113278	C₂	1	1	yes	maximal, characteristic
9	#13113257	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2985	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
14	C₁₃×#446	—	2	2	no	—
15	C₁₃×#446	C₂²	1	2	yes	complemented
16	C₈×Q₈×C₁₃	C₂²	1	2	yes	complemented
17	C₁₃×#433	—	2	2	no	—
18	C₁₃×#432	C₂²	1	1	yes	characteristic
19	C₁₃×#431	C₂²	1	1	yes	characteristic
20	#11718745	C₄	1	2	yes	—
21	#11718744	C₄	1	2	yes	—
22	#11718707	C₄	1	2	yes	—
23	#11718704	C₄	1	2	yes	—
24	C₄×#6106	C₂²	1	2	yes	—
25	C₄×#6106	C₂²	1	2	yes	—
26	#11718586	C₂²	1	1	yes	characteristic
27	#11718586	C₂²	1	1	yes	characteristic
28	C₄×#6099	C₂²	1	1	yes	characteristic
29	#11718495	—	2	2	no	—
30	D₁₃×C₂²⋊C₈	—	2	2	no	—
31	C₈×D₅₂	C₂²	1	2	yes	—
32	C₈×D₅₂	C₂²	1	2	yes	—
33	Q₈⋊#1055	C₂²	1	2	yes	complemented
34	Q₈⋊#1055	C₂²	1	2	yes	complemented
35	#11718075	C₂²	1	2	yes	—
36	#11718075	C₂²	1	2	yes	—
37	#11718072	C₂²	1	2	yes	complemented
38	#11718071	C₂²	1	2	yes	complemented
39	#11718047	C₂²	1	1	yes	characteristic, complemented
40	#11718046	C₂²	1	1	yes	characteristic, complemented
41	#11718046	C₂²	1	1	yes	characteristic, complemented
42	C₈×#1055	C₂²	1	1	yes	characteristic, complemented
43	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₁₃	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
47	C₁₃×#125	—	2	1	no	—
48	C₁₃×C₂²⋊Q₈	—	2	1	no	—
49	C₁₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₁₃	—	2	1	no	—
51	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
52	C₁₃×#117	—	2	1	no	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×#6106	C₂×C₄	1	2	yes	—
55	C₂×#6106	C₂×C₄	1	2	yes	—
56	C₂×#6103	—	2	1	no	—
57	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×#6099	—	2	1	no	—
59	C₂×C₈×D₁₃	—	2	2	no	—
60	C₂×C₈×D₁₃	—	2	2	no	—
61	C₄×D₅₂	—	2	1	no	—
62	C₄×D₅₂	—	2	1	no	—
63	C₄×D₅₂	C₂×C₄	1	2	yes	—
64	C₄×D₅₂	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	—	2	2	no	—
66	C₁₃×C₄⋊C₈	—	2	2	no	—
67	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
68	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
69	C₁₃×C₂²⋊C₈	—	2	1	no	—
70	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
71	C₁₃×#97	—	2	2	no	—
72	C₁₃×#97	—	2	1	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88075	C₂×C₄	1	2	yes	—
77	#88075	C₂×C₄	1	2	yes	—
78	C₂²⋊#1055	—	2	1	no	—
79	C₂²⋊#1055	C₂×C₄	1	2	yes	—
80	#88060	—	2	2	no	—
81	#88060	—	2	2	no	—
82	#88049	C₂×C₄	1	2	yes	—
83	#88049	C₂×C₄	1	2	yes	—
84	#88048	C₂×C₄	1	2	yes	—
85	#88048	C₂×C₄	1	2	yes	—
86	C₄⋊#1055	C₂×C₄	1	2	yes	—
87	C₄⋊#1055	C₂³	1	2	yes	—
88	#88043	—	2	1	no	—
89	C₄×#1055	—	2	1	no	—
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	#17924	—	13	1	no	maximal, Sylow
95	C₂³×D₁₃	—	2	1	no	—
96	C₂×Q₈×C₁₃	—	2	1	no	—
97	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
101	C₂²×Dic₁₃	—	2	1	no	—
102	C₂×D₅₂	—	2	1	no	—
103	C₂×D₅₂	—	2	4	no	—
104	C₂×D₅₂	C₂²×C₄	1	2	yes	—
105	C₂×C₄×D₁₃	—	2	2	no	—
106	C₂×C₄×D₁₃	—	2	2	no	—
107	C₂×C₄×D₁₃	—	8	2	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
117	C₁₃×C₂²⋊C₄	—	2	2	no	—
118	C₄²×C₁₃	—	2	2	no	—
119	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	#41	1	1	yes	characteristic
122	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
123	#6106	—	2	4	no	—
124	#6106	—	2	4	no	—
125	#6103	—	2	2	no	—
126	#6103	—	2	2	no	—
127	C₄⋊Dic₁₃	—	2	1	no	—
128	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
129	#6099	—	4	2	no	—
130	C₂×#1055	—	2	1	no	—
131	C₂×#1055	—	2	2	no	—
132	C₂×#1055	—	2	1	no	—
133	C₂×#1055	C₂²×C₄	1	2	yes	—
134	C₂×#1055	C₂²×C₄	1	2	yes	—
135	C₈×D₁₃	—	8	2	no	—
136	C₈×D₁₃	—	8	2	no	—
137	#3101	—	13	1	no	—
138	#2985	D₁₃	1	1	yes	characteristic, complemented
139	#2972	—	13	1	no	—
140	#1585	—	13	2	no	—
141	#1582	—	13	2	no	—
142	SD₁₆⋊C₈	—	13	2	no	—
143	C₈×SD₁₆	—	13	2	no	—
144	#1572	—	13	1	no	—
145	#1490	—	13	1	no	—
146	#1485	—	13	1	no	—
147	#1453	—	13	1	no	—
148	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₂²×D₁₃	—	2	2	no	—
150	C₂²×D₁₃	—	2	4	no	—
151	C₂²×D₁₃	—	8	1	no	—
152	C₂²×D₁₃	—	8	1	no	—
153	Q₈×C₁₃	—	2	2	no	—
154	Q₈×C₁₃	—	2	2	no	—
155	Q₈×C₁₃	—	8	1	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	—	4	1	no	—
161	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
167	C₂×Dic₁₃	—	2	2	no	—
168	C₂×Dic₁₃	—	8	1	no	—
169	D₅₂	—	2	4	no	—
170	D₅₂	—	2	2	no	—
171	D₅₂	#131	1	4	yes	—
172	C₄×D₁₃	—	4	2	no	—
173	C₄×D₁₃	—	4	2	no	—
174	C₄×D₁₃	—	8	2	no	—
175	C₈×C₁₃	—	2	2	no	—
176	C₈×C₁₃	—	2	2	no	—
177	C₈×C₁₃	—	4	2	no	—
178	#1055	—	2	2	no	—
179	#1055	—	2	2	no	—
180	#1055	—	2	2	no	—
181	#1055	—	4	1	no	—
182	#1055	—	4	2	no	—
183	#533	C₂×D₁₃	1	1	yes	characteristic
184	C₂×C₄×D₄	—	13	1	no	—
185	#483	—	13	2	no	—
186	#476	—	13	2	no	—
187	#461	—	13	2	no	—
188	#450	—	13	2	no	—
189	#446	—	2	2	no	—
190	#446	C₂×D₁₃	1	2	yes	complemented
191	C₈×Q₈	C₂×D₁₃	1	2	yes	complemented
192	C₄×SD₁₆	—	13	2	no	—
193	C₄×SD₁₆	—	13	2	no	—
194	C₈×D₄	—	13	2	no	—
195	C₈×D₄	—	13	2	no	—
196	C₈×D₄	—	26	2	no	—
197	#433	—	2	2	no	—
198	#432	C₂×D₁₃	1	1	yes	characteristic
199	#431	—	13	1	no	—
200	#431	—	13	1	no	—
201	#431	C₂×D₁₃	1	1	yes	characteristic
202	C₂×C₂²⋊C₈	—	26	2	no	—
203	C₄×M₁₆	—	13	1	no	—
204	#335	—	13	2	no	—
205	#334	—	13	2	no	—
206	Q₈⋊C₈	—	13	2	no	—
207	Q₈⋊C₈	—	13	2	no	—
208	D₄⋊C₈	—	13	2	no	—
209	D₄⋊C₈	—	13	2	no	—
210	#322	—	13	1	no	—
211	#322	—	13	1	no	—
212	#322	—	13	1	no	—
213	C₈²	—	13	1	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	—	2	1	no	—
216	C₂²×C₁₃	—	2	1	no	—
217	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×D₁₃	—	2	4	no	—
219	C₂×D₁₃	—	4	2	no	—
220	C₂×D₁₃	—	4	2	no	—
221	C₂×D₁₃	—	8	4	no	—
222	C₄×C₁₃	—	2	1	no	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	4	2	no	—
227	C₄×C₁₃	—	4	2	no	—
228	C₄×C₁₃	C₂×#131	1	2	yes	—
229	C₄×C₁₃	#577	1	2	yes	—
230	Dic₁₃	—	4	2	no	—
231	C₂²×D₄	—	13	1	no	—
232	C₂³×C₄	—	26	1	no	—
233	C₂×SD₁₆	—	13	2	no	—
234	C₂×SD₁₆	—	13	2	no	—
235	C₂×M₁₆	—	26	1	no	—
236	C₂²×C₈	—	26	2	no	—
237	C₂²×C₈	—	26	2	no	—
238	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
239	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
240	#125	—	2	1	no	—
241	C₂²⋊Q₈	—	2	1	no	—
242	C₂²⋊Q₈	C₄×D₁₃	1	2	yes	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
245	C₄×D₄	—	13	2	no	—
246	C₄×D₄	—	13	2	no	—
247	C₄×D₄	—	26	1	no	—
248	C₄×D₄	—	26	1	no	—
249	#117	—	2	1	no	—
250	C₂×C₄⋊C₄	—	13	1	no	—
251	C₂×C₂²⋊C₄	—	26	1	no	—
252	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
253	#106	—	13	2	no	—
254	#106	—	13	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	13	2	no	—
258	C₄⋊C₈	—	13	2	no	—
259	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
260	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
261	#103	—	13	2	no	—
262	#103	—	13	2	no	—
263	#102	—	13	2	no	—
264	#102	—	13	2	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	13	2	no	—
267	C₂²⋊C₈	—	26	1	no	—
268	C₂²⋊C₈	—	26	2	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
271	#97	—	2	2	no	—
272	#97	—	2	1	no	—
273	#97	—	26	1	no	—
274	C₄×C₈	—	2	2	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	13	1	no	—
277	C₄×C₈	—	13	1	no	—
278	C₄×C₈	—	13	1	no	—
279	C₄×C₈	—	26	1	no	—
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
282	C₂×C₁₃	—	4	1	no	—
283	C₂×C₁₃	#2929	1	1	yes	characteristic
284	C₂×C₁₃	#2949	1	1	yes	characteristic
285	C₂×C₁₃	#2956	1	1	yes	characteristic
286	D₁₃	—	4	4	no	—
287	C₂⁴	—	26	1	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
290	C₂×D₄	—	13	2	no	—
291	C₂×D₄	—	26	4	no	—
292	C₂×D₄	—	26	1	no	—
293	C₂²×C₄	—	26	1	no	—
294	C₂²×C₄	—	26	2	no	—
295	C₂²×C₄	—	26	2	no	—
296	C₂²×C₄	—	104	2	no	—
297	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
298	C₂²×C₄	#6127	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
300	SD₁₆	—	26	4	no	—
301	SD₁₆	—	26	4	no	—
302	M₁₆	—	52	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	13	2	no	—
305	C₂×C₈	—	13	2	no	—
306	C₂×C₈	—	26	1	no	—
307	C₂×C₈	—	26	1	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	104	2	no	—
310	C₂×C₈	—	104	2	no	—
311	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
312	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	13	2	no	—
319	C₄⋊C₄	—	26	1	no	—
320	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	26	2	no	—
323	C₂²⋊C₄	—	26	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
326	C₄²	#6127	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
328	C₄²	C₂³×D₁₃	1	1	yes	characteristic
329	C₁₃	#17924	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	26	4	no	—
331	C₂³	—	26	2	no	—
332	C₂³	—	104	1	no	—
333	C₂³	—	104	1	no	—
334	C₂³	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	8	1	no	—
338	D₄	—	13	4	no	—
339	D₄	—	26	4	no	—
340	D₄	—	26	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	—	52	2	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	104	2	no	—
350	C₂×C₄	—	104	1	no	—
351	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
352	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
353	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
355	C₂×C₄	C₂×#6127	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	26	2	no	—
361	C₈	—	26	2	no	—
362	C₈	—	26	2	no	—
363	C₈	—	52	1	no	—
364	C₈	—	52	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	26	4	no	—
369	C₂²	—	52	2	no	—
370	C₂²	—	52	2	no	—
371	C₂²	—	104	4	no	—
372	C₂²	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	52	2	no	—
380	C₄	D₁₃×#131	1	2	yes	—
381	C₄	#11719542	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	52	4	no	—
384	C₂	#13123528	1	1	yes	characteristic
385	C₂	#13123695	1	1	yes	characteristic
386	C₂	#13123743	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2985	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#2985	D₁₃
hypocenter	C₁₃	#17924
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2985

Complements

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

N	H ≅ G/N	type
3. #13113691	C₂	semidirect
4. #13113670	C₂	semidirect
5. #13113646	C₂	semidirect
6. C₈×#6106	C₂	semidirect
7. #13113617	C₂	semidirect
9. #13113257	C₂	semidirect
10. #13113131	C₂	semidirect
11. C₁₃×#2985	C₂	semidirect
15. C₁₃×#446	C₂²	semidirect
16. C₈×Q₈×C₁₃	C₂²	semidirect
33. Q₈⋊#1055	C₂²	semidirect
34. Q₈⋊#1055	C₂²	semidirect
37. #11718072	C₂²	semidirect
38. #11718071	C₂²	semidirect
39. #11718047	C₂²	semidirect
40. #11718046	C₂²	semidirect
41. #11718046	C₂²	semidirect
42. C₈×#1055	C₂²	semidirect
138. #2985	D₁₃	semidirect
190. #446	C₂×D₁₃	semidirect
191. C₈×Q₈	C₂×D₁₃	semidirect
329. C₁₃	#17924	semidirect

Maximal subgroups

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

subgroup	index
#13124840	2
#13123859	2
#13113691	2
#13113670	2
#13113646	2
C₈×#6106	2
#13113617	2
#13113278	2
#13113257	2
#13113131	2
C₁₃×#2985	2
#17924	13