Subgroups — #15762658

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
2	#13115798	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115642	C₂	1	1	yes	maximal, characteristic
4	#13115641	C₂	1	1	yes	maximal, characteristic
5	#13114992	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂²×#6106	—	2	2	no	—
10	C₂×#88128	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#384	C₂²	1	1	yes	characteristic
13	C₂×#88061	—	2	2	no	—
14	#11718746	—	2	2	no	—
15	#11718746	C₄	1	4	yes	complemented
16	C₂×#88049	—	2	2	no	—
17	C₂×#88048	—	2	2	no	—
18	#11718596	C₂²	1	1	yes	characteristic
19	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₁₃×#340	—	2	2	no	—
22	#11718121	—	2	2	no	—
23	C₈×#1055	—	2	2	no	—
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂²×D₅₂	—	2	1	no	—
26	C₂²×D₅₂	—	2	1	no	—
27	C₂²×D₅₂	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₁₃×C₄⋊Q₈	—	2	1	no	—
30	C₁₃×C₄⋊Q₈	—	2	1	no	—
31	C₁₃×C₄⋊Q₈	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6106	—	2	8	no	—
40	C₂×#6106	—	4	4	no	—
41	C₂×#6106	—	4	4	no	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×C₄⋊Dic₁₃	—	2	1	no	—
44	C₂×C₄⋊Dic₁₃	—	2	1	no	—
45	C₂²×#1055	D₄	1	2	yes	—
46	C₂²×#1055	D₄	1	2	yes	—
47	#88128	—	2	1	no	—
48	#88128	—	2	1	no	—
49	#88128	—	2	1	no	—
50	#88128	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	#88061	—	4	4	no	—
54	#88049	—	4	4	no	—
55	#88048	—	4	4	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	—	2	1	no	—
58	C₄×#1055	C₂×C₄	1	2	yes	—
59	C₄×#1055	D₄	1	2	yes	—
60	#9820	—	13	1	no	maximal, Sylow
61	C₂³×D₁₃	—	4	1	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	4	1	no	—
64	C₂×Q₈×C₁₃	—	4	2	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	—	4	1	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	4	1	no	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×D₅₂	—	2	2	no	—
75	C₂×D₅₂	—	4	2	no	—
76	C₂×D₅₂	—	4	2	no	—
77	C₂×D₅₂	—	4	2	no	—
78	C₂×D₅₂	—	4	2	no	—
79	C₂×D₅₂	—	4	2	no	—
80	C₂×D₅₂	—	4	2	no	—
81	C₂×D₅₂	—	4	1	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	4	1	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	#41	1	1	yes	characteristic
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	#6106	—	4	16	no	—
99	C₄⋊Dic₁₃	—	4	2	no	—
100	C₄⋊Dic₁₃	—	4	2	no	—
101	C₄⋊Dic₁₃	—	4	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	D₈	1	4	yes	—
106	C₂×#1055	SD₁₆	1	4	yes	—
107	C₂×#1055	SD₁₆	1	4	yes	complemented
108	C₂×#492	—	13	1	no	—
109	#1990	—	13	1	no	—
110	#1990	D₁₃	1	1	yes	characteristic, complemented
111	#1957	—	13	1	no	—
112	#1957	—	13	1	no	—
113	#1844	—	13	1	no	—
114	C₂×C₈²	—	13	1	no	—
115	C₂³×C₁₃	#127	1	1	yes	characteristic
116	C₂²×D₁₃	—	4	2	no	—
117	C₂²×D₁₃	—	8	1	no	—
118	C₂²×D₁₃	—	8	1	no	—
119	C₂²×D₁₃	—	8	1	no	—
120	C₂²×D₁₃	—	8	1	no	—
121	C₂²×D₁₃	—	8	1	no	—
122	C₂²×D₁₃	—	8	1	no	—
123	Q₈×C₁₃	—	4	4	no	—
124	Q₈×C₁₃	—	8	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	1	no	—
127	C₂×C₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
141	C₂×C₄×C₁₃	#135	1	2	yes	—
142	C₂×C₄×C₁₃	#135	1	2	yes	—
143	C₂×Dic₁₃	—	8	1	no	—
144	C₂×Dic₁₃	—	8	1	no	—
145	C₂×Dic₁₃	—	8	1	no	—
146	D₅₂	—	4	4	no	—
147	D₅₂	—	4	4	no	—
148	D₅₂	—	4	4	no	—
149	D₅₂	—	8	2	no	—
150	D₅₂	—	8	2	no	—
151	D₅₂	—	8	2	no	—
152	C₈×C₁₃	—	2	4	no	—
153	#1055	—	2	4	no	—
154	#1055	#102	1	8	yes	complemented
155	C₂²×SD₁₆	—	26	2	no	—
156	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#127	—	13	1	no	—
158	#492	—	13	4	no	—
159	#492	—	26	2	no	—
160	C₂×#106	—	26	2	no	—
161	C₂×#102	—	26	2	no	—
162	C₂×#102	—	26	2	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#384	—	13	1	no	—
167	#384	C₂×D₁₃	1	1	yes	characteristic
168	#340	—	2	2	no	—
169	#340	—	26	2	no	—
170	C₈²	—	26	2	no	—
171	C₂²×C₁₃	#390	1	1	yes	characteristic
172	C₂²×C₁₃	#492	1	1	yes	characteristic
173	C₂²×C₁₃	#492	1	1	yes	characteristic
174	C₂²×C₁₃	#492	1	1	yes	characteristic
175	C₂²×C₁₃	#493	1	1	yes	characteristic
176	C₂²×C₁₃	#495	1	1	yes	characteristic
177	C₂²×C₁₃	#495	1	1	yes	characteristic
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	8	2	no	—
181	C₂×D₁₃	—	16	1	no	—
182	C₂×D₁₃	—	16	1	no	—
183	C₂×D₁₃	—	16	1	no	—
184	C₂×D₁₃	—	16	1	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	16	1	no	—
187	C₄×C₁₃	C₂×#102	1	2	yes	—
188	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
189	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
190	C₄×C₁₃	#465	1	2	yes	—
191	C₄×C₁₃	#465	1	2	yes	—
192	C₄×C₁₃	#486	1	2	yes	—
193	Dic₁₃	—	16	1	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	26	1	no	—
196	C₂²×D₄	—	26	1	no	—
197	C₂²×D₄	—	26	1	no	—
198	C₂×SD₁₆	—	26	8	no	—
199	C₂×SD₁₆	—	52	4	no	—
200	C₂×SD₁₆	—	52	4	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	C₄×D₁₃	1	2	yes	—
208	#127	—	13	2	no	—
209	#127	—	26	1	no	—
210	#127	—	26	1	no	—
211	#127	—	26	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	26	1	no	—
218	C₂×C₄⋊C₄	—	26	1	no	—
219	C₂×C₄⋊C₄	—	26	1	no	—
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	#106	—	52	4	no	—
222	#102	—	52	4	no	—
223	#102	—	52	4	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	—	26	1	no	—
229	C₄×C₈	—	26	1	no	—
230	C₂×C₁₃	#1705	1	1	yes	characteristic
231	C₂×C₁₃	#1706	1	1	yes	characteristic
232	C₂×C₁₃	#1706	1	1	yes	characteristic
233	C₂×C₁₃	#1708	1	1	yes	characteristic
234	C₂×C₁₃	#1957	1	1	yes	characteristic
235	C₂×C₁₃	#1968	1	1	yes	characteristic
236	C₂×C₁₃	#1968	1	1	yes	characteristic
237	D₁₃	—	16	2	no	—
238	C₂⁴	—	52	1	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×D₄	—	26	2	no	—
244	C₂×D₄	—	26	2	no	—
245	C₂×D₄	—	26	2	no	—
246	C₂×D₄	—	52	1	no	—
247	C₂×D₄	—	52	1	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	2	no	—
250	C₂×D₄	—	52	2	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂×D₄	—	52	1	no	—
253	C₂×D₄	—	52	2	no	—
254	C₂×D₄	—	52	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	52	1	no	—
258	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	SD₁₆	—	52	16	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	13	4	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	13	4	no	—
267	C₂×C₈	—	13	4	no	—
268	C₂×C₈	—	26	2	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	#6106	1	4	yes	complemented
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	2	no	—
280	C₄⋊C₄	—	52	2	no	—
281	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
282	C₄²	#6127	1	1	yes	characteristic
283	C₄²	D₄×D₁₃	1	1	yes	characteristic
284	C₄²	#6129	1	1	yes	characteristic
285	C₁₃	#9820	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	52	2	no	—
287	C₂³	—	104	1	no	—
288	C₂³	—	104	1	no	—
289	C₂³	—	104	1	no	—
290	C₂³	—	104	1	no	—
291	C₂³	—	104	1	no	—
292	C₂³	—	104	1	no	—
293	C₂³	#88194	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	52	4	no	—
297	D₄	—	52	4	no	—
298	D₄	—	52	4	no	—
299	D₄	—	104	2	no	—
300	D₄	—	104	2	no	—
301	D₄	—	104	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	104	1	no	—
309	C₂×C₄	—	104	1	no	—
310	C₂×C₄	—	104	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₄	#88135	1	1	yes	characteristic
313	C₂×C₄	#88138	1	1	yes	characteristic
314	C₂×C₄	#88139	1	1	yes	characteristic
315	C₂×C₄	D₈×D₁₃	1	2	yes	—
316	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
317	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
318	C₂×C₄	#88173	1	2	yes	—
319	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
320	C₂×C₄	#88193	1	1	yes	characteristic
321	C₂×C₄	C₂×#6106	1	2	yes	—
322	C₂×C₄	#88204	1	2	yes	—
323	C₈	—	2	4	no	—
324	C₈	—	13	8	no	—
325	C₈	—	26	4	no	—
326	C₂²	—	104	2	no	—
327	C₂²	—	104	2	no	—
328	C₂²	—	104	2	no	—
329	C₂²	—	208	1	no	—
330	C₂²	—	208	1	no	—
331	C₂²	—	208	1	no	—
332	C₂²	—	208	1	no	—
333	C₂²	#11718618	1	1	yes	characteristic
334	C₂²	#11718720	1	1	yes	characteristic
335	C₂²	#11718746	1	1	yes	characteristic
336	C₂²	#11718812	1	1	yes	characteristic
337	C₂²	#11718836	1	1	yes	characteristic
338	C₂²	#11718836	1	1	yes	characteristic
339	C₂²	#11718852	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	208	1	no	—
343	C₄	#11718421	1	2	yes	—
344	C₄	D₁₃×#102	1	2	yes	—
345	C₄	#11718449	1	2	yes	—
346	C₄	#11718468	1	2	yes	—
347	C₄	C₄×#6106	1	2	yes	—
348	C₄	#11718709	1	2	yes	—
349	C₂	—	208	2	no	—
350	C₂	#13114424	1	1	yes	characteristic
351	C₂	#13114427	1	1	yes	characteristic
352	C₂	#13114429	1	1	yes	characteristic
353	C₂	#13114439	1	1	yes	characteristic
354	C₂	#13115640	1	1	yes	characteristic
355	C₂	#13115687	1	1	yes	characteristic
356	C₂	#13115693	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1990	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1990	D₁₃
hypocenter	C₁₃	#9820
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #13115798	C₂	semidirect
5. #13114992	C₂	semidirect
6. C₂×C₈×#1055	C₂	semidirect
7. C₁₃×#1990	C₂	semidirect
15. #11718746	C₄	semidirect
86. C₂×C₈×C₁₃	SD₁₆	semidirect
107. C₂×#1055	SD₁₆	semidirect
110. #1990	D₁₃	semidirect
154. #1055	#102	semidirect
270. C₂×C₈	#6106	semidirect
285. C₁₃	#9820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718746	2
#13115798	2
#13115642	2
#13115641	2
#13114992	2
C₂×C₈×#1055	2
C₁₃×#1990	2
#9820	13