Subgroups — #15765707

Order 3328.

686 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	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718749	C₂	1	1	yes	maximal, characteristic
3	#13115798	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115642	C₂	1	1	yes	maximal, characteristic
5	#13115639	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112663	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂²×#6106	—	2	2	no	—
11	C₂×#88128	C₂²	1	1	yes	characteristic
12	C₁₃×#500	—	2	2	no	—
13	C₁₃×#500	—	2	2	no	—
14	C₂×C₁₃×#107	—	2	2	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88075	—	2	2	no	—
17	C₂×#88061	—	2	2	no	—
18	#11718749	—	2	2	no	—
19	#11718749	—	2	2	no	—
20	#11718746	—	2	2	no	—
21	#11718746	—	2	2	no	—
22	C₂×#88049	—	2	2	no	—
23	#11718596	C₂²	1	1	yes	characteristic
24	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
26	#11718121	—	2	2	no	—
27	C₂²×Q₈×C₁₃	—	2	1	no	—
28	C₂²×Q₈×C₁₃	—	2	1	no	—
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×D₅₂	—	2	1	no	—
31	C₂²×D₅₂	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₁₃×C₄⋊Q₈	—	2	1	no	—
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×#6106	—	2	8	no	—
47	C₂×#6106	—	4	4	no	—
48	C₂×#6106	—	4	4	no	—
49	C₂×C₄⋊Dic₁₃	—	2	1	no	—
50	C₂×C₄⋊Dic₁₃	—	2	1	no	—
51	C₂×C₄⋊Dic₁₃	—	2	1	no	—
52	C₂²×#1055	D₄	1	2	yes	—
53	C₂²×#1055	D₄	1	2	yes	—
54	#88128	—	2	1	no	—
55	#88128	—	2	1	no	—
56	#88128	—	2	1	no	—
57	#88128	D₄	1	2	yes	—
58	C₁₃×#107	—	2	8	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	#88075	—	4	4	no	—
62	#88061	—	4	4	no	—
63	#88049	—	2	8	no	—
64	C₄×#1055	—	2	1	no	—
65	C₄×#1055	—	2	1	no	—
66	C₄×#1055	—	2	1	no	—
67	C₄×#1055	—	2	1	no	—
68	#10308	—	13	1	no	maximal, Sylow
69	C₂³×D₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	—	4	2	no	—
76	C₂×Q₈×C₁₃	—	4	2	no	—
77	C₂×Q₈×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₁₃	—	4	1	no	—
84	C₂×D₅₂	—	2	2	no	—
85	C₂×D₅₂	—	2	2	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×D₅₂	—	4	2	no	—
88	C₂×D₅₂	—	4	2	no	—
89	C₂×D₅₂	—	4	2	no	—
90	C₂×D₅₂	—	4	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×D₅₂	—	4	1	no	—
94	C₂×D₅₂	—	4	1	no	—
95	C₂×D₅₂	—	4	1	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	#41	1	1	yes	characteristic
111	#6106	—	4	16	no	—
112	C₄⋊Dic₁₃	—	4	2	no	—
113	C₄⋊Dic₁₃	—	4	2	no	—
114	C₄⋊Dic₁₃	—	4	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	2	no	—
120	C₂×#1055	D₈	1	4	yes	—
121	C₂×#500	D₁₃	1	1	yes	characteristic, complemented
122	C₂×#492	—	13	1	no	—
123	C₂×#486	—	13	1	no	—
124	#1990	—	13	1	no	—
125	#1957	—	13	1	no	—
126	#1956	—	13	1	no	—
127	#1273	—	13	1	no	—
128	C₂³×C₁₃	#127	1	1	yes	characteristic
129	C₂²×D₁₃	—	4	2	no	—
130	C₂²×D₁₃	—	8	1	no	—
131	C₂²×D₁₃	—	8	1	no	—
132	C₂²×D₁₃	—	8	1	no	—
133	C₂²×D₁₃	—	8	1	no	—
134	C₂²×D₁₃	—	8	1	no	—
135	C₂²×D₁₃	—	8	1	no	—
136	Q₈×C₁₃	—	4	4	no	—
137	Q₈×C₁₃	—	4	4	no	—
138	Q₈×C₁₃	—	8	2	no	—
139	Q₈×C₁₃	—	8	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	8	1	no	—
143	C₂×C₄×C₁₃	—	8	1	no	—
144	C₂×C₄×C₁₃	—	8	1	no	—
145	C₂×C₄×C₁₃	—	8	1	no	—
146	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₁₃	#135	1	2	yes	—
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₁₃	#137	1	2	yes	—
157	C₂×C₄×C₁₃	#137	1	2	yes	—
158	C₂×Dic₁₃	—	8	1	no	—
159	C₂×Dic₁₃	—	8	1	no	—
160	C₂×Dic₁₃	—	8	1	no	—
161	D₅₂	—	4	4	no	—
162	D₅₂	—	4	4	no	—
163	D₅₂	—	4	4	no	—
164	D₅₂	—	8	2	no	—
165	D₅₂	—	8	2	no	—
166	D₅₂	—	8	2	no	—
167	C₈×C₁₃	—	2	4	no	—
168	#1055	—	2	4	no	—
169	#1055	—	4	2	no	—
170	C₂²×SD₁₆	—	26	2	no	—
171	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×#127	—	13	1	no	—
174	#500	—	2	2	no	—
175	#500	—	2	2	no	—
176	#492	—	26	2	no	—
177	#492	—	26	2	no	—
178	#486	—	26	2	no	—
179	#486	—	26	2	no	—
180	C₂×#107	—	2	2	no	—
181	C₂×#103	—	26	2	no	—
182	C₂×#102	—	26	2	no	—
183	C₂×#102	—	26	2	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	—	13	1	no	—
186	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
187	#384	—	13	1	no	—
188	#340	—	26	2	no	—
189	C₂²×C₁₃	#392	1	1	yes	characteristic
190	C₂²×C₁₃	#492	1	1	yes	characteristic
191	C₂²×C₁₃	#493	1	1	yes	characteristic
192	C₂²×C₁₃	#495	1	1	yes	characteristic
193	C₂²×C₁₃	#496	1	1	yes	characteristic
194	C₂²×C₁₃	#497	1	1	yes	characteristic
195	C₂²×C₁₃	#497	1	1	yes	characteristic
196	C₂×D₁₃	—	8	2	no	—
197	C₂×D₁₃	—	8	2	no	—
198	C₂×D₁₃	—	8	2	no	—
199	C₂×D₁₃	—	16	1	no	—
200	C₂×D₁₃	—	16	1	no	—
201	C₂×D₁₃	—	16	1	no	—
202	C₂×D₁₃	—	16	1	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	#447	1	2	yes	—
206	C₄×C₁₃	#448	1	2	yes	—
207	C₄×C₁₃	#452	1	2	yes	—
208	C₄×C₁₃	#459	1	2	yes	—
209	C₄×C₁₃	#461	1	2	yes	—
210	C₄×C₁₃	#464	1	2	yes	—
211	Dic₁₃	—	16	1	no	—
212	C₂²×Q₈	—	2	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×D₄	—	26	1	no	—
215	C₂²×D₄	—	26	1	no	—
216	C₂²×D₄	—	26	1	no	—
217	C₂×SD₁₆	—	26	8	no	—
218	C₂×SD₁₆	—	52	4	no	—
219	C₂×SD₁₆	—	52	4	no	—
220	C₂²×C₈	—	13	2	no	—
221	C₂²×C₈	—	13	2	no	—
222	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	C₂²⋊D₁₃	1	2	yes	—
230	C₄⋊Q₈	C₂²⋊D₁₃	1	2	yes	—
231	#127	—	13	2	no	—
232	#127	—	26	1	no	—
233	#127	—	26	1	no	—
234	#127	—	26	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	26	1	no	—
240	C₂×C₄⋊C₄	—	26	1	no	—
241	C₂×C₄⋊C₄	—	26	1	no	—
242	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
243	#107	—	2	8	no	—
244	#103	—	52	4	no	—
245	#102	—	26	8	no	—
246	#102	—	52	4	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	26	1	no	—
250	C₄×C₈	—	26	1	no	—
251	C₄×C₈	—	26	1	no	—
252	C₄×C₈	—	26	1	no	—
253	C₂×C₁₃	#1706	1	1	yes	characteristic
254	C₂×C₁₃	#1714	1	1	yes	characteristic
255	C₂×C₁₃	#1719	1	1	yes	characteristic
256	C₂×C₁₃	#1722	1	1	yes	characteristic
257	C₂×C₁₃	#2011	1	1	yes	characteristic
258	C₂×C₁₃	#2013	1	1	yes	characteristic
259	C₂×C₁₃	#2015	1	1	yes	characteristic
260	D₁₃	—	16	2	no	—
261	C₂⁴	—	52	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×D₄	—	26	2	no	—
271	C₂×D₄	—	26	2	no	—
272	C₂×D₄	—	26	2	no	—
273	C₂×D₄	—	52	2	no	—
274	C₂×D₄	—	52	2	no	—
275	C₂×D₄	—	52	2	no	—
276	C₂×D₄	—	52	2	no	—
277	C₂×D₄	—	52	1	no	—
278	C₂×D₄	—	52	1	no	—
279	C₂×D₄	—	52	2	no	—
280	C₂×D₄	—	52	2	no	—
281	C₂×D₄	—	52	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	52	1	no	—
285	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	SD₁₆	—	52	16	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	13	4	no	—
292	C₂×C₈	—	26	2	no	—
293	C₂×C₈	—	26	2	no	—
294	C₂×C₈	—	26	2	no	—
295	C₂×C₈	—	26	2	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	#6106	1	4	yes	complemented
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	52	2	no	—
307	C₄⋊C₄	—	52	2	no	—
308	C₄⋊C₄	—	52	2	no	—
309	C₄²	D₄×D₁₃	1	1	yes	characteristic
310	C₄²	#6129	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
313	C₁₃	#10308	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	52	2	no	—
315	C₂³	—	104	1	no	—
316	C₂³	—	104	1	no	—
317	C₂³	—	104	1	no	—
318	C₂³	—	104	1	no	—
319	C₂³	—	104	1	no	—
320	C₂³	—	104	1	no	—
321	C₂³	#88194	1	1	yes	characteristic
322	Q₈	—	4	4	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	52	4	no	—
327	D₄	—	52	4	no	—
328	D₄	—	52	4	no	—
329	D₄	—	104	2	no	—
330	D₄	—	104	2	no	—
331	D₄	—	104	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	104	1	no	—
339	C₂×C₄	—	104	1	no	—
340	C₂×C₄	—	104	1	no	—
341	C₂×C₄	D₈×D₁₃	1	2	yes	—
342	C₂×C₄	#88165	1	2	yes	—
343	C₂×C₄	#88172	1	2	yes	—
344	C₂×C₄	#88173	1	2	yes	—
345	C₂×C₄	#88191	1	1	yes	characteristic
346	C₂×C₄	#88191	1	1	yes	characteristic
347	C₂×C₄	#88192	1	1	yes	characteristic
348	C₂×C₄	#88193	1	1	yes	characteristic
349	C₂×C₄	#88193	1	1	yes	characteristic
350	C₂×C₄	C₂×#6106	1	2	yes	—
351	C₂×C₄	#88196	1	2	yes	—
352	C₂×C₄	#88207	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	26	4	no	—
355	C₈	—	52	2	no	—
356	C₂²	—	104	2	no	—
357	C₂²	—	104	2	no	—
358	C₂²	—	104	2	no	—
359	C₂²	—	208	1	no	—
360	C₂²	—	208	1	no	—
361	C₂²	—	208	1	no	—
362	C₂²	—	208	1	no	—
363	C₂²	#11718746	1	1	yes	characteristic
364	C₂²	#11718747	1	1	yes	characteristic
365	C₂²	#11718812	1	1	yes	characteristic
366	C₂²	#11718815	1	1	yes	characteristic
367	C₂²	#11718851	1	1	yes	characteristic
368	C₂²	#11718852	1	1	yes	characteristic
369	C₂²	#11718884	1	1	yes	characteristic
370	C₄	—	8	2	no	—
371	C₄	—	8	2	no	—
372	C₄	—	208	1	no	—
373	C₄	#11718744	1	2	yes	—
374	C₄	#11718811	1	2	yes	—
375	C₄	#11718817	1	2	yes	—
376	C₄	#11718846	1	2	yes	—
377	C₄	#11718848	1	2	yes	—
378	C₄	#11718888	1	2	yes	—
379	C₂	—	208	2	no	—
380	C₂	#13114429	1	1	yes	characteristic
381	C₂	#13114459	1	1	yes	characteristic
382	C₂	#13114492	1	1	yes	characteristic
383	C₂	#13114506	1	1	yes	characteristic
384	C₂	#13115905	1	1	yes	characteristic
385	C₂	#13115919	1	1	yes	characteristic
386	C₂	#13115927	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#10308
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #13115798	C₂	semidirect
5. #13115639	C₂	semidirect
6. #13112663	C₂	semidirect
7. C₂×C₁₃×#500	C₂	semidirect
98. C₂×C₈×C₁₃	SD₁₆	semidirect
121. C₂×#500	D₁₃	semidirect
297. C₂×C₈	#6106	semidirect
313. C₁₃	#10308	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
C₂×#11718749	2
#13115798	2
#13115642	2
#13115639	2
#13112663	2
C₂×C₁₃×#500	2
#10308	13