Subgroups — #16017763

Order 3328.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124484	C₂	1	1	yes	maximal, characteristic
2	#13117027	C₂	1	2	yes	maximal, complemented
3	#13117027	C₂	1	2	yes	maximal, complemented
4	#13117021	C₂	1	2	yes	maximal, complemented
5	#13117016	C₂	1	2	yes	maximal, complemented
6	#13116622	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116618	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116551	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718083	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116450	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2913	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
13	C₁₃×#520	—	2	1	no	—
14	#11719535	—	2	1	no	—
15	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
16	C₁₃×#428	—	2	2	no	—
17	C₁₃×#428	—	2	1	no	—
18	C₁₃×#427	C₂²	1	1	yes	characteristic
19	C₁₃×#427	C₂²	1	2	yes	complemented
20	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#107	C₂²	1	2	yes	complemented
22	#11718796	—	2	2	no	—
23	#11718792	—	2	1	no	—
24	#11718776	C₂²	1	2	yes	—
25	#11718776	C₂²	1	2	yes	—
26	#11718767	C₂²	1	1	yes	characteristic
27	C₂×#88052	C₂²	1	1	yes	characteristic
28	C₂²×#6090	C₂²	1	1	yes	characteristic
29	C₂²⋊#6090	C₂²	1	2	yes	complemented
30	C₂²⋊#6090	C₂²	1	2	yes	complemented
31	#11718083	C₂²	1	2	yes	—
32	#11718083	C₂²	1	2	yes	—
33	#11718083	C₂²	1	2	yes	—
34	#11718083	C₂²	1	2	yes	—
35	#11718082	C₂²	1	1	yes	characteristic, complemented
36	#11718082	C₂²	1	2	yes	complemented
37	#11718082	C₂²	1	1	yes	characteristic, complemented
38	#11718081	C₂²	1	1	yes	characteristic, complemented
39	#11718081	C₂²	1	2	yes	complemented
40	#11718081	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
42	C₂²×D₅₂	—	2	1	no	—
43	C₁₃×#131	—	2	2	no	—
44	C₁₃×#131	—	2	2	no	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₁₃	—	4	1	no	—
50	C₄×D₄×C₁₃	—	4	2	no	—
51	C₄×D₄×C₁₃	—	4	1	no	—
52	C₁₃×#117	—	2	1	no	—
53	C₁₃×#117	—	2	2	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
56	#88201	—	4	1	no	—
57	#88194	—	4	1	no	—
58	#88191	—	4	2	no	—
59	#88184	—	2	2	no	—
60	#88184	—	2	2	no	—
61	#88180	—	2	1	no	—
62	C₂×D₁₀₄	—	2	2	no	—
63	C₂×D₁₀₄	—	2	2	no	—
64	C₂×D₁₀₄	—	2	1	no	—
65	C₂×D₁₀₄	C₂³	1	2	yes	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	—	2	2	no	—
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×#106	—	2	1	no	—
72	C₁₃×#106	—	2	2	no	—
73	#88061	—	2	2	no	—
74	#88061	—	2	2	no	—
75	#88058	C₂³	1	1	yes	characteristic
76	#88058	C₂³	1	1	yes	characteristic
77	#88057	—	2	1	no	—
78	#88052	—	2	2	no	—
79	C₂×#6090	—	2	2	no	—
80	C₂×#6090	C₂³	1	1	yes	characteristic
81	C₂×#6090	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	2	yes	—
83	#33650	—	13	1	no	maximal, Sylow
84	C₂³×D₁₃	—	4	1	no	—
85	C₁₃×#41	—	2	2	no	—
86	C₁₃×#41	—	2	2	no	—
87	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
88	C₂×D₄×C₁₃	D₈	1	2	yes	—
89	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	—	4	1	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₂²⋊D₁₃	—	4	2	no	—
95	C₂×C₂²⋊D₁₃	—	4	2	no	—
96	C₂×D₅₂	—	2	2	no	—
97	C₂×D₅₂	—	4	2	no	—
98	C₂×D₅₂	—	4	2	no	—
99	C₂×D₅₂	—	4	1	no	—
100	C₁₃×M₁₆	—	2	2	no	—
101	C₁₃×M₁₆	—	2	4	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	#41	1	2	yes	—
107	C₂×C₈×C₁₃	#41	1	2	yes	—
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	1	no	—
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	—	4	1	no	—
116	C₄²×C₁₃	—	4	2	no	—
117	C₄²×C₁₃	—	4	1	no	—
118	C₂²⋊Dic₁₃	—	4	1	no	—
119	#6103	—	4	2	no	—
120	#6103	—	4	2	no	—
121	C₄⋊Dic₁₃	—	2	1	no	—
122	C₄⋊Dic₁₃	—	2	1	no	—
123	C₄×Dic₁₃	—	4	1	no	—
124	D₁₀₄	—	2	2	no	—
125	D₁₀₄	—	2	2	no	—
126	D₁₀₄	—	4	2	no	—
127	#6090	—	2	1	no	—
128	#6090	—	2	1	no	—
129	#6090	—	2	2	no	—
130	#3058	—	13	1	no	—
131	#2913	D₁₃	1	1	yes	characteristic, complemented
132	#2234	—	13	2	no	—
133	#2234	—	13	2	no	—
134	#2233	—	13	2	no	—
135	#2232	—	13	2	no	—
136	#2159	—	13	1	no	—
137	#2158	—	13	1	no	—
138	#2140	—	13	1	no	—
139	C₂×#357	—	13	1	no	—
140	#2113	—	13	1	no	—
141	C₂³×C₁₃	C₂×D₈	1	2	yes	—
142	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂²×D₁₃	—	4	2	no	—
144	C₂²×D₁₃	—	8	2	no	—
145	C₂²×D₁₃	—	16	1	no	—
146	C₂²×D₁₃	—	16	1	no	—
147	Q₈×C₁₃	—	2	2	no	—
148	D₄×C₁₃	—	2	4	no	—
149	D₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	16	1	no	—
156	C₂×C₄×C₁₃	—	16	2	no	—
157	C₂×C₄×C₁₃	#123	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	#137	1	2	yes	—
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	16	2	no	—
163	C₂²⋊D₁₃	—	16	2	no	—
164	C₂×Dic₁₃	—	4	1	no	—
165	C₂×Dic₁₃	—	4	1	no	—
166	D₅₂	—	4	2	no	—
167	D₅₂	—	4	2	no	—
168	D₅₂	—	8	2	no	—
169	C₈×C₁₃	—	2	2	no	—
170	C₈×C₁₃	#123	1	2	yes	—
171	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
172	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
173	C₂²×D₈	—	13	1	no	—
174	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
175	#535	—	26	1	no	—
176	#520	—	2	1	no	—
177	C₂×M₃₂	—	13	1	no	—
178	C₂²×C₁₆	—	13	1	no	—
179	#469	—	26	2	no	—
180	#466	—	13	2	no	—
181	#466	—	13	2	no	—
182	#428	—	2	2	no	—
183	#428	—	2	1	no	—
184	#427	—	13	1	no	—
185	#427	C₂×D₁₃	1	1	yes	characteristic
186	#427	C₂×D₁₃	1	2	yes	complemented
187	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
188	C₂×#107	C₂×D₁₃	1	2	yes	complemented
189	#418	—	26	1	no	—
190	#367	—	13	1	no	—
191	#367	—	13	1	no	—
192	#367	—	13	2	no	—
193	#366	—	13	1	no	—
194	#366	—	13	1	no	—
195	#366	—	13	2	no	—
196	#357	—	13	2	no	—
197	#357	—	13	2	no	—
198	#357	—	13	2	no	—
199	#357	—	13	2	no	—
200	C₂²⋊C₁₆	—	13	2	no	—
201	C₂²⋊C₁₆	—	13	2	no	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	2	2	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	—	2	2	no	—
207	C₂²×C₁₃	#480	1	2	yes	—
208	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
209	C₂×D₁₃	—	8	2	no	—
210	C₂×D₁₃	—	8	2	no	—
211	C₂×D₁₃	—	8	2	no	—
212	C₂×D₁₃	—	16	4	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	—	8	2	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	#480	1	2	yes	—
218	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
220	Dic₁₃	—	8	1	no	—
221	Dic₁₃	—	8	1	no	—
222	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²×D₄	—	26	1	no	—
224	C₂×D₈	—	13	2	no	—
225	C₂×D₈	—	26	2	no	—
226	C₂×D₈	—	26	2	no	—
227	C₂×D₈	—	26	1	no	—
228	#131	—	2	2	no	—
229	#131	—	2	2	no	—
230	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
232	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
233	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	#127	—	52	1	no	—
235	#124	—	52	1	no	—
236	#121	—	26	2	no	—
237	#121	—	26	2	no	—
238	#120	—	52	2	no	—
239	C₄×Q₈	—	4	1	no	—
240	C₄×D₄	—	4	2	no	—
241	C₄×D₄	—	4	1	no	—
242	#117	—	2	1	no	—
243	#117	—	2	2	no	—
244	#117	—	26	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	2	no	—
247	M₃₂	—	26	2	no	—
248	C₂×C₁₆	—	13	1	no	—
249	C₂×C₁₆	—	13	2	no	—
250	C₂×C₁₆	—	13	1	no	—
251	C₂×C₁₆	—	26	2	no	—
252	#107	—	2	1	no	—
253	#107	—	2	2	no	—
254	#107	—	13	1	no	—
255	#107	—	13	1	no	—
256	#107	C₂²×D₁₃	1	2	yes	—
257	#107	C₂²×D₁₃	1	1	yes	characteristic
258	#107	C₂²×D₁₃	1	1	yes	characteristic
259	#106	—	2	1	no	—
260	#106	—	2	2	no	—
261	#106	—	26	1	no	—
262	#102	—	26	2	no	—
263	#102	—	26	2	no	—
264	C₂×C₁₃	—	2	2	no	—
265	C₂×C₁₃	—	4	2	no	—
266	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
267	C₂×C₁₃	#3415	1	1	yes	characteristic
268	C₂×C₁₃	#3416	1	1	yes	characteristic
269	D₁₃	—	16	2	no	—
270	C₂⁴	—	52	1	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	C₂×Q₈	#6104	1	1	yes	characteristic
274	C₂×D₄	—	26	2	no	—
275	C₂×D₄	—	52	2	no	—
276	C₂×D₄	—	52	2	no	—
277	C₂×D₄	—	52	2	no	—
278	C₂×D₄	—	52	1	no	—
279	C₂×D₄	—	52	2	no	—
280	C₂×D₄	#6104	1	2	yes	—
281	C₂×D₄	#6104	1	1	yes	characteristic
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
285	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	D₈	—	26	2	no	—
287	D₈	—	26	2	no	—
288	D₈	—	52	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	2	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	#6127	1	2	yes	—
294	C₂×C₈	#6127	1	2	yes	—
295	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	26	1	no	—
304	C₄⋊C₄	—	26	1	no	—
305	C₂²⋊C₄	—	4	2	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	52	1	no	—
308	C₂²⋊C₄	—	52	2	no	—
309	C₂²⋊C₄	—	52	2	no	—
310	C₄²	—	4	2	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	52	1	no	—
313	C₁₆	—	26	1	no	—
314	C₁₆	—	26	1	no	—
315	C₁₆	—	26	2	no	—
316	C₁₃	#33650	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	52	2	no	—
318	C₂³	—	104	2	no	—
319	C₂³	—	208	1	no	—
320	C₂³	—	208	1	no	—
321	C₂³	C₂×#6104	1	2	yes	—
322	C₂³	C₂×#6104	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	D₄	—	2	4	no	—
325	D₄	—	2	2	no	—
326	D₄	—	52	2	no	—
327	D₄	—	52	2	no	—
328	D₄	—	104	2	no	—
329	D₄	—	208	2	no	—
330	D₄	—	208	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	16	1	no	—
337	C₂×C₄	—	16	2	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	#88183	1	2	yes	—
341	C₂×C₄	C₂×#6104	1	2	yes	—
342	C₂×C₄	C₂×#6104	1	1	yes	characteristic
343	C₂×C₄	#88196	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	#88183	1	2	yes	—
347	C₈	C₂×#6127	1	1	yes	characteristic
348	C₈	C₂×#6127	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	104	2	no	—
355	C₂²	—	104	2	no	—
356	C₂²	—	104	2	no	—
357	C₂²	—	208	4	no	—
358	C₂²	#11718628	1	2	yes	—
359	C₂²	C₂²×#6104	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	8	2	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	104	1	no	—
365	C₄	—	104	1	no	—
366	C₄	#11718628	1	2	yes	—
367	C₄	C₂×#88183	1	1	yes	characteristic
368	C₄	C₂×#88196	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	4	2	no	—
371	C₂	—	208	2	no	—
372	C₂	C₂×#11718628	1	1	yes	characteristic
373	C₂	#13127975	1	1	yes	characteristic
374	C₂	#13127980	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2913	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13124484	C₂
hypercenter	#2913	D₁₃
hypocenter	C₁₃	#33650
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #2913

Complements

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

N	H ≅ G/N	type
2. #13117027	C₂	semidirect
3. #13117027	C₂	semidirect
4. #13117021	C₂	semidirect
5. #13117016	C₂	semidirect
6. #13116622	C₂	semidirect
7. #13116618	C₂	semidirect
8. #13116551	C₂	semidirect
9. C₂×#11718083	C₂	semidirect
10. #13116450	C₂	semidirect
11. C₁₃×#2913	C₂	semidirect
19. C₁₃×#427	C₂²	semidirect
21. C₂×C₁₃×#107	C₂²	semidirect
29. C₂²⋊#6090	C₂²	semidirect
30. C₂²⋊#6090	C₂²	semidirect
35. #11718082	C₂²	semidirect
36. #11718082	C₂²	semidirect
37. #11718082	C₂²	semidirect
38. #11718081	C₂²	semidirect
39. #11718081	C₂²	semidirect
40. #11718081	C₂²	semidirect
131. #2913	D₁₃	semidirect
186. #427	C₂×D₁₃	semidirect
188. C₂×#107	C₂×D₁₃	semidirect
316. C₁₃	#33650	semidirect

Maximal subgroups

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

subgroup	index
#13124484	2
#13117027	2
#13117021	2
#13117016	2
#13116622	2
#13116618	2
#13116551	2
C₂×#11718083	2
#13116450	2
C₁₃×#2913	2
#33650	13