Subgroups — #15845990

Order 3328.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125751	C₂	1	2	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114315	C₂	1	4	yes	maximal, complemented
4	#13114191	C₂	1	4	yes	maximal, complemented
5	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113505	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#131	—	2	1	no	—
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₂²×D₁₀₄	—	2	1	no	—
11	C₂²×D₁₀₄	—	2	1	no	—
12	C₂×#88128	C₂²	1	2	yes	—
13	C₁₃×#436	—	2	1	no	—
14	C₁₃×#436	C₂²	1	2	yes	complemented
15	C₈×D₄×C₁₃	—	2	1	no	—
16	C₈×D₄×C₁₃	C₂²	1	2	yes	complemented
17	C₁₃×#423	C₂²	1	1	yes	characteristic
18	C₁₃×#408	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718749	—	2	2	no	—
21	#11718734	—	2	2	no	—
22	#11718697	—	2	4	no	—
23	#11718676	C₂²	1	8	yes	complemented
24	#11718586	C₂²	1	2	yes	—
25	#11718356	—	2	1	no	—
26	#11718356	—	2	1	no	—
27	#11718356	C₂²	1	2	yes	—
28	#11718356	C₂²	1	2	yes	—
29	#11718077	C₂²	1	8	yes	complemented
30	#11718050	C₂²	1	2	yes	complemented
31	#11718049	C₂²	1	2	yes	complemented
32	C₂×C₁₃×#41	—	2	1	no	—
33	C₂²×D₅₂	—	2	2	no	—
34	C₂²×D₅₂	—	2	2	no	—
35	C₂²×D₅₂	—	2	2	no	—
36	C₁₃×#131	—	4	4	no	—
37	C₂×C₁₃×M₁₆	—	2	1	no	—
38	C₂×C₁₃×M₁₆	—	2	2	no	—
39	C₂²×C₈×C₁₃	—	2	2	no	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₁₃×#127	D₄	1	1	yes	characteristic
44	C₁₃×#124	—	2	1	no	—
45	C₁₃×#121	—	2	1	no	—
46	C₁₃×#121	D₄	1	2	yes	—
47	C₄×D₄×C₁₃	—	2	1	no	—
48	C₄×D₄×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×#6104	—	2	8	no	—
51	C₂×D₁₀₄	—	2	2	no	—
52	C₂×D₁₀₄	—	2	2	no	—
53	C₂×D₁₀₄	—	2	2	no	—
54	C₂×D₁₀₄	—	2	2	no	—
55	C₂×D₁₀₄	—	4	4	no	—
56	C₂×D₁₀₄	—	4	4	no	—
57	#88128	—	2	2	no	—
58	#88128	—	2	2	no	—
59	#88128	—	2	2	no	—
60	#88128	C₂³	1	4	yes	—
61	C₁₃×C₄⋊C₈	—	2	1	no	—
62	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
63	C₁₃×C₂²⋊C₈	—	2	2	no	—
64	C₁₃×C₂²⋊C₈	—	2	2	no	—
65	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂²⋊#1055	—	2	2	no	—
70	#88049	—	2	8	no	—
71	C₄⋊#1055	C₂³	1	4	yes	—
72	C₄×#1055	—	2	2	no	—
73	#19187	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	—	4	2	no	—
75	C₁₃×#41	—	4	2	no	—
76	C₁₃×#41	—	4	2	no	—
77	C₂×Q₈×C₁₃	—	2	1	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
86	C₂×D₅₂	—	2	4	no	—
87	C₂×D₅₂	—	2	4	no	—
88	C₂×D₅₂	—	2	4	no	—
89	C₂×D₅₂	—	4	4	no	—
90	C₂×D₅₂	—	4	4	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×D₅₂	—	4	2	no	—
94	C₂×D₅₂	—	8	4	no	—
95	C₂×D₅₂	—	8	4	no	—
96	C₁₃×M₁₆	—	4	2	no	—
97	C₁₃×M₁₆	—	4	4	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	4	4	no	—
101	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
108	C₁₃×C₂²⋊C₄	—	2	2	no	—
109	C₁₃×C₂²⋊C₄	—	2	2	no	—
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	#41	1	1	yes	characteristic
113	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
114	#6104	—	8	8	no	—
115	C₂×#1055	—	2	4	no	—
116	D₁₀₄	—	4	4	no	—
117	D₁₀₄	—	4	4	no	—
118	D₁₀₄	—	4	4	no	—
119	D₁₀₄	—	4	4	no	—
120	#3205	—	13	2	no	—
121	C₂×#493	—	13	1	no	—
122	#2931	D₁₃	1	1	yes	characteristic, complemented
123	#1687	—	13	4	no	—
124	#1667	—	13	4	no	—
125	#1564	—	13	1	no	—
126	#1545	—	13	2	no	—
127	C₂³×C₁₃	—	2	2	no	—
128	C₂³×C₁₃	#121	1	1	yes	characteristic
129	C₂²×D₁₃	—	4	4	no	—
130	C₂²×D₁₃	—	8	4	no	—
131	C₂²×D₁₃	—	8	4	no	—
132	C₂²×D₁₃	—	8	4	no	—
133	Q₈×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	2	4	no	—
135	D₄×C₁₃	—	4	4	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	D₅₂	—	4	4	no	—
152	D₅₂	—	4	4	no	—
153	D₅₂	—	4	8	no	—
154	D₅₂	—	8	4	no	—
155	D₅₂	—	8	4	no	—
156	D₅₂	—	8	4	no	—
157	D₅₂	—	8	4	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	—	4	2	no	—
161	C₈×C₁₃	C₂×D₈	1	2	yes	—
162	C₈×C₁₃	C₂×D₈	1	2	yes	—
163	#1055	—	4	4	no	—
164	C₂²×D₈	—	26	1	no	—
165	C₂²×D₈	—	26	1	no	—
166	C₂×#131	—	2	1	no	—
167	#532	C₂×D₁₃	1	1	yes	characteristic
168	C₂×#127	—	13	2	no	—
169	#493	—	13	2	no	—
170	#493	—	13	2	no	—
171	#493	—	26	2	no	—
172	#493	—	26	1	no	—
173	#493	—	26	1	no	—
174	#486	—	26	2	no	—
175	#459	—	13	8	no	—
176	#447	—	26	4	no	—
177	#436	—	2	1	no	—
178	#436	C₂×D₁₃	1	2	yes	complemented
179	C₈×D₄	—	2	1	no	—
180	C₈×D₄	C₂×D₁₃	1	2	yes	complemented
181	#431	—	13	2	no	—
182	#423	C₂×D₁₃	1	1	yes	characteristic
183	#408	—	2	1	no	—
184	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
185	#335	—	13	2	no	—
186	#334	—	13	2	no	—
187	#331	—	13	8	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	4	2	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	4	2	no	—
193	C₂²×C₁₃	#459	1	2	yes	—
194	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
195	C₂×D₁₃	—	8	4	no	—
196	C₂×D₁₃	—	8	4	no	—
197	C₂×D₁₃	—	8	4	no	—
198	C₂×D₁₃	—	16	8	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	#459	1	2	yes	—
203	C₄×C₁₃	#472	1	2	yes	—
204	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄×C₁₃	#576	1	2	yes	—
207	C₂×#41	—	2	1	no	—
208	C₂²×D₄	—	26	2	no	—
209	C₂²×D₄	—	26	2	no	—
210	C₂²×D₄	—	26	2	no	—
211	C₂×D₈	—	26	8	no	—
212	C₂×D₈	—	26	2	no	—
213	C₂×D₈	—	26	2	no	—
214	C₂×D₈	—	26	2	no	—
215	C₂×D₈	—	26	2	no	—
216	C₂×D₈	—	52	4	no	—
217	C₂×D₈	—	52	4	no	—
218	#131	—	4	4	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
225	#127	—	13	4	no	—
226	#127	—	26	2	no	—
227	#127	—	26	2	no	—
228	#127	—	26	2	no	—
229	#127	D₅₂	1	1	yes	characteristic
230	#124	—	2	1	no	—
231	#121	—	2	1	no	—
232	#121	D₅₂	1	2	yes	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	C₂²×D₁₃	1	2	yes	—
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	13	4	no	—
238	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
239	#102	—	26	8	no	—
240	C₂²⋊C₈	—	2	2	no	—
241	C₂²⋊C₈	—	2	2	no	—
242	C₂²⋊C₈	—	26	2	no	—
243	C₄×C₈	—	26	2	no	—
244	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
245	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	—	8	2	no	—
250	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
251	C₂×C₁₃	#3058	1	1	yes	characteristic
252	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
253	D₁₃	—	16	4	no	—
254	C₂⁴	—	52	2	no	—
255	#41	—	4	2	no	—
256	#41	—	4	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	26	4	no	—
262	C₂×D₄	—	26	4	no	—
263	C₂×D₄	—	26	4	no	—
264	C₂×D₄	—	52	2	no	—
265	C₂×D₄	—	52	2	no	—
266	C₂×D₄	—	52	2	no	—
267	C₂×D₄	—	52	4	no	—
268	C₂×D₄	—	52	4	no	—
269	C₂×D₄	—	104	4	no	—
270	C₂×D₄	—	104	4	no	—
271	C₂×D₄	C₂×D₅₂	1	2	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
274	C₂²×C₄	#6127	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	D₈	—	52	4	no	—
277	D₈	—	52	4	no	—
278	D₈	—	52	4	no	—
279	D₈	—	52	4	no	—
280	D₈	—	104	8	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	4	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	4	no	—
286	C₂×C₈	—	26	4	no	—
287	C₂×C₈	#6104	1	4	yes	complemented
288	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₄²	C₂×D₅₂	1	1	yes	characteristic
297	C₄²	#6127	1	1	yes	characteristic
298	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
299	C₄²	C₂³×D₁₃	1	1	yes	characteristic
300	C₁₃	#19187	1	1	yes	characteristic, complemented, Sylow
301	C₂³	—	2	2	no	—
302	C₂³	—	52	4	no	—
303	C₂³	—	104	4	no	—
304	C₂³	—	104	4	no	—
305	C₂³	—	104	4	no	—
306	C₂³	#88184	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	52	8	no	—
313	D₄	—	52	4	no	—
314	D₄	—	52	4	no	—
315	D₄	—	104	4	no	—
316	D₄	—	104	4	no	—
317	D₄	—	104	4	no	—
318	D₄	—	104	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#88184	1	1	yes	characteristic
325	C₂×C₄	#88184	1	1	yes	characteristic
326	C₂×C₄	#88184	1	1	yes	characteristic
327	C₂×C₄	C₂×#6104	1	2	yes	—
328	C₂×C₄	#88203	1	2	yes	—
329	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
330	C₂×C₄	C₂×#6127	1	1	yes	characteristic
331	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	2	no	—
335	C₈	—	52	4	no	—
336	C₈	C₂×#6104	1	2	yes	—
337	C₈	C₂×#6104	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	104	4	no	—
344	C₂²	—	104	4	no	—
345	C₂²	—	104	4	no	—
346	C₂²	—	208	8	no	—
347	C₂²	#11718676	1	2	yes	—
348	C₂²	C₂×#88184	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	#11718676	1	2	yes	—
353	C₄	#11718803	1	2	yes	—
354	C₄	#11719457	1	2	yes	—
355	C₄	C₂²×#6104	1	1	yes	characteristic
356	C₄	C₂×#88203	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	—	208	4	no	—
360	C₂	C₂×#11718676	1	1	yes	characteristic
361	C₂	#13124484	1	1	yes	characteristic
362	C₂	C₂×#11718803	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2931	C₂
socle	C₂²×C₁₃	C₂×#121
hypercenter	#2931	D₁₃
hypocenter	C₁₃	#19187
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #13125751	C₂	semidirect
2. C₂×#11718356	C₂	semidirect
3. #13114315	C₂	semidirect
4. #13114191	C₂	semidirect
5. #13113584	C₂	semidirect
6. #13113505	C₂	semidirect
7. C₁₃×#2931	C₂	semidirect
14. C₁₃×#436	C₂²	semidirect
16. C₈×D₄×C₁₃	C₂²	semidirect
23. #11718676	C₂²	semidirect
29. #11718077	C₂²	semidirect
30. #11718050	C₂²	semidirect
31. #11718049	C₂²	semidirect
101. C₂×C₈×C₁₃	D₈	semidirect
122. #2931	D₁₃	semidirect
178. #436	C₂×D₁₃	semidirect
180. C₈×D₄	C₂×D₁₃	semidirect
287. C₂×C₈	#6104	semidirect
300. C₁₃	#19187	semidirect

Maximal subgroups

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

subgroup	index
#13125751	2
C₂×#11718356	2
#13114315	2
#13114191	2
#13113584	2
#13113505	2
C₁₃×#2931	2
#19187	13