Subgroups — #15845986

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	#13125754	C₂	1	2	yes	maximal, complemented
2	#13123549	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114312	C₂	1	4	yes	maximal, complemented
4	#13114188	C₂	1	4	yes	maximal, complemented
5	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113504	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	—	2	1	no	—
10	C₂×C₁₃×#127	C₂²	1	2	yes	—
11	C₂×#88155	—	2	1	no	—
12	#11719182	C₂²	1	1	yes	characteristic
13	C₁₃×#493	—	2	1	no	—
14	C₁₃×#493	—	2	1	no	—
15	C₁₃×#493	C₂²	1	2	yes	—
16	C₁₃×#493	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#11718878	—	2	4	no	—
19	#11718771	—	2	1	no	—
20	#11718734	—	2	2	no	—
21	#11718733	C₂²	1	8	yes	complemented
22	#11718731	—	2	2	no	—
23	#11718586	C₂²	1	2	yes	—
24	#11718585	C₂²	1	1	yes	characteristic
25	#11718349	—	2	1	no	—
26	#11718349	C₂²	1	2	yes	complemented
27	C₈×D₅₂	—	2	1	no	—
28	C₈×D₅₂	C₂²	1	2	yes	complemented
29	#11718136	C₂²	1	8	yes	complemented
30	#11718050	C₂²	1	2	yes	complemented
31	#11718049	C₂²	1	2	yes	complemented
32	C₂²×D₄×C₁₃	—	2	2	no	—
33	C₂²×D₄×C₁₃	—	2	2	no	—
34	C₂²×D₄×C₁₃	—	2	2	no	—
35	C₂×#6127	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	2	no	—
37	C₂×C₁₃×D₈	—	2	2	no	—
38	C₂×C₁₃×D₈	—	2	2	no	—
39	C₂×C₁₃×D₈	—	2	2	no	—
40	C₂×C₁₃×D₈	—	4	4	no	—
41	C₂×C₁₃×D₈	—	4	4	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×#127	—	2	2	no	—
45	C₁₃×#127	—	2	2	no	—
46	C₁₃×#127	—	2	2	no	—
47	C₁₃×#127	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×#6104	—	2	8	no	—
50	#88184	—	2	1	no	—
51	#88184	D₄	1	2	yes	—
52	C₂×#6099	—	2	1	no	—
53	#88155	—	4	4	no	—
54	C₂×#6094	—	2	2	no	—
55	C₂×C₈×D₁₃	—	2	2	no	—
56	#88129	—	2	1	no	—
57	#88128	D₄	1	1	yes	characteristic
58	C₄×D₅₂	—	2	1	no	—
59	C₄×D₅₂	C₂³	1	2	yes	—
60	#88123	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#88072	—	2	8	no	—
66	C₂²⋊#1055	—	2	2	no	—
67	#88060	—	2	2	no	—
68	#88060	—	2	2	no	—
69	C₄⋊#1055	—	2	1	no	—
70	C₄⋊#1055	C₂³	1	4	yes	—
71	C₄⋊#1055	C₂³	1	2	yes	—
72	C₄×#1055	—	2	2	no	—
73	#19187	—	13	1	no	maximal, Sylow
74	C₂⁴×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	—	2	4	no	—
76	C₂×D₄×C₁₃	—	2	4	no	—
77	C₂×D₄×C₁₃	—	2	4	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	4	no	—
82	C₂×D₄×C₁₃	—	4	4	no	—
83	C₂×D₄×C₁₃	—	8	4	no	—
84	C₂×D₄×C₁₃	—	8	4	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	—	2	2	no	—
89	#6127	—	4	2	no	—
90	#6127	—	4	2	no	—
91	C₂×D₅₂	—	2	1	no	—
92	C₂×D₅₂	—	2	1	no	—
93	C₂×D₅₂	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₁₃	—	2	2	no	—
95	C₂×Dic₂₆	—	2	1	no	—
96	C₁₃×D₈	—	4	4	no	—
97	C₁₃×D₈	—	4	4	no	—
98	C₁₃×D₈	—	4	4	no	—
99	C₁₃×D₈	—	4	4	no	—
100	C₂×C₈×C₁₃	—	2	2	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	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6104	—	8	8	no	—
111	#6103	—	2	2	no	—
112	#6103	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	1	no	—
114	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
115	#6099	—	4	2	no	—
116	C₂×#1055	—	2	4	no	—
117	C₂×#1055	—	2	2	no	—
118	#6094	—	4	4	no	—
119	C₈×D₁₃	—	4	4	no	—
120	#3205	—	13	2	no	—
121	C₂×#493	D₁₃	1	1	yes	characteristic, complemented
122	#2931	—	13	1	no	—
123	#1687	—	13	4	no	—
124	#1667	—	13	4	no	—
125	#1564	—	13	1	no	—
126	#1545	—	13	2	no	—
127	C₂³×C₁₃	—	4	4	no	—
128	C₂³×C₁₃	—	8	4	no	—
129	C₂³×C₁₃	—	8	4	no	—
130	C₂³×C₁₃	—	8	4	no	—
131	C₂³×C₁₃	#121	1	1	yes	characteristic
132	C₂²×D₁₃	—	2	2	no	—
133	D₄×C₁₃	—	4	4	no	—
134	D₄×C₁₃	—	4	4	no	—
135	D₄×C₁₃	—	4	8	no	—
136	D₄×C₁₃	—	8	4	no	—
137	D₄×C₁₃	—	8	4	no	—
138	D₄×C₁₃	—	8	4	no	—
139	D₄×C₁₃	—	8	4	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₁₃	—	4	4	no	—
151	C₂×Dic₁₃	—	2	2	no	—
152	D₅₂	—	2	4	no	—
153	D₅₂	—	4	2	no	—
154	D₅₂	—	4	2	no	—
155	C₄×D₁₃	—	4	2	no	—
156	C₄×D₁₃	—	4	2	no	—
157	Dic₂₆	—	4	2	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	C₂×D₈	1	2	yes	—
161	C₈×C₁₃	C₂×D₈	1	2	yes	—
162	#1055	—	4	2	no	—
163	#1055	—	4	4	no	—
164	C₂²×D₈	—	2	1	no	—
165	C₂²×D₈	—	2	1	no	—
166	C₂×#131	—	26	1	no	—
167	#532	—	13	1	no	—
168	C₂×#127	C₂×D₁₃	1	2	yes	—
169	#493	—	2	1	no	—
170	#493	—	2	1	no	—
171	#493	—	26	2	no	—
172	#493	C₂×D₁₃	1	2	yes	—
173	#493	C₂×D₁₃	1	2	yes	—
174	#486	—	26	2	no	—
175	#459	—	13	8	no	—
176	#447	—	26	4	no	—
177	#436	—	13	2	no	—
178	#436	—	26	1	no	—
179	C₈×D₄	—	13	2	no	—
180	C₈×D₄	—	26	1	no	—
181	#431	—	13	2	no	—
182	#423	—	13	1	no	—
183	#408	—	26	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	—	8	4	no	—
191	C₂²×C₁₃	—	8	4	no	—
192	C₂²×C₁₃	—	8	4	no	—
193	C₂²×C₁₃	—	16	8	no	—
194	C₂²×C₁₃	#459	1	2	yes	—
195	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
196	C₂×D₁₃	—	4	2	no	—
197	C₂×D₁₃	—	4	2	no	—
198	C₂×D₁₃	—	4	2	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	#459	1	2	yes	—
202	C₄×C₁₃	#472	1	2	yes	—
203	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
204	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₄×C₁₃	#576	1	2	yes	—
206	Dic₁₃	—	4	2	no	—
207	C₂×#41	—	26	1	no	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	—	2	2	no	—
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	4	4	no	—
216	C₂×D₈	—	4	4	no	—
217	C₂×D₈	—	26	8	no	—
218	#131	—	52	4	no	—
219	C₂×M₁₆	—	26	1	no	—
220	C₂×M₁₆	—	26	2	no	—
221	C₂²×C₈	—	26	2	no	—
222	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄⋊Q₈	—	13	1	no	—
225	#127	—	2	2	no	—
226	#127	—	2	2	no	—
227	#127	—	2	2	no	—
228	#127	—	13	1	no	—
229	#127	C₂²×D₁₃	1	4	yes	—
230	#124	—	26	1	no	—
231	#121	—	13	2	no	—
232	#121	—	26	1	no	—
233	C₄×D₄	—	13	2	no	—
234	C₄×D₄	—	26	1	no	—
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₄⋊C₈	—	13	2	no	—
237	C₄⋊C₈	—	13	4	no	—
238	C₄⋊C₈	—	26	1	no	—
239	#102	—	26	8	no	—
240	C₂²⋊C₈	—	26	2	no	—
241	C₂²⋊C₈	—	26	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	—	16	4	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	—	8	2	no	—
254	C₂⁴	—	4	2	no	—
255	#41	—	52	2	no	—
256	#41	—	52	2	no	—
257	C₂×Q₈	—	26	1	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	8	4	no	—
267	C₂×D₄	—	8	4	no	—
268	C₂×D₄	—	13	2	no	—
269	C₂×D₄	—	26	2	no	—
270	C₂×D₄	—	26	1	no	—
271	C₂×D₄	—	26	1	no	—
272	C₂²×C₄	—	26	2	no	—
273	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
274	C₂²×C₄	#6129	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	D₈	—	4	4	no	—
277	D₈	—	4	4	no	—
278	D₈	—	4	4	no	—
279	D₈	—	4	4	no	—
280	D₈	—	104	8	no	—
281	M₁₆	—	52	2	no	—
282	M₁₆	—	52	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	26	4	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	—	52	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	—	13	2	no	—
293	C₄⋊C₄	—	26	1	no	—
294	C₂²⋊C₄	—	26	2	no	—
295	C₂²⋊C₄	—	26	2	no	—
296	C₄²	D₄×D₁₃	1	1	yes	characteristic
297	C₄²	#6129	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	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	26	2	no	—
306	C₂³	#88192	1	1	yes	characteristic
307	Q₈	—	52	2	no	—
308	D₄	—	4	8	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	4	no	—
315	D₄	—	26	4	no	—
316	D₄	—	52	4	no	—
317	D₄	—	52	2	no	—
318	D₄	—	52	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	52	2	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	C₂×#6104	1	2	yes	—
325	C₂×C₄	#88186	1	2	yes	—
326	C₂×C₄	#88192	1	1	yes	characteristic
327	C₂×C₄	#88192	1	1	yes	characteristic
328	C₂×C₄	#88192	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#6129	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	—	52	4	no	—
335	C₈	—	52	2	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	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	8	no	—
344	C₂²	—	52	2	no	—
345	C₂²	—	52	2	no	—
346	C₂²	—	52	2	no	—
347	C₂²	#11718733	1	2	yes	—
348	C₂²	C₂×#88192	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	52	2	no	—
352	C₄	#11718733	1	2	yes	—
353	C₄	#11718821	1	2	yes	—
354	C₄	#11719462	1	2	yes	—
355	C₄	C₂²×#6104	1	1	yes	characteristic
356	C₄	C₂×#88186	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	16	4	no	—
359	C₂	—	104	2	no	—
360	C₂	C₂×#11718733	1	1	yes	characteristic
361	C₂	#13124480	1	1	yes	characteristic
362	C₂	C₂×#11718821	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#493	C₂
socle	C₂²×C₁₃	C₂×#121
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	#19187
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #13125754	C₂	semidirect
2. #13123549	C₂	semidirect
3. #13114312	C₂	semidirect
4. #13114188	C₂	semidirect
5. #13113584	C₂	semidirect
6. #13113504	C₂	semidirect
7. C₂×C₁₃×#493	C₂	semidirect
21. #11718733	C₂²	semidirect
26. #11718349	C₂²	semidirect
28. C₈×D₅₂	C₂²	semidirect
29. #11718136	C₂²	semidirect
30. #11718050	C₂²	semidirect
31. #11718049	C₂²	semidirect
101. C₂×C₈×C₁₃	D₈	semidirect
121. C₂×#493	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
#13125754	2
#13123549	2
#13114312	2
#13114188	2
#13113584	2
#13113504	2
C₂×C₁₃×#493	2
#19187	13