Subgroups — #16019846

Order 3328.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127267	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117063	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117041	C₂	1	2	yes	maximal, complemented
4	#13116973	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116925	C₂	1	2	yes	maximal, complemented
6	#13116883	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116776	C₂	1	2	yes	maximal, complemented
8	C₄×#88067	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116543	C₂	1	2	yes	maximal, complemented
11	C₁₃×#3283	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#135	C₂²	1	2	yes	—
13	C₁₃×#547	—	2	1	no	—
14	#11719247	—	2	1	no	—
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#467	C₂²	1	2	yes	—
17	C₁₃×#461	—	2	1	no	—
18	C₁₃×#452	—	2	2	no	—
19	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
20	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88069	C₂²	1	2	yes	—
22	C₂×#88067	C₂²	1	1	yes	characteristic
23	#11718777	C₂²	1	2	yes	—
24	#11718767	C₂²	1	2	yes	—
25	C₂²×#6090	C₂²	1	2	yes	—
26	#11718512	—	2	1	no	—
27	#11718384	—	2	2	no	—
28	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
29	#11718343	C₂²	1	1	yes	characteristic
30	#11718156	C₂²	1	2	yes	—
31	#11718156	C₂²	1	2	yes	—
32	#11718156	C₂²	1	2	yes	—
33	#11718154	C₂²	1	1	yes	characteristic
34	#11718154	C₂²	1	2	yes	—
35	C₂²⋊#6090	C₂²	1	2	yes	—
36	#11718084	C₂²	1	1	yes	characteristic
37	#11718082	C₂²	1	1	yes	characteristic
38	#11718081	C₂²	1	2	yes	—
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×#41	—	2	2	no	—
42	C₁₃×#135	—	2	4	no	—
43	C₁₃×#135	—	4	2	no	—
44	C₂×C₁₃×Q₁₆	—	2	1	no	—
45	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×SD₁₆	—	2	2	no	—
47	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#123	—	4	2	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₁₃×#121	—	4	1	no	—
53	C₄×D₄×C₁₃	—	2	1	no	—
54	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
55	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
56	#88190	—	4	1	no	—
57	#88180	—	2	2	no	—
58	#88178	—	2	2	no	—
59	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
60	#88141	—	4	1	no	—
61	#88133	—	4	2	no	—
62	#88123	—	2	1	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₃×#103	—	2	2	no	—
67	C₁₃×#102	—	2	1	no	—
68	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88069	—	4	2	no	—
71	#88067	—	2	2	no	—
72	#88058	—	2	1	no	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88058	C₂³	1	2	yes	—
75	#88057	—	2	2	no	—
76	#88056	—	2	2	no	—
77	#88056	—	2	1	no	—
78	C₂×#6090	—	4	2	no	—
79	C₂×#6090	C₂³	1	1	yes	characteristic
80	C₂×#6090	C₂³	1	2	yes	—
81	C₂×#6090	C₂³	1	1	yes	characteristic
82	#33869	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	4	4	no	—
84	C₁₃×#41	—	8	2	no	—
85	C₂×Q₈×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	1	no	—
87	C₂×D₄×C₁₃	—	4	2	no	—
88	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×Dic₂₆	—	2	1	no	—
93	C₂×Dic₂₆	—	4	1	no	—
94	C₁₃×Q₁₆	—	2	2	no	—
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×Q₁₆	—	8	1	no	—
97	C₁₃×SD₁₆	—	4	4	no	—
98	C₁₃×D₈	—	2	1	no	—
99	C₁₃×D₈	#41	1	2	yes	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂²⋊Dic₁₃	—	4	2	no	—
115	C₂²⋊Dic₁₃	—	4	2	no	—
116	C₄⋊Dic₁₃	—	2	1	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	—	4	1	no	—
119	#6101	—	4	1	no	—
120	#6101	—	4	2	no	—
121	C₄×Dic₁₃	—	4	1	no	—
122	C₄×Dic₁₃	—	4	2	no	—
123	Dic₅₂	—	2	1	no	—
124	Dic₅₂	—	2	1	no	—
125	#6090	—	2	1	no	—
126	#6090	—	2	2	no	—
127	#6090	—	2	1	no	—
128	#3342	—	13	1	no	—
129	#3283	D₁₃	1	1	yes	characteristic
130	#2241	—	13	1	no	—
131	#2237	—	13	2	no	—
132	#2224	—	13	1	no	—
133	#2216	—	13	2	no	—
134	#2209	—	13	1	no	—
135	#2189	—	13	2	no	—
136	C₄×SD₃₂	—	13	1	no	—
137	D₄×C₁₆	—	13	1	no	—
138	#2138	—	13	2	no	—
139	C₂³×C₁₃	—	4	1	no	—
140	C₂³×C₁₃	C₂×D₈	1	2	yes	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	2	2	no	—
144	D₄×C₁₃	—	2	2	no	—
145	D₄×C₁₃	—	4	1	no	—
146	D₄×C₁₃	—	8	4	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	8	4	no	—
152	C₂×C₄×C₁₃	—	8	2	no	—
153	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×Dic₁₃	—	4	1	no	—
158	C₂×Dic₁₃	—	4	1	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	Dic₂₆	—	4	1	no	—
161	Dic₂₆	—	4	1	no	—
162	Dic₂₆	—	8	2	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	—	2	1	no	—
165	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×#135	C₂×D₁₃	1	2	yes	—
168	#556	—	26	1	no	—
169	#547	—	2	1	no	—
170	C₂×Q₃₂	—	13	2	no	—
171	C₂×SD₃₂	—	13	1	no	—
172	C₂²×C₁₆	—	13	2	no	—
173	#500	—	13	1	no	—
174	#494	C₂×D₁₃	1	1	yes	characteristic
175	#485	—	26	2	no	—
176	#475	—	26	1	no	—
177	#467	—	13	2	no	—
178	#467	C₂×D₁₃	1	2	yes	—
179	#461	—	2	1	no	—
180	#452	—	2	2	no	—
181	C₄×Q₁₆	—	13	1	no	—
182	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
183	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
184	#427	—	13	2	no	—
185	#367	—	13	1	no	—
186	#366	—	13	2	no	—
187	C₄⋊C₁₆	—	13	1	no	—
188	#358	—	13	2	no	—
189	#358	—	13	2	no	—
190	#358	—	13	2	no	—
191	#358	—	13	1	no	—
192	#357	—	13	1	no	—
193	#357	—	13	2	no	—
194	C₂²⋊C₁₆	—	13	2	no	—
195	C₄×C₁₆	—	13	1	no	—
196	C₂²×C₁₃	—	2	2	no	—
197	C₂²×C₁₃	—	2	2	no	—
198	C₂²×C₁₃	—	2	2	no	—
199	C₂²×C₁₃	—	4	2	no	—
200	C₂²×C₁₃	—	8	1	no	—
201	C₂²×C₁₃	—	8	1	no	—
202	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	2	2	no	—
205	C₄×C₁₃	—	8	1	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	#508	1	2	yes	—
208	C₄×C₁₃	#546	1	1	yes	characteristic
209	C₄×C₁₃	#577	1	1	yes	characteristic
210	Dic₁₃	—	8	2	no	—
211	Dic₁₃	—	8	1	no	—
212	Dic₁₃	—	8	1	no	—
213	C₂×#41	—	2	2	no	—
214	#135	—	2	4	no	—
215	#135	—	4	2	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	—	13	1	no	—
218	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	26	1	no	—
224	#126	—	52	2	no	—
225	#124	—	52	1	no	—
226	#123	—	4	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	C₂²⋊Q₈	—	26	2	no	—
229	#121	—	4	1	no	—
230	C₄×Q₈	—	26	1	no	—
231	C₄×Q₈	—	52	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
234	#117	—	26	2	no	—
235	C₂×C₄⋊C₄	—	4	1	no	—
236	Q₃₂	—	52	2	no	—
237	SD₃₂	—	26	2	no	—
238	C₂×C₁₆	—	13	1	no	—
239	C₂×C₁₆	—	13	1	no	—
240	C₂×C₁₆	—	13	2	no	—
241	C₂×C₁₆	—	52	2	no	—
242	#107	—	13	2	no	—
243	#107	—	13	1	no	—
244	#107	—	26	1	no	—
245	#107	C₂²×D₁₃	1	1	yes	characteristic
246	#106	—	26	2	no	—
247	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
248	#103	—	2	2	no	—
249	#103	—	26	1	no	—
250	#103	—	26	2	no	—
251	#102	—	2	1	no	—
252	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
253	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
254	C₂×C₁₃	—	4	2	no	—
255	C₂×C₁₃	—	8	2	no	—
256	C₂×C₁₃	#3294	1	1	yes	characteristic
257	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
258	C₂×C₁₃	#3417	1	1	yes	characteristic
259	#41	—	4	4	no	—
260	#41	—	8	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	26	1	no	—
263	C₂×Q₈	—	52	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	#6104	1	1	yes	characteristic
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	8	1	no	—
273	Q₁₆	—	26	1	no	—
274	Q₁₆	—	26	1	no	—
275	SD₁₆	—	4	4	no	—
276	D₈	—	2	1	no	—
277	D₈	#6129	1	2	yes	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
282	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	26	1	no	—
288	C₄⋊C₄	—	26	2	no	—
289	C₄⋊C₄	—	52	1	no	—
290	C₄⋊C₄	—	52	2	no	—
291	C₄⋊C₄	—	52	1	no	—
292	C₄⋊C₄	#6104	1	1	yes	characteristic
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	52	2	no	—
296	C₂²⋊C₄	—	52	2	no	—
297	C₂²⋊C₄	#6104	1	2	yes	—
298	C₄²	—	52	2	no	—
299	C₄²	—	52	1	no	—
300	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₁₆	—	26	1	no	—
302	C₁₆	—	26	1	no	—
303	C₁₆	—	26	2	no	—
304	C₁₃	#33869	1	1	yes	characteristic, complemented, Sylow
305	C₂³	—	4	1	no	—
306	C₂³	C₂×#6104	1	2	yes	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	52	1	no	—
310	Q₈	—	52	1	no	—
311	Q₈	—	104	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	52	1	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	—	52	1	no	—
325	C₂×C₄	C₂×#6104	1	2	yes	—
326	C₂×C₄	C₂×#6104	1	1	yes	characteristic
327	C₂×C₄	C₂×#6104	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	C₂×#6129	1	1	yes	characteristic
332	C₈	#88251	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×#6104	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	104	2	no	—
345	C₄	—	104	1	no	—
346	C₄	—	104	1	no	—
347	C₄	#11718855	1	2	yes	—
348	C₄	#11719511	1	1	yes	characteristic
349	C₄	#11719541	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	8	2	no	—
352	C₂	#13126649	1	1	yes	characteristic
353	C₂	C₂×#11718855	1	1	yes	characteristic
354	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3283	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3283	D₁₃
hypocenter	C₁₃	#33869
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3283

Complements

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

N	H ≅ G/N	type
1. #13127267	C₂	semidirect
2. #13117063	C₂	semidirect
3. #13117041	C₂	semidirect
4. #13116973	C₂	semidirect
5. #13116925	C₂	semidirect
6. #13116883	C₂	semidirect
7. #13116776	C₂	semidirect
8. C₄×#88067	C₂	semidirect
9. D₄×#6090	C₂	semidirect
10. #13116543	C₂	semidirect
304. C₁₃	#33869	semidirect

Maximal subgroups

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

subgroup	index
#13127267	2
#13117063	2
#13117041	2
#13116973	2
#13116925	2
#13116883	2
#13116776	2
C₄×#88067	2
D₄×#6090	2
#13116543	2
C₁₃×#3283	2
#33869	13