Subgroups — #16019868

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 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	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117024	C₂	1	2	yes	maximal, complemented
4	#13116978	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116930	C₂	1	2	yes	maximal, complemented
6	#13116873	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116776	C₂	1	2	yes	maximal, complemented
8	C₄×#88069	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×#3287	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#135	C₂²	1	2	yes	—
13	C₁₃×#550	—	2	1	no	—
14	#11719247	—	2	1	no	—
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₁₃×#466	C₂²	1	2	yes	—
17	C₁₃×#460	—	2	1	no	—
18	C₁₃×#449	—	2	2	no	—
19	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88069	C₂²	1	1	yes	characteristic
22	C₂×#88067	C₂²	1	2	yes	—
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	1	yes	characteristic
31	#11718156	C₂²	1	2	yes	—
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718154	C₂²	1	2	yes	—
35	C₂²⋊#6090	C₂²	1	2	yes	—
36	#11718084	C₂²	1	1	yes	characteristic
37	#11718082	C₂²	1	2	yes	—
38	#11718081	C₂²	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×SD₁₆	—	2	2	no	—
46	C₂×C₁₃×D₈	—	2	1	no	—
47	C₂×C₁₃×D₈	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#127	—	4	1	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₁₃×#121	—	4	2	no	—
53	C₄×Q₈×C₁₃	—	2	1	no	—
54	C₄×D₄×C₁₃	—	4	1	no	—
55	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
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	1	no	—
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88069	—	2	2	no	—
71	#88067	—	4	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	1	yes	characteristic
81	C₂×#6090	C₂³	1	2	yes	—
82	#33871	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	4	4	no	—
84	C₁₃×#41	—	8	2	no	—
85	C₂×Q₈×C₁₃	—	2	1	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	4	2	no	—
88	C₂×D₄×C₁₃	—	4	2	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	—	4	2	no	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₂₆	—	2	1	no	—
95	C₂×Dic₂₆	—	4	1	no	—
96	C₁₃×Q₁₆	—	2	1	no	—
97	C₁₃×Q₁₆	#41	1	2	yes	—
98	C₁₃×SD₁₆	—	4	4	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₁₃×D₈	—	2	2	no	—
101	C₁₃×D₈	—	8	1	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	2	1	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	D₈	1	2	yes	—
112	C₄²×C₁₃	—	4	1	no	—
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	#3287	D₁₃	1	1	yes	characteristic
130	#2240	—	13	1	no	—
131	#2234	—	13	2	no	—
132	#2225	—	13	1	no	—
133	#2217	—	13	2	no	—
134	#2207	—	13	1	no	—
135	#2189	—	13	2	no	—
136	C₄×Q₃₂	—	13	1	no	—
137	D₄×C₁₆	—	13	1	no	—
138	#2138	—	13	2	no	—
139	C₂³×C₁₃	—	4	2	no	—
140	C₂³×C₁₃	C₂×D₈	1	2	yes	—
141	Q₈×C₁₃	—	2	2	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	D₄×C₁₃	—	2	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	8	4	no	—
147	D₄×C₁₃	—	8	2	no	—
148	D₄×C₁₃	—	16	1	no	—
149	D₄×C₁₃	—	16	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	1	no	—
154	C₂×C₄×C₁₃	—	8	4	no	—
155	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×Dic₁₃	—	4	1	no	—
160	C₂×Dic₁₃	—	4	1	no	—
161	C₂×Dic₁₃	—	4	2	no	—
162	Dic₂₆	—	4	1	no	—
163	Dic₂₆	—	4	1	no	—
164	Dic₂₆	—	8	2	no	—
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×#135	C₂×D₁₃	1	2	yes	—
170	#556	—	26	1	no	—
171	#550	—	2	1	no	—
172	C₂×Q₃₂	—	13	1	no	—
173	C₂×SD₃₂	—	13	2	no	—
174	C₂²×C₁₆	—	13	2	no	—
175	#500	—	13	1	no	—
176	#493	C₂×D₁₃	1	1	yes	characteristic
177	#485	—	26	2	no	—
178	#475	—	26	1	no	—
179	#467	—	13	2	no	—
180	#466	C₂×D₁₃	1	2	yes	—
181	#460	—	2	1	no	—
182	#449	—	2	2	no	—
183	C₄×Q₁₆	—	13	1	no	—
184	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
185	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
186	#427	—	13	2	no	—
187	#367	—	13	2	no	—
188	#366	—	13	1	no	—
189	C₄⋊C₁₆	—	13	1	no	—
190	#358	—	13	1	no	—
191	#358	—	13	2	no	—
192	#358	—	13	1	no	—
193	#357	—	13	2	no	—
194	#357	—	13	2	no	—
195	#357	—	13	2	no	—
196	C₂²⋊C₁₆	—	13	2	no	—
197	C₄×C₁₆	—	13	1	no	—
198	C₂²×C₁₃	—	2	2	no	—
199	C₂²×C₁₃	—	2	2	no	—
200	C₂²×C₁₃	—	2	2	no	—
201	C₂²×C₁₃	—	8	2	no	—
202	C₂²×C₁₃	—	8	2	no	—
203	C₂²×C₁₃	—	8	2	no	—
204	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	2	2	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	8	1	no	—
209	C₄×C₁₃	#508	1	2	yes	—
210	C₄×C₁₃	#546	1	1	yes	characteristic
211	C₄×C₁₃	#577	1	1	yes	characteristic
212	Dic₁₃	—	8	2	no	—
213	Dic₁₃	—	8	1	no	—
214	Dic₁₃	—	8	1	no	—
215	C₂×#41	—	2	2	no	—
216	#135	—	2	4	no	—
217	#135	—	4	2	no	—
218	C₂×Q₁₆	—	13	1	no	—
219	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	C₂²×D₁₃	1	2	yes	—
223	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
224	C₄⋊Q₈	—	26	1	no	—
225	#127	—	2	1	no	—
226	#127	—	4	1	no	—
227	#126	—	52	2	no	—
228	#124	—	52	1	no	—
229	C₂²⋊Q₈	—	26	2	no	—
230	#121	—	2	2	no	—
231	#121	—	4	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	26	1	no	—
234	C₄×Q₈	—	52	1	no	—
235	C₄×D₄	—	4	1	no	—
236	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
237	#117	—	26	2	no	—
238	Q₃₂	—	26	2	no	—
239	SD₃₂	—	52	2	no	—
240	C₂×C₁₆	—	13	1	no	—
241	C₂×C₁₆	—	13	1	no	—
242	C₂×C₁₆	—	13	2	no	—
243	C₂×C₁₆	—	52	2	no	—
244	#107	—	13	1	no	—
245	#107	—	13	2	no	—
246	#107	—	26	1	no	—
247	#107	C₂²×D₁₃	1	1	yes	characteristic
248	#106	—	26	2	no	—
249	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
250	#103	—	2	1	no	—
251	#103	—	26	1	no	—
252	#103	—	26	2	no	—
253	#102	—	2	2	no	—
254	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
255	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
256	C₂×C₁₃	—	4	2	no	—
257	C₂×C₁₃	—	16	2	no	—
258	C₂×C₁₃	#3294	1	1	yes	characteristic
259	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
260	C₂×C₁₃	#3416	1	1	yes	characteristic
261	#41	—	4	4	no	—
262	#41	—	8	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	26	1	no	—
265	C₂×Q₈	—	52	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	#6104	1	1	yes	characteristic
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	26	1	no	—
276	Q₁₆	—	26	1	no	—
277	Q₁₆	#6129	1	2	yes	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	2	2	no	—
280	D₈	—	2	2	no	—
281	D₈	—	8	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
285	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
286	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	26	1	no	—
290	C₄⋊C₄	—	26	2	no	—
291	C₄⋊C₄	—	52	1	no	—
292	C₄⋊C₄	—	52	2	no	—
293	C₄⋊C₄	—	52	1	no	—
294	C₄⋊C₄	#6104	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	52	2	no	—
297	C₂²⋊C₄	—	52	2	no	—
298	C₂²⋊C₄	#6104	1	2	yes	—
299	C₄²	—	4	1	no	—
300	C₄²	—	52	2	no	—
301	C₄²	—	52	1	no	—
302	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
303	C₁₆	—	26	1	no	—
304	C₁₆	—	26	1	no	—
305	C₁₆	—	26	2	no	—
306	C₁₃	#33871	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	4	2	no	—
308	C₂³	C₂×#6104	1	2	yes	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	52	1	no	—
312	Q₈	—	52	1	no	—
313	Q₈	—	104	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	16	1	no	—
320	D₄	—	16	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	52	1	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	—	52	1	no	—
329	C₂×C₄	C₂×#6104	1	2	yes	—
330	C₂×C₄	C₂×#6104	1	1	yes	characteristic
331	C₂×C₄	C₂×#6104	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	C₂×#6129	1	1	yes	characteristic
336	C₈	#88251	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂²×#6104	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	104	2	no	—
349	C₄	—	104	1	no	—
350	C₄	—	104	1	no	—
351	C₄	#11718855	1	2	yes	—
352	C₄	#11719511	1	1	yes	characteristic
353	C₄	#11719541	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	#13126649	1	1	yes	characteristic
357	C₂	C₂×#11718855	1	1	yes	characteristic
358	C₂	#13127983	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×#3287	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3287	D₁₃
hypocenter	C₁₃	#33871
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3287

Complements

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

N	H ≅ G/N	type
1. #13127267	C₂	semidirect
2. #13117059	C₂	semidirect
3. #13117024	C₂	semidirect
4. #13116978	C₂	semidirect
5. #13116930	C₂	semidirect
6. #13116873	C₂	semidirect
7. #13116776	C₂	semidirect
8. C₄×#88069	C₂	semidirect
9. D₄×#6090	C₂	semidirect
10. #13116543	C₂	semidirect
306. C₁₃	#33871	semidirect

Maximal subgroups

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

subgroup	index
#13127267	2
#13117059	2
#13117024	2
#13116978	2
#13116930	2
#13116873	2
#13116776	2
C₄×#88069	2
D₄×#6090	2
#13116543	2
C₁₃×#3287	2
#33871	13