Subgroups — #16017604

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125372	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123688	C₂	1	1	yes	maximal, characteristic
3	#13116995	C₂	1	2	yes	maximal, complemented
4	#13116991	C₂	1	2	yes	maximal, complemented
5	#13116981	C₂	1	2	yes	maximal, complemented
6	#13116971	C₂	1	2	yes	maximal, complemented
7	#13116622	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116594	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116552	C₂	1	1	yes	maximal, characteristic
11	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
13	#11719503	—	2	1	no	—
14	#11719205	—	2	1	no	—
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	#11718876	—	2	1	no	—
18	#11718813	—	2	2	no	—
19	D₈×Dic₁₃	C₂²	1	2	yes	—
20	D₈×Dic₁₃	C₂²	1	2	yes	—
21	#11718767	C₂²	1	1	yes	characteristic
22	#11718767	C₂²	1	1	yes	characteristic
23	#11718762	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	#11718625	—	2	1	no	—
26	#11718539	C₂²	1	2	yes	complemented
27	#11718537	—	2	2	no	—
28	#11718535	C₂²	1	2	yes	complemented
29	C₁₃×#357	C₂²	1	2	yes	—
30	C₁₃×#357	C₂²	1	2	yes	—
31	#11718154	C₂²	1	2	yes	—
32	#11718154	C₂²	1	2	yes	—
33	#11718097	C₂²	1	1	yes	characteristic, complemented
34	#11718097	C₂²	1	1	yes	characteristic, complemented
35	#11718096	C₂²	1	1	yes	characteristic, complemented
36	#11718096	C₂²	1	1	yes	characteristic, complemented
37	#11718093	C₂²	1	2	yes	complemented
38	#11718093	C₂²	1	2	yes	complemented
39	#11718082	C₂²	1	2	yes	complemented
40	#11718081	C₂²	1	2	yes	complemented
41	C₂²×D₄×C₁₃	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	—	2	2	no	—
44	C₂×C₁₃×D₈	Q₈	1	4	yes	complemented
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₂²⋊Dic₁₃	—	4	2	no	—
49	D₄×Dic₁₃	—	2	2	no	—
50	D₄×Dic₁₃	—	2	2	no	—
51	D₄×Dic₁₃	—	4	1	no	—
52	D₄×Dic₁₃	—	4	1	no	—
53	#88180	—	2	1	no	—
54	#88180	—	2	1	no	—
55	#88180	D₄	1	1	yes	characteristic
56	#88178	—	4	2	no	—
57	C₂×#6099	D₄	1	1	yes	characteristic
58	#88144	—	2	2	no	—
59	#88144	—	4	1	no	—
60	#88142	—	2	2	no	—
61	#88142	—	4	1	no	—
62	C₂×C₁₃×C₁₆	—	2	2	no	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	#88072	—	2	2	no	—
68	#88072	—	2	2	no	—
69	#88058	C₂³	1	1	yes	characteristic
70	#88058	C₂³	1	1	yes	characteristic
71	#88058	C₂³	1	1	yes	characteristic
72	#88058	C₂³	1	1	yes	characteristic
73	#88057	—	2	1	no	—
74	#88057	—	2	1	no	—
75	#88055	D₄	1	2	yes	—
76	C₈×Dic₁₃	C₂³	1	2	yes	—
77	#88052	—	2	2	no	—
78	C₂×#6090	C₂³	1	2	yes	—
79	#88048	—	2	2	no	—
80	#88047	—	2	2	no	—
81	#33631	—	13	1	no	maximal, Sylow
82	C₂⁴×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	2	4	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₁₃	—	4	2	no	—
90	C₂²×Dic₁₃	—	4	2	no	—
91	C₂×Dic₂₆	—	4	2	no	—
92	C₁₃×D₈	—	2	4	no	—
93	C₁₃×D₈	—	2	2	no	—
94	C₁₃×D₈	C₂×Q₈	1	4	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₁₃	#41	1	2	yes	—
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₂²⋊Dic₁₃	—	4	1	no	—
102	C₂²⋊Dic₁₃	—	4	1	no	—
103	C₂²⋊Dic₁₃	—	4	2	no	—
104	C₂²⋊Dic₁₃	—	4	2	no	—
105	C₂²⋊Dic₁₃	D₈	1	2	yes	—
106	C₄⋊Dic₁₃	—	2	1	no	—
107	C₄⋊Dic₁₃	—	2	1	no	—
108	C₄⋊Dic₁₃	—	2	1	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄⋊Dic₁₃	—	4	2	no	—
111	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
112	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	C₄×Dic₁₃	—	4	1	no	—
116	C₄×Dic₁₃	—	4	1	no	—
117	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
118	#6099	—	2	2	no	—
119	C₂×#1055	C₂×D₄	1	2	yes	—
120	C₁₃×C₁₆	—	2	1	no	—
121	C₁₃×C₁₆	—	2	1	no	—
122	#6090	—	2	2	no	—
123	#3164	—	13	1	no	—
124	#2948	—	13	1	no	—
125	#2228	—	13	2	no	—
126	#2228	—	13	2	no	—
127	#2226	—	13	2	no	—
128	#2224	—	13	2	no	—
129	#2159	—	13	1	no	—
130	#2158	—	13	1	no	—
131	#2151	—	13	1	no	—
132	#2140	—	13	1	no	—
133	C₂×#357	D₁₃	1	1	yes	characteristic
134	C₂³×C₁₃	—	4	2	no	—
135	C₂³×C₁₃	—	4	4	no	—
136	C₂³×C₁₃	—	16	1	no	—
137	C₂³×C₁₃	—	16	1	no	—
138	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
139	D₄×C₁₃	—	2	4	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	16	1	no	—
144	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#137	1	2	yes	—
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	—	16	2	no	—
153	C₂×Dic₁₃	—	16	2	no	—
154	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
155	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
156	Dic₂₆	—	16	2	no	—
157	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
158	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
159	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
160	#1055	—	2	2	no	—
161	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
162	#544	—	26	1	no	—
163	#518	—	26	1	no	—
164	C₂×M₃₂	—	13	1	no	—
165	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
166	#501	—	26	2	no	—
167	#500	—	13	2	no	—
168	#499	—	13	2	no	—
169	#442	—	26	2	no	—
170	C₄×D₈	—	13	2	no	—
171	C₄×D₈	—	13	2	no	—
172	#428	—	26	1	no	—
173	#427	—	13	1	no	—
174	#427	—	13	1	no	—
175	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
176	#418	—	26	1	no	—
177	#405	—	13	1	no	—
178	#367	—	13	1	no	—
179	#367	—	13	1	no	—
180	#367	—	13	2	no	—
181	#366	—	13	1	no	—
182	#366	—	13	1	no	—
183	#366	—	13	2	no	—
184	C₄⋊C₁₆	—	13	2	no	—
185	C₄⋊C₁₆	—	13	2	no	—
186	#357	—	13	2	no	—
187	#357	—	13	2	no	—
188	#357	C₂×D₁₃	1	2	yes	—
189	#357	C₂×D₁₃	1	2	yes	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	4	4	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	16	4	no	—
196	C₂²×C₁₃	#474	1	2	yes	—
197	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#474	1	2	yes	—
200	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
201	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
202	Dic₁₃	—	2	2	no	—
203	Dic₁₃	—	2	2	no	—
204	Dic₁₃	—	8	1	no	—
205	Dic₁₃	—	8	1	no	—
206	Dic₁₃	—	8	1	no	—
207	Dic₁₃	—	8	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	Dic₂₆	1	4	yes	complemented
212	C₂×D₈	C₂²×D₁₃	1	2	yes	—
213	C₂×M₁₆	—	13	1	no	—
214	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	26	2	no	—
216	C₄⋊Q₈	—	52	1	no	—
217	#125	—	26	2	no	—
218	#125	—	52	1	no	—
219	C₂²⋊Q₈	—	52	2	no	—
220	C₄×D₄	—	26	2	no	—
221	C₄×D₄	—	26	2	no	—
222	C₄×D₄	—	52	1	no	—
223	C₄×D₄	—	52	1	no	—
224	#117	—	13	1	no	—
225	#117	—	26	1	no	—
226	#117	—	26	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₂²⋊C₄	—	52	2	no	—
229	M₃₂	—	26	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	13	2	no	—
232	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	#107	—	2	1	no	—
235	#107	—	13	1	no	—
236	#107	—	13	1	no	—
237	#107	—	13	1	no	—
238	#107	—	13	1	no	—
239	#107	—	26	2	no	—
240	#107	C₂²×D₁₃	1	2	yes	—
241	#106	—	26	2	no	—
242	#106	—	26	1	no	—
243	#106	—	26	1	no	—
244	#102	—	26	2	no	—
245	#102	—	26	2	no	—
246	#97	—	13	2	no	—
247	C₄×C₈	—	13	2	no	—
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	—	8	4	no	—
250	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
251	C₂×C₁₃	#3415	1	1	yes	characteristic
252	C₂×C₁₃	#3416	1	1	yes	characteristic
253	C₂⁴	—	4	1	no	—
254	C₂×Q₈	—	52	2	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	52	2	no	—
261	C₂²×C₄	—	52	2	no	—
262	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
263	D₈	—	2	4	no	—
264	D₈	—	2	2	no	—
265	D₈	C₂×Dic₂₆	1	4	yes	—
266	M₁₆	—	26	2	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
269	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
270	C₂×C₈	#6129	1	2	yes	—
271	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	13	1	no	—
275	C₄⋊C₄	—	13	1	no	—
276	C₄⋊C₄	—	26	1	no	—
277	C₄⋊C₄	—	26	1	no	—
278	C₄⋊C₄	—	26	1	no	—
279	C₄⋊C₄	—	26	1	no	—
280	C₄⋊C₄	—	52	2	no	—
281	C₄⋊C₄	—	52	2	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₂²⋊C₄	—	13	2	no	—
284	C₂²⋊C₄	—	52	2	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	—	52	1	no	—
287	C₂²⋊C₄	—	52	1	no	—
288	C₄²	—	13	2	no	—
289	C₄²	—	52	1	no	—
290	C₄²	—	52	1	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	26	2	no	—
294	C₁₃	#33631	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	16	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	D₈×D₁₃	1	1	yes	characteristic
300	Q₈	—	208	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	52	1	no	—
309	C₂×C₄	—	52	1	no	—
310	C₂×C₄	—	52	1	no	—
311	C₂×C₄	—	52	1	no	—
312	C₂×C₄	—	208	2	no	—
313	C₂×C₄	—	208	2	no	—
314	C₂×C₄	#88132	1	2	yes	—
315	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
316	C₂×C₄	#88169	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
318	C₈	—	26	2	no	—
319	C₈	#88132	1	2	yes	—
320	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
321	C₈	C₂×#6129	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#11718420	1	2	yes	—
329	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	26	2	no	—
332	C₄	—	26	2	no	—
333	C₄	—	104	1	no	—
334	C₄	—	104	1	no	—
335	C₄	—	104	1	no	—
336	C₄	—	104	1	no	—
337	C₄	#11718420	1	2	yes	—
338	C₄	C₂×#88132	1	1	yes	characteristic
339	C₄	C₂×#88169	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#11718420	1	1	yes	characteristic
343	C₂	#13127973	1	1	yes	characteristic
344	C₂	#13127978	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#357	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×C₁₃×D₈	C₂²
hypercenter	C₂×#357	D₁₃
hypocenter	C₁₃	#33631
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. #13125372	C₂	semidirect
3. #13116995	C₂	semidirect
4. #13116991	C₂	semidirect
5. #13116981	C₂	semidirect
6. #13116971	C₂	semidirect
7. #13116622	C₂	semidirect
8. #13116620	C₂	semidirect
9. #13116594	C₂	semidirect
26. #11718539	C₂²	semidirect
28. #11718535	C₂²	semidirect
33. #11718097	C₂²	semidirect
34. #11718097	C₂²	semidirect
35. #11718096	C₂²	semidirect
36. #11718096	C₂²	semidirect
37. #11718093	C₂²	semidirect
38. #11718093	C₂²	semidirect
39. #11718082	C₂²	semidirect
40. #11718081	C₂²	semidirect
44. C₂×C₁₃×D₈	Q₈	semidirect
211. C₂×D₈	Dic₂₆	semidirect
294. C₁₃	#33631	semidirect

Maximal subgroups

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

subgroup	index
#13125372	2
#13123688	2
#13116995	2
#13116991	2
#13116981	2
#13116971	2
#13116622	2
#13116620	2
#13116594	2
#13116552	2
C₂×C₁₃×#357	2
#33631	13