Subgroups — #16017609

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	#13125375	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123689	C₂	1	1	yes	maximal, characteristic
3	#13116994	C₂	1	2	yes	maximal, complemented
4	#13116994	C₂	1	2	yes	maximal, complemented
5	#13116984	C₂	1	2	yes	maximal, complemented
6	#13116974	C₂	1	2	yes	maximal, complemented
7	#13116624	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116553	C₂	1	1	yes	maximal, characteristic
10	C₂×#11718101	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	#11719226	—	2	1	no	—
14	#11719224	—	2	1	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	#11718792	—	2	1	no	—
19	#11718782	—	2	1	no	—
20	#11718767	C₂²	1	1	yes	characteristic
21	#11718767	C₂²	1	1	yes	characteristic
22	C₂×#88058	C₂²	1	1	yes	characteristic
23	#11718372	—	2	2	no	—
24	#11718363	—	2	2	no	—
25	C₄×D₁₀₄	C₂²	1	2	yes	—
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	#11718344	C₂²	1	2	yes	complemented
28	#11718343	C₂²	1	2	yes	complemented
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
31	#11718101	C₂²	1	2	yes	—
32	#11718101	C₂²	1	2	yes	—
33	#11718101	C₂²	1	2	yes	—
34	#11718101	C₂²	1	2	yes	—
35	#11718097	C₂²	1	1	yes	characteristic, complemented
36	#11718097	C₂²	1	2	yes	complemented
37	#11718097	C₂²	1	1	yes	characteristic, complemented
38	#11718096	C₂²	1	1	yes	characteristic, complemented
39	#11718096	C₂²	1	2	yes	complemented
40	#11718096	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₅₂	—	2	1	no	—
42	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#117	D₄	1	1	yes	characteristic
45	#88180	—	2	1	no	—
46	#88180	—	2	1	no	—
47	C₂×C₄⋊Dic₁₃	—	2	1	no	—
48	C₂×D₁₀₄	—	2	1	no	—
49	C₂×D₁₀₄	—	2	2	no	—
50	C₂×D₁₀₄	Q₈	1	4	yes	complemented
51	C₂×D₁₀₄	C₂³	1	2	yes	—
52	#88147	—	4	1	no	—
53	#88147	—	4	1	no	—
54	#88144	—	4	1	no	—
55	#88142	—	4	1	no	—
56	C₂²⋊C₄×D₁₃	—	4	2	no	—
57	#88132	—	4	2	no	—
58	C₄×D₅₂	—	2	2	no	—
59	C₄×D₅₂	—	2	2	no	—
60	#88124	—	2	2	no	—
61	#88123	—	2	2	no	—
62	C₁₃×M₃₂	—	2	2	no	—
63	C₂×C₁₃×C₁₆	—	2	2	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
67	C₁₃×#97	D₄	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	#88061	—	2	2	no	—
70	#88061	—	2	2	no	—
71	#88058	—	2	1	no	—
72	#88058	—	2	2	no	—
73	#88058	C₂³	1	2	yes	—
74	#88058	C₂³	1	1	yes	characteristic
75	#88058	C₂³	1	1	yes	characteristic
76	#88058	C₂³	1	1	yes	characteristic
77	#88058	C₂³	1	1	yes	characteristic
78	#88057	—	2	1	no	—
79	#88057	—	2	1	no	—
80	#88057	—	2	2	no	—
81	#33631	—	13	1	no	maximal, Sylow
82	C₂³×D₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₁₃	—	4	1	no	—
85	C₂×D₅₂	—	2	4	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×D₅₂	—	4	1	no	—
88	C₂×D₅₂	—	4	2	no	—
89	C₂×C₄×D₁₃	—	4	2	no	—
90	C₂×C₄×D₁₃	—	4	2	no	—
91	C₂×Dic₂₆	—	4	2	no	—
92	C₁₃×M₁₆	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₁₃	#41	1	2	yes	—
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
101	C₄²×C₁₃	C₂×D₄	1	2	yes	—
102	C₂²⋊Dic₁₃	—	4	1	no	—
103	C₂²⋊Dic₁₃	—	4	1	no	—
104	#6103	—	4	2	no	—
105	#6103	—	4	2	no	—
106	C₄⋊Dic₁₃	—	2	1	no	—
107	C₄⋊Dic₁₃	—	2	2	no	—
108	C₄⋊Dic₁₃	—	2	1	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄⋊Dic₁₃	—	2	1	no	—
111	C₄⋊Dic₁₃	—	4	2	no	—
112	C₄⋊Dic₁₃	—	4	1	no	—
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	C₄×Dic₁₃	—	4	1	no	—
116	C₄×Dic₁₃	—	4	1	no	—
117	D₁₀₄	—	2	4	no	—
118	D₁₀₄	—	2	2	no	—
119	D₁₀₄	C₂×Q₈	1	4	yes	—
120	C₁₃×C₁₆	—	2	1	no	—
121	C₁₃×C₁₆	—	2	1	no	—
122	C₁₃×C₁₆	—	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	D₁₃	1	1	yes	characteristic, complemented
132	#2140	—	13	1	no	—
133	C₂×#357	—	13	1	no	—
134	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₁₃	—	4	2	no	—
136	C₂²×D₁₃	—	4	4	no	—
137	C₂²×D₁₃	—	16	1	no	—
138	C₂²×D₁₃	—	16	1	no	—
139	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#137	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₁₃	—	4	1	no	—
146	C₂×Dic₁₃	—	4	1	no	—
147	C₂×Dic₁₃	—	4	1	no	—
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	16	1	no	—
151	D₅₂	—	2	4	no	—
152	D₅₂	—	4	2	no	—
153	D₅₂	—	4	4	no	—
154	C₄×D₁₃	—	16	2	no	—
155	C₄×D₁₃	—	16	2	no	—
156	Dic₂₆	—	16	2	no	—
157	C₈×C₁₃	—	2	2	no	—
158	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
159	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
160	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₂²×D₈	—	13	1	no	—
162	#544	—	26	1	no	—
163	#518	—	26	1	no	—
164	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
176	#418	—	26	1	no	—
177	#405	C₂×D₁₃	1	1	yes	characteristic
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	C₂×D₁₃	1	2	yes	complemented
185	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	complemented
186	#357	—	13	2	no	—
187	#357	—	13	2	no	—
188	#357	—	13	2	no	—
189	#357	—	13	2	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	#474	1	2	yes	—
193	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
194	C₂×D₁₃	—	4	4	no	—
195	C₂×D₁₃	—	8	2	no	—
196	C₂×D₁₃	—	8	2	no	—
197	C₂×D₁₃	—	16	4	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	#474	1	2	yes	—
201	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
202	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
203	Dic₁₃	—	8	1	no	—
204	Dic₁₃	—	8	2	no	—
205	Dic₁₃	—	8	1	no	—
206	Dic₁₃	—	8	1	no	—
207	Dic₁₃	—	8	1	no	—
208	C₂²×D₄	—	26	1	no	—
209	C₂×D₈	—	13	4	no	—
210	C₂×D₈	—	13	2	no	—
211	C₂×D₈	—	26	1	no	—
212	C₂×D₈	—	26	2	no	—
213	C₂×M₁₆	D₅₂	1	1	yes	characteristic
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	—	26	1	no	—
225	#117	—	26	1	no	—
226	#117	D₅₂	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	26	1	no	—
228	C₂×C₂²⋊C₄	—	52	2	no	—
229	M₃₂	—	2	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
233	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	#107	—	13	2	no	—
235	#107	—	13	1	no	—
236	#107	—	13	1	no	—
237	#107	—	13	1	no	—
238	#107	—	13	1	no	—
239	#107	—	26	1	no	—
240	#107	—	26	2	no	—
241	#106	—	26	1	no	—
242	#106	—	26	1	no	—
243	#106	—	26	2	no	—
244	#102	—	26	2	no	—
245	#102	—	26	2	no	—
246	#97	D₅₂	1	2	yes	—
247	C₄×C₈	C₂²×D₁₃	1	2	yes	—
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
250	C₂×C₁₃	#3415	1	1	yes	characteristic
251	C₂×C₁₃	#3416	1	1	yes	characteristic
252	D₁₃	—	8	4	no	—
253	C₂⁴	—	52	1	no	—
254	C₂×Q₈	—	52	2	no	—
255	C₂×D₄	—	26	4	no	—
256	C₂×D₄	—	26	2	no	—
257	C₂×D₄	—	52	1	no	—
258	C₂×D₄	—	52	2	no	—
259	C₂²×C₄	—	52	1	no	—
260	C₂²×C₄	—	52	2	no	—
261	C₂²×C₄	—	52	2	no	—
262	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
263	D₈	—	13	4	no	—
264	D₈	—	26	4	no	—
265	D₈	—	26	2	no	—
266	M₁₆	—	2	2	no	—
267	C₂×C₈	C₂×D₅₂	1	2	yes	—
268	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
269	C₂×C₈	#6129	1	2	yes	—
270	C₂×C₈	Q₈×D₁₃	1	2	yes	—
271	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
272	C₄⋊C₄	—	26	1	no	—
273	C₄⋊C₄	—	26	1	no	—
274	C₄⋊C₄	—	26	1	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	26	1	no	—
277	C₄⋊C₄	—	52	1	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	2	no	—
280	C₄⋊C₄	—	52	2	no	—
281	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
282	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
283	C₂²⋊C₄	—	52	1	no	—
284	C₂²⋊C₄	—	52	1	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	—	52	2	no	—
287	C₂²⋊C₄	D₁₀₄	1	2	yes	—
288	C₄²	—	52	1	no	—
289	C₄²	—	52	1	no	—
290	C₄²	C₂×D₅₂	1	2	yes	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	2	no	—
294	C₁₃	#33631	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	52	2	no	—
296	C₂³	—	52	4	no	—
297	C₂³	—	208	1	no	—
298	C₂³	—	208	1	no	—
299	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
300	Q₈	—	208	2	no	—
301	D₄	—	26	4	no	—
302	D₄	—	52	2	no	—
303	D₄	—	52	4	no	—
304	C₂×C₄	—	52	1	no	—
305	C₂×C₄	—	52	1	no	—
306	C₂×C₄	—	52	1	no	—
307	C₂×C₄	—	52	1	no	—
308	C₂×C₄	—	52	2	no	—
309	C₂×C₄	—	208	2	no	—
310	C₂×C₄	—	208	2	no	—
311	C₂×C₄	—	208	1	no	—
312	C₂×C₄	#88151	1	2	yes	—
313	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
314	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
315	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
316	C₂×C₄	#88163	1	2	yes	—
317	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#88151	1	2	yes	—
320	C₈	C₂×#6129	1	1	yes	characteristic
321	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	52	4	no	—
325	C₂²	—	104	2	no	—
326	C₂²	—	104	2	no	—
327	C₂²	—	208	4	no	—
328	C₂²	#11718507	1	2	yes	—
329	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	104	1	no	—
333	C₄	—	104	2	no	—
334	C₄	—	104	1	no	—
335	C₄	—	104	1	no	—
336	C₄	—	104	1	no	—
337	C₄	#11718507	1	2	yes	—
338	C₄	C₂×#88151	1	1	yes	characteristic
339	C₄	C₂×#88163	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	104	4	no	—
342	C₂	C₂×#11718507	1	1	yes	characteristic
343	C₂	#13127977	1	1	yes	characteristic
344	C₂	#13127984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2151	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	#2151	D₁₃
hypocenter	C₁₃	#33631
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

Complements

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

N	H ≅ G/N	type
1. #13125375	C₂	semidirect
3. #13116994	C₂	semidirect
4. #13116994	C₂	semidirect
5. #13116984	C₂	semidirect
6. #13116974	C₂	semidirect
7. #13116624	C₂	semidirect
8. #13116620	C₂	semidirect
11. C₁₃×#2151	C₂	semidirect
27. #11718344	C₂²	semidirect
28. #11718343	C₂²	semidirect
29. C₁₃×C₄⋊C₁₆	C₂²	semidirect
30. C₁₃×C₄⋊C₁₆	C₂²	semidirect
35. #11718097	C₂²	semidirect
36. #11718097	C₂²	semidirect
37. #11718097	C₂²	semidirect
38. #11718096	C₂²	semidirect
39. #11718096	C₂²	semidirect
40. #11718096	C₂²	semidirect
50. C₂×D₁₀₄	Q₈	semidirect
131. #2151	D₁₃	semidirect
184. C₄⋊C₁₆	C₂×D₁₃	semidirect
185. C₄⋊C₁₆	C₂×D₁₃	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
#13125375	2
#13123689	2
#13116994	2
#13116984	2
#13116974	2
#13116624	2
#13116620	2
#13116553	2
C₂×#11718101	2
C₁₃×#2151	2
#33631	13