Subgroups — #16018096

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125495	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123689	C₂	1	1	yes	maximal, characteristic
3	#13117045	C₂	1	2	yes	maximal, complemented
4	#13116994	C₂	1	2	yes	maximal, complemented
5	#13116882	C₂	1	2	yes	maximal, complemented
6	#13116767	C₂	1	2	yes	maximal
7	#13116702	C₂	1	2	yes	maximal, complemented
8	C₄×#88040	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
11	#11719226	—	2	1	no	—
12	#11719223	—	2	1	no	—
13	C₁₃×#405	C₂²	1	1	yes	characteristic
14	#11718798	—	2	1	no	—
15	#11718792	—	2	1	no	—
16	#11718777	C₂²	1	2	yes	—
17	#11718767	C₂²	1	1	yes	characteristic
18	C₂×#88040	C₂²	1	2	yes	—
19	C₂×#88040	C₂²	1	2	yes	—
20	#11718377	—	2	1	no	—
21	#11718372	—	2	1	no	—
22	#11718372	—	2	1	no	—
23	#11718364	—	2	1	no	—
24	#11718360	C₂²	1	1	yes	characteristic, complemented
25	C₄×Dic₅₂	C₂²	1	2	yes	complemented
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	C₄×D₁₀₄	C₂²	1	2	yes	—
28	#11718344	C₂²	1	1	yes	characteristic, complemented
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#346	C₂²	1	1	yes	characteristic, complemented
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
33	#11718101	C₂²	1	2	yes	—
34	#11718101	C₂²	1	2	yes	—
35	#11718097	C₂²	1	2	yes	complemented
36	#11718096	C₂²	1	2	yes	complemented
37	#11718095	C₂²	1	2	yes	complemented
38	#11718095	C₂²	1	2	yes	complemented
39	C₂²×D₅₂	—	2	1	no	—
40	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	D₄	1	1	yes	characteristic
43	#88180	—	2	1	no	—
44	#88178	—	2	2	no	—
45	C₂×Dic₅₂	—	2	1	no	—
46	C₂×Dic₅₂	C₂³	1	2	yes	—
47	C₂×D₁₀₄	—	2	1	no	—
48	C₂×D₁₀₄	—	2	4	no	—
49	C₂×D₁₀₄	C₂³	1	2	yes	—
50	#88147	—	4	2	no	—
51	#88143	—	4	2	no	—
52	C₂²⋊C₄×D₁₃	—	4	2	no	—
53	#88133	—	4	2	no	—
54	C₄×D₅₂	—	2	2	no	—
55	C₄×D₅₂	—	2	2	no	—
56	#88124	—	2	1	no	—
57	#88123	—	2	1	no	—
58	C₄×Dic₂₆	—	2	2	no	—
59	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
61	C₁₃×#97	D₄	1	1	yes	characteristic
62	C₁₃×#97	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#88061	—	2	2	no	—
66	#88061	—	2	2	no	—
67	#88058	C₂³	1	2	yes	—
68	#88058	C₂³	1	2	yes	—
69	#88057	—	2	1	no	—
70	#88057	—	2	1	no	—
71	#88056	—	2	2	no	—
72	#88056	—	2	2	no	—
73	#88040	—	2	4	no	—
74	#88040	—	2	4	no	—
75	#33683	—	13	1	no	maximal, Sylow
76	C₂³×D₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×D₅₂	—	2	4	no	—
79	C₂×D₅₂	—	2	2	no	—
80	C₂×D₅₂	—	4	2	no	—
81	C₂×D₅₂	—	4	1	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×C₄×D₁₃	—	4	2	no	—
84	C₂×Dic₂₆	—	2	2	no	—
85	C₁₃×M₁₆	—	4	1	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
92	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₁₃	—	4	2	no	—
96	C₂²⋊Dic₁₃	—	4	1	no	—
97	#6103	—	4	2	no	—
98	#6103	—	4	2	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	4	2	no	—
102	#6101	—	4	2	no	—
103	#6101	—	4	2	no	—
104	#6101	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	—	4	1	no	—
107	Dic₅₂	—	2	2	no	—
108	Dic₅₂	—	2	2	no	—
109	Dic₅₂	—	4	2	no	—
110	D₁₀₄	—	2	4	no	—
111	D₁₀₄	—	2	2	no	—
112	D₁₀₄	#41	1	4	yes	—
113	C₁₃×C₁₆	—	2	2	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	#3177	—	13	1	no	—
116	#2948	—	13	1	no	—
117	#2237	—	13	2	no	—
118	#2228	—	13	2	no	—
119	#2208	—	13	2	no	—
120	#2187	—	13	2	no	—
121	#2175	—	13	2	no	—
122	C₄×SD₃₂	—	13	2	no	—
123	#2164	D₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₁₃	—	4	2	no	—
126	C₂²×D₁₃	—	4	4	no	—
127	C₂²×D₁₃	—	16	2	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	C₂×Dic₁₃	—	4	2	no	—
137	D₅₂	—	2	4	no	—
138	D₅₂	—	4	4	no	—
139	D₅₂	—	4	2	no	—
140	C₄×D₁₃	—	16	2	no	—
141	C₄×D₁₃	—	16	2	no	—
142	Dic₂₆	—	4	2	no	—
143	Dic₂₆	—	4	2	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	2	yes	—
148	C₂²×D₈	—	13	1	no	—
149	#541	—	26	1	no	—
150	#518	—	26	1	no	—
151	C₂×SD₃₂	—	13	2	no	—
152	C₂×SD₃₂	—	13	2	no	—
153	#499	—	13	1	no	—
154	#494	—	13	1	no	—
155	#491	—	26	1	no	—
156	#470	—	26	1	no	—
157	#467	—	13	2	no	—
158	#442	—	26	1	no	—
159	#442	—	26	1	no	—
160	#441	—	26	1	no	—
161	C₄×Q₁₆	—	13	2	no	—
162	C₄×D₈	—	13	2	no	—
163	C₄×D₈	—	13	2	no	—
164	#427	—	13	1	no	—
165	#418	—	26	1	no	—
166	#405	C₂×D₁₃	1	1	yes	characteristic
167	#367	—	13	2	no	—
168	#366	—	13	2	no	—
169	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	complemented
170	#358	—	13	2	no	—
171	#358	—	13	2	no	—
172	#357	—	13	2	no	—
173	#357	—	13	2	no	—
174	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
175	#346	C₂×D₁₃	1	1	yes	characteristic, complemented
176	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₂×D₁₃	—	4	4	no	—
181	C₂×D₁₃	—	8	2	no	—
182	C₂×D₁₃	—	8	2	no	—
183	C₂×D₁₃	—	16	4	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	#525	1	1	yes	characteristic
188	C₄×C₁₃	#575	1	1	yes	characteristic
189	Dic₁₃	—	8	2	no	—
190	Dic₁₃	—	8	2	no	—
191	Dic₁₃	—	8	2	no	—
192	C₂²×D₄	—	26	1	no	—
193	C₂×Q₁₆	—	13	2	no	—
194	C₂×Q₁₆	—	26	1	no	—
195	C₂×D₈	—	13	2	no	—
196	C₂×D₈	—	26	4	no	—
197	C₂×D₈	—	26	1	no	—
198	C₂×M₁₆	D₅₂	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	26	1	no	—
201	#126	—	52	2	no	—
202	#125	—	26	1	no	—
203	#125	—	52	2	no	—
204	C₂²⋊Q₈	—	26	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×D₄	—	26	2	no	—
207	C₄×D₄	—	26	2	no	—
208	C₄×D₄	—	52	2	no	—
209	#117	—	26	1	no	—
210	#117	D₅₂	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	52	2	no	—
212	SD₃₂	—	26	4	no	—
213	SD₃₂	—	26	4	no	—
214	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
215	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
216	#107	—	13	2	no	—
217	#107	—	13	2	no	—
218	#106	—	26	1	no	—
219	#106	—	26	1	no	—
220	#103	—	26	2	no	—
221	#103	—	26	2	no	—
222	#102	—	26	2	no	—
223	#102	—	26	2	no	—
224	#97	D₅₂	1	1	yes	characteristic
225	#97	D₅₂	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₁₃	—	4	1	no	—
229	C₂×C₁₃	#3100	1	1	yes	characteristic
230	C₂×C₁₃	#3416	1	2	yes	—
231	D₁₃	—	8	4	no	—
232	C₂⁴	—	52	1	no	—
233	C₂×Q₈	—	26	2	no	—
234	C₂×D₄	—	26	4	no	—
235	C₂×D₄	—	26	2	no	—
236	C₂×D₄	—	52	2	no	—
237	C₂×D₄	—	52	1	no	—
238	C₂²×C₄	—	52	2	no	—
239	C₂²×C₄	—	52	2	no	—
240	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
241	Q₁₆	—	26	2	no	—
242	Q₁₆	—	26	2	no	—
243	Q₁₆	—	52	2	no	—
244	D₈	—	13	4	no	—
245	D₈	—	26	4	no	—
246	D₈	—	26	2	no	—
247	M₁₆	—	4	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
250	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
251	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	26	2	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	D₁₀₄	1	2	yes	—
260	C₂²⋊C₄	—	52	1	no	—
261	C₂²⋊C₄	—	52	2	no	—
262	C₂²⋊C₄	—	52	2	no	—
263	C₂²⋊C₄	—	52	2	no	—
264	C₂²⋊C₄	D₁₀₄	1	2	yes	—
265	C₄²	—	52	1	no	—
266	C₄²	—	52	2	no	—
267	C₄²	C₂×D₅₂	1	1	yes	characteristic
268	C₄²	C₂×D₅₂	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₃	#33683	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	52	4	no	—
273	C₂³	—	52	2	no	—
274	C₂³	—	208	2	no	—
275	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
276	Q₈	—	52	2	no	—
277	Q₈	—	52	2	no	—
278	D₄	—	26	4	no	—
279	D₄	—	52	4	no	—
280	D₄	—	52	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	52	2	no	—
284	C₂×C₄	—	52	2	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	—	208	2	no	—
287	C₂×C₄	—	208	2	no	—
288	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
289	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
290	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
291	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#41×D₁₃	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	52	4	no	—
299	C₂²	—	104	2	no	—
300	C₂²	—	104	2	no	—
301	C₂²	—	208	4	no	—
302	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	104	2	no	—
307	C₄	—	104	2	no	—
308	C₄	—	104	2	no	—
309	C₄	#11719229	1	1	yes	characteristic
310	C₄	#11719453	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	104	4	no	—
313	C₂	#13124837	1	1	yes	characteristic
314	C₂	#13127984	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2164	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	#2164	D₁₃
hypocenter	C₁₃	#33683
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2164

Complements

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

N	H ≅ G/N	type
1. #13125495	C₂	semidirect
3. #13117045	C₂	semidirect
4. #13116994	C₂	semidirect
5. #13116882	C₂	semidirect
7. #13116702	C₂	semidirect
8. C₄×#88040	C₂	semidirect
9. C₁₃×#2164	C₂	semidirect
24. #11718360	C₂²	semidirect
25. C₄×Dic₅₂	C₂²	semidirect
28. #11718344	C₂²	semidirect
29. C₁₃×C₄⋊C₁₆	C₂²	semidirect
31. C₁₃×#346	C₂²	semidirect
32. C₄×C₁₃×C₁₆	C₂²	semidirect
35. #11718097	C₂²	semidirect
36. #11718096	C₂²	semidirect
37. #11718095	C₂²	semidirect
38. #11718095	C₂²	semidirect
123. #2164	D₁₃	semidirect
169. C₄⋊C₁₆	C₂×D₁₃	semidirect
175. #346	C₂×D₁₃	semidirect
176. C₄×C₁₆	C₂×D₁₃	semidirect
271. C₁₃	#33683	semidirect

Maximal subgroups

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

subgroup	index
#13125495	2
#13123689	2
#13117045	2
#13116994	2
#13116882	2
#13116767	2
#13116702	2
C₄×#88040	2
C₁₃×#2164	2
#33683	13