Subgroups — #16018838

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117091	C₂	1	2	yes	maximal, complemented
3	#13117059	C₂	1	2	yes	maximal, complemented
4	#13116873	C₂	1	4	yes	maximal, complemented
5	#13116765	C₂	1	4	yes	maximal, complemented
6	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#127	—	2	1	no	—
11	#11719175	—	2	1	no	—
12	C₁₃×#493	—	2	1	no	—
13	C₁₃×#493	—	2	1	no	—
14	C₁₃×#493	C₂²	1	2	yes	—
15	C₁₃×#493	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88067	C₂²	1	8	yes	complemented
18	#11718777	—	2	1	no	—
19	#11718777	C₂²	1	2	yes	—
20	#11718360	C₂²	1	1	yes	characteristic, complemented
21	C₄×Dic₅₂	—	2	1	no	—
22	C₄×Dic₅₂	C₂²	1	2	yes	complemented
23	#11718343	C₂²	1	1	yes	characteristic, complemented
24	#11718342	—	2	1	no	—
25	#11718154	C₂²	1	8	yes	complemented
26	C₂²⋊#6090	C₂²	1	2	yes	—
27	C₄⋊#6090	C₂²	1	2	yes	complemented
28	C₄×#6090	C₂²	1	2	yes	complemented
29	C₂²×D₄×C₁₃	—	2	2	no	—
30	C₂²×D₄×C₁₃	—	4	1	no	—
31	C₂²×D₄×C₁₃	—	4	1	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	—	4	4	no	—
36	C₂×C₁₃×D₈	—	4	4	no	—
37	C₂×C₁₃×D₈	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×#127	—	2	2	no	—
41	C₁₃×#127	—	2	2	no	—
42	C₁₃×#127	—	4	1	no	—
43	C₁₃×#127	—	4	1	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	#88180	—	4	1	no	—
46	#88178	—	2	2	no	—
47	#88178	—	4	1	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	C₂³	1	2	yes	—
50	#88124	—	4	1	no	—
51	#88123	—	2	1	no	—
52	#88123	—	2	1	no	—
53	C₄×Dic₂₆	—	2	2	no	—
54	C₄×Dic₂₆	—	4	1	no	—
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	#88067	—	4	8	no	—
60	#88058	—	2	1	no	—
61	#88058	C₂³	1	2	yes	—
62	#88056	—	2	2	no	—
63	#88056	—	2	2	no	—
64	C₂×#6090	C₂³	1	4	yes	—
65	#33774	—	13	1	no	maximal, Sylow
66	C₂⁴×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	2	4	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	4	4	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	—	8	4	no	—
76	C₂×D₄×C₁₃	—	8	4	no	—
77	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×Dic₂₆	—	4	1	no	—
82	C₁₃×D₈	—	2	8	no	—
83	C₁₃×D₈	—	4	4	no	—
84	C₁₃×D₈	—	4	4	no	—
85	C₁₃×D₈	—	4	4	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₄²×C₁₃	D₈	1	1	yes	characteristic
93	C₄²×C₁₃	D₈	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₁₃	—	4	2	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	#6101	—	4	2	no	—
102	#6101	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	Dic₅₂	—	2	4	no	—
105	Dic₅₂	—	4	2	no	—
106	#6090	—	2	4	no	—
107	#3170	—	13	1	no	—
108	C₂×#493	D₁₃	1	1	yes	characteristic
109	#2248	—	13	2	no	—
110	#2240	—	13	2	no	—
111	#2207	—	13	4	no	—
112	#2187	—	13	4	no	—
113	#2162	—	13	1	no	—
114	C₂³×C₁₃	—	4	4	no	—
115	C₂³×C₁₃	—	8	4	no	—
116	C₂³×C₁₃	—	8	4	no	—
117	C₂³×C₁₃	—	8	4	no	—
118	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
119	D₄×C₁₃	—	4	4	no	—
120	D₄×C₁₃	—	4	4	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	8	4	no	—
123	D₄×C₁₃	—	8	4	no	—
124	D₄×C₁₃	—	8	4	no	—
125	D₄×C₁₃	—	8	4	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	Dic₂₆	—	4	2	no	—
139	Dic₂₆	—	4	2	no	—
140	Dic₂₆	—	8	2	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
145	C₂²×D₈	—	2	1	no	—
146	C₂²×D₈	C₂×D₁₃	1	2	yes	—
147	#533	—	26	1	no	—
148	C₂×#127	—	2	1	no	—
149	C₂×SD₃₂	—	13	8	no	—
150	#500	—	13	1	no	—
151	#494	—	13	1	no	—
152	#493	—	2	1	no	—
153	#493	—	2	1	no	—
154	#493	C₂×D₁₃	1	2	yes	—
155	#493	C₂×D₁₃	1	2	yes	—
156	#487	—	26	1	no	—
157	#467	—	13	2	no	—
158	#467	—	26	1	no	—
159	C₄×Q₁₆	—	13	2	no	—
160	C₄×Q₁₆	—	26	1	no	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₄⋊C₁₆	—	13	2	no	—
163	#357	—	13	8	no	—
164	C₂²⋊C₁₆	—	13	2	no	—
165	C₄×C₁₆	—	13	2	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	8	4	no	—
169	C₂²×C₁₃	—	8	4	no	—
170	C₂²×C₁₃	—	8	4	no	—
171	C₂²×C₁₃	—	16	8	no	—
172	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
173	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
177	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
178	C₄×C₁₃	#509	1	2	yes	—
179	C₄×C₁₃	#535	1	1	yes	characteristic
180	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	Dic₁₃	—	8	2	no	—
182	Dic₁₃	—	8	2	no	—
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	4	1	no	—
185	C₂²×D₄	—	4	1	no	—
186	C₂×Q₁₆	—	13	2	no	—
187	C₂×Q₁₆	—	26	1	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	4	4	no	—
192	C₂×D₈	—	4	4	no	—
193	C₂×D₈	C₂²×D₁₃	1	4	yes	—
194	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	26	1	no	—
197	C₄⋊Q₈	—	26	1	no	—
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	—	4	1	no	—
201	#127	—	4	1	no	—
202	#125	—	52	1	no	—
203	C₂²⋊Q₈	—	26	2	no	—
204	C₂²⋊Q₈	—	52	1	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	52	1	no	—
207	#117	—	52	1	no	—
208	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
209	SD₃₂	—	52	8	no	—
210	C₂×C₁₆	—	13	4	no	—
211	#107	—	13	2	no	—
212	#107	—	26	1	no	—
213	#103	—	26	2	no	—
214	#103	—	26	2	no	—
215	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	16	4	no	—
221	C₂×C₁₃	#3205	1	1	yes	characteristic
222	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
224	C₂⁴	—	4	2	no	—
225	C₂×Q₈	—	26	2	no	—
226	C₂×Q₈	—	52	1	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	8	4	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂²×C₄	#6104	1	1	yes	characteristic
238	C₂²×C₄	#6104	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
240	Q₁₆	—	26	4	no	—
241	Q₁₆	—	52	2	no	—
242	D₈	—	2	8	no	—
243	D₈	—	4	4	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	26	2	no	—
253	C₄⋊C₄	—	52	1	no	—
254	C₄⋊C₄	—	52	1	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	1	no	—
258	C₂²⋊C₄	—	52	2	no	—
259	C₄²	—	52	2	no	—
260	C₄²	#6104	1	1	yes	characteristic
261	C₄²	#6104	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₁₆	—	26	4	no	—
265	C₁₃	#33774	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	4	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	—	8	4	no	—
270	C₂³	C₂×#6104	1	1	yes	characteristic
271	Q₈	—	52	2	no	—
272	Q₈	—	52	2	no	—
273	Q₈	—	104	2	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	52	2	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	#88067	1	4	yes	complemented
287	C₂×C₄	C₂×#6104	1	1	yes	characteristic
288	C₂×C₄	C₂×#6104	1	1	yes	characteristic
289	C₂×C₄	C₂×#6104	1	1	yes	characteristic
290	C₂×C₄	C₂×#6104	1	1	yes	characteristic
291	C₂×C₄	C₂×#6104	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#88186	1	2	yes	—
296	C₈	#88251	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×#88067	1	2	yes	—
304	C₂²	C₂²×#6104	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	104	2	no	—
308	C₄	—	104	2	no	—
309	C₄	#11718808	1	2	yes	—
310	C₄	C₂×#88067	1	2	yes	—
311	C₄	C₂×#88067	1	2	yes	—
312	C₄	C₂×#88186	1	1	yes	characteristic
313	C₄	#11719510	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	16	4	no	—
316	C₂	#13125754	1	1	yes	characteristic
317	C₂	C₂²×#88067	1	1	yes	characteristic
318	C₂	C₂×#11718808	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	C₂×C₁₃×#493	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	#33774
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #13125422	C₂	semidirect
2. #13117091	C₂	semidirect
3. #13117059	C₂	semidirect
4. #13116873	C₂	semidirect
5. #13116765	C₂	semidirect
6. #13116635	C₂	semidirect
17. C₂×#88067	C₂²	semidirect
20. #11718360	C₂²	semidirect
22. C₄×Dic₅₂	C₂²	semidirect
23. #11718343	C₂²	semidirect
25. #11718154	C₂²	semidirect
27. C₄⋊#6090	C₂²	semidirect
28. C₄×#6090	C₂²	semidirect
129. C₂×C₄×C₁₃	SD₃₂	semidirect
265. C₁₃	#33774	semidirect
286. C₂×C₄	#88067	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13117091	2
#13117059	2
#13116873	2
#13116765	2
#13116635	2
C₂×C₁₃×#493	2
#33774	13