Subgroups — #15850846

Order 3328.

1050 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125754	C₂	1	2	yes	maximal, complemented
2	#13125415	C₂	1	1	yes	maximal, characteristic, complemented
3	#13125286	C₂	1	2	yes	maximal, complemented
4	#13114501	C₂	1	4	yes	maximal, complemented
5	#13114450	C₂	1	2	yes	maximal, complemented
6	#13114425	C₂	1	2	yes	maximal, complemented
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	—	2	2	no	—
10	C₂×C₁₃×#127	C₂²	1	2	yes	—
11	C₂×#88186	—	2	4	no	—
12	#11719182	C₂²	1	2	yes	—
13	C₁₃×#493	—	2	2	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	#11718878	—	2	4	no	—
18	#11718776	—	2	2	no	—
19	#11718737	C₂²	1	4	yes	complemented
20	#11718735	—	2	4	no	—
21	#11718734	C₂²	1	4	yes	complemented
22	#11718734	C₂²	1	4	yes	complemented
23	#11718733	—	2	4	no	—
24	#11718731	C₂²	1	4	yes	complemented
25	#11718586	C₂²	1	1	yes	characteristic
26	C₄×#6099	C₂²	1	1	yes	characteristic
27	#11718356	C₂²	1	1	yes	characteristic, complemented
28	#11718355	C₂²	1	2	yes	complemented
29	C₄×D₁₀₄	—	2	2	no	—
30	#11718343	C₂²	1	1	yes	characteristic, complemented
31	#11718046	C₂²	1	2	yes	complemented
32	C₈×#1055	C₂²	1	2	yes	complemented
33	C₂²×D₄×C₁₃	—	2	2	no	—
34	C₂²×D₄×C₁₃	—	2	2	no	—
35	C₂²×D₄×C₁₃	—	2	2	no	—
36	C₂×#6127	—	2	2	no	—
37	C₂×C₁₃×D₈	—	2	4	no	—
38	C₂×C₁₃×D₈	—	2	4	no	—
39	C₂×C₁₃×D₈	—	4	8	no	—
40	C₂²×C₈×C₁₃	D₄	1	2	yes	—
41	C₁₃×#127	—	2	2	no	—
42	C₁₃×#127	—	2	2	no	—
43	C₁₃×#127	—	2	2	no	—
44	C₁₃×#127	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×#6105	—	2	8	no	—
47	#88186	—	4	16	no	—
48	C₂×#6104	—	2	8	no	—
49	C₂×#6104	—	2	8	no	—
50	#88184	—	2	2	no	—
51	#88184	—	2	2	no	—
52	C₂×#6099	D₄	1	2	yes	—
53	C₂×D₁₀₄	—	2	2	no	—
54	#88129	—	2	2	no	—
55	#88128	C₂³	1	2	yes	—
56	C₄×D₅₂	—	2	2	no	—
57	C₄×D₅₂	—	2	2	no	—
58	#88123	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	D₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	#88072	—	2	8	no	—
63	C₂²⋊#1055	D₄	1	2	yes	—
64	#88061	—	2	4	no	—
65	#88058	—	2	2	no	—
66	C₄⋊#1055	D₄	1	2	yes	—
67	#88043	D₄	1	2	yes	—
68	C₄×#1055	C₂³	1	2	yes	—
69	C₄×#1055	C₂³	1	2	yes	—
70	#19629	—	13	1	no	maximal, Sylow
71	C₂⁴×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	2	4	no	—
73	C₂×D₄×C₁₃	—	2	4	no	—
74	C₂×D₄×C₁₃	—	2	4	no	—
75	C₂×D₄×C₁₃	—	4	8	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂×D₄×C₁₃	—	4	8	no	—
80	C₂×D₄×C₁₃	—	8	4	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₃	—	4	2	no	—
85	#6127	—	4	8	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×D₅₂	—	2	2	no	—
88	C₂×D₅₂	—	2	2	no	—
89	C₂×C₄×D₁₃	—	4	2	no	—
90	C₂×Dic₂₆	—	2	2	no	—
91	C₁₃×D₈	—	4	8	no	—
92	C₁₃×D₈	—	4	8	no	—
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
100	#6105	—	4	16	no	—
101	#6104	—	4	16	no	—
102	#6104	—	8	8	no	—
103	#6103	—	4	2	no	—
104	#6103	—	4	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	#6099	D₈	1	8	yes	complemented
108	C₂×#1055	C₂×D₄	1	4	yes	—
109	C₂×#1055	C₂×D₄	1	4	yes	—
110	D₁₀₄	—	4	4	no	—
111	#3205	—	13	2	no	—
112	#3169	—	13	1	no	—
113	#3151	—	13	2	no	—
114	C₂×#493	D₁₃	1	1	yes	characteristic, complemented
115	#1721	—	13	4	no	—
116	#1712	—	13	2	no	—
117	#1706	—	13	2	no	—
118	C₈×M₁₆	—	13	1	no	—
119	C₂³×C₁₃	—	4	4	no	—
120	C₂³×C₁₃	—	8	4	no	—
121	C₂³×C₁₃	—	8	4	no	—
122	C₂³×C₁₃	—	8	4	no	—
123	C₂³×C₁₃	#127	1	1	yes	characteristic
124	C₂²×D₁₃	—	4	2	no	—
125	D₄×C₁₃	—	4	8	no	—
126	D₄×C₁₃	—	4	8	no	—
127	D₄×C₁₃	—	4	8	no	—
128	D₄×C₁₃	—	8	4	no	—
129	D₄×C₁₃	—	8	4	no	—
130	D₄×C₁₃	—	8	4	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₁₃	—	8	4	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	D₅₂	—	4	4	no	—
143	D₅₂	—	4	4	no	—
144	D₅₂	—	4	4	no	—
145	C₄×D₁₃	—	8	4	no	—
146	Dic₂₆	—	4	4	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	complemented
149	#1055	—	2	4	no	—
150	#1055	C₂×D₈	1	8	yes	complemented
151	C₂×AGL(1,ℤ/8)	—	26	4	no	—
152	C₂²×D₈	—	2	2	no	—
153	#532	—	13	2	no	—
154	C₂×#127	C₂×D₁₃	1	2	yes	—
155	#500	—	13	1	no	—
156	#496	—	26	4	no	—
157	#493	—	2	2	no	—
158	#493	—	13	4	no	—
159	#493	—	13	4	no	—
160	#493	—	13	1	no	—
161	#493	C₂×D₁₃	1	2	yes	—
162	#493	C₂×D₁₃	1	2	yes	—
163	#492	—	13	4	no	—
164	#486	—	13	4	no	—
165	#486	—	13	2	no	—
166	#466	—	26	2	no	—
167	#459	—	26	4	no	—
168	#447	—	26	4	no	—
169	C₄×D₈	—	26	2	no	—
170	#431	—	13	1	no	—
171	C₄×M₁₆	—	13	1	no	—
172	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
173	#322	—	13	2	no	—
174	C₈²	—	13	2	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	8	4	no	—
178	C₂²×C₁₃	—	8	4	no	—
179	C₂²×C₁₃	—	8	4	no	—
180	C₂²×C₁₃	—	16	8	no	—
181	C₂²×C₁₃	#493	1	2	yes	—
182	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
183	C₂×D₁₃	—	8	2	no	—
184	C₂×D₁₃	—	8	2	no	—
185	C₂×D₁₃	—	8	2	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	#493	1	2	yes	—
189	C₄×C₁₃	C₂²×D₈	1	2	yes	—
190	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
191	C₄×C₁₃	#576	1	2	yes	—
192	Dic₁₃	—	8	2	no	—
193	C₂×#41	—	26	2	no	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	—	2	2	no	—
197	AGL(1,ℤ/8)	—	52	16	no	—
198	C₂×SD₁₆	—	26	8	no	—
199	C₂×D₈	—	2	4	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	4	8	no	—
202	C₂×D₈	—	26	8	no	—
203	C₂×D₈	—	26	8	no	—
204	C₂×D₈	—	26	2	no	—
205	C₂×M₁₆	—	13	2	no	—
206	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
207	C₄⋊Q₈	—	13	2	no	—
208	#127	—	2	2	no	—
209	#127	—	2	2	no	—
210	#127	—	2	2	no	—
211	#127	—	13	2	no	—
212	#127	C₂²×D₁₃	1	4	yes	—
213	#124	—	26	2	no	—
214	#121	—	26	2	no	—
215	#121	—	26	2	no	—
216	C₄×D₄	—	26	2	no	—
217	C₄×D₄	—	26	2	no	—
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	#107	—	26	2	no	—
220	C₄⋊C₈	—	13	2	no	—
221	#102	—	26	8	no	—
222	#102	—	26	4	no	—
223	C₂²⋊C₈	—	13	2	no	—
224	#97	—	13	2	no	—
225	C₄×C₈	—	13	2	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
228	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×C₁₃	—	2	2	no	—
231	C₂×C₁₃	—	16	4	no	—
232	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
233	C₂×C₁₃	#3151	1	1	yes	characteristic
234	C₂×C₁₃	#3155	1	1	yes	characteristic
235	D₁₃	—	16	2	no	—
236	C₂⁴	—	4	2	no	—
237	#41	—	52	8	no	—
238	C₂×Q₈	—	26	2	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	4	8	no	—
243	C₂×D₄	—	4	8	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	8	4	no	—
248	C₂×D₄	—	26	2	no	—
249	C₂×D₄	—	26	2	no	—
250	C₂×D₄	—	26	2	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂²×C₄	—	52	2	no	—
253	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	SD₁₆	—	52	16	no	—
257	D₈	—	4	8	no	—
258	D₈	—	4	8	no	—
259	D₈	—	52	16	no	—
260	D₈	—	52	4	no	—
261	D₈	—	104	8	no	—
262	M₁₆	—	13	8	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	13	4	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
267	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₂²⋊C₄	—	52	2	no	—
271	C₂²⋊C₄	—	52	2	no	—
272	C₄²	D₄×D₁₃	1	1	yes	characteristic
273	C₄²	D₄×D₁₃	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₄²	C₂³×D₁₃	1	1	yes	characteristic
276	C₁₃	#19629	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	4	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	52	2	no	—
282	C₂³	#88194	1	1	yes	characteristic
283	Q₈	—	52	4	no	—
284	D₄	—	4	8	no	—
285	D₄	—	4	8	no	—
286	D₄	—	4	8	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	4	no	—
290	D₄	—	52	4	no	—
291	D₄	—	52	4	no	—
292	D₄	—	52	4	no	—
293	D₄	—	104	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	104	4	no	—
298	C₂×C₄	D₈×D₁₃	1	4	yes	—
299	C₂×C₄	#88194	1	1	yes	characteristic
300	C₂×C₄	#88194	1	1	yes	characteristic
301	C₂×C₄	#88194	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	13	8	no	—
307	C₈	—	26	4	no	—
308	C₈	#88186	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	4	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	16	8	no	—
315	C₂²	—	104	2	no	—
316	C₂²	—	104	2	no	—
317	C₂²	—	104	2	no	—
318	C₂²	#11718812	1	2	yes	—
319	C₂²	C₂×#88194	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	104	2	no	—
323	C₄	#11718812	1	2	yes	—
324	C₄	C₂×D₈×D₁₃	1	2	yes	—
325	C₄	#11719462	1	2	yes	—
326	C₄	C₂×#88186	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	16	4	no	—
329	C₂	—	208	2	no	—
330	C₂	C₂×#11718812	1	1	yes	characteristic
331	C₂	#13125286	1	1	yes	characteristic
332	C₂	#13125317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#493	C₂
socle	C₂²×C₁₃	C₂×#127
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	#19629
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #13125754	C₂	semidirect
2. #13125415	C₂	semidirect
3. #13125286	C₂	semidirect
4. #13114501	C₂	semidirect
5. #13114450	C₂	semidirect
6. #13114425	C₂	semidirect
7. C₈×#6099	C₂	semidirect
8. C₂×C₁₃×#493	C₂	semidirect
19. #11718737	C₂²	semidirect
21. #11718734	C₂²	semidirect
22. #11718734	C₂²	semidirect
24. #11718731	C₂²	semidirect
27. #11718356	C₂²	semidirect
28. #11718355	C₂²	semidirect
30. #11718343	C₂²	semidirect
31. #11718046	C₂²	semidirect
32. C₈×#1055	C₂²	semidirect
107. #6099	D₈	semidirect
114. C₂×#493	D₁₃	semidirect
148. C₈×C₁₃	AGL(1,ℤ/8)	semidirect
150. #1055	C₂×D₈	semidirect
276. C₁₃	#19629	semidirect
308. C₈	#88186	semidirect

Maximal subgroups

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

subgroup	index
#13125754	2
#13125415	2
#13125286	2
#13114501	2
#13114450	2
#13114425	2
C₈×#6099	2
C₂×C₁₃×#493	2
#19629	13