Subgroups — #13124457

Order 1664.

548 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	C₁₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88203	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6099	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718893	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718710	C₂	1	2	yes	maximal, complemented
7	#11718709	C₂	1	2	yes	maximal, complemented
8	#11718700	C₂	1	2	yes	maximal, complemented
9	#11718699	C₂	1	2	yes	maximal, complemented
10	#11718662	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718656	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₃	—	2	1	no	—
15	C₂²×D₅₂	C₂²	1	1	yes	characteristic
16	C₁₃×#123	—	2	1	no	—
17	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#121	C₂²	1	1	yes	characteristic, complemented
20	C₁₃×#121	C₂²	1	1	yes	characteristic, complemented
21	C₁₃×#120	—	2	1	no	—
22	C₄×D₄×C₁₃	—	2	1	no	—
23	C₄×D₄×C₁₃	—	2	1	no	—
24	C₁₃×#117	C₂²	1	1	yes	characteristic
25	#88203	—	2	4	no	—
26	C₂×#6106	C₂²	1	2	yes	complemented
27	C₂×#6104	C₂²	1	2	yes	complemented
28	#88184	—	2	1	no	—
29	#88184	—	2	1	no	—
30	#88184	C₂²	1	2	yes	—
31	#88184	C₂²	1	2	yes	—
32	C₂×#6103	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
34	C₂×#6099	—	2	4	no	—
35	C₂×#6099	C₂²	1	1	yes	characteristic
36	C₂×#6099	C₂²	1	1	yes	characteristic
37	C₂×#6099	C₂²	1	2	yes	complemented
38	C₂²×#1055	C₂²	1	2	yes	complemented
39	#88075	C₂²	1	2	yes	complemented
40	#88072	C₂²	1	2	yes	complemented
41	#88049	C₂²	1	2	yes	complemented
42	#88049	C₂²	1	2	yes	complemented
43	#88048	C₂²	1	2	yes	complemented
44	#88047	C₂²	1	2	yes	complemented
45	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
46	C₂³×D₁₃	—	2	1	no	—
47	C₁₃×#41	—	2	2	no	—
48	C₁₃×#41	—	2	2	no	—
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	—	2	1	no	—
52	C₂×D₄×C₁₃	—	2	1	no	—
53	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	2	no	—
64	C₂×C₂²⋊D₁₃	—	2	2	no	—
65	C₂×C₂²⋊D₁₃	—	4	4	no	—
66	C₂²×Dic₁₃	—	2	1	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	2	1	no	—
70	C₂×D₅₂	C₂³	1	2	yes	—
71	C₁₃×C₄⋊C₄	—	2	1	no	—
72	C₁₃×C₄⋊C₄	—	2	1	no	—
73	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₁₃×C₂²⋊C₄	—	2	1	no	—
76	C₁₃×C₂²⋊C₄	—	2	1	no	—
77	C₁₃×C₂²⋊C₄	—	2	1	no	—
78	C₁₃×C₂²⋊C₄	—	2	1	no	—
79	C₁₃×C₂²⋊C₄	—	2	1	no	—
80	C₄²×C₁₃	—	2	1	no	—
81	#6106	—	2	4	no	—
82	#6104	—	2	4	no	—
83	#6103	—	2	2	no	—
84	#6103	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	1	no	—
86	C₄⋊Dic₁₃	C₂³	1	2	yes	—
87	#6099	—	2	2	no	—
88	#6099	—	2	4	no	—
89	#6099	D₄	1	4	yes	complemented
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	C₂³	1	2	yes	—
92	C₂×#1055	C₂³	1	2	yes	—
93	#3055	—	13	1	no	maximal, Sylow
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	—	2	2	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	4	1	no	—
98	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×D₁₃	—	2	2	no	—
102	C₂²×D₁₃	—	4	2	no	—
103	C₂²×D₁₃	—	4	2	no	—
104	C₂²×D₁₃	—	4	2	no	—
105	Q₈×C₁₃	—	2	1	no	—
106	Q₈×C₁₃	—	2	1	no	—
107	D₄×C₁₃	—	2	1	no	—
108	D₄×C₁₃	—	2	1	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	#41	1	2	yes	—
128	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₃	—	4	4	no	—
130	C₂²⋊D₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	4	2	no	—
133	D₅₂	—	2	2	no	—
134	D₅₂	—	2	2	no	—
135	D₅₂	—	4	2	no	—
136	#1055	—	2	2	no	—
137	#1055	C₂×D₄	1	4	yes	complemented
138	C₂×AGL(1,ℤ/8)	—	13	1	no	—
139	C₂²×M₁₆	—	13	1	no	—
140	#525	D₁₃	1	1	yes	characteristic, complemented
141	C₂×#121	—	13	1	no	—
142	#469	—	13	2	no	—
143	#468	—	13	2	no	—
144	#466	—	13	2	no	—
145	#465	—	13	2	no	—
146	#428	—	13	1	no	—
147	#418	—	13	1	no	—
148	#417	—	13	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	#121	1	2	yes	—
161	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	complemented
162	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₁₃	—	4	2	no	—
164	C₂×D₁₃	—	4	2	no	—
165	C₂×D₁₃	—	4	2	no	—
166	C₂×D₁₃	—	8	4	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	—	4	1	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	#121	1	2	yes	—
173	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
175	Dic₁₃	—	4	2	no	—
176	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
177	C₂²×D₄	—	13	1	no	—
178	C₂²×D₄	—	26	1	no	—
179	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
180	AGL(1,ℤ/8)	—	26	4	no	—
181	C₂×SD₁₆	—	13	2	no	—
182	C₂×D₈	—	13	2	no	—
183	C₂×M₁₆	—	13	1	no	—
184	C₂×M₁₆	—	13	1	no	—
185	C₂×M₁₆	—	13	2	no	—
186	C₂×M₁₆	—	26	4	no	—
187	C₂²×C₈	—	13	2	no	—
188	#123	—	2	1	no	—
189	C₂²⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
190	C₂²⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
191	#121	—	13	2	no	—
192	#121	—	13	2	no	—
193	#121	—	26	1	no	—
194	#121	—	26	1	no	—
195	#121	C₂×D₁₃	1	1	yes	characteristic, complemented
196	#121	C₂×D₁₃	1	1	yes	characteristic, complemented
197	#120	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	#117	C₂×D₁₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	13	1	no	—
202	C₂×C₂²⋊C₄	—	26	1	no	—
203	#107	—	13	2	no	—
204	#106	—	13	2	no	—
205	#103	—	13	2	no	—
206	#102	—	13	2	no	—
207	#102	—	13	2	no	—
208	#102	—	13	2	no	—
209	C₂×C₁₃	—	2	2	no	—
210	C₂×C₁₃	—	2	2	no	—
211	C₂×C₁₃	—	4	1	no	—
212	C₂×C₁₃	—	8	1	no	—
213	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
214	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂×C₁₃	#575	1	1	yes	characteristic
216	D₁₃	—	8	2	no	—
217	C₂⁴	—	26	1	no	—
218	C₂⁴	C₂²⋊D₁₃	1	1	yes	characteristic
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	13	2	no	—
226	C₂×D₄	—	26	2	no	—
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂×D₄	—	26	2	no	—
230	C₂×D₄	—	26	1	no	—
231	C₂×D₄	—	52	4	no	—
232	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	26	1	no	—
237	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
243	SD₁₆	—	26	4	no	—
244	D₈	—	26	4	no	—
245	M₁₆	—	13	4	no	—
246	M₁₆	—	26	4	no	—
247	M₁₆	—	26	2	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	26	4	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	13	2	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	26	2	no	—
263	C₂²⋊C₄	—	26	2	no	—
264	C₄²	—	2	1	no	—
265	C₁₃	#3055	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	26	2	no	—
271	C₂³	—	52	2	no	—
272	C₂³	—	52	2	no	—
273	C₂³	—	52	2	no	—
274	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	26	2	no	—
285	D₄	—	26	2	no	—
286	D₄	—	52	4	no	—
287	D₄	—	52	4	no	—
288	D₄	—	52	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	D₄×D₁₃	1	2	yes	—
303	C₂×C₄	#6129	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
308	C₈	—	13	4	no	—
309	C₈	—	26	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	52	2	no	—
322	C₂²	—	52	2	no	—
323	C₂²	—	52	2	no	—
324	C₂²	—	104	4	no	—
325	C₂²	#88193	1	2	yes	—
326	C₂²	#88203	1	2	yes	complemented
327	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	52	2	no	—
334	C₄	#88193	1	2	yes	—
335	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
336	C₄	C₂×#6129	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	104	2	no	—
342	C₂	C₂×#88193	1	1	yes	characteristic
343	C₂	C₂×#88203	1	1	yes	characteristic
344	C₂	#11719527	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#525	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#525	D₁₃
hypocenter	C₁₃	#3055
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₁₃×#525	C₂	semidirect
2. C₂×#88203	C₂	semidirect
3. C₂×#88184	C₂	semidirect
4. C₂²×#6099	C₂	semidirect
5. #11718893	C₂	semidirect
6. #11718710	C₂	semidirect
7. #11718709	C₂	semidirect
8. #11718700	C₂	semidirect
9. #11718699	C₂	semidirect
10. #11718662	C₂	semidirect
11. #11718656	C₂	semidirect
17. C₁₃×C₂²⋊Q₈	C₂²	semidirect
18. C₁₃×C₂²⋊Q₈	C₂²	semidirect
19. C₁₃×#121	C₂²	semidirect
20. C₁₃×#121	C₂²	semidirect
26. C₂×#6106	C₂²	semidirect
27. C₂×#6104	C₂²	semidirect
37. C₂×#6099	C₂²	semidirect
38. C₂²×#1055	C₂²	semidirect
39. #88075	C₂²	semidirect
40. #88072	C₂²	semidirect
41. #88049	C₂²	semidirect
42. #88049	C₂²	semidirect
43. #88048	C₂²	semidirect
44. #88047	C₂²	semidirect
89. #6099	D₄	semidirect
137. #1055	C₂×D₄	semidirect
140. #525	D₁₃	semidirect
161. C₂²×C₁₃	AGL(1,ℤ/8)	semidirect
189. C₂²⋊Q₈	C₂×D₁₃	semidirect
190. C₂²⋊Q₈	C₂×D₁₃	semidirect
195. #121	C₂×D₁₃	semidirect
196. #121	C₂×D₁₃	semidirect
265. C₁₃	#3055	semidirect
326. C₂²	#88203	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#525	2
C₂×#88203	2
C₂×#88184	2
C₂²×#6099	2
#11718893	2
#11718710	2
#11718709	2
#11718700	2
#11718699	2
#11718662	2
#11718656	2
#3055	13