Subgroups — #13127680

Order 1664.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719289	C₂	1	1	yes	maximal, characteristic
2	#11719272	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718511	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718510	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718508	C₂	1	1	yes	maximal, characteristic
7	#11718506	C₂	1	1	yes	maximal, characteristic
8	#11718506	C₂	1	1	yes	maximal, characteristic
9	#11718490	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718377	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718375	C₂	1	1	yes	maximal, characteristic
12	#11718371	C₂	1	1	yes	maximal, characteristic
13	#11718364	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718358	C₂	1	1	yes	maximal, characteristic
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	#88200	—	2	1	no	—
18	#88198	—	2	1	no	—
19	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
20	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
21	C₂×#6095	C₂²	1	1	yes	characteristic
22	#88152	—	2	1	no	—
23	#88152	—	2	1	no	—
24	#88146	—	2	1	no	—
25	#88143	—	2	1	no	—
26	#88143	—	2	1	no	—
27	#88141	—	2	1	no	—
28	#88129	C₂²	1	1	yes	characteristic
29	#88129	C₂²	1	1	yes	characteristic
30	C₄×D₅₂	C₂²	1	1	yes	characteristic
31	#88124	C₂²	1	1	yes	characteristic
32	#88124	C₂²	1	1	yes	characteristic
33	#88123	C₂²	1	1	yes	characteristic
34	#88123	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₃×#97	C₂²	1	1	yes	characteristic
44	C₁₃×#97	C₂²	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
46	#88061	C₂²	1	1	yes	characteristic
47	#88061	C₂²	1	1	yes	characteristic
48	#88061	C₂²	1	1	yes	characteristic
49	#88058	C₂²	1	1	yes	characteristic
50	#88058	C₂²	1	1	yes	characteristic
51	#88057	C₂²	1	1	yes	characteristic
52	#88056	C₂²	1	1	yes	characteristic
53	#88056	C₂²	1	1	yes	characteristic
54	#88056	C₂²	1	1	yes	characteristic
55	#88056	C₂²	1	1	yes	characteristic
56	#88056	C₂²	1	1	yes	characteristic
57	#88056	C₂²	1	1	yes	characteristic
58	#88056	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
60	C₂×D₅₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₃	—	2	1	no	—
62	C₂×Dic₂₆	—	2	1	no	—
63	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	#6103	—	2	1	no	—
77	#6103	—	2	1	no	—
78	#6103	—	2	1	no	—
79	#6103	—	2	1	no	—
80	#6103	—	2	1	no	—
81	C₄⋊Dic₁₃	—	2	1	no	—
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
88	#6101	—	2	1	no	—
89	#6101	—	2	1	no	—
90	#6101	—	2	1	no	—
91	#6101	—	2	1	no	—
92	#6101	—	2	1	no	—
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	Dic₅₂	—	2	2	no	—
103	Dic₅₂	—	2	2	no	—
104	#6095	—	2	2	no	—
105	#3377	—	13	1	no	maximal, Sylow
106	C₂²×D₁₃	—	2	1	no	—
107	Q₈×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	1	no	—
116	C₂×Dic₁₃	—	2	1	no	—
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	1	no	—
121	C₂×Dic₁₃	—	2	1	no	—
122	D₅₂	—	2	1	no	—
123	D₅₂	—	2	1	no	—
124	C₄×D₁₃	—	4	2	no	—
125	Dic₂₆	—	2	1	no	—
126	Dic₂₆	—	2	1	no	—
127	Dic₂₆	—	2	1	no	—
128	Dic₂₆	—	2	1	no	—
129	Dic₂₆	—	2	1	no	—
130	Dic₂₆	—	4	2	no	—
131	Dic₂₆	#41	1	2	yes	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	#557	—	13	1	no	—
137	#556	—	13	1	no	—
138	#491	—	13	1	no	—
139	#489	—	13	1	no	—
140	#488	—	13	1	no	—
141	#479	—	13	1	no	—
142	#478	—	13	1	no	—
143	#477	—	13	1	no	—
144	#464	—	13	1	no	—
145	#464	—	13	1	no	—
146	#462	—	13	1	no	—
147	#446	D₁₃	1	1	yes	characteristic, complemented
148	#441	—	13	1	no	—
149	#440	—	13	1	no	—
150	C₄×Q₁₆	—	13	1	no	—
151	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₃	—	4	1	no	—
153	C₂×D₁₃	—	4	1	no	—
154	C₂×D₁₃	—	4	1	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₁₃	—	2	2	no	—
164	Dic₁₃	—	4	1	no	—
165	Dic₁₃	—	4	1	no	—
166	Dic₁₃	—	4	1	no	—
167	Dic₁₃	—	4	1	no	—
168	Dic₁₃	—	4	1	no	—
169	Dic₁₃	—	4	1	no	—
170	C₂×Q₁₆	—	13	1	no	—
171	C₂×Q₁₆	—	13	1	no	—
172	C₂×SD₁₆	—	13	1	no	—
173	C₄⋊Q₈	—	13	1	no	—
174	C₄⋊Q₈	—	13	1	no	—
175	C₄⋊Q₈	—	26	1	no	—
176	#126	—	26	1	no	—
177	#126	—	26	1	no	—
178	#125	—	13	1	no	—
179	#125	—	13	1	no	—
180	#125	—	26	1	no	—
181	#125	—	26	1	no	—
182	#124	—	13	1	no	—
183	#124	—	13	1	no	—
184	C₂²⋊Q₈	—	26	1	no	—
185	C₄×Q₈	—	13	1	no	—
186	C₄×Q₈	—	13	1	no	—
187	C₄×Q₈	—	13	1	no	—
188	C₄×Q₈	—	13	1	no	—
189	C₄×Q₈	—	13	1	no	—
190	C₄×Q₈	—	26	1	no	—
191	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
192	C₄×D₄	—	13	1	no	—
193	#117	—	26	1	no	—
194	#107	—	13	1	no	—
195	#107	—	13	1	no	—
196	#106	—	13	1	no	—
197	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
198	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
200	#103	—	13	1	no	—
201	#103	—	13	1	no	—
202	#103	—	13	1	no	—
203	#103	—	13	1	no	—
204	#103	—	13	1	no	—
205	#103	—	13	1	no	—
206	#103	—	13	1	no	—
207	#102	—	13	1	no	—
208	#102	—	13	1	no	—
209	#102	—	13	1	no	—
210	#97	C₂×D₁₃	1	1	yes	characteristic
211	#97	C₂×D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
213	C₂×C₁₃	#548	1	1	yes	characteristic
214	C₂×C₁₃	#575	1	1	yes	characteristic
215	C₂×C₁₃	#578	1	1	yes	characteristic
216	D₁₃	—	8	1	no	—
217	C₂×Q₈	—	13	1	no	—
218	C₂×Q₈	—	13	1	no	—
219	C₂×Q₈	—	13	1	no	—
220	C₂×Q₈	—	26	1	no	—
221	C₂×Q₈	D₅₂	1	1	yes	characteristic
222	C₂×D₄	—	13	1	no	—
223	C₂²×C₄	—	26	1	no	—
224	Q₁₆	—	26	2	no	—
225	Q₁₆	—	26	2	no	—
226	SD₁₆	—	26	2	no	—
227	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	13	1	no	—
232	C₄⋊C₄	—	13	1	no	—
233	C₄⋊C₄	—	13	1	no	—
234	C₄⋊C₄	—	13	1	no	—
235	C₄⋊C₄	—	26	1	no	—
236	C₄⋊C₄	—	26	1	no	—
237	C₄⋊C₄	—	26	1	no	—
238	C₄⋊C₄	—	26	1	no	—
239	C₄⋊C₄	—	26	1	no	—
240	C₄⋊C₄	—	26	1	no	—
241	C₄⋊C₄	—	26	1	no	—
242	C₄⋊C₄	—	26	1	no	—
243	C₄⋊C₄	—	26	1	no	—
244	C₄⋊C₄	—	26	1	no	—
245	C₄⋊C₄	—	26	1	no	—
246	C₄⋊C₄	—	26	1	no	—
247	C₄⋊C₄	D₅₂	1	1	yes	characteristic
248	C₄⋊C₄	D₅₂	1	1	yes	characteristic
249	C₄⋊C₄	D₅₂	1	1	yes	characteristic
250	C₂²⋊C₄	—	26	1	no	—
251	C₂²⋊C₄	—	26	1	no	—
252	C₂²⋊C₄	—	26	1	no	—
253	C₂²⋊C₄	—	26	1	no	—
254	C₂²⋊C₄	—	26	1	no	—
255	C₄²	—	26	1	no	—
256	C₄²	—	26	1	no	—
257	C₄²	—	26	1	no	—
258	C₄²	—	26	1	no	—
259	C₄²	—	26	1	no	—
260	C₄²	C₂²×D₁₃	1	1	yes	characteristic
261	C₄²	C₂²×D₁₃	1	1	yes	characteristic
262	C₄²	C₂²×D₁₃	1	1	yes	characteristic
263	C₁₃	#3377	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	26	1	no	—
265	Q₈	—	4	1	no	—
266	Q₈	—	13	2	no	—
267	Q₈	—	26	1	no	—
268	Q₈	—	26	1	no	—
269	Q₈	—	26	1	no	—
270	Q₈	—	26	1	no	—
271	Q₈	—	26	1	no	—
272	Q₈	—	52	2	no	—
273	D₄	—	26	1	no	—
274	D₄	—	26	1	no	—
275	C₂×C₄	—	26	1	no	—
276	C₂×C₄	—	26	1	no	—
277	C₂×C₄	—	26	1	no	—
278	C₂×C₄	—	26	1	no	—
279	C₂×C₄	—	26	1	no	—
280	C₂×C₄	—	26	1	no	—
281	C₂×C₄	—	26	1	no	—
282	C₂×C₄	—	52	2	no	—
283	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
284	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
285	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
286	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
287	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
288	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₂²	—	52	1	no	—
295	C₂²	—	52	1	no	—
296	C₂²	—	52	1	no	—
297	C₂²	C₂²×D₅₂	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	26	2	no	—
305	C₄	—	52	1	no	—
306	C₄	—	52	1	no	—
307	C₄	—	52	1	no	—
308	C₄	—	52	1	no	—
309	C₄	—	52	1	no	—
310	C₄	—	52	1	no	—
311	C₄	#88254	1	1	yes	characteristic
312	C₄	#41×D₁₃	1	1	yes	characteristic
313	C₂	—	104	1	no	—
314	C₂	#11719281	1	1	yes	characteristic
315	C₂	#11719453	1	1	yes	characteristic
316	C₂	#11719458	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#446	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₅₂	C₂³
hypercenter	#446	D₁₃
hypocenter	C₁₃	#3377
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11719272	C₂	semidirect
3. C₁₃×#446	C₂	semidirect
4. #11718511	C₂	semidirect
5. #11718510	C₂	semidirect
9. #11718490	C₂	semidirect
10. #11718377	C₂	semidirect
13. #11718364	C₂	semidirect
14. C₄×Dic₅₂	C₂	semidirect
147. #446	D₁₃	semidirect
263. C₁₃	#3377	semidirect

Maximal subgroups

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

subgroup	index
#11719289	2
#11719272	2
C₁₃×#446	2
#11718511	2
#11718510	2
#11718508	2
#11718506	2
#11718490	2
#11718377	2
#11718375	2
#11718371	2
#11718364	2
C₄×Dic₅₂	2
#11718358	2
#3377	13