Subgroups — #13127747

Order 1664.

340 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	Q₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719244	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#440	C₂	1	1	yes	maximal, characteristic
4	#11718825	C₂	1	1	yes	maximal, characteristic
5	SD₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	#11718683	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718668	C₂	1	1	yes	maximal, characteristic
8	#11718537	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718535	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718460	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718455	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718422	C₂	1	1	yes	maximal, characteristic
13	#11718420	C₂	1	1	yes	maximal, characteristic
14	#11718363	C₂	1	1	yes	maximal, characteristic, complemented
15	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	#88178	—	2	1	no	—
24	C₂×#6101	—	2	1	no	—
25	#88144	C₂²	1	1	yes	characteristic
26	#88142	—	2	1	no	—
27	#88142	—	2	1	no	—
28	#88142	C₂²	1	1	yes	characteristic
29	#88142	C₂²	1	1	yes	characteristic
30	#88142	C₂²	1	1	yes	characteristic
31	#88132	—	2	1	no	—
32	#88132	—	2	1	no	—
33	#88124	C₂²	1	1	yes	characteristic
34	#88123	—	2	1	no	—
35	#88123	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₆	—	2	1	no	—
37	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
38	C₁₃×#107	C₂²	1	1	yes	characteristic
39	C₁₃×#103	C₂²	1	1	yes	characteristic
40	C₁₃×#102	C₂²	1	1	yes	characteristic
41	C₁₃×#97	C₂²	1	1	yes	characteristic
42	#88075	C₂²	1	1	yes	characteristic
43	#88075	C₂²	1	1	yes	characteristic
44	#88072	C₂²	1	1	yes	characteristic
45	#88072	C₂²	1	1	yes	characteristic
46	#88058	C₂²	1	1	yes	characteristic
47	#88058	C₂²	1	1	yes	characteristic
48	#88057	C₂²	1	1	yes	characteristic
49	#88057	C₂²	1	1	yes	characteristic
50	#88055	C₂²	1	1	yes	characteristic
51	#88054	C₂²	1	1	yes	characteristic
52	#88054	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
54	#88048	C₂²	1	1	yes	characteristic
55	#88047	C₂²	1	1	yes	characteristic
56	#88047	C₂²	1	1	yes	characteristic
57	#88047	C₂²	1	1	yes	characteristic
58	C₄⋊#1055	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂×Dic₂₆	—	2	1	no	—
65	C₂×Dic₂₆	—	2	1	no	—
66	C₂×Dic₂₆	—	2	1	no	—
67	C₂×Dic₂₆	—	2	1	no	—
68	C₂×Dic₂₆	D₄	1	1	yes	characteristic
69	C₁₃×SD₁₆	Q₈	1	4	yes	complemented
70	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₁₃	—	2	1	no	—
79	C₂²⋊Dic₁₃	—	2	1	no	—
80	C₄⋊Dic₁₃	—	2	1	no	—
81	C₄⋊Dic₁₃	—	2	1	no	—
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	D₄	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	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
96	#6101	D₄	1	1	yes	characteristic
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
101	C₂×#1055	C₂³	1	1	yes	characteristic
102	C₂×#1055	C₂³	1	1	yes	characteristic
103	#3385	—	13	1	no	maximal, Sylow
104	C₂³×C₁₃	—	2	1	no	—
105	Q₈×C₁₃	—	2	1	no	—
106	Q₈×C₁₃	C₂×Q₈	1	2	yes	—
107	D₄×C₁₃	—	2	1	no	—
108	D₄×C₁₃	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	8	1	no	—
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	—	4	2	no	—
121	C₂×Dic₁₃	—	8	1	no	—
122	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	Dic₂₆	—	2	2	no	—
127	Dic₂₆	—	8	1	no	—
128	Dic₂₆	—	8	1	no	—
129	Dic₂₆	—	8	1	no	—
130	Dic₂₆	—	8	1	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	C₂×Q₈	1	2	yes	—
133	#1055	—	2	1	no	—
134	#1055	—	2	1	no	—
135	Q₈²	—	13	1	no	—
136	#554	—	13	1	no	—
137	#501	—	13	1	no	—
138	#501	—	13	1	no	—
139	#500	—	13	1	no	—
140	#479	—	13	1	no	—
141	#479	—	13	1	no	—
142	#477	—	13	1	no	—
143	#476	—	13	1	no	—
144	#474	—	13	1	no	—
145	#474	—	13	1	no	—
146	#446	—	13	1	no	—
147	#440	—	13	1	no	—
148	#440	D₁₃	1	1	yes	characteristic
149	C₄×SD₁₆	—	13	1	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	4	1	no	—
158	C₄×C₁₃	—	4	1	no	—
159	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
160	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₃	—	2	1	no	—
162	Dic₁₃	—	2	1	no	—
163	Dic₁₃	—	2	1	no	—
164	Dic₁₃	—	4	1	no	—
165	Dic₁₃	—	4	1	no	—
166	Dic₁₃	—	4	1	no	—
167	Dic₁₃	—	4	1	no	—
168	Dic₁₃	#137	1	2	yes	—
169	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	13	1	no	—
171	C₄⋊Q₈	—	13	1	no	—
172	C₄⋊Q₈	—	13	1	no	—
173	C₄⋊Q₈	—	13	1	no	—
174	C₄⋊Q₈	—	26	1	no	—
175	C₄⋊Q₈	—	26	1	no	—
176	C₄⋊Q₈	—	26	1	no	—
177	#125	—	13	1	no	—
178	#125	—	13	1	no	—
179	C₂²⋊Q₈	—	26	1	no	—
180	C₂²⋊Q₈	—	26	1	no	—
181	C₂²⋊Q₈	—	26	1	no	—
182	C₄×Q₈	—	13	1	no	—
183	C₄×Q₈	—	13	1	no	—
184	C₄×Q₈	—	13	1	no	—
185	C₄×Q₈	—	26	1	no	—
186	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
187	C₄×D₄	—	13	1	no	—
188	C₄×D₄	—	13	1	no	—
189	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	26	1	no	—
191	#107	—	13	1	no	—
192	#107	—	13	1	no	—
193	#107	—	13	1	no	—
194	#107	—	13	1	no	—
195	#107	—	13	1	no	—
196	#107	C₂×D₁₃	1	1	yes	characteristic
197	#106	—	13	1	no	—
198	#106	—	13	1	no	—
199	#106	—	13	1	no	—
200	C₄⋊C₈	—	13	1	no	—
201	C₄⋊C₈	—	13	1	no	—
202	C₄⋊C₈	—	13	1	no	—
203	#103	—	13	1	no	—
204	#103	—	13	1	no	—
205	#103	C₂×D₁₃	1	1	yes	characteristic
206	#102	—	13	1	no	—
207	#102	—	13	1	no	—
208	#102	C₂×D₁₃	1	1	yes	characteristic
209	#97	—	13	1	no	—
210	#97	C₂×D₁₃	1	1	yes	characteristic
211	C₄×C₈	—	13	1	no	—
212	C₂×C₁₃	—	4	2	no	—
213	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
215	C₂×C₁₃	#576	1	1	yes	characteristic
216	C₂×Q₈	—	13	1	no	—
217	C₂×Q₈	—	26	1	no	—
218	C₂×Q₈	—	26	1	no	—
219	C₂×Q₈	—	26	1	no	—
220	C₂×Q₈	—	26	1	no	—
221	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	26	1	no	—
225	C₂²×C₄	—	26	1	no	—
226	SD₁₆	Dic₂₆	1	4	yes	complemented
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	13	1	no	—
233	C₄⋊C₄	—	13	1	no	—
234	C₄⋊C₄	—	13	1	no	—
235	C₄⋊C₄	—	13	1	no	—
236	C₄⋊C₄	—	13	1	no	—
237	C₄⋊C₄	—	13	1	no	—
238	C₄⋊C₄	—	13	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	—	26	1	no	—
248	C₄⋊C₄	—	26	1	no	—
249	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	26	1	no	—
253	C₂²⋊C₄	—	26	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	13	1	no	—
256	C₄²	—	13	1	no	—
257	C₄²	—	26	1	no	—
258	C₄²	—	26	1	no	—
259	C₄²	C₂²×D₁₃	1	1	yes	characteristic
260	C₁₃	#3385	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	2	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	—	26	2	no	—
264	Q₈	—	104	1	no	—
265	Q₈	—	104	1	no	—
266	Q₈	—	104	1	no	—
267	Q₈	—	104	1	no	—
268	Q₈	C₂×Dic₂₆	1	2	yes	—
269	D₄	—	2	1	no	—
270	D₄	C₂×Dic₂₆	1	2	yes	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	13	1	no	—
276	C₂×C₄	—	13	1	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	26	1	no	—
280	C₂×C₄	—	26	1	no	—
281	C₂×C₄	—	26	1	no	—
282	C₂×C₄	—	26	1	no	—
283	C₂×C₄	—	52	2	no	—
284	C₂×C₄	—	104	1	no	—
285	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
286	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	26	1	no	—
290	C₈	—	26	1	no	—
291	C₈	C₂×Dic₂₆	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	13	2	no	—
302	C₄	—	26	1	no	—
303	C₄	—	26	1	no	—
304	C₄	—	26	1	no	—
305	C₄	—	52	1	no	—
306	C₄	—	52	1	no	—
307	C₄	—	52	1	no	—
308	C₄	—	52	1	no	—
309	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
310	C₄	#88257	1	1	yes	characteristic
311	C₂	—	4	2	no	—
312	C₂	D₄×Dic₂₆	1	1	yes	characteristic
313	C₂	#11719479	1	1	yes	characteristic
314	C₂	D₁₃×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#440	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₄×C₁₃	C₂³
hypercenter	#440	D₁₃
hypocenter	C₁₃	#3385
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₆	C₂	semidirect
6. #11718683	C₂	semidirect
8. #11718537	C₂	semidirect
9. #11718535	C₂	semidirect
10. #11718460	C₂	semidirect
11. #11718455	C₂	semidirect
14. #11718363	C₂	semidirect
15. #11718362	C₂	semidirect
69. C₁₃×SD₁₆	Q₈	semidirect
226. SD₁₆	Dic₂₆	semidirect
260. C₁₃	#3385	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₆	2
#11719244	2
C₁₃×#440	2
#11718825	2
SD₁₆×Dic₁₃	2
#11718683	2
#11718668	2
#11718537	2
#11718535	2
#11718460	2
#11718455	2
#11718422	2
#11718420	2
#11718363	2
#11718362	2
#3385	13