Subgroups — #13128641

Order 1664.

964 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	#11719427	C₂	1	2	yes	maximal, complemented
3	#11719426	C₂	1	2	yes	maximal, complemented
4	#11719390	C₂	1	4	yes	maximal, complemented
5	#11719328	C₂	1	4	yes	maximal
6	#11719327	C₂	1	4	yes	maximal
7	#11719272	C₂	1	2	yes	maximal, complemented
8	#11719272	C₂	1	4	yes	maximal, complemented
9	Q₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
10	#11719224	C₂	1	2	yes	maximal
11	#11719221	C₂	1	2	yes	maximal
12	#11719175	C₂	1	1	yes	maximal, characteristic
13	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
14	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#125	C₂²	1	2	yes	—
18	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
22	C₁₃×#117	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₃	C₂²	1	4	yes	—
25	#88198	C₂²	1	4	yes	—
26	#88198	C₂²	1	8	yes	—
27	#88180	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	#88178	C₂²	1	2	yes	—
30	#88178	C₂²	1	2	yes	—
31	C₂×#6101	C₂²	1	2	yes	—
32	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
33	#88144	C₂²	1	4	yes	—
34	#88143	C₂²	1	8	yes	—
35	#88143	C₂²	1	8	yes	—
36	#88143	C₂²	1	8	yes	—
37	#88142	C₂²	1	4	yes	—
38	#88142	C₂²	1	4	yes	—
39	#88142	C₂²	1	8	yes	—
40	#88141	C₂²	1	4	yes	—
41	#88141	C₂²	1	4	yes	—
42	#88141	C₂²	1	4	yes	—
43	#88141	C₂²	1	8	yes	—
44	#88132	C₂²	1	8	yes	—
45	#88131	C₂²	1	8	yes	—
46	#88124	C₂²	1	2	yes	—
47	#88124	C₂²	1	4	yes	—
48	#88123	C₂²	1	1	yes	characteristic
49	#88123	C₂²	1	1	yes	characteristic
50	#88123	C₂²	1	4	yes	—
51	C₄×Dic₂₆	C₂²	1	2	yes	—
52	C₄×Dic₂₆	C₂²	1	2	yes	—
53	C₄×Dic₂₆	C₂²	1	2	yes	—
54	C₄×Dic₂₆	C₂²	1	2	yes	—
55	C₄×Dic₂₆	C₂²	1	2	yes	—
56	C₄×Dic₂₆	C₂²	1	2	yes	—
57	C₄×Dic₂₆	C₂²	1	4	yes	—
58	C₄×Dic₂₆	C₂²	1	4	yes	—
59	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
60	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂³	1	2	yes	—
65	C₂²×Dic₁₃	C₂³	1	2	yes	—
66	C₂×Dic₂₆	Q₈	1	8	yes	complemented
67	C₂×Dic₂₆	C₂³	1	2	yes	—
68	C₂×Dic₂₆	C₂³	1	2	yes	—
69	C₂×Dic₂₆	C₂³	1	8	yes	—
70	C₂×Dic₂₆	C₂³	1	2	yes	—
71	C₂×Dic₂₆	C₂³	1	2	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
75	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
76	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
77	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
78	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
79	C₄²×C₁₃	C₂³	1	1	yes	characteristic
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	C₄²×C₁₃	C₂³	1	1	yes	characteristic
82	C₄²×C₁₃	C₂³	1	4	yes	—
83	C₄²×C₁₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
85	C₄⋊Dic₁₃	C₂³	1	2	yes	—
86	C₄⋊Dic₁₃	C₂³	1	2	yes	—
87	C₄⋊Dic₁₃	C₂³	1	2	yes	—
88	C₄⋊Dic₁₃	C₂³	1	2	yes	—
89	C₄⋊Dic₁₃	C₂³	1	2	yes	—
90	C₄⋊Dic₁₃	C₂³	1	2	yes	—
91	C₄⋊Dic₁₃	C₂³	1	8	yes	—
92	#6101	C₂³	1	4	yes	—
93	#6101	C₂³	1	4	yes	—
94	#6101	C₂³	1	4	yes	—
95	#6101	C₂³	1	4	yes	—
96	#6101	C₂³	1	8	yes	—
97	#6101	C₂³	1	8	yes	—
98	#6101	C₂³	1	8	yes	—
99	#6101	C₂³	1	8	yes	—
100	C₄×Dic₁₃	C₂³	1	4	yes	—
101	C₄×Dic₁₃	C₂³	1	4	yes	—
102	C₄×Dic₁₃	C₂³	1	4	yes	—
103	C₄×Dic₁₃	C₂³	1	8	yes	—
104	C₄×Dic₁₃	C₂³	1	8	yes	—
105	#3502	—	13	1	no	maximal, Sylow
106	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	4	no	—
108	Q₈×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
114	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₁₃	—	4	2	no	—
121	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
122	C₂×Dic₁₃	C₂⁴	1	4	yes	—
123	C₂×Dic₁₃	C₂⁴	1	4	yes	—
124	C₂×Dic₁₃	C₂⁴	1	4	yes	—
125	C₂×Dic₁₃	C₂⁴	1	4	yes	—
126	Dic₂₆	—	2	4	no	—
127	Dic₂₆	—	4	2	no	—
128	Dic₂₆	—	4	8	no	—
129	Dic₂₆	C₂×Q₈	1	8	yes	—
130	Dic₂₆	C₂×Q₈	1	8	yes	—
131	Q₈²	—	13	2	no	—
132	Q₈²	—	13	2	no	—
133	#557	—	13	2	no	—
134	#557	—	13	4	no	—
135	#557	—	13	4	no	—
136	#557	—	13	2	no	—
137	#544	—	13	4	no	—
138	#544	—	13	2	no	—
139	#533	—	13	4	no	—
140	#533	—	13	1	no	—
141	#533	D₁₃	1	1	yes	characteristic
142	#519	—	13	2	no	—
143	C₂×C₄×Q₈	—	13	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
147	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	4	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
154	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
155	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
156	Dic₁₃	—	2	4	no	—
157	Dic₁₃	—	2	4	no	—
158	Dic₁₃	—	2	4	no	—
159	Dic₁₃	C₂²×Q₈	1	8	yes	—
160	C₂²×Q₈	—	13	1	no	—
161	C₄⋊Q₈	—	13	4	no	—
162	C₄⋊Q₈	—	13	8	no	—
163	C₄⋊Q₈	—	13	4	no	—
164	C₄⋊Q₈	—	13	8	no	—
165	C₄⋊Q₈	—	13	4	no	—
166	C₄⋊Q₈	—	13	1	no	—
167	C₄⋊Q₈	—	13	1	no	—
168	C₄⋊Q₈	—	13	4	no	—
169	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
170	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
171	#125	—	13	8	no	—
172	#125	—	13	8	no	—
173	#125	—	13	8	no	—
174	#125	—	13	4	no	—
175	#125	—	13	2	no	—
176	#125	—	13	4	no	—
177	#125	C₂×D₁₃	1	2	yes	—
178	C₂²⋊Q₈	—	13	2	no	—
179	C₂²⋊Q₈	—	13	2	no	—
180	C₂²⋊Q₈	—	13	8	no	—
181	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
182	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
183	C₄×Q₈	—	13	4	no	—
184	C₄×Q₈	—	13	4	no	—
185	C₄×Q₈	—	13	4	no	—
186	C₄×Q₈	—	13	8	no	—
187	C₄×Q₈	—	13	4	no	—
188	C₄×Q₈	—	13	2	no	—
189	C₄×Q₈	—	13	2	no	—
190	C₄×Q₈	—	13	2	no	—
191	C₄×Q₈	—	13	2	no	—
192	C₄×Q₈	—	13	4	no	—
193	C₄×Q₈	—	13	4	no	—
194	C₄×Q₈	—	13	2	no	—
195	C₄×Q₈	—	13	2	no	—
196	C₄×Q₈	C₂×D₁₃	1	2	yes	—
197	C₄×Q₈	C₂×D₁₃	1	2	yes	—
198	#117	—	13	2	no	—
199	#117	—	13	8	no	—
200	#117	C₂×D₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	—	13	2	no	—
202	C₂×C₄⋊C₄	—	13	1	no	—
203	C₂×C₄²	—	13	2	no	—
204	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
207	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
208	C₂×C₁₃	#585	1	1	yes	characteristic
209	C₂×Q₈	—	13	8	no	—
210	C₂×Q₈	—	13	2	no	—
211	C₂×Q₈	—	13	2	no	—
212	C₂×Q₈	—	13	2	no	—
213	C₂×Q₈	—	13	2	no	—
214	C₂×Q₈	—	13	8	no	—
215	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
216	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
222	C₄⋊C₄	—	13	4	no	—
223	C₄⋊C₄	—	13	4	no	—
224	C₄⋊C₄	—	13	2	no	—
225	C₄⋊C₄	—	13	2	no	—
226	C₄⋊C₄	—	13	8	no	—
227	C₄⋊C₄	—	13	2	no	—
228	C₄⋊C₄	—	13	8	no	—
229	C₄⋊C₄	—	13	2	no	—
230	C₄⋊C₄	—	13	4	no	—
231	C₄⋊C₄	—	13	2	no	—
232	C₄⋊C₄	—	13	8	no	—
233	C₄⋊C₄	—	13	2	no	—
234	C₄⋊C₄	—	13	4	no	—
235	C₄⋊C₄	—	13	8	no	—
236	C₄⋊C₄	—	13	8	no	—
237	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
238	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
239	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
240	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
241	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
242	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
243	C₂²⋊C₄	—	13	4	no	—
244	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
245	C₄²	—	13	4	no	—
246	C₄²	—	13	4	no	—
247	C₄²	—	13	8	no	—
248	C₄²	—	13	8	no	—
249	C₄²	—	13	4	no	—
250	C₄²	C₂²×D₁₃	1	1	yes	characteristic
251	C₄²	C₂²×D₁₃	1	1	yes	characteristic
252	C₄²	C₂²×D₁₃	1	1	yes	characteristic
253	C₄²	C₂²×D₁₃	1	1	yes	characteristic
254	C₄²	C₂²×D₁₃	1	4	yes	—
255	C₁₃	#3502	1	1	yes	characteristic, complemented, Sylow
256	C₂³	C₂³×D₁₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	13	8	no	—
260	Q₈	—	13	8	no	—
261	Q₈	—	26	4	no	—
262	Q₈	—	52	2	no	—
263	Q₈	—	52	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	13	8	no	—
267	C₂×C₄	—	13	4	no	—
268	C₂×C₄	—	13	4	no	—
269	C₂×C₄	—	13	4	no	—
270	C₂×C₄	—	13	4	no	—
271	C₂×C₄	—	52	2	no	—
272	C₂×C₄	Q₈×D₁₃	1	4	yes	—
273	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₁₃	1	4	yes	—
275	C₂×C₄	C₂³×D₁₃	1	4	yes	—
276	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
284	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	4	no	—
290	C₄	—	13	8	no	—
291	C₄	—	26	4	no	—
292	C₄	—	26	4	no	—
293	C₄	—	26	4	no	—
294	C₄	C₂×Q₈×D₁₃	1	2	yes	—
295	C₄	C₂×Q₈×D₁₃	1	2	yes	—
296	C₄	#88257	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
299	C₂	C₂×#88257	1	1	yes	characteristic
300	C₂	#11719648	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
prime radical	C₂³×C₁₃	C₂⁴
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3502
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #11719427	C₂	semidirect
3. #11719426	C₂	semidirect
4. #11719390	C₂	semidirect
7. #11719272	C₂	semidirect
8. #11719272	C₂	semidirect
9. Q₈×Dic₂₆	C₂	semidirect
66. C₂×Dic₂₆	Q₈	semidirect
255. C₁₃	#3502	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11719427	2
#11719426	2
#11719390	2
#11719328	2
#11719327	2
#11719272	2
Q₈×Dic₂₆	2
#11719224	2
#11719221	2
#11719175	2
C₂×C₄×Dic₂₆	2
#3502	13