Subgroups — #13124375

Order 1664.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88204	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88203	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719230	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718689	C₂	1	2	yes	maximal, complemented
6	#11718688	C₂	1	2	yes	maximal, complemented
7	#11718677	C₂	1	2	yes	maximal, complemented
8	#11718676	C₂	1	2	yes	maximal, complemented
9	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718627	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88049	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×#6127	C₂²	1	1	yes	characteristic
14	C₂²×D₅₂	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
16	C₄×D₄×C₁₃	—	2	2	no	—
17	C₄×D₄×C₁₃	C₂²	1	2	yes	complemented
18	C₁₃×#117	—	2	1	no	—
19	C₁₃×#117	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	#88204	—	2	4	no	—
23	#88203	—	2	4	no	—
24	C₂×#6107	C₂²	1	1	yes	characteristic, complemented
25	C₂×#6106	C₂²	1	2	yes	—
26	C₂×#6106	C₂²	1	1	yes	characteristic, complemented
27	C₂×#6105	C₂²	1	1	yes	characteristic, complemented
28	C₂×#6104	C₂²	1	2	yes	—
29	C₂×#6104	C₂²	1	1	yes	characteristic, complemented
30	C₂×#6099	C₂²	1	1	yes	characteristic
31	C₂²×#1055	C₂²	1	1	yes	characteristic
32	#88149	—	2	1	no	—
33	#88149	—	2	1	no	—
34	#88136	—	2	2	no	—
35	#88129	C₂²	1	2	yes	—
36	#88128	C₂²	1	2	yes	—
37	#88050	C₂²	1	2	yes	complemented
38	#88049	C₂²	1	2	yes	—
39	#88049	C₂²	1	2	yes	—
40	#88049	C₂²	1	2	yes	complemented
41	C₄⋊#1055	C₂²	1	2	yes	complemented
42	C₄⋊#1055	C₂²	1	2	yes	complemented
43	C₂³×D₁₃	—	2	1	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₁₃×#41	D₄	1	4	yes	complemented
46	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₃	—	2	1	no	—
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₃	—	2	1	no	—
54	#6127	—	2	2	no	—
55	#6127	—	2	2	no	—
56	C₂×D₅₂	—	2	2	no	—
57	C₂×D₅₂	—	2	2	no	—
58	C₂×D₅₂	—	2	1	no	—
59	C₂×D₅₂	—	2	2	no	—
60	C₂×D₅₂	—	2	2	no	—
61	C₂×D₅₂	C₂³	1	1	yes	characteristic
62	C₂×D₅₂	C₂³	1	2	yes	—
63	C₂×C₄×D₁₃	—	2	1	no	—
64	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	C₂³	1	2	yes	—
74	#6107	—	2	2	no	—
75	#6106	—	2	4	no	—
76	#6106	—	2	2	no	—
77	#6105	—	2	2	no	—
78	#6104	—	2	4	no	—
79	#6104	—	2	2	no	—
80	#6103	—	2	2	no	—
81	#6103	—	2	2	no	—
82	#6099	—	2	2	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	C₂³	1	1	yes	characteristic
85	C₂×#1055	C₂³	1	1	yes	characteristic
86	C₂×#1055	C₂³	1	2	yes	—
87	#3045	—	13	1	no	maximal, Sylow
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₃	—	2	2	no	—
91	C₂²×D₁₃	—	2	1	no	—
92	C₂²×D₁₃	—	4	2	no	—
93	C₂²×D₁₃	—	8	1	no	—
94	C₂²×D₁₃	—	8	1	no	—
95	Q₈×C₁₃	—	2	1	no	—
96	Q₈×C₁₃	C₂×D₄	1	2	yes	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	2	1	no	—
99	D₄×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	8	1	no	—
105	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃	#41	1	2	yes	—
110	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₂²⋊D₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	2	1	no	—
113	D₅₂	—	2	2	no	—
114	D₅₂	—	2	1	no	—
115	D₅₂	—	2	2	no	—
116	D₅₂	—	2	1	no	—
117	D₅₂	—	4	2	no	—
118	D₅₂	—	8	2	no	—
119	D₅₂	—	8	2	no	—
120	C₄×D₁₃	—	4	2	no	—
121	Dic₂₆	—	2	1	no	—
122	Dic₂₆	—	2	1	no	—
123	#1055	—	2	1	no	—
124	#1055	—	2	1	no	—
125	#1055	—	2	2	no	—
126	C₂×AGL(1,ℤ/8)	—	13	1	no	—
127	C₂×#135	—	13	1	no	—
128	#535	—	13	1	no	—
129	#520	D₁₃	1	1	yes	characteristic, complemented
130	#464	—	13	2	no	—
131	#463	—	13	2	no	—
132	#460	—	13	2	no	—
133	#459	—	13	2	no	—
134	#424	—	13	1	no	—
135	#417	—	13	1	no	—
136	C₂×#102	—	13	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	1	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	#121	1	2	yes	—
143	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₃	—	4	1	no	—
145	C₂×D₁₃	—	4	2	no	—
146	C₂×D₁₃	—	4	1	no	—
147	C₂×D₁₃	—	4	2	no	—
148	C₂×D₁₃	—	4	1	no	—
149	C₂×D₁₃	—	4	2	no	—
150	C₂×D₁₃	—	8	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	#121	1	2	yes	—
156	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
158	Dic₁₃	—	4	1	no	—
159	C₂×#41	—	13	1	no	—
160	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
161	C₂²×D₄	—	13	1	no	—
162	AGL(1,ℤ/8)	—	26	4	no	—
163	#135	—	26	4	no	—
164	C₂×Q₁₆	—	13	1	no	—
165	C₂×SD₁₆	—	13	1	no	—
166	C₂×SD₁₆	—	13	2	no	—
167	C₂×SD₁₆	—	13	1	no	—
168	C₂×D₈	—	13	1	no	—
169	C₂×D₈	—	13	2	no	—
170	C₂×M₁₆	—	13	1	no	—
171	C₂²×C₈	—	13	1	no	—
172	#127	—	13	2	no	—
173	#124	—	13	2	no	—
174	#121	—	26	1	no	—
175	#121	—	26	1	no	—
176	#120	—	26	2	no	—
177	C₄×Q₈	C₂×D₁₃	1	2	yes	complemented
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×D₁₃	1	2	yes	complemented
180	#117	—	2	1	no	—
181	#117	C₂×D₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
184	C₄⋊C₈	—	13	2	no	—
185	C₄⋊C₈	—	13	2	no	—
186	#103	—	13	2	no	—
187	#102	—	13	2	no	—
188	#102	—	13	2	no	—
189	#102	—	13	2	no	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	—	4	2	no	—
192	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
193	C₂×C₁₃	#576	1	1	yes	characteristic
194	C₂×C₁₃	#577	1	1	yes	characteristic
195	D₁₃	—	8	1	no	—
196	D₁₃	—	8	2	no	—
197	C₂⁴	—	26	1	no	—
198	#41	—	2	2	no	—
199	#41	—	26	2	no	—
200	#41	—	26	2	no	—
201	#41	D₅₂	1	4	yes	complemented
202	C₂×Q₈	—	13	1	no	—
203	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	13	2	no	—
206	C₂×D₄	—	13	1	no	—
207	C₂×D₄	—	26	1	no	—
208	C₂×D₄	—	26	2	no	—
209	C₂×D₄	—	26	2	no	—
210	C₂×D₄	—	26	1	no	—
211	C₂×D₄	—	26	2	no	—
212	C₂×D₄	—	26	2	no	—
213	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	26	1	no	—
218	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
219	Q₁₆	—	26	2	no	—
220	SD₁₆	—	26	2	no	—
221	SD₁₆	—	26	4	no	—
222	SD₁₆	—	26	2	no	—
223	D₈	—	26	2	no	—
224	D₈	—	26	4	no	—
225	M₁₆	—	26	2	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
233	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
234	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	26	2	no	—
237	C₂²⋊C₄	—	26	2	no	—
238	C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
239	C₄²	—	2	2	no	—
240	C₄²	C₂²×D₁₃	1	2	yes	—
241	C₁₃	#3045	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	1	no	—
243	C₂³	—	26	2	no	—
244	C₂³	—	26	1	no	—
245	C₂³	—	52	2	no	—
246	C₂³	—	104	1	no	—
247	C₂³	—	104	1	no	—
248	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	26	1	no	—
251	Q₈	—	26	1	no	—
252	Q₈	C₂×D₅₂	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	26	2	no	—
256	D₄	—	26	1	no	—
257	D₄	—	26	2	no	—
258	D₄	—	26	1	no	—
259	D₄	—	52	2	no	—
260	D₄	—	52	2	no	—
261	D₄	—	104	2	no	—
262	D₄	—	104	2	no	—
263	D₄	C₂×D₅₂	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	26	1	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	C₂×D₅₂	1	2	yes	—
272	C₂×C₄	#6127	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
274	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
277	C₈	—	26	1	no	—
278	C₈	—	26	1	no	—
279	C₈	—	26	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	52	1	no	—
286	C₂²	—	52	2	no	—
287	C₂²	—	52	1	no	—
288	C₂²	—	52	2	no	—
289	C₂²	—	52	1	no	—
290	C₂²	—	52	2	no	—
291	C₂²	—	104	4	no	—
292	C₂²	#88184	1	2	yes	—
293	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	52	1	no	—
299	C₄	#88184	1	2	yes	—
300	C₄	C₂²×D₅₂	1	1	yes	characteristic
301	C₄	C₂×#6127	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	104	2	no	—
305	C₂	—	104	1	no	—
306	C₂	C₂×#88184	1	1	yes	characteristic
307	C₂	#11719540	1	1	yes	characteristic
308	C₂	#11719542	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×#520	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#520	D₁₃
hypocenter	C₁₃	#3045
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₁₃×#520	C₂	semidirect
2. C₂×#88204	C₂	semidirect
3. C₂×#88203	C₂	semidirect
4. #11719230	C₂	semidirect
5. #11718689	C₂	semidirect
6. #11718688	C₂	semidirect
7. #11718677	C₂	semidirect
8. #11718676	C₂	semidirect
9. #11718660	C₂	semidirect
10. #11718627	C₂	semidirect
11. C₂×#88049	C₂	semidirect
15. C₄×Q₈×C₁₃	C₂²	semidirect
17. C₄×D₄×C₁₃	C₂²	semidirect
24. C₂×#6107	C₂²	semidirect
26. C₂×#6106	C₂²	semidirect
27. C₂×#6105	C₂²	semidirect
29. C₂×#6104	C₂²	semidirect
37. #88050	C₂²	semidirect
40. #88049	C₂²	semidirect
41. C₄⋊#1055	C₂²	semidirect
42. C₄⋊#1055	C₂²	semidirect
45. C₁₃×#41	D₄	semidirect
129. #520	D₁₃	semidirect
177. C₄×Q₈	C₂×D₁₃	semidirect
179. C₄×D₄	C₂×D₁₃	semidirect
201. #41	D₅₂	semidirect
241. C₁₃	#3045	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#520	2
C₂×#88204	2
C₂×#88203	2
#11719230	2
#11718689	2
#11718688	2
#11718677	2
#11718676	2
#11718660	2
#11718627	2
C₂×#88049	2
#3045	13