Subgroups — #13124373

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	C₂×C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719535	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719532	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88203	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718893	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718846	C₂	1	2	yes	maximal, complemented
7	#11718826	C₂	1	2	yes	maximal, complemented
8	#11718824	C₂	1	2	yes	maximal, complemented
9	#11718811	C₂	1	2	yes	maximal, complemented
10	C₂×#88061	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718763	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×D₅₂	C₂²	1	1	yes	characteristic
15	C₁₃×#135	—	2	4	no	—
16	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#88203	—	2	4	no	—
22	#88201	C₂²	1	2	yes	—
23	Q₈×Dic₁₃	C₂²	1	2	yes	complemented
24	C₂×#6106	C₂²	1	2	yes	—
25	#88194	C₂²	1	2	yes	—
26	#88191	—	2	2	no	—
27	D₄×Dic₁₃	—	2	2	no	—
28	D₄×Dic₁₃	C₂²	1	2	yes	complemented
29	C₂×#6104	C₂²	1	2	yes	—
30	#88184	—	2	1	no	—
31	#88184	—	2	1	no	—
32	#88180	—	2	1	no	—
33	#88180	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
36	C₂×#6099	C₂²	1	1	yes	characteristic
37	#88075	C₂²	1	2	yes	complemented
38	#88072	C₂²	1	2	yes	complemented
39	#88061	C₂²	1	2	yes	—
40	#88061	C₂²	1	2	yes	—
41	#88054	C₂²	1	2	yes	complemented
42	#88054	C₂²	1	2	yes	complemented
43	C₂³×D₁₃	—	2	1	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₁₃×#41	—	2	2	no	—
46	C₁₃×#41	—	2	2	no	—
47	C₁₃×#41	D₄	1	4	yes	complemented
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₃	—	2	2	no	—
58	C₂×C₂²⋊D₁₃	—	2	2	no	—
59	C₂²×Dic₁₃	—	2	1	no	—
60	C₂²×Dic₁₃	—	2	2	no	—
61	C₂×D₅₂	—	2	2	no	—
62	C₂×D₅₂	—	2	2	no	—
63	C₂×D₅₂	—	2	1	no	—
64	C₂×D₅₂	C₂³	1	2	yes	—
65	C₁₃×Q₁₆	—	2	2	no	—
66	C₁₃×SD₁₆	—	2	2	no	—
67	C₁₃×SD₁₆	—	2	2	no	—
68	C₁₃×D₈	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₂²⋊Dic₁₃	—	2	2	no	—
73	C₂²⋊Dic₁₃	D₄	1	2	yes	—
74	#6106	—	2	4	no	—
75	#6104	—	2	4	no	—
76	#6103	—	2	2	no	—
77	#6103	—	2	2	no	—
78	C₄⋊Dic₁₃	—	2	1	no	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
81	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
82	C₄⋊Dic₁₃	C₂³	1	2	yes	—
83	C₄×Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	C₂³	1	2	yes	—
85	#6099	—	2	2	no	—
86	C₂×#1055	C₂³	1	2	yes	—
87	#3045	—	13	1	no	maximal, Sylow
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	2	1	no	—
90	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×D₁₃	—	2	2	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	—	2	1	no	—
97	Q₈×C₁₃	—	2	1	no	—
98	Q₈×C₁₃	C₂×D₄	1	2	yes	—
99	D₄×C₁₃	—	2	4	no	—
100	D₄×C₁₃	—	2	1	no	—
101	D₄×C₁₃	—	2	1	no	—
102	D₄×C₁₃	—	2	1	no	—
103	D₄×C₁₃	—	4	2	no	—
104	D₄×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	#41	1	2	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₁₃	—	8	2	no	—
114	C₂²⋊D₁₃	—	8	2	no	—
115	C₂×Dic₁₃	—	2	2	no	—
116	C₂×Dic₁₃	—	8	2	no	—
117	C₂×Dic₁₃	—	8	1	no	—
118	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
119	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
120	D₅₂	—	2	2	no	—
121	D₅₂	—	2	2	no	—
122	D₅₂	—	4	2	no	—
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	#1055	—	2	2	no	—
126	C₂×AGL(1,ℤ/8)	—	13	1	no	—
127	C₂×#135	D₁₃	1	1	yes	characteristic, complemented
128	#535	—	13	1	no	—
129	#520	—	13	1	no	—
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	—	4	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	#121	1	2	yes	—
146	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₃	—	4	2	no	—
148	C₂×D₁₃	—	4	2	no	—
149	C₂×D₁₃	—	4	2	no	—
150	C₂×D₁₃	—	8	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	#121	1	2	yes	—
154	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	Dic₁₃	—	2	2	no	—
157	Dic₁₃	—	2	2	no	—
158	Dic₁₃	—	4	2	no	—
159	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
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	—	2	4	no	—
164	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
165	C₂×SD₁₆	—	13	2	no	—
166	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
168	C₂×D₈	—	13	2	no	—
169	C₂×D₈	C₂×D₁₃	1	1	yes	characteristic, complemented
170	C₂×M₁₆	—	13	1	no	—
171	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
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	—	13	2	no	—
178	C₄×D₄	—	13	2	no	—
179	C₄×D₄	—	26	2	no	—
180	#117	—	13	1	no	—
181	#117	—	26	1	no	—
182	C₂×C₄⋊C₄	—	13	1	no	—
183	C₂×C₄⋊C₄	—	26	1	no	—
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	—	8	1	no	—
193	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
194	C₂×C₁₃	#576	1	1	yes	characteristic
195	C₂×C₁₃	#577	1	1	yes	characteristic
196	D₁₃	—	8	2	no	—
197	C₂⁴	—	26	1	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	C₂²⋊D₁₃	1	4	yes	complemented
202	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	13	2	no	—
207	C₂×D₄	—	26	2	no	—
208	C₂×D₄	—	26	2	no	—
209	C₂×D₄	—	26	1	no	—
210	C₂×D₄	—	26	2	no	—
211	C₂×D₄	—	26	2	no	—
212	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
213	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	26	2	no	—
217	C₂²×C₄	—	26	1	no	—
218	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
219	Q₁₆	—	2	2	no	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	26	4	no	—
223	D₈	—	2	2	no	—
224	D₈	—	26	4	no	—
225	M₁₆	—	26	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	13	1	no	—
231	C₄⋊C₄	—	13	1	no	—
232	C₄⋊C₄	—	13	2	no	—
233	C₄⋊C₄	—	26	2	no	—
234	C₄⋊C₄	—	26	1	no	—
235	C₂²⋊C₄	—	13	2	no	—
236	C₂²⋊C₄	—	26	2	no	—
237	C₂²⋊C₄	—	26	2	no	—
238	C₂²⋊C₄	—	26	2	no	—
239	C₄²	—	13	2	no	—
240	C₄²	—	26	2	no	—
241	C₁₃	#3045	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	26	2	no	—
245	C₂³	—	52	2	no	—
246	C₂³	—	104	1	no	—
247	C₂³	—	104	1	no	—
248	C₂³	D₄×D₁₃	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	2	1	no	—
252	Q₈	C₂×C₂²⋊D₁₃	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	2	1	no	—
256	D₄	—	2	1	no	—
257	D₄	—	4	2	no	—
258	D₄	—	26	2	no	—
259	D₄	—	26	2	no	—
260	D₄	—	52	2	no	—
261	D₄	—	104	2	no	—
262	D₄	—	104	2	no	—
263	D₄	C₂×C₂²⋊D₁₃	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	—	104	2	no	—
272	C₂×C₄	—	104	1	no	—
273	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
274	C₂×C₄	#6129	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
276	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	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	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	52	2	no	—
289	C₂²	—	52	2	no	—
290	C₂²	—	52	2	no	—
291	C₂²	—	104	4	no	—
292	C₂²	#88193	1	2	yes	—
293	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	26	2	no	—
297	C₄	—	26	2	no	—
298	C₄	—	52	2	no	—
299	C₄	#88193	1	2	yes	—
300	C₄	C₂×#6129	1	1	yes	characteristic
301	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	1	no	—
305	C₂	—	104	2	no	—
306	C₂	#11719474	1	1	yes	characteristic
307	C₂	#11719475	1	1	yes	characteristic
308	C₂	C₂×#88193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₂×#135	D₁₃
hypocenter	C₁₃	#3045
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#135	C₂	semidirect
2. #11719535	C₂	semidirect
3. #11719532	C₂	semidirect
4. C₂×#88203	C₂	semidirect
5. #11718893	C₂	semidirect
6. #11718846	C₂	semidirect
7. #11718826	C₂	semidirect
8. #11718824	C₂	semidirect
9. #11718811	C₂	semidirect
10. C₂×#88061	C₂	semidirect
11. #11718763	C₂	semidirect
16. C₂×C₁₃×Q₁₆	C₂²	semidirect
17. C₂×C₁₃×SD₁₆	C₂²	semidirect
18. C₂×C₁₃×SD₁₆	C₂²	semidirect
19. C₂×C₁₃×D₈	C₂²	semidirect
23. Q₈×Dic₁₃	C₂²	semidirect
28. D₄×Dic₁₃	C₂²	semidirect
37. #88075	C₂²	semidirect
38. #88072	C₂²	semidirect
41. #88054	C₂²	semidirect
42. #88054	C₂²	semidirect
47. C₁₃×#41	D₄	semidirect
127. C₂×#135	D₁₃	semidirect
164. C₂×Q₁₆	C₂×D₁₃	semidirect
166. C₂×SD₁₆	C₂×D₁₃	semidirect
167. C₂×SD₁₆	C₂×D₁₃	semidirect
169. C₂×D₈	C₂×D₁₃	semidirect
201. #41	C₂²⋊D₁₃	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
C₂×C₁₃×#135	2
#11719535	2
#11719532	2
C₂×#88203	2
#11718893	2
#11718846	2
#11718826	2
#11718824	2
#11718811	2
C₂×#88061	2
#11718763	2
#3045	13