Subgroups — #13123643

Order 1664.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88162	C₂	1	1	yes	maximal, characteristic
2	#11719227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718662	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718627	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718550	C₂	1	2	yes	maximal, complemented
9	#11718533	C₂	1	2	yes	maximal, complemented
10	#11718530	C₂	1	2	yes	maximal, complemented
11	#11718513	C₂	1	2	yes	maximal, complemented
12	C₂×#6127	C₂²	1	1	yes	characteristic
13	C₂²×D₅₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	—	2	1	no	—
15	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#117	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×#6103	—	2	1	no	—
19	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×#6099	C₂²	1	1	yes	characteristic
21	#88162	C₄	1	8	yes	complemented
22	C₂×D₁₀₄	C₂²	1	2	yes	—
23	C₂×#6095	C₂²	1	2	yes	—
24	#88147	—	2	1	no	—
25	#88147	—	2	1	no	—
26	#88147	C₂²	1	2	yes	—
27	#88147	C₂²	1	2	yes	—
28	#88147	C₂²	1	1	yes	characteristic, complemented
29	#88147	C₂²	1	1	yes	characteristic, complemented
30	#88141	C₂²	1	1	yes	characteristic, complemented
31	#88141	C₂²	1	1	yes	characteristic, complemented
32	#88135	—	2	1	no	—
33	#88135	—	2	1	no	—
34	#88126	—	2	1	no	—
35	C₄²×D₁₃	—	2	1	no	—
36	C₁₃×#107	C₂²	1	2	yes	complemented
37	C₁₃×#106	C₂²	1	2	yes	complemented
38	#88055	C₂²	1	2	yes	complemented
39	C₈×Dic₁₃	C₂²	1	2	yes	complemented
40	#88050	C₂²	1	2	yes	complemented
41	#88049	C₂²	1	2	yes	—
42	#88049	C₂²	1	2	yes	—
43	#88049	C₂²	1	2	yes	complemented
44	C₂³×D₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₃	—	2	1	no	—
48	C₂²×Dic₁₃	D₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	D₄	1	1	yes	characteristic
50	#6127	—	2	2	no	—
51	#6127	C₂×C₄	1	4	yes	complemented
52	C₂×D₅₂	—	2	1	no	—
53	C₂×D₅₂	—	2	2	no	—
54	C₂×D₅₂	C₂×C₄	1	4	yes	complemented
55	C₂×D₅₂	C₂³	1	2	yes	—
56	C₂×D₅₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₃	—	2	1	no	—
58	C₂×C₄×D₁₃	—	2	1	no	—
59	C₂×C₄×D₁₃	—	2	1	no	—
60	C₂×C₄×D₁₃	—	2	2	no	—
61	C₂×C₄×D₁₃	—	2	2	no	—
62	C₂×C₄×D₁₃	—	4	4	no	—
63	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
64	C₁₃×M₁₆	C₂×C₄	1	4	yes	complemented
65	C₂×C₈×C₁₃	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	#6103	—	2	1	no	—
73	#6103	—	2	1	no	—
74	#6103	—	2	2	no	—
75	#6103	—	2	2	no	—
76	#6101	—	2	1	no	—
77	#6101	—	2	1	no	—
78	C₄×Dic₁₃	—	2	1	no	—
79	C₄×Dic₁₃	—	2	1	no	—
80	C₄×Dic₁₃	D₄	1	1	yes	characteristic
81	C₄×Dic₁₃	D₄	1	1	yes	characteristic
82	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
84	#6099	—	2	2	no	—
85	C₂×#1055	C₂³	1	2	yes	—
86	D₁₀₄	C₂×C₄	1	8	yes	complemented
87	#6095	C₂×C₄	1	8	yes	complemented
88	C₄×AGL(1,ℤ/8)	—	13	1	no	maximal, Sylow
89	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₃	—	2	4	no	—
91	C₂²×D₁₃	—	2	1	no	—
92	C₂²×D₁₃	—	2	2	no	—
93	C₂²×D₁₃	—	4	2	no	—
94	C₂²×D₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#41	1	2	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₁₃	—	2	4	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	4	1	no	—
107	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	D₅₂	—	2	4	no	—
112	D₅₂	—	2	2	no	—
113	D₅₂	—	2	1	no	—
114	D₅₂	C₂²×C₄	1	4	yes	—
115	D₅₂	C₂²×C₄	1	2	yes	complemented
116	C₄×D₁₃	—	2	4	no	—
117	C₄×D₁₃	—	4	2	no	—
118	C₄×D₁₃	—	4	4	no	—
119	C₄×D₁₃	—	4	2	no	—
120	C₄×D₁₃	—	4	4	no	—
121	Dic₂₆	—	2	1	no	—
122	Dic₂₆	C₂²×C₄	1	2	yes	complemented
123	C₈×C₁₃	C₂²×C₄	1	4	yes	complemented
124	#1055	—	2	2	no	—
125	C₂×AGL(1,ℤ/8)	—	13	1	no	—
126	C₄×#41	—	13	1	no	—
127	C₂×C₄×D₄	—	13	1	no	—
128	#442	—	13	2	no	—
129	#440	—	13	2	no	—
130	C₄×SD₁₆	—	13	2	no	—
131	C₄×D₈	—	13	2	no	—
132	#428	D₁₃	1	1	yes	characteristic, complemented
133	#418	—	13	1	no	—
134	#417	—	13	1	no	—
135	C₄×M₁₆	—	13	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	C₄×D₄	1	2	yes	—
139	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	2	4	no	—
142	C₂×D₁₃	—	4	8	no	—
143	C₂×D₁₃	—	4	2	no	—
144	C₂×D₁₃	—	4	1	no	—
145	C₂×D₁₃	—	4	1	no	—
146	C₂×D₁₃	—	4	2	no	—
147	C₄×C₁₃	—	4	2	no	—
148	C₄×C₁₃	—	4	1	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	C₄×D₄	1	2	yes	—
151	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	—	2	1	no	—
155	Dic₁₃	—	2	1	no	—
156	Dic₁₃	—	2	1	no	—
157	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	complemented
158	C₂×#41	—	13	1	no	—
159	C₂²×D₄	—	13	1	no	—
160	C₂³×C₄	—	26	1	no	—
161	AGL(1,ℤ/8)	—	13	8	no	—
162	C₂×SD₁₆	—	13	2	no	—
163	C₂×D₈	—	13	2	no	—
164	C₂×M₁₆	—	13	1	no	—
165	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
166	C₄×Q₈	—	13	1	no	—
167	C₄×Q₈	—	13	1	no	—
168	C₄×D₄	—	13	2	no	—
169	C₄×D₄	—	13	2	no	—
170	C₄×D₄	—	13	1	no	—
171	C₄×D₄	—	13	1	no	—
172	C₄×D₄	—	26	1	no	—
173	C₄×D₄	—	26	1	no	—
174	C₄×D₄	—	26	1	no	—
175	C₄×D₄	—	26	1	no	—
176	#117	—	26	1	no	—
177	#117	C₂×D₁₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	—	26	1	no	—
180	C₂×C₄²	—	13	1	no	—
181	C₂×C₄²	—	26	1	no	—
182	#107	C₂×D₁₃	1	2	yes	complemented
183	#106	C₂×D₁₃	1	2	yes	complemented
184	#103	—	13	2	no	—
185	#102	—	13	2	no	—
186	#102	—	13	2	no	—
187	#102	—	13	2	no	—
188	#97	—	13	2	no	—
189	C₄×C₈	—	13	2	no	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₂×C₁₃	#575	1	1	yes	characteristic
194	D₁₃	—	4	2	no	—
195	D₁₃	—	4	4	no	—
196	C₂⁴	—	26	1	no	—
197	#41	—	13	4	no	—
198	#41	—	26	2	no	—
199	C₂×Q₈	—	13	1	no	—
200	C₂×D₄	—	13	4	no	—
201	C₂×D₄	—	13	1	no	—
202	C₂×D₄	—	13	2	no	—
203	C₂×D₄	—	26	1	no	—
204	C₂×D₄	—	26	1	no	—
205	C₂×D₄	—	26	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	13	1	no	—
209	C₂²×C₄	—	26	1	no	—
210	C₂²×C₄	—	26	1	no	—
211	C₂²×C₄	—	26	2	no	—
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	—	26	1	no	—
214	C₂²×C₄	—	52	4	no	—
215	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
216	SD₁₆	—	13	8	no	—
217	D₈	—	13	8	no	—
218	M₁₆	—	26	2	no	—
219	M₁₆	C₄×D₁₃	1	4	yes	complemented
220	C₂×C₈	—	13	2	no	—
221	C₂×C₈	C₂²×D₁₃	1	2	yes	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	26	1	no	—
224	C₄⋊C₄	—	26	1	no	—
225	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	26	2	no	—
230	C₂²⋊C₄	—	26	2	no	—
231	C₂²⋊C₄	—	26	1	no	—
232	C₂²⋊C₄	—	26	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	13	1	no	—
235	C₄²	—	13	1	no	—
236	C₄²	—	13	1	no	—
237	C₄²	—	13	1	no	—
238	C₄²	—	26	1	no	—
239	C₄²	—	26	1	no	—
240	C₁₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	26	4	no	—
242	C₂³	—	26	1	no	—
243	C₂³	—	26	2	no	—
244	C₂³	—	52	2	no	—
245	C₂³	—	52	2	no	—
246	C₂³	D₄×D₁₃	1	1	yes	characteristic
247	Q₈	—	13	2	no	—
248	Q₈	—	26	1	no	—
249	D₄	—	13	2	no	—
250	D₄	—	13	4	no	—
251	D₄	—	26	4	no	—
252	D₄	—	26	4	no	—
253	D₄	—	26	2	no	—
254	D₄	—	26	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	1	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	26	1	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	—	52	2	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	—	52	2	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	52	1	no	—
271	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
272	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	#6131	1	2	yes	—
274	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
275	C₈	—	26	2	no	—
276	C₈	C₂×C₄×D₁₃	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	26	2	no	—
280	C₂²	—	26	4	no	—
281	C₂²	—	52	8	no	—
282	C₂²	—	52	2	no	—
283	C₂²	—	52	1	no	—
284	C₂²	—	52	1	no	—
285	C₂²	—	52	2	no	—
286	C₂²	#88147	1	2	yes	—
287	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	13	2	no	—
292	C₄	—	26	2	no	—
293	C₄	—	26	1	no	—
294	C₄	—	26	1	no	—
295	C₄	—	26	1	no	—
296	C₄	#88147	1	2	yes	—
297	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
298	C₄	C₂×#6131	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	52	4	no	—
301	C₂	—	52	2	no	—
302	C₂	C₂×#88147	1	1	yes	characteristic
303	C₂	D₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂	#11719482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#428	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#88162	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#428	D₁₃
hypocenter	C₁₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #11719227	C₂	semidirect
3. C₂×#88147	C₂	semidirect
4. C₁₃×#428	C₂	semidirect
5. M₁₆×Dic₁₃	C₂	semidirect
6. #11718662	C₂	semidirect
7. #11718627	C₂	semidirect
8. #11718550	C₂	semidirect
9. #11718533	C₂	semidirect
10. #11718530	C₂	semidirect
11. #11718513	C₂	semidirect
21. #88162	C₄	semidirect
28. #88147	C₂²	semidirect
29. #88147	C₂²	semidirect
30. #88141	C₂²	semidirect
31. #88141	C₂²	semidirect
36. C₁₃×#107	C₂²	semidirect
37. C₁₃×#106	C₂²	semidirect
38. #88055	C₂²	semidirect
39. C₈×Dic₁₃	C₂²	semidirect
40. #88050	C₂²	semidirect
43. #88049	C₂²	semidirect
51. #6127	C₂×C₄	semidirect
54. C₂×D₅₂	C₂×C₄	semidirect
64. C₁₃×M₁₆	C₂×C₄	semidirect
86. D₁₀₄	C₂×C₄	semidirect
87. #6095	C₂×C₄	semidirect
115. D₅₂	C₂²×C₄	semidirect
122. Dic₂₆	C₂²×C₄	semidirect
123. C₈×C₁₃	C₂²×C₄	semidirect
132. #428	D₁₃	semidirect
157. Dic₁₃	AGL(1,ℤ/8)	semidirect
182. #107	C₂×D₁₃	semidirect
183. #106	C₂×D₁₃	semidirect
219. M₁₆	C₄×D₁₃	semidirect
240. C₁₃	C₄×AGL(1,ℤ/8)	semidirect
276. C₈	C₂×C₄×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88162	2
#11719227	2
C₂×#88147	2
C₁₃×#428	2
M₁₆×Dic₁₃	2
#11718662	2
#11718627	2
#11718550	2
#11718533	2
#11718530	2
#11718513	2
C₄×AGL(1,ℤ/8)	13