Subgroups — #13123735

Order 1664.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719527	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88161	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718785	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718666	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718658	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718485	C₂	1	4	yes	maximal, complemented
9	#11718484	C₂	1	4	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂×#6127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#117	C₂²	1	1	yes	characteristic
14	#88205	C₄	1	2	yes	complemented
15	#88204	C₂²	1	4	yes	—
16	#88203	C₄	1	2	yes	complemented
17	#88202	—	2	1	no	—
18	#88202	—	2	1	no	—
19	#88202	C₂²	1	2	yes	complemented
20	#88202	C₂²	1	2	yes	complemented
21	#88196	C₄	1	2	yes	complemented
22	#88186	C₄	1	2	yes	complemented
23	C₂×#6099	—	2	1	no	—
24	C₂×#6099	C₂²	1	1	yes	characteristic
25	C₂×#6099	C₂²	1	1	yes	characteristic
26	C₂²×#1055	—	2	1	no	—
27	C₂²×#1055	C₂²	1	1	yes	characteristic
28	#88161	—	2	1	no	—
29	#88161	—	2	1	no	—
30	#88161	C₂²	1	2	yes	complemented
31	#88161	C₂²	1	2	yes	complemented
32	C₂×#6094	—	2	1	no	—
33	C₂×C₈×D₁₃	—	2	1	no	—
34	C₁₃×#104	C₂²	1	4	yes	complemented
35	#88062	C₂²	1	4	yes	complemented
36	#88045	C₂²	1	4	yes	complemented
37	#88043	C₂²	1	2	yes	complemented
38	C₄×#1055	C₂²	1	2	yes	complemented
39	C₁₃×#41	—	2	1	no	—
40	C₁₃×#41	—	2	1	no	—
41	C₁₃×#41	C₂×C₄	1	2	yes	—
42	C₁₃×#41	C₂³	1	2	yes	—
43	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₃	—	2	1	no	—
49	#6127	—	2	1	no	—
50	#6127	—	2	1	no	—
51	#6127	C₂×C₄	1	2	yes	—
52	#6127	C₂³	1	2	yes	—
53	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	—	2	1	no	—
55	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
56	C₁₃×M₁₆	—	2	1	no	—
57	C₁₃×M₁₆	C₂³	1	2	yes	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₂²⋊C₄	—	2	1	no	—
61	C₄²×C₁₃	C₂³	1	2	yes	—
62	#6107	C₂×C₄	1	4	yes	—
63	#6106	C₂×C₄	1	4	yes	—
64	#6105	C₂×C₄	1	4	yes	—
65	#6104	C₂×C₄	1	4	yes	—
66	#6099	—	2	1	no	—
67	#6099	—	2	2	no	—
68	#6099	—	2	2	no	—
69	#6099	C₂×C₄	1	2	yes	—
70	#6099	C₂×C₄	1	2	yes	—
71	#6099	C₂³	1	2	yes	—
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	—	2	2	no	—
75	C₂×#1055	D₄	1	1	yes	characteristic
76	C₂×#1055	D₄	1	1	yes	characteristic
77	C₂×#1055	D₄	1	1	yes	characteristic
78	C₂×#1055	D₄	1	1	yes	characteristic
79	C₂×#1055	C₂³	1	2	yes	—
80	C₂×#1055	C₂³	1	1	yes	characteristic
81	C₂×#1055	C₂³	1	1	yes	characteristic
82	#6094	—	2	2	no	—
83	#6094	—	2	2	no	—
84	C₈×D₁₃	—	2	2	no	—
85	C₈×D₁₃	—	2	2	no	—
86	#2955	—	13	1	no	maximal, Sylow
87	C₂³×C₁₃	—	2	1	no	—
88	C₂³×C₁₃	#41	1	1	yes	characteristic
89	C₂²×D₁₃	—	2	1	no	—
90	Q₈×C₁₃	—	2	1	no	—
91	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
92	D₄×C₁₃	—	2	2	no	—
93	D₄×C₁₃	—	2	2	no	—
94	D₄×C₁₃	—	2	1	no	—
95	D₄×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₃	—	2	2	no	—
107	C₂²⋊D₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	D₅₂	—	2	1	no	—
110	D₅₂	C₂²×C₄	1	2	yes	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	—	2	2	no	—
113	Dic₂₆	—	2	1	no	—
114	Dic₂₆	C₂²×C₄	1	2	yes	—
115	C₈×C₁₃	—	2	2	no	—
116	#1055	—	2	2	no	—
117	#1055	C₂²×C₄	1	4	yes	—
118	#1055	C₂×D₄	1	1	yes	characteristic
119	#1055	C₂×D₄	1	1	yes	characteristic
120	#1055	C₂×D₄	1	1	yes	characteristic
121	#1055	C₂×D₄	1	1	yes	characteristic
122	#575	—	13	1	no	—
123	C₂×#131	—	13	1	no	—
124	C₂×#131	—	13	1	no	—
125	#444	—	13	4	no	—
126	#443	—	13	4	no	—
127	#430	—	13	1	no	—
128	#421	—	13	1	no	—
129	#421	D₁₃	1	1	yes	characteristic, complemented
130	#405	—	13	1	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	4	1	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×D₁₃	—	2	2	no	—
138	C₂×D₁₃	—	4	2	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	Dic₁₃	—	2	2	no	—
146	C₂×#41	—	13	1	no	—
147	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
148	#137	—	13	2	no	—
149	#137	—	13	2	no	—
150	AGL(1,ℤ/8)	—	13	2	no	—
151	AGL(1,ℤ/8)	—	13	2	no	—
152	#135	—	13	4	no	—
153	#131	—	13	2	no	—
154	#131	—	13	2	no	—
155	#131	—	13	2	no	—
156	#131	—	13	2	no	—
157	#131	—	26	1	no	—
158	#131	—	26	1	no	—
159	#131	—	26	1	no	—
160	#131	—	26	1	no	—
161	C₂×M₁₆	—	13	1	no	—
162	C₂×M₁₆	—	13	1	no	—
163	C₂×M₁₆	—	26	1	no	—
164	C₂×M₁₆	—	26	1	no	—
165	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	—	13	1	no	—
167	C₂²×C₈	—	26	1	no	—
168	C₂²×C₈	—	26	1	no	—
169	#117	C₂×D₁₃	1	1	yes	characteristic
170	#108	—	13	4	no	—
171	#104	—	13	4	no	—
172	#104	C₂×D₁₃	1	4	yes	complemented
173	#97	—	13	2	no	—
174	C₄×C₈	—	13	2	no	—
175	C₂×C₁₃	—	2	1	no	—
176	C₂×C₁₃	—	2	1	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	4	2	no	—
179	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	D₁₃	—	4	2	no	—
181	#41	—	2	1	no	—
182	#41	—	2	1	no	—
183	#41	—	13	2	no	—
184	#41	—	13	2	no	—
185	#41	—	26	1	no	—
186	#41	—	26	1	no	—
187	#41	C₄×D₁₃	1	2	yes	—
188	#41	C₂²×D₁₃	1	2	yes	—
189	C₂×Q₈	—	13	1	no	—
190	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	13	1	no	—
193	C₂×D₄	—	26	1	no	—
194	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	26	1	no	—
197	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
198	Q₁₆	—	13	4	no	—
199	SD₁₆	—	13	4	no	—
200	SD₁₆	—	13	4	no	—
201	D₈	—	13	4	no	—
202	M₁₆	—	2	1	no	—
203	M₁₆	—	13	2	no	—
204	M₁₆	—	13	2	no	—
205	M₁₆	—	13	2	no	—
206	M₁₆	—	26	2	no	—
207	M₁₆	—	26	2	no	—
208	M₁₆	—	26	2	no	—
209	M₁₆	—	26	2	no	—
210	M₁₆	—	26	1	no	—
211	M₁₆	C₂²×D₁₃	1	2	yes	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	13	1	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	26	2	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₄²	C₂²×D₁₃	1	2	yes	—
228	C₁₃	#2955	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	2	1	no	—
230	C₂³	—	26	1	no	—
231	C₂³	#6129	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	13	2	no	—
234	Q₈	—	26	1	no	—
235	Q₈	C₂×C₄×D₁₃	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	13	2	no	—
240	D₄	—	26	2	no	—
241	D₄	—	26	2	no	—
242	D₄	—	26	1	no	—
243	D₄	C₂×C₄×D₁₃	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
254	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂×C₄	#6129	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	13	4	no	—
259	C₈	—	13	1	no	—
260	C₈	—	13	1	no	—
261	C₈	—	13	1	no	—
262	C₈	—	13	1	no	—
263	C₈	—	26	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	26	2	no	—
268	C₂²	—	52	2	no	—
269	C₂²	#88135	1	1	yes	characteristic
270	C₂²	#88135	1	1	yes	characteristic
271	C₂²	C₂×#6129	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	26	2	no	—
275	C₄	#88135	1	1	yes	characteristic
276	C₄	#88135	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
278	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	52	2	no	—
284	C₂	C₂×#88135	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#421	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#11719527	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#421	D₁₃
hypocenter	C₁₃	#2955
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×#88161	C₂	semidirect
4. C₁₃×#421	C₂	semidirect
5. #11718785	C₂	semidirect
6. #11718666	C₂	semidirect
7. #11718658	C₂	semidirect
8. #11718485	C₂	semidirect
9. #11718484	C₂	semidirect
14. #88205	C₄	semidirect
16. #88203	C₄	semidirect
19. #88202	C₂²	semidirect
20. #88202	C₂²	semidirect
21. #88196	C₄	semidirect
22. #88186	C₄	semidirect
30. #88161	C₂²	semidirect
31. #88161	C₂²	semidirect
34. C₁₃×#104	C₂²	semidirect
35. #88062	C₂²	semidirect
36. #88045	C₂²	semidirect
37. #88043	C₂²	semidirect
38. C₄×#1055	C₂²	semidirect
129. #421	D₁₃	semidirect
172. #104	C₂×D₁₃	semidirect
228. C₁₃	#2955	semidirect

Maximal subgroups

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

subgroup	index
#11719527	2
C₂×#88202	2
C₂×#88161	2
C₁₃×#421	2
#11718785	2
#11718666	2
#11718658	2
#11718485	2
#11718484	2
#2955	13