Subgroups — #13123682

Order 1664.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719532	C₂	1	1	yes	maximal, characteristic
2	C₂×#88205	C₂	1	1	yes	maximal, characteristic
3	#11719221	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#416	C₂	1	1	yes	maximal, characteristic
5	C₂×#88056	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718762	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718656	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718451	C₂	1	2	yes	maximal, complemented
9	#11718450	C₂	1	2	yes	maximal, complemented
10	#11718419	C₂	1	2	yes	maximal, complemented
11	#11718418	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#117	C₂²	1	1	yes	characteristic
16	#88205	C₄	1	8	yes	complemented
17	Q₈×Dic₁₃	C₂²	1	2	yes	—
18	C₂×#6107	C₂²	1	2	yes	—
19	D₄×Dic₁₃	—	2	2	no	—
20	D₄×Dic₁₃	C₂²	1	2	yes	—
21	C₂×#6105	C₂²	1	2	yes	—
22	#88180	—	2	1	no	—
23	#88180	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	1	no	—
25	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#6099	C₂²	1	1	yes	characteristic
27	#88141	C₂²	1	2	yes	—
28	#88141	C₂²	1	2	yes	—
29	#88131	—	2	2	no	—
30	C₄×Dic₂₆	—	2	1	no	—
31	C₄×Dic₂₆	—	2	1	no	—
32	C₁₃×#103	C₂²	1	1	yes	characteristic
33	C₁₃×#103	C₂²	1	1	yes	characteristic
34	C₁₃×#102	C₂²	1	1	yes	characteristic
35	C₁₃×#102	C₂²	1	1	yes	characteristic
36	#88056	C₂²	1	2	yes	—
37	#88056	C₂²	1	2	yes	—
38	#88055	C₂²	1	2	yes	—
39	C₈×Dic₁₃	C₂²	1	2	yes	—
40	#88048	C₂²	1	2	yes	—
41	#88047	C₂²	1	2	yes	—
42	C₁₃×#41	—	2	2	no	—
43	C₁₃×#41	C₂×C₄	1	4	yes	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	—	2	1	no	—
50	C₂²×Dic₁₃	—	2	1	no	—
51	C₂²×Dic₁₃	—	2	2	no	—
52	C₂×Dic₂₆	—	2	1	no	—
53	C₂×Dic₂₆	—	2	2	no	—
54	C₂×Dic₂₆	C₂×C₄	1	4	yes	complemented
55	C₂×Dic₂₆	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₄	—	2	1	no	—
62	C₄²×C₁₃	—	2	1	no	—
63	C₂²⋊Dic₁₃	—	2	2	no	—
64	C₂²⋊Dic₁₃	D₄	1	2	yes	—
65	#6107	C₂×C₄	1	8	yes	complemented
66	#6105	C₂×C₄	1	8	yes	complemented
67	C₄⋊Dic₁₃	—	2	1	no	—
68	C₄⋊Dic₁₃	—	2	2	no	—
69	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
70	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
71	C₄⋊Dic₁₃	C₂³	1	2	yes	—
72	#6101	—	2	2	no	—
73	#6101	—	2	2	no	—
74	C₄×Dic₁₃	—	2	2	no	—
75	C₄×Dic₁₃	—	2	2	no	—
76	C₄×Dic₁₃	—	2	2	no	—
77	C₄×Dic₁₃	C₂³	1	2	yes	—
78	#6099	C₂×C₄	1	4	yes	complemented
79	C₂×#1055	C₂³	1	2	yes	—
80	#2947	—	13	1	no	maximal, Sylow
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₁₃	—	2	1	no	—
84	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
85	D₄×C₁₃	—	2	4	no	—
86	D₄×C₁₃	—	2	1	no	—
87	D₄×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#41	1	2	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×Dic₁₃	—	2	4	no	—
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	1	no	—
101	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
102	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
103	Dic₂₆	—	2	4	no	—
104	Dic₂₆	—	2	2	no	—
105	Dic₂₆	C₂²×C₄	1	4	yes	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	—	2	1	no	—
108	#1055	C₂²×C₄	1	4	yes	—
109	C₂×#137	—	13	1	no	—
110	#520	—	13	1	no	—
111	#519	—	13	1	no	—
112	#441	—	13	2	no	—
113	#440	—	13	2	no	—
114	C₄×Q₁₆	—	13	2	no	—
115	C₄×SD₁₆	—	13	2	no	—
116	#428	—	13	1	no	—
117	#416	D₁₃	1	1	yes	characteristic
118	C₂×#103	—	13	1	no	—
119	#405	—	13	1	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	C₄×D₄	1	2	yes	—
126	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	—	4	1	no	—
129	C₄×C₁₃	—	4	1	no	—
130	C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	Dic₁₃	—	2	4	no	—
134	Dic₁₃	—	2	2	no	—
135	Dic₁₃	—	2	2	no	—
136	Dic₁₃	—	4	2	no	—
137	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
138	C₂²×Q₈	—	13	1	no	—
139	#137	—	13	8	no	—
140	C₂×Q₁₆	—	13	2	no	—
141	C₂×SD₁₆	—	13	2	no	—
142	C₂×M₁₆	—	13	1	no	—
143	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
144	C₄×Q₈	—	13	2	no	—
145	C₄×Q₈	—	13	2	no	—
146	C₄×Q₈	—	13	2	no	—
147	C₄×Q₈	—	26	1	no	—
148	C₄×Q₈	—	26	1	no	—
149	C₄×D₄	—	13	2	no	—
150	C₄×D₄	—	26	2	no	—
151	#117	—	13	1	no	—
152	#117	—	26	2	no	—
153	#117	—	26	1	no	—
154	#117	C₂×D₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	13	1	no	—
156	C₂×C₄⋊C₄	—	26	1	no	—
157	#107	—	13	2	no	—
158	#106	—	13	2	no	—
159	#103	—	13	2	no	—
160	#103	—	13	2	no	—
161	#103	C₂×D₁₃	1	1	yes	characteristic
162	#103	C₂×D₁₃	1	1	yes	characteristic
163	#102	C₂×D₁₃	1	1	yes	characteristic
164	#102	C₂×D₁₃	1	1	yes	characteristic
165	#97	—	13	2	no	—
166	C₄×C₈	—	13	2	no	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	4	2	no	—
169	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×C₁₃	#577	1	1	yes	characteristic
171	C₂×C₁₃	#578	1	1	yes	characteristic
172	#41	—	2	2	no	—
173	#41	C₄×D₁₃	1	4	yes	—
174	C₂×Q₈	—	13	4	no	—
175	C₂×Q₈	—	13	2	no	—
176	C₂×Q₈	—	26	1	no	—
177	C₂×Q₈	—	26	2	no	—
178	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	26	2	no	—
183	C₂²×C₄	—	26	1	no	—
184	C₂²×C₄	—	26	1	no	—
185	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
186	Q₁₆	—	13	8	no	—
187	SD₁₆	—	13	8	no	—
188	M₁₆	—	13	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	13	1	no	—
194	C₄⋊C₄	—	13	1	no	—
195	C₄⋊C₄	—	13	2	no	—
196	C₄⋊C₄	—	26	2	no	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	2	no	—
199	C₄⋊C₄	—	26	1	no	—
200	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	13	2	no	—
204	C₂²⋊C₄	—	26	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	13	2	no	—
207	C₄²	—	26	2	no	—
208	C₄²	—	26	2	no	—
209	C₄²	—	26	2	no	—
210	C₁₃	#2947	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	1	no	—
212	C₂³	D₄×D₁₃	1	1	yes	characteristic
213	Q₈	—	2	1	no	—
214	Q₈	—	13	4	no	—
215	Q₈	—	26	4	no	—
216	Q₈	—	26	2	no	—
217	Q₈	C₂×C₄×D₁₃	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂×C₄×D₁₃	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	26	4	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	—	104	2	no	—
231	C₂×C₄	—	104	1	no	—
232	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
233	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	#6129	1	2	yes	—
235	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	13	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#88135	1	2	yes	—
245	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	26	4	no	—
250	C₄	—	26	2	no	—
251	C₄	—	26	2	no	—
252	C₄	—	52	2	no	—
253	C₄	#88135	1	2	yes	—
254	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
255	C₄	C₂×#6129	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×#88135	1	1	yes	characteristic
259	C₂	#11719475	1	1	yes	characteristic
260	C₂	#11719476	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#416	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#416	D₁₃
hypocenter	C₁₃	#2947
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
3. #11719221	C₂	semidirect
5. C₂×#88056	C₂	semidirect
6. #11718762	C₂	semidirect
7. #11718656	C₂	semidirect
8. #11718451	C₂	semidirect
9. #11718450	C₂	semidirect
10. #11718419	C₂	semidirect
11. #11718418	C₂	semidirect
16. #88205	C₄	semidirect
54. C₂×Dic₂₆	C₂×C₄	semidirect
65. #6107	C₂×C₄	semidirect
66. #6105	C₂×C₄	semidirect
78. #6099	C₂×C₄	semidirect
210. C₁₃	#2947	semidirect

Maximal subgroups

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

subgroup	index
#11719532	2
C₂×#88205	2
#11719221	2
C₁₃×#416	2
C₂×#88056	2
#11718762	2
#11718656	2
#11718451	2
#11718450	2
#11718419	2
#11718418	2
#2947	13