Subgroups — #15756139

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123770	C₂	1	1	yes	maximal, characteristic
2	#13115000	C₂	1	1	yes	maximal, characteristic
3	#13113401	C₂	1	2	yes	maximal
4	#13112695	C₂	1	2	yes	maximal
5	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#11719175	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	—	2	1	no	—
10	C₂×#88057	—	2	1	no	—
11	#11718763	—	2	1	no	—
12	#11718762	—	2	1	no	—
13	#11718597	C₂²	1	2	yes	—
14	#11718585	C₂²	1	1	yes	characteristic
15	#11718340	—	2	1	no	—
16	#11718340	C₄	1	2	yes	complemented
17	C₈×Dic₂₆	—	2	1	no	—
18	C₈×Dic₂₆	C₄	1	2	yes	complemented
19	C₁₃×#335	—	2	1	no	—
20	C₁₃×#334	—	2	1	no	—
21	#11718059	—	2	4	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	#88180	—	2	1	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	#88178	—	2	1	no	—
28	#88178	D₄	1	2	yes	—
29	C₂×#6099	—	2	1	no	—
30	#88124	—	2	1	no	—
31	#88123	Q₈	1	1	yes	characteristic
32	#88123	Q₈	1	1	yes	characteristic
33	C₄×Dic₂₆	—	2	1	no	—
34	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₂²⋊C₈	—	2	2	no	—
37	C₄×C₈×C₁₃	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	#88071	—	2	2	no	—
43	#88071	—	2	2	no	—
44	#88058	—	4	2	no	—
45	#88057	—	4	2	no	—
46	#88055	—	2	2	no	—
47	#88054	—	2	2	no	—
48	#88054	—	2	2	no	—
49	C₈×Dic₁₃	—	2	2	no	—
50	C₄⋊#1055	—	2	1	no	—
51	C₄⋊#1055	C₂×C₄	1	2	yes	—
52	#8677	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	4	2	no	—
57	C₂×Dic₂₆	—	2	1	no	—
58	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₂²⋊Dic₁₃	—	2	2	no	—
73	C₄⋊Dic₁₃	—	2	1	no	—
74	C₄⋊Dic₁₃	—	2	1	no	—
75	C₄⋊Dic₁₃	—	2	1	no	—
76	C₄⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
79	#6101	—	2	2	no	—
80	#6101	—	2	2	no	—
81	C₄×Dic₁₃	—	2	2	no	—
82	#6099	—	4	1	no	—
83	#6099	—	4	1	no	—
84	C₂×#1055	—	2	2	no	—
85	#2960	—	13	1	no	—
86	#1846	—	13	1	no	—
87	#1564	D₁₃	1	1	yes	characteristic
88	#1518	—	13	2	no	—
89	#1286	—	13	2	no	—
90	C₂³×C₁₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	4	1	no	—
92	C₂×C₄×C₁₃	—	4	1	no	—
93	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	8	2	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	C₂×Dic₁₃	—	8	2	no	—
108	Dic₂₆	—	4	2	no	—
109	Dic₂₆	—	4	2	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	8	2	no	—
113	C₈×C₁₃	#103	1	2	yes	—
114	C₈×C₁₃	#103	1	2	yes	—
115	#1055	—	4	2	no	—
116	#533	—	13	1	no	—
117	#446	—	13	2	no	—
118	#446	—	26	1	no	—
119	C₈×Q₈	—	13	2	no	—
120	C₈×Q₈	—	26	1	no	—
121	#431	C₂×D₁₃	1	2	yes	—
122	C₂×#107	—	26	1	no	—
123	C₂×#106	—	26	1	no	—
124	#424	—	26	1	no	—
125	#423	—	13	1	no	—
126	#405	—	26	1	no	—
127	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
128	#382	—	13	2	no	—
129	#335	—	2	1	no	—
130	#334	—	2	1	no	—
131	#328	—	26	4	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	#380	1	1	yes	characteristic
135	C₂²×C₁₃	#460	1	1	yes	characteristic
136	C₂²×C₁₃	#462	1	1	yes	characteristic
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
142	C₄×C₁₃	#417	1	1	yes	characteristic
143	C₄×C₁₃	#475	1	1	yes	characteristic
144	C₄×C₁₃	#477	1	1	yes	characteristic
145	Dic₁₃	—	4	2	no	—
146	Dic₁₃	—	4	2	no	—
147	Dic₁₃	—	16	2	no	—
148	C₂×M₁₆	—	26	1	no	—
149	C₂²×C₈	D₅₂	1	1	yes	characteristic
150	C₂²×C₈	D₅₂	1	1	yes	characteristic
151	C₄⋊Q₈	—	13	1	no	—
152	C₄⋊Q₈	—	13	1	no	—
153	#125	—	26	1	no	—
154	C₂²⋊Q₈	—	13	2	no	—
155	C₂²⋊Q₈	—	26	1	no	—
156	C₄×Q₈	—	13	2	no	—
157	C₄×Q₈	—	26	1	no	—
158	#117	—	26	1	no	—
159	C₂×C₄⋊C₄	—	26	2	no	—
160	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	#107	—	52	2	no	—
162	#106	—	52	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	13	2	no	—
165	C₄⋊C₈	—	26	2	no	—
166	C₄⋊C₈	—	26	2	no	—
167	C₄⋊C₈	—	26	1	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	#97	—	26	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	26	2	no	—
172	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
173	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	#95	—	26	2	no	—
177	#95	—	26	2	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	#1803	1	1	yes	characteristic
181	C₂×C₁₃	#1812	1	1	yes	characteristic
182	C₂×C₁₃	#1814	1	1	yes	characteristic
183	C₂×Q₈	—	13	2	no	—
184	C₂×Q₈	—	26	1	no	—
185	C₂²×C₄	—	52	2	no	—
186	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
188	C₂²×C₄	#6131	1	1	yes	characteristic
189	M₁₆	—	52	1	no	—
190	M₁₆	—	52	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	26	2	no	—
195	C₂×C₈	#6095	1	2	yes	—
196	C₂×C₈	Dic₅₂	1	2	yes	—
197	C₂×C₈	#6103	1	1	yes	characteristic
198	C₂×C₈	#6103	1	1	yes	characteristic
199	C₂×C₈	#6103	1	1	yes	characteristic
200	C₂×C₈	#6103	1	1	yes	characteristic
201	C₄⋊C₄	—	13	2	no	—
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	26	1	no	—
204	C₄⋊C₄	—	26	1	no	—
205	C₄⋊C₄	—	26	2	no	—
206	C₄⋊C₄	—	26	1	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₂²⋊C₄	—	26	2	no	—
210	C₄²	—	26	2	no	—
211	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₄²	#6129	1	1	yes	characteristic
213	C₄²	Q₈×D₁₃	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₁₃	#8677	1	1	yes	characteristic, complemented, Sylow
216	C₂³	#88149	1	1	yes	characteristic
217	Q₈	—	52	2	no	—
218	Q₈	—	52	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	2	no	—
225	C₂×C₄	—	104	2	no	—
226	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₄	#88146	1	1	yes	characteristic
228	C₂×C₄	#88151	1	1	yes	characteristic
229	C₂×C₄	C₂×#6095	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
231	C₂×C₄	#88162	1	1	yes	characteristic
232	C₂×C₄	#88163	1	1	yes	characteristic
233	C₂×C₄	C₂×#6103	1	1	yes	characteristic
234	C₂×C₄	#88192	1	1	yes	characteristic
235	C₂×C₄	#88200	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	52	2	no	—
240	C₈	#88056	1	2	yes	—
241	C₈	#88056	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#11718509	1	1	yes	characteristic
245	C₂²	#11718510	1	1	yes	characteristic
246	C₂²	#11718648	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	52	2	no	—
252	C₄	—	52	2	no	—
253	C₄	—	208	2	no	—
254	C₄	#11718511	1	1	yes	characteristic
255	C₄	#11718512	1	1	yes	characteristic
256	C₄	C₂×#88056	1	1	yes	characteristic
257	C₄	#11718794	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#13114842	1	1	yes	characteristic
261	C₂	#13114879	1	1	yes	characteristic
262	C₂	#13114891	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718648
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#8677
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. #11718340	C₄	semidirect
18. C₈×Dic₂₆	C₄	semidirect
215. C₁₃	#8677	semidirect

Maximal subgroups

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

subgroup	index
#13123770	2
#13115000	2
#13113401	2
#13112695	2
C₁₃×#1564	2
#8677	13