Subgroups — #15756138

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	#13115000	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13113400	C₂	1	2	yes	maximal
4	#13112696	C₂	1	2	yes	maximal
5	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₁₃×#446	—	2	1	no	—
8	C₁₃×#446	C₄	1	2	yes	complemented
9	C₈×Q₈×C₁₃	—	2	1	no	—
10	C₈×Q₈×C₁₃	C₄	1	2	yes	complemented
11	C₁₃×#424	—	2	1	no	—
12	C₁₃×#423	C₂²	1	1	yes	characteristic
13	C₁₃×#405	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88058	—	2	1	no	—
16	C₂×#88057	—	2	1	no	—
17	#11718597	C₂²	1	2	yes	—
18	#11718586	C₂²	1	2	yes	—
19	#11718130	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	1	no	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₁₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₁₃×#125	—	2	1	no	—
28	C₁₃×C₂²⋊Q₈	—	2	1	no	—
29	C₁₃×C₂²⋊Q₈	D₄	1	2	yes	—
30	C₄×Q₈×C₁₃	—	2	1	no	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₁₃×#117	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	1	no	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₃×#97	—	2	2	no	—
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	#88071	—	2	2	no	—
46	#88071	—	2	2	no	—
47	C₂²⋊#1055	—	2	2	no	—
48	#88058	—	4	2	no	—
49	#88057	—	4	2	no	—
50	C₄⋊#1055	—	2	2	no	—
51	C₄×#1055	—	2	2	no	—
52	#8677	—	13	1	no	maximal, Sylow
53	C₂×Q₈×C₁₃	—	2	1	no	—
54	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	4	2	no	—
59	C₁₃×M₁₆	—	4	1	no	—
60	C₁₃×M₁₆	—	4	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
69	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	1	no	—
73	C₁₃×C₄⋊C₄	—	2	1	no	—
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	C₄⋊Dic₁₃	—	4	2	no	—
83	C₄⋊Dic₁₃	—	4	2	no	—
84	C₂×#1055	—	4	2	no	—
85	#2960	D₁₃	1	1	yes	characteristic
86	#1846	—	13	1	no	—
87	#1564	—	13	1	no	—
88	#1518	—	13	2	no	—
89	#1286	—	13	2	no	—
90	C₂³×C₁₃	#121	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	2	no	—
92	Q₈×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	4	1	no	—
96	C₂×C₄×C₁₃	—	4	1	no	—
97	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
107	C₂×Dic₁₃	—	8	2	no	—
108	C₂×Dic₁₃	—	8	2	no	—
109	C₂×Dic₁₃	—	8	2	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	4	2	no	—
113	C₈×C₁₃	#103	1	2	yes	—
114	C₈×C₁₃	#103	1	2	yes	—
115	#1055	—	8	2	no	—
116	#533	C₂×D₁₃	1	1	yes	characteristic
117	#446	—	2	1	no	—
118	#446	Dic₁₃	1	2	yes	complemented
119	C₈×Q₈	—	2	1	no	—
120	C₈×Q₈	Dic₁₃	1	2	yes	complemented
121	#431	—	13	2	no	—
122	C₂×#107	—	26	1	no	—
123	C₂×#106	—	26	1	no	—
124	#424	—	2	1	no	—
125	#423	C₂×D₁₃	1	1	yes	characteristic
126	#405	—	2	1	no	—
127	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
128	#382	—	13	2	no	—
129	#335	—	26	1	no	—
130	#334	—	26	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	—	4	2	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
144	C₄×C₁₃	#417	1	1	yes	characteristic
145	C₄×C₁₃	#475	1	1	yes	characteristic
146	C₄×C₁₃	#477	1	1	yes	characteristic
147	Dic₁₃	—	16	2	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
151	C₄⋊Q₈	Dic₂₆	1	1	yes	characteristic
152	C₄⋊Q₈	Dic₂₆	1	1	yes	characteristic
153	#125	—	2	1	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	D₅₂	1	2	yes	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
158	#117	—	2	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	—	2	2	no	—
165	C₄⋊C₈	—	2	1	no	—
166	C₄⋊C₈	—	26	2	no	—
167	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
168	C₂²⋊C₈	—	26	2	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	26	2	no	—
172	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
173	C₄×C₈	C₂×Dic₁₃	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	—	2	1	no	—
184	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
185	C₂²×C₄	—	52	2	no	—
186	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
187	C₂²×C₄	#6127	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	52	2	no	—
195	C₂×C₈	#6106	1	2	yes	—
196	C₂×C₈	#6107	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
212	C₄²	#6127	1	1	yes	characteristic
213	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₁₃	#8677	1	1	yes	characteristic, complemented, Sylow
216	C₂³	#88184	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	2	no	—
225	C₂×C₄	—	104	2	no	—
226	C₂×C₄	#88178	1	1	yes	characteristic
227	C₂×C₄	#88178	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
229	C₂×C₄	#88180	1	1	yes	characteristic
230	C₂×C₄	#88184	1	1	yes	characteristic
231	C₂×C₄	C₂×#6106	1	1	yes	characteristic
232	C₂×C₄	C₂×#6107	1	1	yes	characteristic
233	C₂×C₄	#88203	1	1	yes	characteristic
234	C₂×C₄	#88205	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	104	2	no	—
240	C₈	#88075	1	2	yes	—
241	C₈	#88075	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#11718688	1	1	yes	characteristic
245	C₂²	#11718692	1	1	yes	characteristic
246	C₂²	#11718873	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	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	208	2	no	—
254	C₄	#11718681	1	1	yes	characteristic
255	C₄	#11718682	1	1	yes	characteristic
256	C₄	C₂×#88075	1	1	yes	characteristic
257	C₄	#11718893	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#13114841	1	1	yes	characteristic
261	C₂	#13114882	1	1	yes	characteristic
262	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718873
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2960	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#8677
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
8. C₁₃×#446	C₄	semidirect
10. C₈×Q₈×C₁₃	C₄	semidirect
118. #446	Dic₁₃	semidirect
120. C₈×Q₈	Dic₁₃	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
#13115000	2
#13113584	2
#13113400	2
#13112696	2
C₁₃×#2960	2
#8677	13