Subgroups — #15756015

Order 3328.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
2	#13114838	C₂	1	2	yes	maximal
3	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
4	#13112719	C₂	1	2	yes	maximal
5	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	—	2	1	no	—
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#384	C₂²	1	2	yes	—
11	C₂×#88055	—	2	1	no	—
12	C₂×#88054	—	2	1	no	—
13	C₂×#88050	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
16	#11718340	—	2	2	no	—
17	#11718340	C₄	1	4	yes	complemented
18	#11718121	—	2	2	no	—
19	#11718049	—	2	2	no	—
20	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
28	C₂×#6101	—	2	1	no	—
29	C₂×C₄×Dic₁₃	—	2	1	no	—
30	C₂²×#1055	—	2	1	no	—
31	C₂²×#1055	—	2	2	no	—
32	C₄×Dic₂₆	—	2	1	no	—
33	C₄×Dic₂₆	—	2	1	no	—
34	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₆	Q₈	1	2	yes	—
36	C₁₃×#107	—	4	2	no	—
37	C₁₃×#107	—	4	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	#88055	—	2	4	no	—
41	#88054	—	2	4	no	—
42	#88050	—	4	4	no	—
43	C₄⋊#1055	—	2	1	no	—
44	C₄⋊#1055	—	2	2	no	—
45	C₄⋊#1055	—	2	2	no	—
46	C₄⋊#1055	C₂×C₄	1	2	yes	—
47	#8653	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	2	1	no	—
53	C₂²×Dic₁₃	—	2	1	no	—
54	C₂×Dic₂₆	—	2	2	no	—
55	C₂×Dic₂₆	—	2	2	no	—
56	C₂×Dic₂₆	—	2	1	no	—
57	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
63	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₄⋊C₄	—	8	2	no	—
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₁₃	#41	1	1	yes	characteristic
73	C₄⋊Dic₁₃	—	2	1	no	—
74	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
75	#6101	—	2	2	no	—
76	#6101	—	2	2	no	—
77	C₄×Dic₁₃	—	2	2	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	4	1	no	—
81	C₂×#1055	—	4	1	no	—
82	C₂×#1055	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	2	no	—
85	C₂×#446	—	13	1	no	—
86	#1845	D₁₃	1	1	yes	characteristic
87	#1803	—	13	2	no	—
88	C₂×#334	—	13	1	no	—
89	#1296	—	13	2	no	—
90	C₂³×C₁₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	8	2	no	—
94	C₂×C₄×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	—	8	2	no	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#137	1	2	yes	—
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	4	1	no	—
111	C₂×Dic₁₃	—	4	1	no	—
112	C₂×Dic₁₃	—	4	1	no	—
113	C₂×Dic₁₃	—	4	1	no	—
114	Dic₂₆	—	2	4	no	—
115	Dic₂₆	—	4	2	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	#103	1	4	yes	complemented
118	#1055	—	4	2	no	—
119	#1055	—	8	2	no	—
120	C₂×C₄×Q₈	—	13	1	no	—
121	#446	—	13	4	no	—
122	#446	—	26	2	no	—
123	C₂×#107	—	2	1	no	—
124	C₂×#107	—	2	1	no	—
125	C₂×C₄⋊C₈	—	13	1	no	—
126	C₂×C₄⋊C₈	—	13	2	no	—
127	C₂×C₄⋊C₈	—	26	1	no	—
128	C₂×#103	—	26	2	no	—
129	C₂×#97	—	26	1	no	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	#384	C₂×D₁₃	1	2	yes	—
132	#340	—	26	2	no	—
133	#334	—	26	2	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	C₂²×C₁₃	#469	1	1	yes	characteristic
138	C₂²×C₁₃	#470	1	1	yes	characteristic
139	C₂²×C₁₃	#472	1	1	yes	characteristic
140	C₂²×C₁₃	#473	1	1	yes	characteristic
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	16	2	no	—
144	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
145	C₄×C₁₃	#417	1	1	yes	characteristic
146	C₄×C₁₃	#428	1	2	yes	—
147	C₄×C₁₃	#468	1	2	yes	—
148	C₄×C₁₃	#475	1	1	yes	characteristic
149	C₄×C₁₃	#477	1	1	yes	characteristic
150	Dic₁₃	—	4	2	no	—
151	Dic₁₃	—	4	2	no	—
152	C₂²×Q₈	—	13	1	no	—
153	C₂²×C₈	—	26	1	no	—
154	C₂²×C₈	—	26	2	no	—
155	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
157	C₄×Q₈	—	13	2	no	—
158	C₄×Q₈	—	13	2	no	—
159	C₄×Q₈	—	26	1	no	—
160	C₄×Q₈	—	26	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	13	1	no	—
165	C₂×C₄⋊C₄	—	26	1	no	—
166	C₂×C₄²	—	26	1	no	—
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	#107	—	4	2	no	—
169	#107	—	4	2	no	—
170	C₄⋊C₈	—	13	2	no	—
171	C₄⋊C₈	—	26	4	no	—
172	C₄⋊C₈	—	26	1	no	—
173	C₄⋊C₈	—	26	2	no	—
174	C₄⋊C₈	—	26	2	no	—
175	#103	—	52	4	no	—
176	#97	—	26	4	no	—
177	C₄×C₈	C₄×D₁₃	1	2	yes	—
178	C₄×C₈	D₅₂	1	2	yes	—
179	C₂×C₁₃	#1688	1	1	yes	characteristic
180	C₂×C₁₃	#1690	1	1	yes	characteristic
181	C₂×C₁₃	#1694	1	1	yes	characteristic
182	C₂×C₁₃	#1696	1	1	yes	characteristic
183	C₂×C₁₃	#1803	1	1	yes	characteristic
184	C₂×C₁₃	#1809	1	1	yes	characteristic
185	C₂×C₁₃	#1816	1	1	yes	characteristic
186	C₂×Q₈	—	13	2	no	—
187	C₂×Q₈	—	26	1	no	—
188	C₂×Q₈	—	26	2	no	—
189	C₂×Q₈	—	26	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	26	1	no	—
192	C₂²×C₄	—	26	1	no	—
193	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	#6129	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	—	52	1	no	—
200	C₂×C₈	—	52	1	no	—
201	C₂×C₈	—	52	2	no	—
202	C₂×C₈	—	52	2	no	—
203	C₂×C₈	—	52	2	no	—
204	C₂×C₈	#6103	1	2	yes	—
205	C₂×C₈	#6103	1	2	yes	—
206	C₂×C₈	#6105	1	2	yes	—
207	C₂×C₈	#6107	1	2	yes	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	8	2	no	—
213	C₄⋊C₄	—	13	2	no	—
214	C₄⋊C₄	—	26	2	no	—
215	C₄⋊C₄	—	26	1	no	—
216	C₄⋊C₄	—	26	2	no	—
217	C₄²	—	26	2	no	—
218	C₄²	—	26	2	no	—
219	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
220	C₄²	C₂×D₅₂	1	1	yes	characteristic
221	C₄²	Q₈×D₁₃	1	1	yes	characteristic
222	C₄²	#6131	1	1	yes	characteristic
223	C₁₃	#8653	1	1	yes	characteristic, complemented, Sylow
224	C₂³	#88192	1	1	yes	characteristic
225	Q₈	—	26	4	no	—
226	Q₈	—	52	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	—	52	1	no	—
235	C₂×C₄	—	52	1	no	—
236	C₂×C₄	—	52	1	no	—
237	C₂×C₄	—	52	1	no	—
238	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₄	#88146	1	1	yes	characteristic
240	C₂×C₄	#88149	1	1	yes	characteristic
241	C₂×C₄	#88151	1	1	yes	characteristic
242	C₂×C₄	#88165	1	2	yes	—
243	C₂×C₄	#88172	1	2	yes	—
244	C₂×C₄	C₂×#6103	1	1	yes	characteristic
245	C₂×C₄	#88186	1	1	yes	characteristic
246	C₂×C₄	C₂×#6105	1	1	yes	characteristic
247	C₂×C₄	#88196	1	1	yes	characteristic
248	C₂×C₄	C₂×#6107	1	1	yes	characteristic
249	C₂×C₄	#88200	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	52	2	no	—
252	C₈	—	104	2	no	—
253	C₈	#88050	1	4	yes	complemented
254	C₂²	#11718648	1	1	yes	characteristic
255	C₂²	#11718736	1	1	yes	characteristic
256	C₂²	#11718751	1	1	yes	characteristic
257	C₂²	#11718820	1	1	yes	characteristic
258	C₂²	#11718821	1	1	yes	characteristic
259	C₂²	#11718850	1	1	yes	characteristic
260	C₂²	#11718853	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	16	2	no	—
264	C₄	—	52	2	no	—
265	C₄	—	52	2	no	—
266	C₄	#11718542	1	2	yes	—
267	C₄	#11718547	1	2	yes	—
268	C₄	#11718627	1	1	yes	characteristic
269	C₄	C₂×#88050	1	1	yes	characteristic
270	C₄	#11718752	1	1	yes	characteristic
271	C₄	#11718755	1	1	yes	characteristic
272	C₂	#13114317	1	1	yes	characteristic
273	C₂	#13114327	1	1	yes	characteristic
274	C₂	#13114351	1	1	yes	characteristic
275	C₂	#13114361	1	1	yes	characteristic
276	C₂	#13114838	1	1	yes	characteristic
277	C₂	#13114870	1	1	yes	characteristic
278	C₂	#13114901	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#8653
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
17. #11718340	C₄	semidirect
117. C₈×C₁₃	#103	semidirect
223. C₁₃	#8653	semidirect
253. C₈	#88050	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718340	2
#13114838	2
C₂×#11718049	2
#13112719	2
C₁₃×#1845	2
#8653	13