Subgroups — #15780681

Order 3328.

324 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116594	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718100	C₂	1	1	yes	maximal, characteristic
3	#13116449	C₂	1	1	yes	maximal, characteristic
4	#13114673	C₂	1	1	yes	maximal, characteristic
5	#13112924	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112918	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718774	C₄	1	1	yes	characteristic
13	#11718762	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	C₂×#88045	C₄	1	1	yes	characteristic
17	C₁₃×#346	—	2	2	no	—
18	#11718144	C₄	1	1	yes	characteristic
19	#11718115	C₄	1	1	yes	characteristic
20	#11718100	—	2	2	no	—
21	#11718099	—	2	2	no	—
22	#11718093	—	2	2	no	—
23	C₂×#6127	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	#88180	C₂×C₄	1	1	yes	characteristic
28	C₂×#6099	C₂×C₄	1	1	yes	characteristic
29	C₂×#6099	C₂×C₄	1	1	yes	characteristic
30	#88155	—	2	1	no	—
31	#88155	—	2	1	no	—
32	#88155	—	2	1	no	—
33	#88155	—	2	1	no	—
34	C₂×#6094	—	2	1	no	—
35	C₂×C₈×D₁₃	—	2	1	no	—
36	C₂×C₁₃×C₁₆	—	2	1	no	—
37	C₂×C₁₃×C₁₆	—	2	1	no	—
38	C₂×C₁₃×C₁₆	—	2	1	no	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
41	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	#88061	C₈	1	2	yes	—
45	#88056	C₈	1	2	yes	—
46	#88055	—	2	1	no	—
47	C₈×Dic₁₃	—	2	1	no	—
48	#88052	—	2	1	no	—
49	#88052	—	2	1	no	—
50	#88052	—	2	1	no	—
51	#88052	—	2	1	no	—
52	C₂×#6090	—	2	1	no	—
53	C₂×#6090	—	2	1	no	—
54	#88045	—	2	2	no	—
55	#13125	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₃	—	2	1	no	—
59	#6127	—	2	1	no	—
60	#6127	—	2	1	no	—
61	#6127	—	2	1	no	—
62	#6127	—	2	1	no	—
63	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₁₃	—	2	1	no	—
65	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	M₁₆	1	2	yes	—
77	C₂²⋊Dic₁₃	—	2	1	no	—
78	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
79	C₄×Dic₁₃	—	2	1	no	—
80	#6099	—	2	1	no	—
81	#6099	—	2	1	no	—
82	#6099	—	2	1	no	—
83	#6099	—	2	1	no	—
84	C₂×#1055	—	2	1	no	—
85	C₂×#1055	—	2	1	no	—
86	#6094	—	4	1	no	—
87	#6094	—	4	1	no	—
88	C₈×D₁₃	—	4	1	no	—
89	C₈×D₁₃	—	4	1	no	—
90	C₁₃×C₁₆	—	2	2	no	—
91	C₁₃×C₁₆	—	2	2	no	—
92	#6090	—	4	1	no	—
93	#6090	—	4	1	no	—
94	#2151	—	13	1	no	—
95	C₂×#350	—	13	1	no	—
96	#2113	—	13	1	no	—
97	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
98	#1764	—	13	1	no	—
99	#1379	—	13	1	no	—
100	#1376	—	13	1	no	—
101	C₂³×C₁₃	#131	1	1	yes	characteristic
102	C₂²×D₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
111	C₂²⋊D₁₃	—	4	1	no	—
112	C₂²⋊D₁₃	—	4	1	no	—
113	C₂×Dic₁₃	—	2	2	no	—
114	C₂×Dic₁₃	—	2	1	no	—
115	D₅₂	—	2	1	no	—
116	D₅₂	—	2	1	no	—
117	C₄×D₁₃	—	4	1	no	—
118	C₄×D₁₃	—	4	1	no	—
119	Dic₂₆	—	2	1	no	—
120	Dic₂₆	—	2	1	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	#121	1	1	yes	characteristic
123	C₈×C₁₃	#123	1	1	yes	characteristic
124	C₈×C₁₃	#124	1	1	yes	characteristic
125	C₈×C₁₃	#126	1	1	yes	characteristic
126	#1055	—	4	1	no	—
127	#1055	—	4	1	no	—
128	C₂×#131	—	13	1	no	—
129	C₂×M₃₂	—	13	1	no	—
130	C₂×M₃₂	—	13	1	no	—
131	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
132	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
133	C₂×#104	—	13	1	no	—
134	#416	—	13	1	no	—
135	#405	—	13	1	no	—
136	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₄⋊C₁₆	—	26	2	no	—
138	#350	—	26	2	no	—
139	C₂²⋊C₁₆	—	26	2	no	—
140	#346	—	2	2	no	—
141	#341	—	13	1	no	—
142	#339	—	13	1	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	#433	1	1	yes	characteristic
145	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
146	C₂²×C₁₃	#435	1	1	yes	characteristic
147	C₂×D₁₃	—	4	2	no	—
148	C₂×D₁₃	—	4	1	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	#388	1	1	yes	characteristic
151	C₄×C₁₃	#431	1	1	yes	characteristic
152	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
153	C₄×C₁₃	#436	1	1	yes	characteristic
154	Dic₁₃	—	4	2	no	—
155	Dic₁₃	—	4	1	no	—
156	C₂×#41	—	13	1	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	C₂²×C₈	—	26	1	no	—
165	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
167	#117	—	13	1	no	—
168	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
169	M₃₂	—	26	1	no	—
170	M₃₂	—	26	1	no	—
171	M₃₂	—	26	1	no	—
172	M₃₂	—	26	1	no	—
173	C₂×C₁₆	—	2	1	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	26	1	no	—
178	C₂×C₁₆	—	26	1	no	—
179	C₂×C₁₆	D₅₂	1	2	yes	—
180	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
181	#104	—	26	2	no	—
182	#103	—	13	2	no	—
183	#102	—	13	2	no	—
184	#97	—	26	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	26	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	#1923	1	1	yes	characteristic
191	C₂×C₁₃	#2171	1	2	yes	—
192	D₁₃	—	8	1	no	—
193	#41	—	26	1	no	—
194	#41	—	26	1	no	—
195	#41	—	26	1	no	—
196	#41	—	26	1	no	—
197	C₂×Q₈	—	13	1	no	—
198	C₂×D₄	—	13	1	no	—
199	C₂×D₄	—	26	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	26	1	no	—
202	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
203	M₁₆	—	26	1	no	—
204	M₁₆	—	26	1	no	—
205	M₁₆	—	26	1	no	—
206	M₁₆	—	26	1	no	—
207	M₁₆	—	52	1	no	—
208	M₁₆	—	52	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	26	1	no	—
212	C₂×C₈	—	26	1	no	—
213	C₂×C₈	—	52	1	no	—
214	C₂×C₈	—	52	1	no	—
215	C₂×C₈	C₈×D₁₃	1	2	yes	—
216	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
217	C₂×C₈	#6127	1	1	yes	characteristic
218	C₂×C₈	#6127	1	1	yes	characteristic
219	C₂×C₈	#6127	1	1	yes	characteristic
220	C₂×C₈	#6127	1	1	yes	characteristic
221	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₄⋊C₄	—	13	2	no	—
223	C₂²⋊C₄	—	26	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	26	1	no	—
226	C₄²	#6094	1	2	yes	—
227	C₁₆	—	2	2	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	52	1	no	—
230	C₁₆	—	52	1	no	—
231	C₁₃	#13125	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	26	1	no	—
233	C₂³	#88155	1	1	yes	characteristic
234	Q₈	—	26	1	no	—
235	Q₈	—	26	1	no	—
236	D₄	—	26	1	no	—
237	D₄	—	26	1	no	—
238	D₄	—	52	1	no	—
239	D₄	—	52	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	26	1	no	—
244	C₂×C₄	—	52	1	no	—
245	C₂×C₄	—	52	1	no	—
246	C₂×C₄	#88126	1	1	yes	characteristic
247	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#6094	1	1	yes	characteristic
250	C₂×C₄	#88155	1	1	yes	characteristic
251	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	52	1	no	—
254	C₈	—	52	1	no	—
255	C₈	#88129	1	1	yes	characteristic
256	C₈	#88130	1	1	yes	characteristic
257	C₈	#88183	1	1	yes	characteristic
258	C₈	#88184	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	52	2	no	—
261	C₂²	—	52	1	no	—
262	C₂²	C₈×D₅₂	1	1	yes	characteristic
263	C₂²	#11718351	1	1	yes	characteristic
264	C₂²	#11718772	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	52	2	no	—
267	C₄	—	52	1	no	—
268	C₄	#11718346	1	1	yes	characteristic
269	C₄	#11718349	1	1	yes	characteristic
270	C₄	#11718601	1	1	yes	characteristic
271	C₄	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	104	1	no	—
275	C₂	#13115442	1	1	yes	characteristic
276	C₂	#13116679	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#13125
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. #13116594	C₂	semidirect
5. #13112924	C₂	semidirect
6. #13112918	C₂	semidirect
7. C₂×C₁₃×#346	C₂	semidirect
97. C₂×#346	D₁₃	semidirect
231. C₁₃	#13125	semidirect

Maximal subgroups

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

subgroup	index
#13116594	2
C₂×#11718100	2
#13116449	2
#13114673	2
#13112924	2
#13112918	2
C₂×C₁₃×#346	2
#13125	13