Subgroups — #15871105

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125182	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123549	C₂	1	1	yes	maximal, characteristic
4	#13115817	C₂	1	1	yes	maximal, characteristic, complemented
5	#13115668	C₂	1	1	yes	maximal, characteristic
6	#13115547	C₂	1	4	yes	maximal, complemented
7	#13114977	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114673	C₂	1	4	yes	maximal, complemented
9	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88155	C₂²	1	2	yes	—
11	#11719182	C₂²	1	1	yes	characteristic
12	#11719175	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	C₂²	1	2	yes	—
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#376	C₂²	1	1	yes	characteristic
18	#11718774	C₂²	1	4	yes	—
19	#11718771	C₂²	1	2	yes	—
20	#11718763	C₂²	1	2	yes	—
21	#11718762	C₂²	1	2	yes	—
22	C₂×#88045	C₂²	1	4	yes	—
23	#11718585	C₂²	1	1	yes	characteristic
24	#11718585	C₂²	1	1	yes	characteristic
25	#11718375	C₄	1	4	yes	complemented
26	#11718374	C₄	1	2	yes	complemented
27	#11718364	C₄	1	2	yes	complemented
28	#11718349	—	2	2	no	—
29	C₈×D₅₂	—	2	2	no	—
30	#11718340	—	2	2	no	—
31	C₈×Dic₂₆	—	2	2	no	—
32	#11718144	C₂²	1	4	yes	—
33	#11718115	C₂²	1	4	yes	—
34	C₂×#6127	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	#88184	—	2	2	no	—
41	#88180	C₂³	1	2	yes	—
42	#88178	—	2	2	no	—
43	C₂×#6099	C₂³	1	2	yes	—
44	C₂×#6099	C₂³	1	2	yes	—
45	#88155	—	2	8	no	—
46	C₂×#6094	—	2	2	no	—
47	C₂×C₈×D₁₃	—	2	2	no	—
48	#88129	C₂×C₄	1	2	yes	—
49	#88128	C₂×C₄	1	1	yes	characteristic
50	C₄×D₅₂	—	2	2	no	—
51	#88124	C₂×C₄	1	2	yes	—
52	#88123	C₂×C₄	1	1	yes	characteristic
53	#88123	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₆	—	2	2	no	—
55	C₁₃×#97	—	2	4	no	—
56	C₁₃×#97	C₂×C₄	1	2	yes	—
57	C₁₃×#97	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	—	2	4	no	—
59	C₄×C₈×C₁₃	D₄	1	2	yes	—
60	C₄×C₈×C₁₃	D₄	1	2	yes	—
61	C₁₃×#95	—	2	2	no	—
62	#88061	C₂×C₄	1	8	yes	—
63	#88060	—	2	4	no	—
64	#88056	C₂×C₄	1	8	yes	—
65	#88055	—	2	2	no	—
66	#88054	—	2	4	no	—
67	C₈×Dic₁₃	—	2	2	no	—
68	#88045	—	2	8	no	—
69	C₄⋊#1055	—	2	2	no	—
70	C₄⋊#1055	—	2	2	no	—
71	#21661	—	13	1	no	maximal, Sylow
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₃	—	2	2	no	—
76	#6127	—	2	4	no	—
77	#6127	—	2	4	no	—
78	C₂×D₅₂	—	2	2	no	—
79	C₂×D₅₂	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₁₃	—	2	2	no	—
81	C₂×Dic₂₆	—	2	2	no	—
82	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	#41	1	4	yes	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	2	yes	—
95	C₄²×C₁₃	#41	1	4	yes	—
96	C₂²⋊Dic₁₃	—	2	2	no	—
97	#6103	—	2	4	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
101	#6101	—	2	4	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	#6099	—	2	4	no	—
104	#6099	—	2	4	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	#6094	—	4	4	no	—
108	C₈×D₁₃	—	4	4	no	—
109	#3135	—	13	1	no	—
110	#2960	—	13	1	no	—
111	#2931	—	13	1	no	—
112	#1995	—	13	1	no	—
113	#1963	—	13	1	no	—
114	#1938	—	13	4	no	—
115	#1838	—	13	1	no	—
116	#1764	—	13	4	no	—
117	#1726	D₁₃	1	1	yes	characteristic, complemented
118	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×D₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂²⋊D₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	2	2	no	—
132	D₅₂	—	2	4	no	—
133	D₅₂	—	4	4	no	—
134	C₄×D₁₃	—	4	2	no	—
135	C₄×D₁₃	—	4	2	no	—
136	Dic₂₆	—	2	4	no	—
137	Dic₂₆	—	4	4	no	—
138	C₈×C₁₃	—	2	4	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	#124	1	2	yes	—
141	C₈×C₁₃	#124	1	2	yes	—
142	#1055	—	4	2	no	—
143	#1055	—	4	2	no	—
144	C₂×#131	—	13	2	no	—
145	#533	—	13	1	no	—
146	#532	—	13	1	no	—
147	#491	—	13	2	no	—
148	#490	—	13	2	no	—
149	#489	—	13	4	no	—
150	#446	—	26	2	no	—
151	C₈×Q₈	—	26	2	no	—
152	#436	—	26	2	no	—
153	C₈×D₄	—	26	2	no	—
154	#424	—	13	2	no	—
155	#423	—	13	1	no	—
156	#423	—	13	1	no	—
157	C₂×#104	—	13	4	no	—
158	#416	—	13	4	no	—
159	#408	—	13	2	no	—
160	#405	—	13	2	no	—
161	C₂×#97	C₂×D₁₃	1	2	yes	—
162	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	#376	C₂×D₁₃	1	1	yes	characteristic
166	#341	—	13	4	no	—
167	#339	—	13	4	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₁₃	#529	1	2	yes	—
171	C₂×D₁₃	—	4	4	no	—
172	C₂×D₁₃	—	4	2	no	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	C₄×#41	1	2	yes	—
177	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
178	C₄×C₁₃	#532	1	1	yes	characteristic
179	Dic₁₃	—	4	4	no	—
180	Dic₁₃	—	4	2	no	—
181	C₂×#41	—	13	2	no	—
182	#131	—	26	8	no	—
183	C₂×M₁₆	—	13	2	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂²×C₈	—	26	2	no	—
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	13	2	no	—
191	C₄⋊Q₈	—	13	1	no	—
192	#127	—	13	1	no	—
193	#125	—	13	2	no	—
194	#124	—	13	2	no	—
195	C₂²⋊Q₈	—	26	2	no	—
196	#121	—	26	2	no	—
197	C₄×Q₈	—	26	2	no	—
198	C₄×D₄	—	26	2	no	—
199	#117	—	13	2	no	—
200	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
201	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
202	C₄⋊C₈	—	26	4	no	—
203	C₄⋊C₈	—	26	2	no	—
204	C₄⋊C₈	—	26	2	no	—
205	#104	—	26	8	no	—
206	#103	—	13	8	no	—
207	#102	—	13	8	no	—
208	C₂²⋊C₈	—	26	4	no	—
209	#97	—	2	4	no	—
210	#97	—	26	2	no	—
211	#97	C₄×D₁₃	1	2	yes	—
212	#97	C₄×D₁₃	1	2	yes	—
213	C₄×C₈	—	2	4	no	—
214	C₄×C₈	—	26	2	no	—
215	C₄×C₈	D₅₂	1	2	yes	—
216	C₄×C₈	D₅₂	1	2	yes	—
217	#95	—	2	2	no	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
220	C₂×C₁₃	#2955	1	2	yes	—
221	D₁₃	—	8	2	no	—
222	#41	—	26	4	no	—
223	#41	—	26	4	no	—
224	C₂×Q₈	—	13	2	no	—
225	C₂×Q₈	—	26	2	no	—
226	C₂×D₄	—	13	2	no	—
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	26	2	no	—
231	C₂²×C₄	#6127	1	2	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
233	M₁₆	—	26	4	no	—
234	M₁₆	—	26	4	no	—
235	M₁₆	—	52	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	26	2	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	52	4	no	—
242	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
243	C₂×C₈	C₂×D₅₂	1	2	yes	—
244	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
245	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
246	C₂×C₈	#6127	1	4	yes	—
247	C₄⋊C₄	—	13	4	no	—
248	C₄⋊C₄	—	26	2	no	—
249	C₄⋊C₄	—	26	4	no	—
250	C₄⋊C₄	—	26	2	no	—
251	C₂²⋊C₄	—	26	2	no	—
252	C₂²⋊C₄	—	26	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	26	2	no	—
255	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₄²	C₂×D₅₂	1	2	yes	—
258	C₄²	#6127	1	4	yes	—
259	C₁₃	#21661	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	26	2	no	—
261	C₂³	C₂×#6127	1	1	yes	characteristic
262	Q₈	—	26	4	no	—
263	Q₈	—	52	4	no	—
264	D₄	—	26	4	no	—
265	D₄	—	52	4	no	—
266	D₄	—	52	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	—	26	2	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	52	2	no	—
274	C₂×C₄	C₄×D₅₂	1	2	yes	—
275	C₂×C₄	C₄×D₅₂	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
278	C₂×C₄	C₂×#6127	1	2	yes	—
279	C₂×C₄	C₂×#6127	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	2	no	—
282	C₈	—	52	2	no	—
283	C₈	—	52	2	no	—
284	C₈	#88129	1	2	yes	—
285	C₈	#88129	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	52	4	no	—
288	C₂²	—	52	2	no	—
289	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
290	C₂²	#11719184	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	52	4	no	—
295	C₄	—	52	2	no	—
296	C₄	C₄×#6127	1	2	yes	—
297	C₄	C₂×#88129	1	1	yes	characteristic
298	C₄	#11719182	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	104	2	no	—
301	C₂	C₄×#88129	1	1	yes	characteristic
302	C₂	#13123740	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1726	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1726	D₁₃
hypocenter	C₁₃	#21661
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	semidirect
4. #13115817	C₂	semidirect
6. #13115547	C₂	semidirect
7. #13114977	C₂	semidirect
8. #13114673	C₂	semidirect
9. C₁₃×#1726	C₂	semidirect
25. #11718375	C₄	semidirect
26. #11718374	C₄	semidirect
27. #11718364	C₄	semidirect
117. #1726	D₁₃	semidirect
259. C₁₃	#21661	semidirect

Maximal subgroups

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

subgroup	index
#13125182	2
#13123770	2
#13123549	2
#13115817	2
#13115668	2
#13115547	2
#13114977	2
#13114673	2
C₁₃×#1726	2
#21661	13