Subgroups — #15871106

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	#13123771	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123548	C₂	1	1	yes	maximal, characteristic
3	#13115818	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115669	C₂	1	1	yes	maximal, characteristic
5	#13115549	C₂	1	4	yes	maximal, complemented
6	#13114977	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114676	C₂	1	4	yes	maximal, complemented
8	#13114524	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3135	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#533	C₂²	1	1	yes	characteristic
11	C₁₃×#532	C₂²	1	1	yes	characteristic
12	C₂×#88202	C₂²	1	2	yes	—
13	C₁₃×#491	C₄	1	2	yes	complemented
14	C₁₃×#490	C₄	1	2	yes	complemented
15	C₁₃×#489	C₄	1	4	yes	complemented
16	C₁₃×#416	C₂²	1	4	yes	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×#88077	C₂²	1	4	yes	—
19	#11718895	C₂²	1	2	yes	—
20	C₂×#88055	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	#11718685	—	2	2	no	—
23	Q₈×#1055	—	2	2	no	—
24	#11718673	—	2	2	no	—
25	D₄×#1055	—	2	2	no	—
26	#11718660	C₂²	1	2	yes	—
27	#11718658	C₂²	1	2	yes	—
28	#11718593	C₂²	1	1	yes	characteristic
29	#11718585	C₂²	1	1	yes	characteristic
30	#11718585	C₂²	1	1	yes	characteristic
31	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
32	#11718144	C₂²	1	4	yes	—
33	#11718124	C₂²	1	4	yes	—
34	C₂×C₁₃×#41	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
39	C₁₃×#125	C₂×C₄	1	2	yes	—
40	C₁₃×#124	C₂×C₄	1	2	yes	—
41	C₁₃×C₂²⋊Q₈	—	2	2	no	—
42	C₁₃×#121	—	2	2	no	—
43	C₄×Q₈×C₁₃	—	2	2	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₁₃×#117	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88202	—	2	8	no	—
48	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	C₂³	1	2	yes	—
51	C₂×#6099	C₂³	1	2	yes	—
52	C₂²×#1055	—	2	2	no	—
53	C₂²×#1055	C₂³	1	1	yes	characteristic
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₁₃×#103	C₂×C₄	1	8	yes	—
56	C₁₃×#102	C₂×C₄	1	8	yes	—
57	C₁₃×#97	C₂×C₄	1	2	yes	—
58	C₁₃×#97	C₂×C₄	1	2	yes	—
59	#88077	—	2	8	no	—
60	#88071	—	2	2	no	—
61	C₂²⋊#1055	—	2	4	no	—
62	#88055	—	2	4	no	—
63	C₈×Dic₁₃	—	2	4	no	—
64	C₄⋊#1055	—	2	2	no	—
65	C₄⋊#1055	—	2	2	no	—
66	C₄⋊#1055	—	2	4	no	—
67	#88043	—	2	2	no	—
68	C₄×#1055	—	2	2	no	—
69	C₄×#1055	D₄	1	2	yes	—
70	C₄×#1055	D₄	1	2	yes	—
71	#21661	—	13	1	no	maximal, Sylow
72	C₁₃×#41	—	2	4	no	—
73	C₁₃×#41	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	#41	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₁₃×C₂²⋊C₄	—	2	2	no	—
91	C₁₃×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	2	yes	—
96	C₄×Dic₁₃	—	2	2	no	—
97	C₄×Dic₁₃	#41	1	4	yes	—
98	#6099	—	2	4	no	—
99	#6099	—	2	4	no	—
100	#6099	—	4	4	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	4	4	no	—
105	C₂×#1055	C₂×D₄	1	2	yes	—
106	C₂×#1055	C₂×D₄	1	1	yes	characteristic
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	#41	1	4	yes	—
109	#3135	D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	4	no	—
121	Q₈×C₁₃	—	4	4	no	—
122	D₄×C₁₃	—	2	4	no	—
123	D₄×C₁₃	—	4	4	no	—
124	D₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
136	C₂×Dic₁₃	—	2	4	no	—
137	C₂×Dic₁₃	—	4	4	no	—
138	C₈×C₁₃	—	2	4	no	—
139	#1055	—	2	2	no	—
140	#1055	—	4	2	no	—
141	#1055	—	4	2	no	—
142	#1055	#124	1	2	yes	—
143	#1055	#124	1	2	yes	—
144	C₂×#131	—	13	2	no	—
145	#533	C₂×D₁₃	1	1	yes	characteristic
146	#532	C₂×D₁₃	1	1	yes	characteristic
147	#491	Dic₁₃	1	2	yes	complemented
148	#490	Dic₁₃	1	2	yes	complemented
149	#489	Dic₁₃	1	4	yes	complemented
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	C₂×D₁₃	1	4	yes	—
159	#408	—	13	2	no	—
160	#405	—	13	2	no	—
161	C₂×#97	—	13	2	no	—
162	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	—	13	1	no	—
164	C₂×C₄×C₈	—	13	2	no	—
165	#376	—	13	1	no	—
166	#341	—	13	4	no	—
167	#339	—	13	4	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	4	4	no	—
170	C₂²×C₁₃	—	4	2	no	—
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₃	#529	1	2	yes	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	C₄×#41	1	2	yes	—
178	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
179	C₄×C₁₃	#532	1	1	yes	characteristic
180	Dic₁₃	—	4	4	no	—
181	C₂×#41	C₂²×D₁₃	1	2	yes	—
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	—	13	1	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
190	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
192	#127	C₂×Dic₁₃	1	1	yes	characteristic
193	#125	C₂×Dic₁₃	1	2	yes	—
194	#124	C₂×Dic₁₃	1	2	yes	—
195	C₂²⋊Q₈	—	2	2	no	—
196	#121	—	2	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	#117	C₂²×D₁₃	1	2	yes	—
200	C₂×C₄²	—	13	2	no	—
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	C₂×Dic₁₃	1	8	yes	—
207	#102	C₂×Dic₁₃	1	8	yes	—
208	C₂²⋊C₈	—	26	4	no	—
209	#97	—	26	2	no	—
210	#97	—	26	4	no	—
211	#97	C₂×Dic₁₃	1	2	yes	—
212	#97	C₂×Dic₁₃	1	2	yes	—
213	C₄×C₈	—	13	2	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	—	26	2	no	—
216	C₄×C₈	—	26	4	no	—
217	#95	—	26	2	no	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	—	8	2	no	—
220	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
221	C₂×C₁₃	#2955	1	2	yes	—
222	#41	—	2	4	no	—
223	#41	—	2	4	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	26	2	no	—
231	C₂²×C₄	#6129	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	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	52	4	no	—
246	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	13	4	no	—
255	C₄²	—	26	2	no	—
256	C₄²	D₄×D₁₃	1	2	yes	—
257	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
258	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
259	C₁₃	#21661	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	2	2	no	—
261	C₂³	C₂×#6129	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	52	4	no	—
274	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
275	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
276	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
277	C₂×C₄	#41×D₁₃	1	2	yes	—
278	C₂×C₄	#41×D₁₃	1	2	yes	—
279	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	—	13	2	no	—
282	C₈	—	13	2	no	—
283	C₈	—	26	2	no	—
284	C₈	—	52	2	no	—
285	C₈	—	52	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#11719385	1	2	yes	—
290	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	2	no	—
295	C₄	—	52	4	no	—
296	C₄	#11719373	1	1	yes	characteristic
297	C₄	D₁₃×#124	1	1	yes	characteristic
298	C₄	#41×Dic₁₃	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	8	2	no	—
301	C₂	#124×Dic₁₃	1	1	yes	characteristic
302	C₂	#13123739	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#3135	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#532	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#3135	D₁₃
hypocenter	C₁₃	#21661
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

Complements

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

N	H ≅ G/N	type
1. #13123771	C₂	semidirect
3. #13115818	C₂	semidirect
5. #13115549	C₂	semidirect
6. #13114977	C₂	semidirect
7. #13114676	C₂	semidirect
8. #13114524	C₂	semidirect
13. C₁₃×#491	C₄	semidirect
14. C₁₃×#490	C₄	semidirect
15. C₁₃×#489	C₄	semidirect
147. #491	Dic₁₃	semidirect
148. #490	Dic₁₃	semidirect
149. #489	Dic₁₃	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
#13123771	2
#13123548	2
#13115818	2
#13115669	2
#13115549	2
#13114977	2
#13114676	2
#13114524	2
C₁₃×#3135	2
#21661	13