Subgroups — #15871094

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125158	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115812	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115665	C₂	1	1	yes	maximal, characteristic
5	#13115550	C₂	1	4	yes	maximal, complemented
6	#13114973	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114678	C₂	1	4	yes	maximal, complemented
8	C₄×#88055	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#131	C₂²	1	2	yes	—
11	C₁₃×#532	C₂²	1	1	yes	characteristic
12	#11719175	C₂²	1	1	yes	characteristic
13	C₁₃×#436	—	2	2	no	—
14	C₈×D₄×C₁₃	—	2	2	no	—
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#408	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88077	C₂²	1	4	yes	—
19	#11718894	C₂²	1	4	yes	—
20	#11718763	C₂²	1	2	yes	—
21	#11718762	C₂²	1	2	yes	—
22	C₂×#88055	C₂²	1	2	yes	—
23	C₂×#88055	C₂²	1	2	yes	—
24	#11718742	C₄	1	2	yes	complemented
25	#11718731	C₄	1	2	yes	complemented
26	#11718717	C₄	1	4	yes	complemented
27	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
28	#11718585	C₂²	1	1	yes	characteristic
29	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
30	#11718340	—	2	2	no	—
31	C₈×Dic₂₆	—	2	2	no	—
32	#11718150	C₂²	1	4	yes	—
33	#11718147	C₂²	1	4	yes	—
34	C₂×C₁₃×#41	C₂³	1	2	yes	—
35	C₁₃×#131	—	2	8	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	—	2	2	no	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
43	C₁₃×#124	C₂×C₄	1	2	yes	—
44	C₁₃×#121	—	2	2	no	—
45	C₄×D₄×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88180	C₂³	1	2	yes	—
48	#88178	—	2	2	no	—
49	C₂×C₄×Dic₁₃	—	2	2	no	—
50	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
51	C₂×#6099	C₂³	1	2	yes	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	#88124	C₂×C₄	1	2	yes	—
54	#88123	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₆	—	2	2	no	—
56	C₁₃×C₄⋊C₈	—	2	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	4	no	—
58	C₄×C₈×C₁₃	D₄	1	2	yes	—
59	C₄×C₈×C₁₃	D₄	1	2	yes	—
60	#88077	—	2	8	no	—
61	#88075	C₂×C₄	1	8	yes	—
62	#88072	C₂×C₄	1	8	yes	—
63	#88055	—	2	2	no	—
64	#88055	—	2	4	no	—
65	#88055	—	2	4	no	—
66	#88054	—	2	4	no	—
67	C₈×Dic₁₃	—	2	2	no	—
68	C₄⋊#1055	—	2	2	no	—
69	C₄×#1055	C₂×C₄	1	2	yes	—
70	C₄×#1055	C₂×C₄	1	2	yes	—
71	#21660	—	13	1	no	maximal, Sylow
72	C₁₃×#41	—	2	4	no	—
73	C₁₃×#41	—	2	4	no	—
74	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂×Dic₂₆	—	2	2	no	—
83	C₁₃×M₁₆	—	2	4	no	—
84	C₁₃×M₁₆	—	4	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	4	4	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	4	yes	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₂²⋊C₄	—	2	4	no	—
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	2	yes	—
97	C₂²⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
100	#6101	—	2	4	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	—	4	4	no	—
104	C₄×Dic₁₃	#41	1	4	yes	—
105	#6099	—	2	4	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	2	no	—
109	C₂×#1055	C₂²×C₄	1	4	yes	—
110	#3131	—	13	1	no	—
111	#2960	—	13	1	no	—
112	#2931	D₁₃	1	1	yes	characteristic
113	#1994	—	13	1	no	—
114	#1962	—	13	1	no	—
115	#1938	—	13	4	no	—
116	#1837	—	13	1	no	—
117	#1765	—	13	4	no	—
118	C₄×#97	—	13	1	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	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	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
135	C₂×Dic₁₃	—	2	4	no	—
136	C₂×Dic₁₃	—	2	4	no	—
137	C₂×Dic₁₃	—	4	4	no	—
138	Dic₂₆	—	4	4	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	—	4	2	no	—
141	C₈×C₁₃	#124	1	2	yes	—
142	C₈×C₁₃	#124	1	2	yes	—
143	#1055	—	2	4	no	—
144	#1055	—	4	2	no	—
145	C₂×#131	C₂×D₁₃	1	2	yes	—
146	#533	—	13	1	no	—
147	#532	C₂×D₁₃	1	1	yes	characteristic
148	#488	—	13	4	no	—
149	#487	—	13	2	no	—
150	#486	—	13	2	no	—
151	#446	—	26	2	no	—
152	C₈×Q₈	—	26	2	no	—
153	#436	—	2	2	no	—
154	C₈×D₄	—	2	2	no	—
155	#424	—	13	2	no	—
156	#423	—	13	1	no	—
157	#423	C₂×D₁₃	1	1	yes	characteristic
158	C₂×#104	—	13	4	no	—
159	#416	—	13	4	no	—
160	#408	C₂×D₁₃	1	2	yes	—
161	#405	—	13	2	no	—
162	C₂×#97	—	13	2	no	—
163	C₂×#97	—	13	2	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₄³	—	13	1	no	—
167	#341	—	13	4	no	—
168	#339	—	13	4	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	4	2	no	—
172	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₁₃	#529	1	2	yes	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	#444	1	4	yes	—
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	Dic₁₃	—	4	4	no	—
182	C₂×#41	C₂²×D₁₃	1	2	yes	—
183	#131	—	2	8	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	13	2	no	—
186	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	13	2	no	—
192	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
193	#127	C₄×D₁₃	1	1	yes	characteristic
194	#125	—	13	2	no	—
195	#124	C₄×D₁₃	1	2	yes	—
196	C₂²⋊Q₈	—	26	2	no	—
197	#121	—	2	2	no	—
198	C₄×Q₈	—	26	2	no	—
199	C₄×D₄	—	2	2	no	—
200	#117	—	13	2	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	26	2	no	—
203	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	26	4	no	—
206	C₄⋊C₈	—	26	2	no	—
207	#104	—	26	8	no	—
208	#103	—	13	8	no	—
209	#102	—	13	8	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	#97	—	26	4	no	—
212	#97	—	26	4	no	—
213	#97	—	26	2	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	—	13	2	no	—
216	C₄×C₈	—	26	2	no	—
217	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
218	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	8	2	no	—
221	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#444	1	2	yes	—
223	#41	—	2	4	no	—
224	#41	—	2	4	no	—
225	C₂×Q₈	—	26	2	no	—
226	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	26	2	no	—
232	C₂²×C₄	#6127	1	2	yes	—
233	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	4	4	no	—
236	M₁₆	—	26	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	26	2	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	#6129	1	4	yes	—
245	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
246	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	13	4	no	—
250	C₄⋊C₄	—	26	4	no	—
251	C₄⋊C₄	—	26	2	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	26	2	no	—
254	C₄²	—	13	4	no	—
255	C₄²	—	26	2	no	—
256	C₄²	—	26	2	no	—
257	C₄²	—	52	4	no	—
258	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
261	C₁₃	#21660	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#6127	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	52	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	—	52	4	no	—
276	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
277	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#6129	1	2	yes	—
280	C₂×C₄	#41×D₁₃	1	2	yes	—
281	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	26	4	no	—
285	C₈	—	52	2	no	—
286	C₈	#88190	1	2	yes	—
287	C₈	#88190	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	#11719262	1	2	yes	—
292	C₂²	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	52	4	no	—
296	C₄	—	52	4	no	—
297	C₄	#11718806	1	4	yes	—
298	C₄	C₄×#6129	1	2	yes	—
299	C₄	C₂×#88190	1	1	yes	characteristic
300	C₄	#11719537	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	8	2	no	—
303	C₂	C₄×#88190	1	1	yes	characteristic
304	C₂	C₂×#11718806	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2931	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#532	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2931	D₁₃
hypocenter	C₁₃	#21660
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	semidirect
3. #13115812	C₂	semidirect
5. #13115550	C₂	semidirect
6. #13114973	C₂	semidirect
7. #13114678	C₂	semidirect
8. C₄×#88055	C₂	semidirect
24. #11718742	C₄	semidirect
25. #11718731	C₄	semidirect
26. #11718717	C₄	semidirect
261. C₁₃	#21660	semidirect

Maximal subgroups

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

subgroup	index
#13125158	2
#13123770	2
#13115812	2
#13115665	2
#13115550	2
#13114973	2
#13114678	2
C₄×#88055	2
C₁₃×#2931	2
#21660	13