Subgroups — #15871096

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	#13125159	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123549	C₂	1	1	yes	maximal, characteristic
4	#13115813	C₂	1	1	yes	maximal, characteristic, complemented
5	#13115664	C₂	1	1	yes	maximal, characteristic
6	#13115547	C₂	1	4	yes	maximal, complemented
7	#13114974	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114680	C₂	1	4	yes	maximal, complemented
9	C₄×C₁₃×#97	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	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄³×C₁₃	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	#11718358	C₄	1	4	yes	complemented
26	#11718355	C₄	1	2	yes	complemented
27	#11718349	—	2	2	no	—
28	C₈×D₅₂	—	2	2	no	—
29	#11718342	C₄	1	2	yes	complemented
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	—	2	2	no	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	#88184	—	2	2	no	—
42	#88180	C₂³	1	2	yes	—
43	#88178	—	2	2	no	—
44	C₂×#6099	C₂³	1	2	yes	—
45	C₂×#6099	C₂³	1	2	yes	—
46	#88155	—	2	8	no	—
47	C₂×#6094	—	2	2	no	—
48	C₂×C₈×D₁₃	—	2	2	no	—
49	#88129	C₂×C₄	1	2	yes	—
50	#88128	C₂×C₄	1	1	yes	characteristic
51	C₄×D₅₂	—	2	2	no	—
52	#88124	C₂×C₄	1	2	yes	—
53	#88123	C₂×C₄	1	1	yes	characteristic
54	#88123	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₆	—	2	2	no	—
56	C₁₃×#97	—	2	4	no	—
57	C₁₃×#97	—	2	4	no	—
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	D₄	1	2	yes	—
61	C₄×C₈×C₁₃	D₄	1	2	yes	—
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	#21660	—	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	—	4	4	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	C₄²×C₁₃	#41	1	4	yes	—
97	C₂²⋊Dic₁₃	—	2	2	no	—
98	#6103	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
102	#6101	—	2	4	no	—
103	C₄×Dic₁₃	—	2	2	no	—
104	#6099	—	2	4	no	—
105	#6099	—	2	4	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	2	no	—
108	#6094	—	4	4	no	—
109	C₈×D₁₃	—	4	4	no	—
110	#3131	—	13	1	no	—
111	#2960	—	13	1	no	—
112	#2931	—	13	1	no	—
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	D₁₃	1	1	yes	characteristic, complemented
119	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂²×D₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
130	C₂²⋊D₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	2	4	no	—
132	C₂×Dic₁₃	—	2	2	no	—
133	D₅₂	—	2	4	no	—
134	D₅₂	—	4	4	no	—
135	C₄×D₁₃	—	4	2	no	—
136	C₄×D₁₃	—	4	2	no	—
137	Dic₂₆	—	2	4	no	—
138	Dic₂₆	—	4	4	no	—
139	C₈×C₁₃	—	2	4	no	—
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	#124	1	2	yes	—
142	C₈×C₁₃	#124	1	2	yes	—
143	#1055	—	4	2	no	—
144	#1055	—	4	2	no	—
145	C₂×#131	—	13	2	no	—
146	#533	—	13	1	no	—
147	#532	—	13	1	no	—
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	—	26	2	no	—
154	C₈×D₄	—	26	2	no	—
155	#424	—	13	2	no	—
156	#423	—	13	1	no	—
157	#423	—	13	1	no	—
158	C₂×#104	—	13	4	no	—
159	#416	—	13	4	no	—
160	#408	—	13	2	no	—
161	#405	—	13	2	no	—
162	C₂×#97	C₂×D₁₃	1	2	yes	—
163	C₂×#97	C₂×D₁₃	1	2	yes	—
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₄³	C₂×D₁₃	1	1	yes	characteristic
167	#341	—	13	4	no	—
168	#339	—	13	4	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₁₃	#529	1	2	yes	—
172	C₂×D₁₃	—	4	4	no	—
173	C₂×D₁₃	—	4	2	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	4	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	2	no	—
182	C₂×#41	—	13	2	no	—
183	#131	—	26	8	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂×M₁₆	—	13	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
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	—	13	1	no	—
193	#127	—	13	1	no	—
194	#125	—	13	2	no	—
195	#124	—	13	2	no	—
196	C₂²⋊Q₈	—	26	2	no	—
197	#121	—	26	2	no	—
198	C₄×Q₈	—	26	2	no	—
199	C₄×D₄	—	26	2	no	—
200	#117	—	13	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
203	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊C₈	—	26	4	no	—
205	C₄⋊C₈	—	26	2	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	—	26	4	no	—
211	#97	—	2	4	no	—
212	#97	—	2	4	no	—
213	#97	—	26	2	no	—
214	C₄×C₈	—	26	2	no	—
215	C₄×C₈	C₄×D₁₃	1	2	yes	—
216	C₄×C₈	C₄×D₁₃	1	2	yes	—
217	C₄×C₈	D₅₂	1	2	yes	—
218	C₄×C₈	D₅₂	1	2	yes	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#444	1	2	yes	—
222	D₁₃	—	8	2	no	—
223	#41	—	26	4	no	—
224	#41	—	26	4	no	—
225	C₂×Q₈	—	13	2	no	—
226	C₂×Q₈	—	26	2	no	—
227	C₂×D₄	—	13	2	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂×D₄	—	26	2	no	—
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	—	26	4	no	—
235	M₁₆	—	26	4	no	—
236	M₁₆	—	52	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	2	no	—
242	C₂×C₈	—	52	4	no	—
243	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
244	C₂×C₈	C₂×D₅₂	1	2	yes	—
245	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
246	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
247	C₂×C₈	#6127	1	4	yes	—
248	C₄⋊C₄	—	13	4	no	—
249	C₄⋊C₄	—	26	2	no	—
250	C₄⋊C₄	—	26	4	no	—
251	C₄⋊C₄	—	26	2	no	—
252	C₂²⋊C₄	—	26	2	no	—
253	C₂²⋊C₄	—	26	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	4	no	—
256	C₄²	—	26	2	no	—
257	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₄²	C₂×D₅₂	1	2	yes	—
260	C₄²	#6127	1	4	yes	—
261	C₁₃	#21660	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	26	2	no	—
263	C₂³	C₂×#6127	1	1	yes	characteristic
264	Q₈	—	26	4	no	—
265	Q₈	—	52	4	no	—
266	D₄	—	26	4	no	—
267	D₄	—	52	4	no	—
268	D₄	—	52	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	26	2	no	—
274	C₂×C₄	—	52	2	no	—
275	C₂×C₄	—	52	2	no	—
276	C₂×C₄	C₄×D₅₂	1	2	yes	—
277	C₂×C₄	C₄×D₅₂	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
280	C₂×C₄	C₂×#6127	1	2	yes	—
281	C₂×C₄	C₂×#6127	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	2	2	no	—
284	C₈	—	52	2	no	—
285	C₈	—	52	2	no	—
286	C₈	#88129	1	2	yes	—
287	C₈	#88129	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	52	4	no	—
290	C₂²	—	52	2	no	—
291	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
292	C₂²	#11719184	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	52	4	no	—
296	C₄	—	52	2	no	—
297	C₄	#11718378	1	4	yes	—
298	C₄	C₄×#6127	1	2	yes	—
299	C₄	C₂×#88129	1	1	yes	characteristic
300	C₄	#11719182	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	104	2	no	—
303	C₂	C₄×#88129	1	1	yes	characteristic
304	C₂	C₂×#11718378	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#97	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#21660
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	semidirect
4. #13115813	C₂	semidirect
6. #13115547	C₂	semidirect
7. #13114974	C₂	semidirect
8. #13114680	C₂	semidirect
9. C₄×C₁₃×#97	C₂	semidirect
25. #11718358	C₄	semidirect
26. #11718355	C₄	semidirect
29. #11718342	C₄	semidirect
118. C₄×#97	D₁₃	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
#13125159	2
#13123770	2
#13123549	2
#13115813	2
#13115664	2
#13115547	2
#13114974	2
#13114680	2
C₄×C₁₃×#97	2
#21660	13