Subgroups — #15780659

Order 3328.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718107	C₂	1	1	yes	maximal, characteristic
2	#13116451	C₂	1	1	yes	maximal, characteristic
3	#13116435	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114673	C₂	1	1	yes	maximal, characteristic
5	#13112926	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112914	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	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	C₂²×#6090	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	C₂×#88045	C₄	1	1	yes	characteristic
17	C₁₃×#364	—	2	2	no	—
18	#11718144	C₄	1	1	yes	characteristic
19	#11718115	C₄	1	1	yes	characteristic
20	#11718107	—	2	2	no	—
21	#11718099	—	2	1	no	—
22	#11718099	—	2	1	no	—
23	#11718094	—	2	1	no	—
24	C₁₆×Dic₁₃	—	2	1	no	—
25	C₂×#6127	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#88180	C₂×C₄	1	1	yes	characteristic
30	C₂×#6099	C₂×C₄	1	1	yes	characteristic
31	C₂×#6099	C₂×C₄	1	1	yes	characteristic
32	#88155	—	2	1	no	—
33	#88155	—	2	1	no	—
34	#88155	—	2	1	no	—
35	#88155	—	2	1	no	—
36	C₂×#6094	—	2	1	no	—
37	C₂×C₈×D₁₃	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	—	2	1	no	—
40	C₁₃×M₃₂	—	2	1	no	—
41	C₁₃×M₃₂	D₄	1	2	yes	—
42	C₂×C₁₃×C₁₆	—	2	1	no	—
43	C₂×C₁₃×C₁₆	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	—	2	1	no	—
46	#88061	C₈	1	1	yes	characteristic
47	#88061	C₈	1	1	yes	characteristic
48	#88056	C₈	1	1	yes	characteristic
49	#88056	C₈	1	1	yes	characteristic
50	#88055	—	2	1	no	—
51	C₈×Dic₁₃	—	2	1	no	—
52	C₂×#6090	—	2	1	no	—
53	C₂×#6090	—	2	1	no	—
54	C₂×#6090	—	2	1	no	—
55	C₂×#6090	—	2	1	no	—
56	C₂×#6090	—	2	1	no	—
57	C₂×#6090	D₄	1	1	yes	characteristic
58	C₂×#6090	D₄	1	1	yes	characteristic
59	#88045	—	2	2	no	—
60	#13122	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	1	no	—
64	#6127	—	2	1	no	—
65	#6127	—	2	1	no	—
66	#6127	—	2	1	no	—
67	#6127	—	2	1	no	—
68	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
81	C₄²×C₁₃	—	2	1	no	—
82	C₄²×C₁₃	—	2	1	no	—
83	C₂²⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
86	C₄×Dic₁₃	—	2	1	no	—
87	#6099	—	2	1	no	—
88	#6099	—	2	1	no	—
89	#6099	—	2	1	no	—
90	#6099	M₁₆	1	2	yes	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	—	2	1	no	—
93	#6094	—	4	1	no	—
94	#6094	—	4	1	no	—
95	C₈×D₁₃	—	4	1	no	—
96	C₈×D₁₃	—	4	1	no	—
97	C₁₃×C₁₆	—	2	2	no	—
98	C₁₃×C₁₆	—	4	1	no	—
99	#6090	—	2	1	no	—
100	#6090	—	2	1	no	—
101	#6090	—	4	1	no	—
102	C₂×#364	D₁₃	1	1	yes	characteristic, complemented
103	C₂×#350	—	13	1	no	—
104	#2113	—	13	1	no	—
105	#2108	—	13	1	no	—
106	#1764	—	13	1	no	—
107	#1379	—	13	1	no	—
108	#1374	—	13	1	no	—
109	C₂³×C₁₃	#131	1	1	yes	characteristic
110	C₂²×D₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
120	C₂²⋊D₁₃	—	4	1	no	—
121	C₂²⋊D₁₃	—	4	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	—	2	1	no	—
125	D₅₂	—	2	1	no	—
126	D₅₂	—	2	1	no	—
127	C₄×D₁₃	—	4	1	no	—
128	C₄×D₁₃	—	4	1	no	—
129	Dic₂₆	—	2	1	no	—
130	Dic₂₆	—	2	1	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	#121	1	1	yes	characteristic
134	C₈×C₁₃	#123	1	1	yes	characteristic
135	C₈×C₁₃	#124	1	1	yes	characteristic
136	C₈×C₁₃	#126	1	1	yes	characteristic
137	#1055	—	2	2	no	—
138	#1055	—	4	1	no	—
139	C₂×#131	—	13	1	no	—
140	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
141	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
142	C₂²×C₁₆	—	13	1	no	—
143	C₂²×C₁₆	—	13	1	no	—
144	C₂×#104	—	13	1	no	—
145	#416	—	13	1	no	—
146	#405	—	13	1	no	—
147	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	#364	—	2	2	no	—
149	#350	—	26	2	no	—
150	C₂²⋊C₁₆	—	26	1	no	—
151	C₂²⋊C₁₆	—	26	1	no	—
152	#346	—	26	1	no	—
153	C₄×C₁₆	—	26	1	no	—
154	#341	—	13	1	no	—
155	#339	—	13	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	#433	1	1	yes	characteristic
159	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
160	C₂²×C₁₃	#436	1	1	yes	characteristic
161	C₂×D₁₃	—	4	1	no	—
162	C₂×D₁₃	—	4	1	no	—
163	C₂×D₁₃	—	4	1	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	#388	1	1	yes	characteristic
166	C₄×C₁₃	#431	1	1	yes	characteristic
167	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
168	C₄×C₁₃	#435	1	1	yes	characteristic
169	Dic₁₃	—	4	1	no	—
170	Dic₁₃	—	4	1	no	—
171	Dic₁₃	—	4	1	no	—
172	C₂×#41	—	13	1	no	—
173	#131	—	26	1	no	—
174	#131	—	26	1	no	—
175	#131	—	26	1	no	—
176	#131	—	26	1	no	—
177	C₂×M₁₆	—	13	1	no	—
178	C₂×M₁₆	—	13	1	no	—
179	C₂×M₁₆	—	26	1	no	—
180	C₂²×C₈	—	26	1	no	—
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	#117	—	13	1	no	—
184	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	C₂²⋊D₁₃	1	2	yes	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	13	1	no	—
192	C₂×C₁₆	—	13	1	no	—
193	C₂×C₁₆	—	26	1	no	—
194	C₂×C₁₆	—	26	1	no	—
195	C₂×C₁₆	—	26	1	no	—
196	C₂×C₁₆	—	26	1	no	—
197	C₂×C₁₆	—	26	1	no	—
198	#104	—	26	2	no	—
199	#103	—	13	1	no	—
200	#103	—	13	1	no	—
201	#102	—	13	1	no	—
202	#102	—	13	1	no	—
203	#97	—	26	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	26	1	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	#1923	1	1	yes	characteristic
210	C₂×C₁₃	#2170	1	1	yes	characteristic
211	C₂×C₁₃	#2171	1	1	yes	characteristic
212	D₁₃	—	8	1	no	—
213	#41	—	26	1	no	—
214	#41	—	26	1	no	—
215	#41	—	26	1	no	—
216	#41	—	26	1	no	—
217	C₂×Q₈	—	13	1	no	—
218	C₂×D₄	—	13	1	no	—
219	C₂×D₄	—	26	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	26	1	no	—
222	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
223	M₁₆	—	13	2	no	—
224	M₁₆	—	26	1	no	—
225	M₁₆	—	26	1	no	—
226	M₁₆	—	26	1	no	—
227	M₁₆	—	52	1	no	—
228	M₁₆	—	52	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	1	no	—
233	C₂×C₈	—	52	1	no	—
234	C₂×C₈	—	52	1	no	—
235	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
237	C₂×C₈	#6127	1	1	yes	characteristic
238	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
239	C₂×C₈	#6129	1	1	yes	characteristic
240	C₂×C₈	#6131	1	1	yes	characteristic
241	C₂×C₈	#6131	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄⋊C₄	—	13	1	no	—
244	C₄⋊C₄	—	13	1	no	—
245	C₂²⋊C₄	—	26	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	26	1	no	—
249	C₁₆	—	2	2	no	—
250	C₁₆	—	4	1	no	—
251	C₁₆	—	26	1	no	—
252	C₁₆	—	26	1	no	—
253	C₁₆	—	52	1	no	—
254	C₁₃	#13122	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	26	1	no	—
256	C₂³	#88161	1	1	yes	characteristic
257	Q₈	—	26	1	no	—
258	Q₈	—	26	1	no	—
259	D₄	—	26	1	no	—
260	D₄	—	26	1	no	—
261	D₄	—	52	1	no	—
262	D₄	—	52	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	26	1	no	—
267	C₂×C₄	—	26	1	no	—
268	C₂×C₄	—	26	1	no	—
269	C₂×C₄	—	52	1	no	—
270	C₂×C₄	—	52	1	no	—
271	C₂×C₄	#88146	1	1	yes	characteristic
272	C₂×C₄	#88147	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	#88155	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	26	2	no	—
280	C₈	—	52	1	no	—
281	C₈	#88148	1	1	yes	characteristic
282	C₈	#88152	1	1	yes	characteristic
283	C₈	#88193	1	1	yes	characteristic
284	C₈	#88201	1	1	yes	characteristic
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	52	1	no	—
288	C₂²	—	52	1	no	—
289	C₂²	—	52	1	no	—
290	C₂²	#11718495	1	1	yes	characteristic
291	C₂²	#11718501	1	1	yes	characteristic
292	C₂²	#11718789	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	52	1	no	—
295	C₄	—	52	1	no	—
296	C₄	—	52	1	no	—
297	C₄	#11718494	1	1	yes	characteristic
298	C₄	#11718497	1	1	yes	characteristic
299	C₄	#11718651	1	1	yes	characteristic
300	C₄	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	104	1	no	—
304	C₂	#13115441	1	1	yes	characteristic
305	C₂	#13116672	1	1	yes	characteristic
306	C₂	#13116677	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	#13122
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #13116435	C₂	semidirect
5. #13112926	C₂	semidirect
6. #13112914	C₂	semidirect
7. C₂×C₁₃×#364	C₂	semidirect
102. C₂×#364	D₁₃	semidirect
254. C₁₃	#13122	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718107	2
#13116451	2
#13116435	2
#13114673	2
#13112926	2
#13112914	2
C₂×C₁₃×#364	2
#13122	13