Subgroups — #15768620

Order 3328.

626 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×#11718762	C₂	1	1	yes	maximal, characteristic
2	#13114574	C₂	1	2	yes	maximal
3	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718152	C₂	1	2	yes	maximal
5	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	1	yes	characteristic
11	#11718762	—	2	2	no	—
12	#11718762	—	2	2	no	—
13	C₂×#88055	C₄	1	2	yes	—
14	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
15	C₁₃×#341	—	2	2	no	—
16	#11718152	C₄	1	8	yes	complemented
17	#11718146	—	2	4	no	—
18	#11718144	—	2	2	no	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
34	#88180	—	2	1	no	—
35	#88180	—	2	1	no	—
36	#88180	—	2	1	no	—
37	#88180	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×#6099	—	2	1	no	—
41	C₂×#6099	—	2	1	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	C₂×C₄	1	4	yes	—
49	C₂×#6099	C₂×C₄	1	4	yes	—
50	C₂²×#1055	—	2	2	no	—
51	C₂²×#1055	C₂×C₄	1	2	yes	—
52	#88055	—	2	4	no	—
53	C₈×Dic₁₃	—	2	4	no	—
54	#10815	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₂²⋊Dic₁₃	—	2	2	no	—
83	C₂²⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄×Dic₁₃	—	2	2	no	—
87	C₄×Dic₁₃	—	2	2	no	—
88	#6099	—	2	4	no	—
89	#6099	—	2	4	no	—
90	#6099	—	2	4	no	—
91	#6099	—	2	4	no	—
92	#6099	—	4	2	no	—
93	#6099	—	4	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	4	2	no	—
98	C₂×#1055	—	4	2	no	—
99	C₂×#1055	C₄²	1	4	yes	—
100	C₂×#405	—	13	1	no	—
101	#1742	—	13	2	no	—
102	C₂×#341	—	13	1	no	—
103	C₂×#341	D₁₃	1	1	yes	characteristic
104	C₂×#338	—	13	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₁₃	#125	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	2	yes	—
116	C₂×C₄×C₁₃	#117	1	2	yes	—
117	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#123	1	2	yes	—
125	C₂×C₄×C₁₃	#123	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#126	1	2	yes	—
128	C₂×C₄×C₁₃	#126	1	2	yes	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₈×C₁₃	—	2	2	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	—	4	4	no	—
134	#1055	—	2	4	no	—
135	#1055	—	4	4	no	—
136	C₂²×M₁₆	—	13	1	no	—
137	C₂²×M₁₆	—	13	2	no	—
138	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
139	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#117	—	13	1	no	—
141	#405	—	26	2	no	—
142	#405	—	26	2	no	—
143	C₂×#97	—	13	2	no	—
144	C₂×C₄×C₈	—	13	2	no	—
145	#343	—	26	4	no	—
146	#341	—	2	2	no	—
147	#341	—	26	2	no	—
148	#338	—	13	8	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₃	#380	1	1	yes	characteristic
156	C₂²×C₁₃	#382	1	1	yes	characteristic
157	C₂²×C₁₃	#387	1	2	yes	—
158	C₂²×C₁₃	#387	1	2	yes	—
159	C₄×C₁₃	#376	1	1	yes	characteristic
160	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	#381	1	1	yes	characteristic
162	C₄×C₁₃	#383	1	1	yes	characteristic
163	C₄×C₁₃	#388	1	2	yes	—
164	C₄×C₁₃	#388	1	2	yes	—
165	Dic₁₃	—	4	4	no	—
166	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	13	4	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	1	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	1	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	26	2	no	—
182	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₈	D₅₂	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
192	#117	—	26	1	no	—
193	#117	—	26	1	no	—
194	#117	—	26	1	no	—
195	#117	—	26	1	no	—
196	C₂×C₄⋊C₄	—	13	2	no	—
197	C₂×C₂²⋊C₄	—	13	2	no	—
198	C₂×C₄²	—	13	2	no	—
199	#97	—	26	4	no	—
200	C₄×C₈	—	26	4	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	#1434	1	1	yes	characteristic
206	C₂×C₁₃	#1813	1	2	yes	—
207	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	—	13	4	no	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
211	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
212	C₂²×C₄	#6127	1	2	yes	—
213	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	#6129	1	1	yes	characteristic
215	C₂²×C₄	#6129	1	2	yes	—
216	C₂²×C₄	#6129	1	2	yes	—
217	C₂²×C₄	#6131	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	26	4	no	—
224	M₁₆	—	26	4	no	—
225	M₁₆	—	26	4	no	—
226	M₁₆	—	26	4	no	—
227	M₁₆	—	52	2	no	—
228	M₁₆	—	52	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	26	2	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	4	no	—
242	C₂×C₈	—	52	2	no	—
243	C₂×C₈	—	52	2	no	—
244	C₂×C₈	#6103	1	2	yes	—
245	C₂×C₈	#6103	1	2	yes	—
246	C₄⋊C₄	—	26	2	no	—
247	C₄⋊C₄	—	26	2	no	—
248	C₂²⋊C₄	—	26	2	no	—
249	C₂²⋊C₄	—	26	2	no	—
250	C₄²	—	26	2	no	—
251	C₄²	—	26	2	no	—
252	C₁₃	#10815	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	4	2	no	—
254	C₂³	#88141	1	2	yes	—
255	C₂³	#88143	1	2	yes	—
256	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
257	C₂³	#88151	1	1	yes	characteristic
258	C₂³	#88200	1	1	yes	characteristic
259	C₂×C₄	—	26	4	no	—
260	C₂×C₄	—	52	4	no	—
261	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
262	C₂×C₄	#88126	1	1	yes	characteristic
263	C₂×C₄	C₄×D₅₂	1	2	yes	—
264	C₂×C₄	#88131	1	1	yes	characteristic
265	C₂×C₄	#88131	1	2	yes	—
266	C₂×C₄	#88131	1	1	yes	characteristic
267	C₂×C₄	#88133	1	2	yes	—
268	C₂×C₄	#88135	1	2	yes	—
269	C₂×C₄	#88139	1	2	yes	—
270	C₂×C₄	#88140	1	2	yes	—
271	C₂×C₄	#88146	1	2	yes	—
272	C₂×C₄	#88146	1	1	yes	characteristic
273	C₂×C₄	#88149	1	1	yes	characteristic
274	C₂×C₄	#88152	1	2	yes	—
275	C₂×C₄	C₂×#6103	1	1	yes	characteristic
276	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
277	C₂×C₄	#88189	1	2	yes	—
278	C₂×C₄	#88192	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	4	4	no	—
282	C₈	—	26	4	no	—
283	C₈	—	52	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#11718310	1	1	yes	characteristic
290	C₂²	#11718311	1	1	yes	characteristic
291	C₂²	#11718314	1	2	yes	—
292	C₂²	#11718639	1	2	yes	—
293	C₂²	#11718648	1	1	yes	characteristic
294	C₄	—	52	4	no	—
295	C₄	#11718309	1	1	yes	characteristic
296	C₄	#11718312	1	1	yes	characteristic
297	C₄	#11718331	1	2	yes	—
298	C₄	C₄×#6103	1	1	yes	characteristic
299	C₄	#11718604	1	1	yes	characteristic
300	C₄	#11718605	1	2	yes	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#13113060	1	1	yes	characteristic
306	C₂	#13114883	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88146
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#10815
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
16. #11718152	C₄	semidirect
252. C₁₃	#10815	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718762	2
#13114574	2
C₂×#11718144	2
C₂×#11718152	2
C₂×C₁₃×#341	2
#10815	13