Subgroups — #15758933

Order 3328.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123400	C₂	1	1	yes	maximal, characteristic
2	#13113521	C₂	1	2	yes	maximal, complemented
3	#13113150	C₂	1	1	yes	maximal, characteristic
4	#13112694	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2628	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂²×#6099	—	2	1	no	—
9	C₁₃×#400	C₄	1	2	yes	—
10	C₁₃×#399	C₄	1	2	yes	—
11	C₁₃×#395	—	2	1	no	—
12	C₁₃×#393	—	2	1	no	—
13	C₁₃×#389	—	2	1	no	—
14	C₁₃×#382	C₂²	1	1	yes	characteristic
15	C₁₃×#380	—	2	1	no	—
16	#11718660	—	2	4	no	—
17	#11718586	C₂²	1	2	yes	—
18	#11718585	—	2	1	no	—
19	#11718585	C₂²	1	2	yes	—
20	#11718139	—	2	4	no	—
21	#11718120	—	2	1	no	—
22	#11718117	—	2	1	no	—
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	2	2	no	—
27	C₁₃×#117	—	2	2	no	—
28	C₁₃×#117	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	4	2	no	—
43	C₂×#6099	—	4	2	no	—
44	C₂²×#1055	—	2	2	no	—
45	C₁₃×#95	—	2	2	no	—
46	C₁₃×#95	—	2	2	no	—
47	C₁₃×#95	—	2	1	no	—
48	C₁₃×#95	C₂×C₄	1	2	yes	—
49	C₂²⋊#1055	—	2	2	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	—	2	2	no	—
53	C₄⋊#1055	D₄	1	4	yes	—
54	C₄×#1055	—	2	2	no	—
55	#9193	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Q₈×C₁₃	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	8	2	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
69	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	4	1	no	—
77	C₁₃×C₄⋊C₄	—	4	1	no	—
78	C₁₃×C₄⋊C₄	—	4	1	no	—
79	C₁₃×C₄⋊C₄	—	4	1	no	—
80	C₁₃×C₄⋊C₄	—	8	2	no	—
81	C₁₃×C₂²⋊C₄	—	2	2	no	—
82	C₁₃×C₂²⋊C₄	—	2	2	no	—
83	C₁₃×C₂²⋊C₄	—	8	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃	#41	1	2	yes	—
88	#6099	—	4	4	no	—
89	#6099	—	4	2	no	—
90	#6099	—	4	2	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	4	4	no	—
93	C₂×#1055	—	4	2	no	—
94	#2905	—	13	1	no	—
95	#2628	D₁₃	1	1	yes	characteristic
96	#1549	—	13	2	no	—
97	#1458	—	13	1	no	—
98	#1286	—	13	2	no	—
99	C₂³×C₁₃	—	2	1	no	—
100	C₂³×C₁₃	—	2	1	no	—
101	C₂³×C₁₃	—	4	1	no	—
102	C₂³×C₁₃	—	4	1	no	—
103	C₂³×C₁₃	#103	1	2	yes	—
104	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	Q₈×C₁₃	—	8	4	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	#103	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#121	1	2	yes	—
128	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
133	#1055	—	4	4	no	—
134	#1055	—	8	2	no	—
135	C₂²×M₁₆	—	26	1	no	—
136	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
138	#431	—	13	2	no	—
139	#424	—	26	4	no	—
140	#423	—	13	2	no	—
141	#423	—	26	1	no	—
142	#400	Dic₁₃	1	2	yes	—
143	#399	Dic₁₃	1	2	yes	—
144	#395	—	2	1	no	—
145	#393	—	2	1	no	—
146	#389	—	2	1	no	—
147	#382	C₂×D₁₃	1	1	yes	characteristic
148	#380	—	2	1	no	—
149	#332	—	26	4	no	—
150	#324	—	26	1	no	—
151	#323	—	26	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	—	8	1	no	—
159	C₂²×C₁₃	#386	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
161	C₂²×C₁₃	#417	1	1	yes	characteristic
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	4	1	no	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	#450	1	1	yes	characteristic
170	C₄×C₁₃	#451	1	1	yes	characteristic
171	C₄×C₁₃	#483	1	1	yes	characteristic
172	C₄×C₁₃	#484	1	1	yes	characteristic
173	C₂²×Q₈	—	2	1	no	—
174	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	52	2	no	—
178	C₂×M₁₆	—	52	2	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊C₈	—	13	4	no	—
196	C₄⋊C₈	—	26	4	no	—
197	C₄⋊C₈	—	26	2	no	—
198	C₄⋊C₈	—	26	2	no	—
199	C₂²⋊C₈	—	26	2	no	—
200	C₄×C₈	—	26	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	1	no	—
204	#95	C₂×Dic₁₃	1	2	yes	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	—	4	2	no	—
208	C₂×C₁₃	#1872	1	1	yes	characteristic
209	C₂×C₁₃	#1910	1	1	yes	characteristic
210	C₂×C₁₃	#1915	1	1	yes	characteristic
211	C₂⁴	C₂²⋊Dic₁₃	1	1	yes	characteristic
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	8	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	#6106	1	2	yes	—
221	C₂²×C₄	#6107	1	2	yes	—
222	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	M₁₆	—	52	4	no	—
229	M₁₆	—	52	2	no	—
230	M₁₆	—	52	2	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	52	4	no	—
233	C₂×C₈	—	52	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	8	2	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	D₄×D₁₃	1	2	yes	—
248	C₄²	#6129	1	2	yes	—
249	C₁₃	#9193	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	#88075	1	2	yes	—
255	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
256	Q₈	—	8	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#88075	1	2	yes	—
276	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
277	C₂×C₄	#88189	1	1	yes	characteristic
278	C₂×C₄	#88191	1	1	yes	characteristic
279	C₂×C₄	#88193	1	2	yes	—
280	C₂×C₄	C₂×#6106	1	1	yes	characteristic
281	C₂×C₄	C₂×#6107	1	1	yes	characteristic
282	C₂×C₄	#88203	1	1	yes	characteristic
283	C₂×C₄	#88205	1	1	yes	characteristic
284	C₈	—	52	4	no	—
285	C₈	—	104	2	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	#11718881	1	1	yes	characteristic
294	C₂²	C₂×#88075	1	1	yes	characteristic
295	C₂²	#11718893	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	#11718704	1	1	yes	characteristic
304	C₄	#11718705	1	1	yes	characteristic
305	C₄	#11718707	1	1	yes	characteristic
306	C₄	#11718711	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#13115144	1	1	yes	characteristic
311	C₂	#13115364	1	1	yes	characteristic
312	C₂	#13115389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718881
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2628	C₂
socle	C₂²×C₁₃	#386
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2628	D₁₃
hypocenter	C₁₃	#9193
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2628

Complements

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

N	H ≅ G/N	type
2. #13113521	C₂	semidirect
4. #13112694	C₂	semidirect
249. C₁₃	#9193	semidirect

Maximal subgroups

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

subgroup	index
#13123400	2
#13113521	2
#13113150	2
#13112694	2
C₁₃×#2628	2
#9193	13