Subgroups — #15846183

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13126178	C₂	1	2	yes	maximal
2	#13123770	C₂	1	1	yes	maximal, characteristic
3	#13114339	C₂	1	4	yes	maximal, complemented
4	#13114237	C₂	1	2	yes	maximal, complemented
5	#13114227	C₂	1	2	yes	maximal, complemented
6	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
7	#13113526	C₂	1	2	yes	maximal
8	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₂²	1	2	yes	—
10	#11719175	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₂²	1	1	yes	characteristic
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#499	—	2	1	no	—
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#498	C₂²	1	2	yes	—
16	C₁₃×#427	—	2	1	no	—
17	C₁₃×#427	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718763	—	2	1	no	—
20	#11718762	—	2	1	no	—
21	#11718752	C₂²	1	4	yes	—
22	#11718751	C₂²	1	4	yes	—
23	#11718712	—	2	4	no	—
24	#11718692	—	2	2	no	—
25	#11718682	—	2	2	no	—
26	#11718585	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	2	yes	—
28	#11718340	—	2	1	no	—
29	#11718340	C₂²	1	2	yes	—
30	C₈×Dic₂₆	—	2	1	no	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	#11718139	C₂²	1	4	yes	—
33	#11718138	C₂²	1	4	yes	—
34	#11718050	C₂²	1	2	yes	—
35	#11718050	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₁₃×#125	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	—	2	2	no	—
42	C₁₃×C₂²⋊Q₈	—	2	2	no	—
43	C₄×Q₈×C₁₃	—	2	2	no	—
44	C₄×Q₈×C₁₃	—	2	2	no	—
45	C₁₃×#117	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×#6107	—	2	4	no	—
48	#88180	—	2	1	no	—
49	#88178	—	2	1	no	—
50	#88178	D₄	1	2	yes	—
51	C₂×#6099	—	2	1	no	—
52	C₂²×#1055	—	2	2	no	—
53	#88124	—	2	1	no	—
54	#88123	D₄	1	1	yes	characteristic
55	#88123	D₄	1	1	yes	characteristic
56	C₄×Dic₂₆	—	2	1	no	—
57	C₄×Dic₂₆	C₂³	1	2	yes	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#106	—	2	2	no	—
63	C₁₃×#106	—	2	2	no	—
64	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	#88075	—	2	4	no	—
69	#88055	—	2	2	no	—
70	#88054	—	2	2	no	—
71	#88054	—	2	2	no	—
72	C₈×Dic₁₃	—	2	2	no	—
73	#88050	—	2	4	no	—
74	#88047	—	2	4	no	—
75	C₄⋊#1055	—	2	2	no	—
76	C₄⋊#1055	—	2	1	no	—
77	C₄⋊#1055	C₂³	1	4	yes	—
78	C₄⋊#1055	C₂³	1	2	yes	—
79	#19211	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
85	C₂×Dic₂₆	—	2	1	no	—
86	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₁₃	—	4	2	no	—
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	#41	1	1	yes	characteristic
104	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊Dic₁₃	—	2	2	no	—
106	#6107	—	8	4	no	—
107	C₄⋊Dic₁₃	—	2	1	no	—
108	C₄⋊Dic₁₃	—	2	1	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
111	#6101	—	2	2	no	—
112	#6101	—	2	2	no	—
113	C₄×Dic₁₃	—	2	2	no	—
114	#6099	—	4	2	no	—
115	C₂×#1055	—	2	4	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	8	2	no	—
118	#3249	—	13	2	no	—
119	#3160	D₁₃	1	1	yes	characteristic
120	#2960	—	13	1	no	—
121	#1692	—	13	4	no	—
122	#1676	—	13	2	no	—
123	#1674	—	13	2	no	—
124	C₂×#335	—	13	1	no	—
125	#1550	—	13	2	no	—
126	C₂³×C₁₃	#121	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	#137	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	Dic₂₆	—	2	2	no	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	4	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
151	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
152	#1055	—	4	2	no	—
153	#1055	—	4	4	no	—
154	#533	—	13	1	no	—
155	#533	C₂×D₁₃	1	2	yes	—
156	#500	C₂×D₁₃	1	1	yes	characteristic
157	#500	C₂×D₁₃	1	1	yes	characteristic
158	#499	—	2	1	no	—
159	#499	—	2	1	no	—
160	#498	C₂×D₁₃	1	2	yes	—
161	#477	—	13	4	no	—
162	#477	—	26	2	no	—
163	#467	—	26	4	no	—
164	#462	—	13	4	no	—
165	#462	—	26	2	no	—
166	#446	—	13	2	no	—
167	#446	—	26	1	no	—
168	C₈×Q₈	—	13	2	no	—
169	C₈×Q₈	—	26	1	no	—
170	#427	—	2	1	no	—
171	#427	—	2	1	no	—
172	#424	—	26	1	no	—
173	#423	—	13	1	no	—
174	C₂×C₄⋊C₈	—	13	2	no	—
175	#405	—	26	1	no	—
176	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
177	#335	—	13	2	no	—
178	#335	—	13	2	no	—
179	#333	—	13	4	no	—
180	#332	—	13	4	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	#462	1	2	yes	—
184	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	#462	1	2	yes	—
191	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
192	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
193	C₄×C₁₃	#576	1	1	yes	characteristic
194	C₄×C₁₃	#578	1	1	yes	characteristic
195	Dic₁₃	—	4	2	no	—
196	Dic₁₃	—	4	2	no	—
197	C₂×Q₁₆	—	26	4	no	—
198	C₂×M₁₆	—	26	1	no	—
199	C₂²×C₈	—	26	2	no	—
200	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₄⋊Q₈	—	13	1	no	—
203	C₄⋊Q₈	—	13	1	no	—
204	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
205	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
206	#125	—	2	2	no	—
207	#125	—	26	1	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	13	2	no	—
211	C₂²⋊Q₈	—	26	1	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	13	2	no	—
215	C₄×Q₈	—	26	1	no	—
216	#117	—	2	2	no	—
217	#117	—	26	1	no	—
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	—	26	4	no	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	—	13	4	no	—
228	C₄⋊C₈	—	26	2	no	—
229	C₄⋊C₈	—	26	2	no	—
230	C₄⋊C₈	—	26	2	no	—
231	C₄⋊C₈	—	26	1	no	—
232	#103	—	26	4	no	—
233	#103	—	26	4	no	—
234	#97	—	26	2	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
242	C₂×C₁₃	#3061	1	1	yes	characteristic
243	C₂×C₁₃	#3067	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	13	2	no	—
247	C₂×Q₈	—	26	1	no	—
248	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
249	C₂²×C₄	#6129	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	Q₁₆	—	104	4	no	—
252	M₁₆	—	52	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	26	4	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	104	2	no	—
257	C₂×C₈	#6107	1	4	yes	complemented
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	13	2	no	—
269	C₄⋊C₄	—	26	1	no	—
270	C₄⋊C₄	—	26	1	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	1	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	26	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	26	2	no	—
278	C₄²	D₄×D₁₃	1	1	yes	characteristic
279	C₄²	#6129	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
281	C₄²	C₂³×D₁₃	1	1	yes	characteristic
282	C₁₃	#19211	1	1	yes	characteristic, complemented, Sylow
283	C₂³	#88192	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	26	2	no	—
288	Q₈	—	26	2	no	—
289	Q₈	—	52	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	#88192	1	1	yes	characteristic
298	C₂×C₄	#88192	1	1	yes	characteristic
299	C₂×C₄	#88192	1	1	yes	characteristic
300	C₂×C₄	#88196	1	2	yes	—
301	C₂×C₄	C₂×#6107	1	2	yes	—
302	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#6129	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	52	4	no	—
308	C₈	—	52	2	no	—
309	C₈	C₂×#6107	1	2	yes	—
310	C₈	C₂×#6107	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#11718751	1	2	yes	—
314	C₂²	C₂×#88192	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	52	2	no	—
321	C₄	—	52	2	no	—
322	C₄	#11718751	1	2	yes	—
323	C₄	#11719474	1	1	yes	characteristic
324	C₄	#11719476	1	1	yes	characteristic
325	C₄	C₂×#88196	1	1	yes	characteristic
326	C₄	C₂²×#6107	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	C₂×#11718751	1	1	yes	characteristic
329	C₂	#13124502	1	1	yes	characteristic
330	C₂	#13124552	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#19211
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #13114339	C₂	semidirect
4. #13114237	C₂	semidirect
5. #13114227	C₂	semidirect
88. C₂×C₈×C₁₃	Q₁₆	semidirect
257. C₂×C₈	#6107	semidirect
282. C₁₃	#19211	semidirect

Maximal subgroups

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

subgroup	index
#13126178	2
#13123770	2
#13114339	2
#13114237	2
#13114227	2
C₂×#11718050	2
#13113526	2
C₁₃×#3160	2
#19211	13