Subgroups — #15846179

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#13125350	C₂	1	1	yes	maximal, characteristic
3	#13114364	C₂	1	2	yes	maximal, complemented
4	#13114354	C₂	1	2	yes	maximal, complemented
5	#13114218	C₂	1	4	yes	maximal, complemented
6	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
7	#13113527	C₂	1	2	yes	maximal
8	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₂²	1	1	yes	characteristic
10	#11719175	C₂²	1	2	yes	—
11	C₁₃×#446	—	2	1	no	—
12	C₁₃×#446	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₃	—	2	1	no	—
14	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#424	—	2	1	no	—
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₁₃×#405	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718767	—	2	1	no	—
20	#11718767	—	2	1	no	—
21	#11718752	—	2	2	no	—
22	#11718751	—	2	2	no	—
23	#11718712	—	2	4	no	—
24	#11718692	C₂²	1	4	yes	—
25	#11718682	C₂²	1	4	yes	—
26	C₂×C₄⋊#1055	C₂²	1	2	yes	—
27	#11718344	—	2	1	no	—
28	#11718344	—	2	1	no	—
29	#11718343	C₂²	1	1	yes	characteristic
30	#11718343	C₂²	1	1	yes	characteristic
31	#11718341	C₂²	1	2	yes	—
32	#11718078	C₂²	1	4	yes	—
33	#11718073	C₂²	1	4	yes	—
34	#11718049	C₂²	1	2	yes	—
35	#11718049	C₂²	1	2	yes	—
36	C₂×C₁₃×M₁₆	—	2	1	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₁₃×#125	—	2	1	no	—
42	C₁₃×C₂²⋊Q₈	—	2	1	no	—
43	C₁₃×C₂²⋊Q₈	D₄	1	2	yes	—
44	C₄×Q₈×C₁₃	—	2	1	no	—
45	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₁₃×#117	—	2	1	no	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×#6107	—	2	4	no	—
49	#88180	—	2	2	no	—
50	#88178	—	2	2	no	—
51	#88178	—	2	2	no	—
52	C₂²×#1055	—	2	2	no	—
53	#88124	—	2	2	no	—
54	#88123	C₂³	1	2	yes	—
55	#88123	C₂³	1	2	yes	—
56	C₄×Dic₂₆	—	2	2	no	—
57	C₄×Dic₂₆	—	2	2	no	—
58	C₁₃×C₄⋊C₈	—	2	2	no	—
59	C₁₃×C₄⋊C₈	—	2	2	no	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
62	C₁₃×#97	—	2	2	no	—
63	C₄×C₈×C₁₃	—	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	#88058	—	2	1	no	—
70	#88058	—	2	1	no	—
71	#88058	—	2	1	no	—
72	#88058	—	2	1	no	—
73	#88057	—	2	2	no	—
74	#88057	—	2	2	no	—
75	#88050	—	2	4	no	—
76	#88047	—	2	4	no	—
77	C₄⋊#1055	—	2	2	no	—
78	C₄⋊#1055	C₂³	1	4	yes	—
79	#19210	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
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	2	no	—
86	C₂×Dic₂₆	—	2	2	no	—
87	C₁₃×M₁₆	—	4	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	#41	1	1	yes	characteristic
106	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₁₃	—	4	2	no	—
108	#6107	—	8	4	no	—
109	C₄⋊Dic₁₃	—	2	2	no	—
110	C₄⋊Dic₁₃	—	2	2	no	—
111	C₄⋊Dic₁₃	—	2	2	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	C₄×Dic₁₃	—	4	2	no	—
116	C₂×#1055	—	2	4	no	—
117	C₂×#1055	—	8	2	no	—
118	#3249	—	13	2	no	—
119	#3160	—	13	1	no	—
120	#2960	D₁₃	1	1	yes	characteristic
121	#1696	—	13	2	no	—
122	#1694	—	13	2	no	—
123	#1672	—	13	4	no	—
124	C₂×#334	—	13	1	no	—
125	#1550	—	13	2	no	—
126	C₂³×C₁₃	#121	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	4	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
137	C₂×C₄×C₁₃	#137	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	Dic₂₆	—	4	2	no	—
144	Dic₂₆	—	4	2	no	—
145	Dic₂₆	—	8	2	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
150	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
151	#1055	—	4	4	no	—
152	#533	—	13	2	no	—
153	#533	C₂×D₁₃	1	1	yes	characteristic
154	#500	—	13	1	no	—
155	#500	—	13	1	no	—
156	#499	—	26	1	no	—
157	#499	—	26	1	no	—
158	#498	—	13	2	no	—
159	#477	—	13	4	no	—
160	#477	—	26	2	no	—
161	#467	—	26	4	no	—
162	#462	—	13	4	no	—
163	#462	—	26	2	no	—
164	#446	—	2	1	no	—
165	#446	C₂×D₁₃	1	2	yes	—
166	C₈×Q₈	—	2	1	no	—
167	C₈×Q₈	C₂×D₁₃	1	2	yes	—
168	#427	—	26	1	no	—
169	#427	—	26	1	no	—
170	#424	—	2	1	no	—
171	#423	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	13	2	no	—
173	#405	—	2	1	no	—
174	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
175	#334	—	13	2	no	—
176	#334	—	13	2	no	—
177	#333	—	13	4	no	—
178	#332	—	13	4	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	#462	1	2	yes	—
182	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	#462	1	2	yes	—
189	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
190	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
191	C₄×C₁₃	#577	1	2	yes	—
192	Dic₁₃	—	8	2	no	—
193	Dic₁₃	—	8	2	no	—
194	C₂×Q₁₆	—	26	4	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂²×C₈	—	26	2	no	—
197	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
199	C₄⋊Q₈	—	13	2	no	—
200	C₄⋊Q₈	—	13	2	no	—
201	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
202	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
203	#125	—	2	1	no	—
204	#125	—	26	2	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	C₂²⋊Q₈	—	26	2	no	—
207	C₂²⋊Q₈	—	26	2	no	—
208	C₂²⋊Q₈	D₅₂	1	2	yes	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×Q₈	—	26	2	no	—
211	C₄×Q₈	—	26	2	no	—
212	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
213	#117	—	2	1	no	—
214	#117	—	26	2	no	—
215	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	#107	—	26	1	no	—
217	#107	—	26	1	no	—
218	#107	—	26	1	no	—
219	#107	—	26	1	no	—
220	#107	—	26	4	no	—
221	#106	—	26	2	no	—
222	#106	—	26	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	13	4	no	—
227	C₄⋊C₈	—	26	2	no	—
228	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
229	#103	—	26	4	no	—
230	#103	—	26	4	no	—
231	#97	—	2	2	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₂×C₁₃	—	2	2	no	—
238	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
239	C₂×C₁₃	#3060	1	1	yes	characteristic
240	C₂×C₁₃	#3067	1	1	yes	characteristic
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	26	2	no	—
243	C₂×Q₈	—	26	2	no	—
244	C₂×Q₈	C₂×D₅₂	1	2	yes	—
245	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
246	C₂²×C₄	#6127	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	Q₁₆	—	104	4	no	—
249	M₁₆	—	4	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	26	4	no	—
253	C₂×C₈	—	104	2	no	—
254	C₂×C₈	#6107	1	4	yes	complemented
255	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	52	2	no	—
269	C₄⋊C₄	—	52	2	no	—
270	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	52	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	52	2	no	—
275	C₄²	C₂×D₅₂	1	1	yes	characteristic
276	C₄²	#6127	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
278	C₄²	C₂³×D₁₃	1	1	yes	characteristic
279	C₁₃	#19210	1	1	yes	characteristic, complemented, Sylow
280	C₂³	#88184	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	52	2	no	—
284	Q₈	—	52	2	no	—
285	Q₈	—	104	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	52	2	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	#88184	1	1	yes	characteristic
293	C₂×C₄	#88184	1	1	yes	characteristic
294	C₂×C₄	#88184	1	1	yes	characteristic
295	C₂×C₄	C₂×#6107	1	2	yes	—
296	C₂×C₄	#88205	1	2	yes	—
297	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
298	C₂×C₄	C₂×#6127	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	52	4	no	—
304	C₈	C₂×#6107	1	2	yes	—
305	C₈	C₂×#6107	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#11718692	1	2	yes	—
309	C₂²	C₂×#88184	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	104	2	no	—
316	C₄	—	104	2	no	—
317	C₄	#11718692	1	2	yes	—
318	C₄	#11719456	1	2	yes	—
319	C₄	C₂²×#6107	1	1	yes	characteristic
320	C₄	C₂×#88205	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	C₂×#11718692	1	1	yes	characteristic
323	C₂	#13124498	1	1	yes	characteristic
324	C₂	#13124556	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2960	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#19210
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #13114364	C₂	semidirect
4. #13114354	C₂	semidirect
5. #13114218	C₂	semidirect
90. C₂×C₈×C₁₃	Q₁₆	semidirect
254. C₂×C₈	#6107	semidirect
279. C₁₃	#19210	semidirect

Maximal subgroups

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

subgroup	index
#13126178	2
#13125350	2
#13114364	2
#13114354	2
#13114218	2
C₂×#11718049	2
#13113527	2
C₁₃×#2960	2
#19210	13