Subgroups — #16807286

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	2	yes	maximal, complemented
2	#13131221	C₂	1	2	yes	maximal, complemented
3	#13131218	C₂	1	2	yes	maximal, complemented
4	#13131216	C₂	1	2	yes	maximal, complemented
5	#13131202	C₂	1	2	yes	maximal, complemented
6	#13131200	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131047	C₂	1	2	yes	maximal, complemented
8	#13131043	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123814	C₂	1	1	yes	maximal, characteristic
10	#11719262	C₄	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	2	yes	—
12	C₂²×#6092	C₂²	1	2	yes	—
13	C₂×#88098	C₂²	1	2	yes	—
14	C₁₃×#436	C₄	1	1	yes	characteristic
15	#11718788	C₄	1	2	yes	—
16	#11718783	C₂²	1	2	yes	—
17	D₄×#1055	C₂²	1	1	yes	characteristic
18	#11718496	C₄	1	1	yes	characteristic
19	#11718491	C₂²	1	1	yes	characteristic
20	#11718386	C₂²	1	2	yes	—
21	#11718381	C₄	1	2	yes	—
22	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#11718340	C₄	1	1	yes	characteristic
24	C₂²⋊#6092	C₂²	1	2	yes	—
25	C₂²⋊#6092	C₂²	1	2	yes	—
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718259	C₂²	1	1	yes	characteristic
29	#11718259	C₂²	1	1	yes	characteristic
30	#11718258	C₂²	1	2	yes	—
31	#11718258	C₂²	1	2	yes	—
32	#11718257	C₂²	1	1	yes	characteristic
33	#11718257	C₂²	1	1	yes	characteristic
34	#11718257	C₂²	1	2	yes	—
35	#11718231	C₂²	1	2	yes	—
36	C₄⋊#6092	C₂²	1	2	yes	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
38	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	#88192	C₈	1	1	yes	characteristic
40	#88190	C₈	1	1	yes	characteristic
41	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
42	#88180	C₂×C₄	1	2	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
46	#88151	C₈	1	1	yes	characteristic
47	#88144	C₈	1	1	yes	characteristic
48	#88137	C₈	1	2	yes	—
49	#88133	C₈	1	2	yes	—
50	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
52	#88106	—	2	4	no	—
53	C₂×#6092	—	2	4	no	—
54	C₂×#6092	C₂³	1	2	yes	—
55	C₂×#6092	C₂³	1	2	yes	—
56	C₂×#6092	C₂³	1	2	yes	—
57	C₂×#6092	C₂³	1	2	yes	—
58	#88098	—	2	4	no	—
59	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊#1055	C₂×C₄	1	2	yes	—
63	#88060	C₂×C₄	1	2	yes	—
64	#88055	C₂×C₄	1	2	yes	—
65	#88055	C₂×C₄	1	2	yes	—
66	#88054	C₂×C₄	1	2	yes	—
67	#88054	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	2	yes	—
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
72	C₄×#1055	C₂³	1	1	yes	characteristic
73	#33481	—	13	1	no	maximal, Sylow
74	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
81	C₁₃×M₁₆	—	2	4	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
90	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
91	#6103	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
94	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
95	#6101	C₂×C₈	1	2	yes	—
96	#6101	C₂×C₈	1	2	yes	—
97	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
104	C₂×#1055	C₂²×C₄	1	2	yes	—
105	C₂×#1055	C₂⁴	1	1	yes	characteristic
106	C₈×D₁₃	—	2	2	no	—
107	C₈×D₁₃	—	2	2	no	—
108	#6092	—	2	2	no	—
109	#6092	—	2	2	no	—
110	#6092	—	2	2	no	—
111	#6092	#41	1	4	yes	—
112	#2966	—	13	1	no	—
113	#2183	—	13	2	no	—
114	#2169	—	13	2	no	—
115	#2168	—	13	2	no	—
116	#2168	—	13	2	no	—
117	#2164	—	13	2	no	—
118	#2163	—	13	1	no	—
119	#2108	—	13	2	no	—
120	C₄×M₃₂	—	13	1	no	—
121	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
122	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₁₃	—	4	2	no	—
131	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
136	C₄×D₁₃	—	2	1	no	—
137	C₄×D₁₃	—	2	1	no	—
138	C₄×D₁₃	—	4	1	no	—
139	C₄×D₁₃	—	4	1	no	—
140	Dic₂₆	—	4	1	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	#131	1	2	yes	—
144	#1055	—	2	2	no	—
145	#1055	—	2	1	no	—
146	#1055	C₂×#41	1	2	yes	—
147	#529	—	13	1	no	—
148	C₂×M₃₂	—	13	2	no	—
149	C₂×M₃₂	—	13	2	no	—
150	C₂²×C₁₆	—	13	2	no	—
151	#446	—	13	1	no	—
152	C₈×Q₈	—	13	1	no	—
153	#436	#261	1	1	yes	characteristic
154	#435	—	13	2	no	—
155	C₈×D₄	—	13	1	no	—
156	C₈×D₄	—	13	2	no	—
157	#433	—	13	2	no	—
158	#432	—	13	1	no	—
159	#405	—	13	2	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₄⋊C₁₆	—	13	2	no	—
162	C₄⋊C₁₆	—	13	2	no	—
163	C₄⋊C₁₆	—	13	2	no	—
164	C₂²⋊C₁₆	—	13	1	no	—
165	C₂²⋊C₁₆	—	13	1	no	—
166	C₂²⋊C₁₆	—	13	2	no	—
167	C₂²⋊C₁₆	—	13	2	no	—
168	#346	—	13	2	no	—
169	#346	—	13	2	no	—
170	#346	—	13	2	no	—
171	C₄×C₁₆	—	13	1	no	—
172	C₄×C₁₆	—	13	1	no	—
173	C₄×C₁₆	—	13	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
178	C₂×D₁₃	—	2	1	no	—
179	C₂×D₁₃	—	2	1	no	—
180	C₂×D₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
185	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
186	Dic₁₃	—	2	2	no	—
187	Dic₁₃	—	2	1	no	—
188	Dic₁₃	—	2	1	no	—
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	1	no	—
191	C₂×M₁₆	C₂×#261	1	2	yes	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	1	no	—
195	#126	—	13	2	no	—
196	#125	—	13	1	no	—
197	#124	—	13	1	no	—
198	#123	—	13	2	no	—
199	C₂²⋊Q₈	—	13	1	no	—
200	#121	—	13	1	no	—
201	C₄×Q₈	—	13	1	no	—
202	C₄×D₄	—	13	2	no	—
203	C₄×D₄	C₂×#261	1	1	yes	characteristic
204	#117	—	13	2	no	—
205	C₂×C₄²	—	13	1	no	—
206	M₃₂	—	26	4	no	—
207	M₃₂	—	26	4	no	—
208	C₂×C₁₆	—	13	2	no	—
209	C₂×C₁₆	—	13	2	no	—
210	C₂×C₁₆	—	13	2	no	—
211	C₂×C₁₆	—	13	2	no	—
212	C₂×C₁₆	—	26	4	no	—
213	C₄⋊C₈	—	13	2	no	—
214	C₄⋊C₈	—	13	2	no	—
215	C₄⋊C₈	—	13	1	no	—
216	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
217	C₂²⋊C₈	—	13	2	no	—
218	C₂²⋊C₈	—	13	2	no	—
219	C₂²⋊C₈	C₂×#261	1	2	yes	—
220	#97	—	13	2	no	—
221	#97	—	13	2	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	—	13	2	no	—
224	C₄×C₈	—	13	1	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	C₂×#261	1	1	yes	characteristic
227	C₂×C₁₃	—	4	2	no	—
228	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
229	C₂×C₁₃	#3407	1	1	yes	characteristic
230	C₂×C₁₃	#3407	1	1	yes	characteristic
231	D₁₃	—	4	1	no	—
232	C₂×Q₈	—	13	1	no	—
233	C₂×D₄	—	13	2	no	—
234	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	C₂²×#261	1	2	yes	—
239	M₁₆	—	2	4	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	1	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	C₂²×#261	1	1	yes	characteristic
248	C₂×C₈	C₂²×#261	1	2	yes	—
249	C₂×C₈	C₂²×#261	1	1	yes	characteristic
250	C₄⋊C₄	—	13	2	no	—
251	C₄⋊C₄	—	13	2	no	—
252	C₄⋊C₄	—	13	2	no	—
253	C₄⋊C₄	—	13	1	no	—
254	C₄⋊C₄	—	13	2	no	—
255	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
256	C₂²⋊C₄	—	13	2	no	—
257	C₂²⋊C₄	—	13	2	no	—
258	C₂²⋊C₄	—	13	2	no	—
259	C₂²⋊C₄	—	13	2	no	—
260	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
261	C₄²	—	13	1	no	—
262	C₄²	—	13	1	no	—
263	C₄²	—	13	1	no	—
264	C₄²	—	13	2	no	—
265	C₄²	C₂²×#261	1	1	yes	characteristic
266	C₁₆	—	13	4	no	—
267	C₁₆	—	26	2	no	—
268	C₁₆	—	26	2	no	—
269	C₁₆	—	26	2	no	—
270	C₁₃	#33481	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	13	1	no	—
272	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
273	Q₈	—	52	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	52	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	26	1	no	—
284	C₂×C₄	—	26	1	no	—
285	C₂×C₄	—	52	1	no	—
286	C₂×C₄	—	52	1	no	—
287	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
290	C₂×C₄	C₂³×#261	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	1	no	—
293	C₈	—	13	2	no	—
294	C₈	—	26	1	no	—
295	C₈	—	26	2	no	—
296	C₈	#88238	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	26	1	no	—
301	C₂²	—	26	1	no	—
302	C₂²	—	26	1	no	—
303	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	26	1	no	—
308	C₄	—	26	2	no	—
309	C₄	—	26	1	no	—
310	C₄	—	26	2	no	—
311	C₄	—	26	1	no	—
312	C₄	C₂×#88238	1	1	yes	characteristic
313	C₄	#41×#261	1	1	yes	characteristic
314	C₂	—	4	2	no	—
315	C₂	—	52	1	no	—
316	C₂	#13131641	1	1	yes	characteristic
317	C₂	#13131804	1	1	yes	characteristic
318	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#436	C₄
prime radical	#88192	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#436	#261
hypocenter	C₁₃	#33481
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131221	C₂	semidirect
3. #13131218	C₂	semidirect
4. #13131216	C₂	semidirect
5. #13131202	C₂	semidirect
6. #13131200	C₂	semidirect
7. #13131047	C₂	semidirect
8. #13131043	C₂	semidirect
270. C₁₃	#33481	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131221	2
#13131218	2
#13131216	2
#13131202	2
#13131200	2
#13131047	2
#13131043	2
#13123814	2
#33481	13