Subgroups — #16792738

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131645	C₂	1	2	yes	maximal
2	#13130059	C₂	1	2	yes	maximal, complemented
3	#13130055	C₂	1	2	yes	maximal, complemented
4	#13130043	C₂	1	2	yes	maximal, complemented
5	#13130039	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#6122	C₂	1	4	yes	maximal, complemented
7	C₂×C₈×#1057	C₂	1	1	yes	maximal, characteristic, complemented
8	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
10	#41×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88202	C₄	1	1	yes	characteristic
12	#11719158	C₄	1	4	yes	—
13	#11719145	C₄	1	4	yes	—
14	#11719144	C₄	1	8	yes	—
15	#11719125	C₂²	1	4	yes	—
16	C₄×#6122	C₂²	1	4	yes	—
17	#11719119	C₂²	1	2	yes	—
18	C₂×C₄×#1057	C₂²	1	2	yes	—
19	#11718783	C₄	1	2	yes	—
20	#11718783	C₄	1	1	yes	characteristic
21	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
22	M₁₆×Dic₁₃	C₂²	1	2	yes	—
23	#11718762	C₄	1	2	yes	—
24	#11718762	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
27	#1057⋊C₈	C₂²	1	4	yes	—
28	#1057⋊C₈	C₂²	1	4	yes	—
29	#11718251	C₂²	1	2	yes	—
30	C₈×#1057	C₂²	1	2	yes	—
31	C₈×#1057	C₂²	1	1	yes	characteristic
32	C₈×#1057	C₂²	1	1	yes	characteristic
33	C₈×#1057	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂×#6122	C₂×C₄	1	8	yes	—
36	C₂²×#1057	C₂×C₄	1	4	yes	—
37	C₁₃×#131	—	2	4	no	—
38	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88202	—	2	4	no	—
43	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
44	D₄×Dic₁₃	C₂×C₄	1	4	yes	—
45	D₄×Dic₁₃	C₂×C₄	1	2	yes	—
46	#88180	C₂×C₄	1	2	yes	—
47	#88180	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
50	C₂×#6099	C₂×C₄	1	1	yes	characteristic
51	C₂×#6099	C₂×C₄	1	2	yes	—
52	C₂²×#1055	C₂×C₄	1	2	yes	—
53	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊#1057	C₂×C₄	1	4	yes	—
55	C₂²⋊#1057	C₂×C₄	1	8	yes	—
56	#88112	C₂×C₄	1	8	yes	—
57	#88110	C₂×C₄	1	8	yes	—
58	C₄⋊#1057	C₂×C₄	1	4	yes	—
59	C₄×#1057	C₂×C₄	1	4	yes	—
60	C₄×#1057	C₂³	1	2	yes	—
61	C₄×#1057	C₂³	1	2	yes	—
62	C₄×#1057	C₂³	1	4	yes	—
63	#88055	C₂×C₄	1	2	yes	—
64	#88055	C₂×C₄	1	2	yes	—
65	#88055	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	2	yes	—
73	#17335	—	13	1	no	maximal, Sylow
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₄²	1	2	yes	—
78	C₂×D₄×C₁₃	C₄²	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
83	#6122	—	2	16	no	—
84	C₂×#1057	—	2	8	no	—
85	C₂×#1057	C₂²×C₄	1	4	yes	—
86	C₂×#1057	C₂²×C₄	1	8	yes	—
87	C₂×#1057	C₂²×C₄	1	4	yes	—
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	2	4	no	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²⋊Dic₁₃	C₄²	1	4	yes	—
96	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
97	C₄⋊Dic₁₃	C₄²	1	4	yes	—
98	C₄⋊Dic₁₃	C₄²	1	2	yes	—
99	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
102	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
103	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
104	#6099	—	2	2	no	—
105	#6099	—	2	4	no	—
106	C₂×#1055	—	2	4	no	—
107	C₂×#1055	M₁₆	1	4	yes	complemented
108	C₂×#1055	C₂²×C₄	1	2	yes	—
109	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
110	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
111	#2965	—	13	2	no	—
112	C₄×#131	—	13	1	no	—
113	#1455	—	13	2	no	—
114	#1453	—	13	2	no	—
115	#1451	—	13	2	no	—
116	#1450	—	13	1	no	—
117	C₈×M₁₆	—	13	4	no	—
118	C₂×C₈²	—	13	1	no	—
119	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
120	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	2	no	—
122	D₄×C₁₃	—	2	4	no	—
123	D₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×Dic₁₃	—	2	4	no	—
132	C₂×Dic₁₃	—	2	8	no	—
133	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
134	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
135	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
136	#1057	—	2	4	no	—
137	#1057	—	2	8	no	—
138	#1057	#131	1	8	yes	complemented
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
141	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
142	#1055	—	2	2	no	—
143	#1055	C₂×M₁₆	1	2	yes	—
144	#1055	C₂×M₁₆	1	2	yes	—
145	C₂×#131	—	13	1	no	—
146	C₂×#131	#261	1	1	yes	characteristic
147	C₄×#41	—	13	1	no	—
148	#446	—	13	4	no	—
149	#436	—	13	4	no	—
150	#435	—	13	8	no	—
151	#431	—	13	4	no	—
152	#431	—	13	2	no	—
153	#405	—	13	1	no	—
154	#405	—	13	2	no	—
155	#405	—	13	2	no	—
156	#405	—	13	1	no	—
157	C₄×M₁₆	—	13	4	no	—
158	C₄×M₁₆	—	13	1	no	—
159	C₄×M₁₆	—	13	2	no	—
160	C₂×C₄×C₈	—	13	2	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	—	13	2	no	—
163	#322	—	13	2	no	—
164	#322	—	13	4	no	—
165	#322	—	13	4	no	—
166	C₈²	—	13	2	no	—
167	C₈²	—	13	1	no	—
168	C₈²	—	13	1	no	—
169	C₈²	—	13	2	no	—
170	C₂²×C₁₃	—	2	2	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	4	no	—
173	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
174	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
177	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
178	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
179	Dic₁₃	—	2	2	no	—
180	Dic₁₃	—	2	4	no	—
181	Dic₁₃	C₂×#131	1	4	yes	—
182	C₂×#41	C₂×#261	1	1	yes	characteristic
183	#131	—	2	4	no	—
184	#131	—	26	4	no	—
185	C₂×M₁₆	—	13	8	no	—
186	C₂×M₁₆	—	13	2	no	—
187	C₂×M₁₆	—	13	1	no	—
188	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
189	C₂×M₁₆	C₂×#261	1	2	yes	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	4	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	C₂×#261	1	2	yes	—
194	C₂²×C₈	C₂×#261	1	1	yes	characteristic
195	C₄×Q₈	—	13	2	no	—
196	C₄×D₄	—	13	4	no	—
197	C₄×D₄	—	13	2	no	—
198	#117	—	13	2	no	—
199	#117	—	13	1	no	—
200	C₂×C₄²	—	13	1	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₄⋊C₈	—	13	4	no	—
203	C₄⋊C₈	—	13	8	no	—
204	C₂²⋊C₈	—	13	4	no	—
205	C₂²⋊C₈	—	13	8	no	—
206	#97	—	13	8	no	—
207	#97	—	13	2	no	—
208	#97	—	13	2	no	—
209	#97	—	13	2	no	—
210	C₄×C₈	—	13	4	no	—
211	C₄×C₈	—	13	2	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	—	13	4	no	—
214	C₄×C₈	—	13	1	no	—
215	C₄×C₈	—	13	2	no	—
216	C₄×C₈	—	13	2	no	—
217	C₄×C₈	—	13	1	no	—
218	C₄×C₈	—	13	1	no	—
219	C₄×C₈	—	13	1	no	—
220	C₄×C₈	—	13	2	no	—
221	C₂×C₁₃	—	2	2	no	—
222	C₂×C₁₃	—	4	2	no	—
223	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
224	C₂×C₁₃	C₄×#131	1	2	yes	—
225	#41	—	2	2	no	—
226	#41	—	2	2	no	—
227	C₂×Q₈	C₄×#261	1	1	yes	characteristic
228	C₂×D₄	C₄×#261	1	2	yes	—
229	C₂×D₄	C₄×#261	1	1	yes	characteristic
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	13	4	no	—
232	C₂²×C₄	C₂²×#261	1	2	yes	—
233	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	26	4	no	—
237	M₁₆	—	26	16	no	—
238	M₁₆	—	26	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	4	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	—	13	4	no	—
245	C₂×C₈	—	13	8	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	26	4	no	—
248	C₂×C₈	—	26	8	no	—
249	C₂×C₈	#6122	1	4	yes	complemented
250	C₂×C₈	C₂²×#261	1	2	yes	—
251	C₂×C₈	C₂²×#261	1	1	yes	characteristic
252	C₂×C₈	C₂²×#261	1	1	yes	characteristic
253	C₄⋊C₄	—	13	4	no	—
254	C₄⋊C₄	—	13	2	no	—
255	C₂²⋊C₄	—	13	4	no	—
256	C₂²⋊C₄	—	13	2	no	—
257	C₄²	—	13	2	no	—
258	C₄²	—	13	4	no	—
259	C₄²	—	13	1	no	—
260	C₄²	—	13	1	no	—
261	C₄²	—	13	2	no	—
262	C₁₃	#17335	1	1	yes	characteristic, complemented, Sylow
263	C₂³	C₂×C₄×#261	1	2	yes	—
264	C₂³	C₂×C₄×#261	1	1	yes	characteristic
265	Q₈	—	2	2	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	13	4	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	8	no	—
275	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
276	C₂×C₄	C₂×C₄×#261	1	2	yes	—
277	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
278	C₂×C₄	C₂×#6122	1	2	yes	—
279	C₂×C₄	C₂³×#261	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	—	13	8	no	—
282	C₈	—	13	2	no	—
283	C₈	—	13	2	no	—
284	C₈	—	26	4	no	—
285	C₈	—	26	8	no	—
286	C₈	—	26	2	no	—
287	C₈	C₂×#6122	1	2	yes	—
288	C₈	C₂×#6122	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	C₄×#6122	1	2	yes	—
293	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	13	4	no	—
296	C₄	—	26	2	no	—
297	C₄	—	26	4	no	—
298	C₄	C₄×#6122	1	2	yes	—
299	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
300	C₄	C₂²×#6122	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	C₂×C₄×#6122	1	1	yes	characteristic
304	C₂	#131×#261	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#131	C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#17335
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #13130059	C₂	semidirect
3. #13130055	C₂	semidirect
4. #13130043	C₂	semidirect
5. #13130039	C₂	semidirect
6. C₈×#6122	C₂	semidirect
7. C₂×C₈×#1057	C₂	semidirect
91. C₂×C₈×C₁₃	M₁₆	semidirect
107. C₂×#1055	M₁₆	semidirect
138. #1057	#131	semidirect
249. C₂×C₈	#6122	semidirect
262. C₁₃	#17335	semidirect

Maximal subgroups

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

subgroup	index
#13131645	2
#13130059	2
#13130055	2
#13130043	2
#13130039	2
C₈×#6122	2
C₂×C₈×#1057	2
#131×Dic₁₃	2
#17335	13