Subgroups — #16780503

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130129	C₂	1	2	yes	maximal
2	#13130039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718251	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129669	C₂	1	2	yes	maximal, complemented
5	#13123258	C₂	1	1	yes	maximal, characteristic
6	C₂×#88161	C₄	1	1	yes	characteristic
7	C₂×#88155	C₄	1	1	yes	characteristic
8	#11719227	C₂²	1	1	yes	characteristic
9	#11719119	C₂²	1	2	yes	—
10	C₂×C₄×#1057	C₂²	1	2	yes	—
11	C₁₃×#405	C₄	1	1	yes	characteristic
12	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	#11718658	C₄	1	1	yes	characteristic
15	#11718369	—	2	1	no	—
16	#11718368	—	2	1	no	—
17	#11718350	—	2	1	no	—
18	C₄×C₈×D₁₃	—	2	1	no	—
19	#11718251	—	2	1	no	—
20	#11718251	—	2	1	no	—
21	#11718251	—	2	1	no	—
22	C₈×#1057	—	2	1	no	—
23	#11718244	C₄	1	4	yes	—
24	#11718243	C₄	1	4	yes	—
25	C₂×#6127	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1057	—	2	2	no	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
31	C₂×#6099	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	#88161	—	2	2	no	—
34	#88161	C₈	1	4	yes	complemented
35	#88155	—	2	2	no	—
36	#88155	C₈	1	4	yes	complemented
37	C₂×#6094	—	2	1	no	—
38	C₂×#6094	—	2	1	no	—
39	C₂×C₈×D₁₃	—	2	1	no	—
40	C₂×C₈×D₁₃	—	2	1	no	—
41	#88147	C₂×C₄	1	2	yes	—
42	#88141	C₂×C₄	1	2	yes	—
43	#88135	—	2	2	no	—
44	#88126	—	2	1	no	—
45	C₄²×D₁₃	—	2	1	no	—
46	C₂²⋊#1057	—	2	2	no	—
47	C₄⋊#1057	C₂×C₄	1	4	yes	—
48	C₄×#1057	—	2	2	no	—
49	C₄×#1057	—	2	2	no	—
50	C₄×#1057	C₂×C₄	1	4	yes	—
51	C₁₃×#97	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	#88055	—	2	1	no	—
54	C₈×Dic₁₃	—	2	1	no	—
55	C₈×Dic₁₃	D₄	1	1	yes	characteristic
56	C₈×Dic₁₃	D₄	1	1	yes	characteristic
57	C₈×Dic₁₃	D₄	1	1	yes	characteristic
58	C₈×Dic₁₃	D₄	1	1	yes	characteristic
59	#88043	—	2	1	no	—
60	C₄×#1055	—	2	1	no	—
61	#8185	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	#41	1	2	yes	—
65	#6127	—	2	1	no	—
66	#6127	—	2	1	no	—
67	#6127	C₂×C₈	1	2	yes	—
68	#6127	C₂×C₈	1	2	yes	—
69	C₂×D₅₂	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	—	2	2	no	—
72	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
73	C₂×#1057	—	2	4	no	—
74	C₂×#1057	—	2	4	no	—
75	C₂×#1057	—	4	4	no	—
76	C₁₃×M₁₆	—	2	1	no	—
77	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
83	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
84	C₁₃×C₂²⋊C₄	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	#6103	—	2	2	no	—
87	#6101	—	2	2	no	—
88	C₄×Dic₁₃	—	2	2	no	—
89	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
90	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
92	#6099	—	2	1	no	—
93	#6099	C₂×C₈	1	2	yes	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×#1055	C₂×C₈	1	2	yes	—
98	C₂×#1055	M₁₆	1	2	yes	—
99	#6094	—	2	4	no	—
100	#6094	—	2	4	no	—
101	C₈×D₁₃	—	2	4	no	—
102	C₈×D₁₃	—	2	4	no	—
103	C₄×#131	—	13	1	no	—
104	#1476	—	13	2	no	—
105	#1450	—	13	1	no	—
106	C₂×#322	—	13	1	no	—
107	#1278	—	13	2	no	—
108	C₂³×C₁₃	#117	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂²⋊D₁₃	—	2	2	no	—
118	C₂²⋊D₁₃	—	2	2	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
122	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
123	D₅₂	—	2	1	no	—
124	D₅₂	C₄×C₈	1	2	yes	—
125	C₄×D₁₃	—	2	2	no	—
126	C₄×D₁₃	—	2	2	no	—
127	C₄×D₁₃	—	4	4	no	—
128	Dic₂₆	—	2	1	no	—
129	Dic₂₆	C₄×C₈	1	2	yes	—
130	#1057	—	4	4	no	—
131	#1057	—	4	4	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
134	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
135	#1055	—	2	2	no	—
136	#1055	C₂²⋊C₈	1	2	yes	—
137	#1055	C₂²⋊C₈	1	2	yes	—
138	C₂×#131	—	13	1	no	—
139	C₂×#131	—	13	1	no	—
140	C₄×#41	—	13	1	no	—
141	#431	—	13	2	no	—
142	#405	—	13	1	no	—
143	#405	—	26	1	no	—
144	#405	—	26	1	no	—
145	#405	#261	1	1	yes	characteristic
146	C₄×M₁₆	—	13	1	no	—
147	C₄×M₁₆	—	26	1	no	—
148	C₂×C₄×C₈	—	13	1	no	—
149	C₂×C₄×C₈	—	13	2	no	—
150	C₂×C₄×C₈	—	26	1	no	—
151	Q₈⋊C₈	—	13	4	no	—
152	D₄⋊C₈	—	13	4	no	—
153	#322	—	26	1	no	—
154	#322	—	26	1	no	—
155	#322	—	26	1	no	—
156	C₈²	—	26	1	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃	#431	1	1	yes	characteristic
160	C₂²×C₁₃	#431	1	1	yes	characteristic
161	C₂×D₁₃	—	2	2	no	—
162	C₂×D₁₃	—	4	2	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
165	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
166	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	Dic₁₃	—	2	2	no	—
169	Dic₁₃	—	2	2	no	—
170	Dic₁₃	#444	1	4	yes	—
171	C₂×#41	—	13	1	no	—
172	#131	—	13	4	no	—
173	#131	—	13	4	no	—
174	#131	—	26	2	no	—
175	#131	—	26	2	no	—
176	C₂×M₁₆	—	13	1	no	—
177	C₂×M₁₆	—	26	1	no	—
178	C₂×M₁₆	—	26	1	no	—
179	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	26	1	no	—
182	C₂²×C₈	—	26	1	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	C₂×#261	1	1	yes	characteristic
185	C₄×Q₈	—	13	2	no	—
186	C₄×D₄	—	13	2	no	—
187	C₄×D₄	—	26	2	no	—
188	#117	—	26	1	no	—
189	#117	C₂×#261	1	1	yes	characteristic
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	—	26	1	no	—
192	C₄⋊C₈	—	13	4	no	—
193	C₂²⋊C₈	—	26	2	no	—
194	#97	—	2	1	no	—
195	#97	—	26	1	no	—
196	#97	—	26	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	13	1	no	—
201	C₄×C₈	—	13	1	no	—
202	C₄×C₈	—	13	4	no	—
203	C₄×C₈	—	26	1	no	—
204	C₄×C₈	—	26	1	no	—
205	C₄×C₈	—	26	2	no	—
206	C₄×C₈	—	26	2	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
210	C₂×C₁₃	#1765	1	2	yes	—
211	D₁₃	—	4	2	no	—
212	#41	—	13	2	no	—
213	#41	—	13	2	no	—
214	#41	—	26	1	no	—
215	#41	—	26	1	no	—
216	C₂×Q₈	—	13	1	no	—
217	C₂×D₄	—	13	1	no	—
218	C₂×D₄	—	26	1	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	26	2	no	—
221	C₂²×C₄	—	26	1	no	—
222	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
223	M₁₆	—	2	1	no	—
224	M₁₆	—	13	2	no	—
225	M₁₆	—	26	4	no	—
226	M₁₆	—	26	1	no	—
227	M₁₆	—	26	4	no	—
228	M₁₆	D₁₃⋊C₈	1	2	yes	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	13	2	no	—
231	C₂×C₈	—	13	2	no	—
232	C₂×C₈	—	13	1	no	—
233	C₂×C₈	—	13	1	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	1	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	52	4	no	—
240	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
241	C₂×C₈	#6118	1	2	yes	—
242	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
244	C₄⋊C₄	—	26	2	no	—
245	C₄⋊C₄	C₄×#261	1	2	yes	—
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	26	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	13	2	no	—
250	C₄²	—	13	1	no	—
251	C₄²	—	13	1	no	—
252	C₄²	—	26	2	no	—
253	C₁₃	#8185	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	26	1	no	—
255	C₂³	#88237	1	1	yes	characteristic
256	Q₈	—	13	2	no	—
257	Q₈	—	26	1	no	—
258	D₄	—	13	2	no	—
259	D₄	—	26	2	no	—
260	D₄	—	26	2	no	—
261	D₄	—	26	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	—	26	1	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#6118	1	1	yes	characteristic
273	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	13	2	no	—
278	C₈	—	13	2	no	—
279	C₈	—	26	2	no	—
280	C₈	—	52	4	no	—
281	C₈	—	52	4	no	—
282	C₈	#88111	1	2	yes	—
283	C₈	#88111	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	26	2	no	—
286	C₂²	—	52	2	no	—
287	C₂²	#11719049	1	1	yes	characteristic
288	C₂²	#11719051	1	1	yes	characteristic
289	C₂²	C₄×C₂²⋊#261	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	13	4	no	—
292	C₄	—	26	2	no	—
293	C₄	—	26	2	no	—
294	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
295	C₄	C₄×#6118	1	1	yes	characteristic
296	C₄	C₂×#88111	1	1	yes	characteristic
297	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	52	2	no	—
301	C₂	C₄×#88111	1	1	yes	characteristic
302	C₂	#13130614	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#405	C₄
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#405	#261
hypocenter	C₁₃	#8185
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13130039	C₂	semidirect
3. C₂×#11718251	C₂	semidirect
4. #13129669	C₂	semidirect
34. #88161	C₈	semidirect
36. #88155	C₈	semidirect
253. C₁₃	#8185	semidirect

Maximal subgroups

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

subgroup	index
#13130129	2
#13130039	2
C₂×#11718251	2
#13129669	2
#13123258	2
#8185	13