Subgroups — #16789349

Order 3328.

510 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	C₂²×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718259	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718257	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
5	#13129875	C₂	1	1	yes	maximal, characteristic, complemented
6	#13129874	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114606	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	1	yes	characteristic
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	C₂²×#6092	C₂²	1	1	yes	characteristic
13	C₂×D₁₃⋊C₁₆	—	2	2	no	—
14	C₂×D₁₃⋊C₁₆	—	2	2	no	—
15	C₂×D₁₃⋊C₁₆	C₄	1	4	yes	complemented
16	C₂×#88060	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₄×#6103	C₄	1	1	yes	characteristic
19	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
20	#11718259	C₄	1	4	yes	complemented
21	#11718257	—	2	2	no	—
22	C₄×#6092	—	2	2	no	—
23	C₁₃×#336	C₄	1	1	yes	characteristic
24	#11718142	C₄	1	1	yes	characteristic
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#6103	C₈	1	2	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂³	1	1	yes	characteristic
33	C₂×C₈×D₁₃	—	2	1	no	—
34	C₂×C₈×D₁₃	—	2	1	no	—
35	C₂×C₈×D₁₃	—	2	1	no	—
36	C₂×C₈×D₁₃	—	2	1	no	—
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×#6092	—	2	1	no	—
40	C₂×#6092	—	2	1	no	—
41	C₂×#6092	—	2	1	no	—
42	C₂×#6092	—	2	1	no	—
43	C₂×#6092	—	2	1	no	—
44	C₂×#6092	—	2	1	no	—
45	C₂×#6092	C₂×C₄	1	2	yes	—
46	C₂×#6092	C₂×C₄	1	2	yes	—
47	C₂×#6092	C₂×C₄	1	2	yes	—
48	C₂×#6092	C₂×C₄	1	2	yes	—
49	C₂×#6092	D₄	1	2	yes	—
50	C₂×#6092	D₄	1	2	yes	—
51	D₁₃⋊C₁₆	—	2	8	no	—
52	#88071	C₈	1	2	yes	—
53	#88060	C₈	1	4	yes	—
54	C₈×Dic₁₃	—	2	1	no	—
55	C₈×Dic₁₃	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	—	2	1	no	—
58	C₁₆×C₂²⋊C₄	—	13	1	no	maximal, Sylow
59	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₁₃	—	2	1	no	—
65	C₂×C₄×D₁₃	—	2	1	no	—
66	C₂×C₄×D₁₃	—	2	1	no	—
67	C₂×C₄×D₁₃	—	2	1	no	—
68	C₂×C₄×D₁₃	C₄²	1	2	yes	—
69	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	C₄²	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
76	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	—	2	1	no	—
78	#6103	C₁₆	1	8	yes	complemented
79	C₄×Dic₁₃	—	2	1	no	—
80	C₄×Dic₁₃	—	2	1	no	—
81	C₂×#1055	—	2	1	no	—
82	C₂×#1055	—	2	1	no	—
83	C₂×#1055	C₂×C₈	1	2	yes	—
84	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
85	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
86	C₂×#1055	C₂×D₄	1	1	yes	characteristic
87	C₂×#1055	C₂×D₄	1	1	yes	characteristic
88	C₂×#1055	#41	1	1	yes	characteristic
89	C₂×#1055	#41	1	1	yes	characteristic
90	C₈×D₁₃	—	2	2	no	—
91	C₈×D₁₃	—	2	2	no	—
92	C₈×D₁₃	—	2	2	no	—
93	C₈×D₁₃	—	2	2	no	—
94	#6092	—	2	2	no	—
95	#6092	—	2	2	no	—
96	#6092	—	2	2	no	—
97	#6092	C₂²⋊C₄	1	4	yes	complemented
98	C₂³×C₁₆	—	13	1	no	—
99	C₂×C₂²⋊C₁₆	—	13	1	no	—
100	C₂×C₄×C₁₆	—	13	1	no	—
101	C₂×C₄×C₁₆	—	13	1	no	—
102	C₈×C₂²⋊C₄	—	13	1	no	—
103	#1374	—	13	1	no	—
104	#1374	—	13	1	no	—
105	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂²×D₁₃	—	2	1	no	—
107	C₂²×D₁₃	—	2	1	no	—
108	C₂²×D₁₃	—	2	2	no	—
109	C₂²×D₁₃	C₄×C₈	1	2	yes	—
110	C₂²×D₁₃	C₂×C₁₆	1	4	yes	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
114	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
125	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	complemented
126	C₄×D₁₃	—	2	2	no	—
127	C₄×D₁₃	—	2	2	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	—	2	2	no	—
132	#1055	—	2	2	no	—
133	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	#1055	#117	1	1	yes	characteristic
135	#1055	C₄×D₄	1	1	yes	characteristic
136	#1055	C₄×D₄	1	1	yes	characteristic
137	C₂³×C₈	—	13	1	no	—
138	C₂²×C₁₆	—	13	1	no	—
139	C₂²×C₁₆	—	13	1	no	—
140	C₂²×C₁₆	—	13	1	no	—
141	C₂²×C₁₆	—	13	1	no	—
142	C₂²×C₁₆	—	13	4	no	—
143	C₂²×C₁₆	—	26	2	no	—
144	C₂²×C₁₆	—	26	2	no	—
145	C₂×C₂²⋊C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₄×C₂²⋊C₄	—	13	1	no	—
149	C₂²⋊C₁₆	—	13	4	no	—
150	C₄×C₁₆	—	26	2	no	—
151	C₄×C₁₆	—	26	2	no	—
152	#336	—	13	1	no	—
153	#336	#261	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
155	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
158	C₂²×C₁₃	#504	1	2	yes	—
159	C₂×D₁₃	—	2	2	no	—
160	C₂×D₁₃	—	2	4	no	—
161	C₂×D₁₃	—	2	4	no	—
162	C₂×D₁₃	—	2	2	no	—
163	C₂×D₁₃	C₄×C₁₆	1	4	yes	complemented
164	C₄×C₁₃	—	2	4	no	—
165	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₁₃	#405	1	1	yes	characteristic
167	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
168	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
169	Dic₁₃	—	2	4	no	—
170	Dic₁₃	—	2	2	no	—
171	C₂³×C₄	—	13	1	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	—	26	1	no	—
177	C₂²×C₈	—	26	1	no	—
178	C₂²×C₈	—	26	1	no	—
179	C₂²×C₈	—	26	1	no	—
180	C₂²×C₈	C₂×#261	1	1	yes	characteristic
181	C₂²×C₈	C₂×#261	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	13	2	no	—
183	C₂×C₄²	—	13	1	no	—
184	C₂×C₄²	C₂×#261	1	1	yes	characteristic
185	C₂×C₁₆	—	13	2	no	—
186	C₂×C₁₆	—	13	2	no	—
187	C₂×C₁₆	—	13	2	no	—
188	C₂×C₁₆	—	13	2	no	—
189	C₂×C₁₆	—	13	2	no	—
190	C₂×C₁₆	—	13	2	no	—
191	C₂×C₁₆	—	26	8	no	—
192	C₂×C₁₆	—	26	1	no	—
193	C₂×C₁₆	—	26	1	no	—
194	C₂×C₁₆	—	26	1	no	—
195	C₂×C₁₆	—	26	1	no	—
196	C₂×C₁₆	—	26	1	no	—
197	C₂×C₁₆	—	26	1	no	—
198	C₂²⋊C₈	—	13	4	no	—
199	C₄×C₈	—	26	1	no	—
200	C₄×C₈	—	26	1	no	—
201	C₄×C₈	—	26	1	no	—
202	C₄×C₈	—	26	1	no	—
203	#95	—	13	2	no	—
204	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
206	C₂×C₁₃	#2108	1	1	yes	characteristic
207	C₂×C₁₃	D₄×C₁₆	1	2	yes	—
208	C₂×C₁₃	D₄×C₁₆	1	2	yes	—
209	D₁₃	—	2	4	no	—
210	C₂⁴	—	13	1	no	—
211	C₂²×C₄	—	13	1	no	—
212	C₂²×C₄	—	13	2	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	26	1	no	—
216	C₂²×C₄	—	26	1	no	—
217	C₂²×C₄	—	26	1	no	—
218	C₂²×C₄	—	26	1	no	—
219	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
220	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	13	1	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	2	no	—
232	C₂×C₈	—	26	2	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	—	26	1	no	—
237	C₂×C₈	—	26	1	no	—
238	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
239	C₂×C₈	C₄×#261	1	2	yes	—
240	C₂²⋊C₄	—	13	8	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	26	1	no	—
244	C₄²	—	26	1	no	—
245	C₁₆	—	13	4	no	—
246	C₁₆	—	26	2	no	—
247	C₁₆	—	26	2	no	—
248	C₁₆	—	26	2	no	—
249	C₁₃	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	13	2	no	—
251	C₂³	—	13	4	no	—
252	C₂³	—	26	1	no	—
253	C₂³	—	26	1	no	—
254	C₂³	—	26	2	no	—
255	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	4	no	—
260	C₂×C₄	—	26	1	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	1	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	26	2	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	D₁₃⋊C₁₆	1	4	yes	complemented
269	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
271	C₂×C₄	#88238	1	1	yes	characteristic
272	C₂×C₄	D₄×#261	1	1	yes	characteristic
273	C₂×C₄	D₄×#261	1	1	yes	characteristic
274	C₂×C₄	#88240	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	13	1	no	—
278	C₈	—	13	1	no	—
279	C₈	—	13	1	no	—
280	C₈	—	13	1	no	—
281	C₈	—	26	2	no	—
282	C₂²	—	13	4	no	—
283	C₂²	—	26	2	no	—
284	C₂²	—	26	2	no	—
285	C₂²	—	26	4	no	—
286	C₂²	—	26	4	no	—
287	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
288	C₂²	#11719063	1	1	yes	characteristic
289	C₂²	#11719073	1	1	yes	characteristic
290	C₂²	C₂×D₁₃⋊C₁₆	1	2	yes	—
291	C₂²	#11719096	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	26	2	no	—
294	C₄	—	26	4	no	—
295	C₄	#11719063	1	1	yes	characteristic
296	C₄	C₂²⋊C₄×#261	1	1	yes	characteristic
297	C₄	#11719072	1	1	yes	characteristic
298	C₄	#11719073	1	1	yes	characteristic
299	C₂	—	26	4	no	—
300	C₂	#13130570	1	1	yes	characteristic
301	C₂	C₄×D₁₃⋊C₁₆	1	1	yes	characteristic
302	C₂	#13131046	1	1	yes	characteristic
303	C₂	#13131214	1	2	yes	—
304	C₂	#13131215	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₁₃×#336	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	#336	#261
hypocenter	C₁₃	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
3. C₂×#11718257	C₂	semidirect
4. C₂×C₄×#6092	C₂	semidirect
5. #13129875	C₂	semidirect
6. #13129874	C₂	semidirect
15. C₂×D₁₃⋊C₁₆	C₄	semidirect
20. #11718259	C₄	semidirect
78. #6103	C₁₆	semidirect
97. #6092	C₂²⋊C₄	semidirect
113. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
125. C₂×Dic₁₃	C₂×C₁₆	semidirect
163. C₂×D₁₃	C₄×C₁₆	semidirect
249. C₁₃	C₁₆×C₂²⋊C₄	semidirect
268. C₂×C₄	D₁₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₃⋊C₁₆	2
C₂×#11718259	2
C₂×#11718257	2
C₂×C₄×#6092	2
#13129875	2
#13129874	2
#13114606	2
C₁₆×C₂²⋊C₄	13