Subgroups — #16780331

Order 3328.

642 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	C₄×#88111	C₂	1	2	yes	maximal
2	C₂×#11718251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×#1057	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129658	C₂	1	2	yes	maximal, complemented
5	C₂×#11718368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×#6094	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88111	C₄	1	4	yes	—
10	C₂×C₄×#1057	C₂²	1	2	yes	—
11	C₂×C₄×#1057	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88043	C₄	1	1	yes	characteristic
16	#11718368	—	2	2	no	—
17	#11718368	—	2	2	no	—
18	#11718290	C₄	1	4	yes	—
19	#11718251	—	2	2	no	—
20	C₈×#1057	—	2	2	no	—
21	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1057	C₂×C₄	1	4	yes	—
24	C₂²×#1057	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂×#6094	—	2	2	no	—
33	C₂×#6094	—	2	2	no	—
34	C₂×#6094	—	2	2	no	—
35	C₂×#6094	—	2	2	no	—
36	C₂×#6094	C₈	1	4	yes	complemented
37	C₂×#6094	C₈	1	4	yes	complemented
38	C₄²×D₁₃	—	2	1	no	—
39	C₄²×D₁₃	—	2	1	no	—
40	C₄²×D₁₃	—	2	2	no	—
41	#88111	—	2	8	no	—
42	C₄×#1057	—	2	2	no	—
43	C₄×#1057	—	2	2	no	—
44	C₄×#1057	—	2	2	no	—
45	C₄×#1057	—	2	2	no	—
46	C₁₃×#97	—	2	2	no	—
47	C₈×Dic₁₃	—	2	1	no	—
48	C₈×Dic₁₃	—	2	1	no	—
49	C₈×Dic₁₃	D₄	1	2	yes	—
50	C₈×Dic₁₃	D₄	1	2	yes	—
51	#88043	—	2	2	no	—
52	#8146	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×D₁₃	—	2	1	no	—
59	C₂×C₄×D₁₃	—	2	1	no	—
60	C₂×C₄×D₁₃	—	2	1	no	—
61	C₂×C₄×D₁₃	—	2	1	no	—
62	C₂×C₄×D₁₃	—	2	2	no	—
63	C₂×C₄×D₁₃	—	2	2	no	—
64	C₂×C₄×D₁₃	—	2	4	no	—
65	C₂×C₄×D₁₃	—	2	4	no	—
66	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
68	C₂×#1057	—	2	4	no	—
69	C₂×#1057	—	2	4	no	—
70	C₂×#1057	—	2	4	no	—
71	C₂×#1057	—	2	4	no	—
72	C₂×#1057	—	2	4	no	—
73	C₂×#1057	M₁₆	1	8	yes	complemented
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄×Dic₁₃	—	2	2	no	—
83	C₄×Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄×Dic₁₃	#41	1	2	yes	—
87	C₂×#1055	—	2	1	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	C₂²⋊C₄	1	2	yes	—
90	C₂×#1055	C₂×C₈	1	2	yes	—
91	C₂×#1055	C₂×C₈	1	2	yes	—
92	C₂×#1055	M₁₆	1	2	yes	—
93	#6094	—	2	8	no	—
94	#6094	—	2	8	no	—
95	C₂×C₄×M₁₆	—	13	1	no	—
96	C₄×C₂²⋊C₈	—	13	2	no	—
97	C₂×#322	—	13	1	no	—
98	C₂×C₈²	—	13	1	no	—
99	#1273	—	13	2	no	—
100	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
101	C₂²×D₁₃	—	2	1	no	—
102	C₂²×D₁₃	—	2	2	no	—
103	C₂²×D₁₃	—	2	1	no	—
104	C₂²×D₁₃	—	2	2	no	—
105	C₂²×D₁₃	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	1	no	—
116	C₂×Dic₁₃	—	2	1	no	—
117	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
118	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
119	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
120	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
121	C₂×Dic₁₃	#131	1	4	yes	—
122	C₄×D₁₃	—	2	2	no	—
123	C₄×D₁₃	—	2	2	no	—
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	2	2	no	—
126	C₄×D₁₃	—	4	4	no	—
127	C₄×D₁₃	—	4	4	no	—
128	#1057	—	2	8	no	—
129	#1057	—	4	4	no	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	C₂²⋊C₈	1	4	yes	complemented
132	#1055	—	2	2	no	—
133	#1055	C₂²⋊C₈	1	4	yes	complemented
134	C₂²×M₁₆	—	13	1	no	—
135	C₂²×M₁₆	—	13	1	no	—
136	C₂²×C₄²	—	13	1	no	—
137	C₂×C₂²⋊C₈	—	13	4	no	—
138	C₄×M₁₆	—	26	2	no	—
139	C₄×M₁₆	—	26	2	no	—
140	C₂×#97	—	13	1	no	—
141	C₂×#97	#261	1	1	yes	characteristic
142	C₂×C₄×C₈	—	13	1	no	—
143	C₂×C₄×C₈	—	13	2	no	—
144	C₂×C₄×C₈	—	13	2	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	#336	—	13	4	no	—
147	#322	—	26	2	no	—
148	C₈²	—	26	2	no	—
149	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
151	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
152	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
153	C₂²×C₁₃	#405	1	2	yes	—
154	C₂×D₁₃	—	2	4	no	—
155	C₂×D₁₃	—	2	2	no	—
156	C₂×D₁₃	—	4	1	no	—
157	C₂×D₁₃	—	4	1	no	—
158	C₂×D₁₃	—	4	2	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₁₃	#431	1	1	yes	characteristic
163	C₄×C₁₃	#431	1	1	yes	characteristic
164	Dic₁₃	—	2	2	no	—
165	Dic₁₃	#435	1	4	yes	—
166	Dic₁₃	#436	1	4	yes	—
167	C₂³×C₄	—	13	1	no	—
168	C₂³×C₄	—	13	2	no	—
169	C₂×M₁₆	—	13	4	no	—
170	C₂×M₁₆	—	13	4	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂×M₁₆	—	26	2	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	4	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	4	no	—
179	C₂²×C₈	C₂×#261	1	1	yes	characteristic
180	C₂²×C₈	C₂×#261	1	1	yes	characteristic
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	—	13	2	no	—
183	C₂×C₄²	—	26	1	no	—
184	C₂×C₄²	—	26	1	no	—
185	C₂×C₄²	—	26	2	no	—
186	C₂×C₄²	C₂×#261	1	1	yes	characteristic
187	C₂²⋊C₈	—	26	8	no	—
188	#97	—	2	2	no	—
189	#97	—	26	2	no	—
190	C₄×C₈	—	13	2	no	—
191	C₄×C₈	—	13	2	no	—
192	C₄×C₈	—	26	1	no	—
193	C₄×C₈	—	26	1	no	—
194	C₄×C₈	—	26	2	no	—
195	C₄×C₈	—	26	2	no	—
196	C₄×C₈	—	26	2	no	—
197	C₄×C₈	—	26	2	no	—
198	C₂×C₁₃	C₈×M₁₆	1	2	yes	—
199	C₂×C₁₃	#1453	1	1	yes	characteristic
200	C₂×C₁₃	#1455	1	1	yes	characteristic
201	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
202	C₂×C₁₃	#1753	1	2	yes	—
203	D₁₃	—	4	2	no	—
204	C₂⁴	—	13	1	no	—
205	C₂²×C₄	—	13	2	no	—
206	C₂²×C₄	—	13	1	no	—
207	C₂²×C₄	—	13	2	no	—
208	C₂²×C₄	—	13	2	no	—
209	C₂²×C₄	—	26	4	no	—
210	C₂²×C₄	—	26	2	no	—
211	C₂²×C₄	—	26	2	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	—	26	1	no	—
214	C₂²×C₄	—	26	1	no	—
215	C₂²×C₄	—	26	1	no	—
216	C₂²×C₄	—	26	1	no	—
217	C₂²×C₄	C₄×#261	1	2	yes	—
218	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
219	M₁₆	—	26	8	no	—
220	M₁₆	—	26	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	8	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
236	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
237	C₂×C₈	#6118	1	2	yes	—
238	C₂×C₈	C₂²⋊#261	1	2	yes	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	13	2	no	—
242	C₄²	—	13	1	no	—
243	C₄²	—	13	1	no	—
244	C₄²	—	26	2	no	—
245	C₄²	—	26	2	no	—
246	C₁₃	#8146	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	13	2	no	—
248	C₂³	—	26	1	no	—
249	C₂³	—	26	2	no	—
250	C₂³	—	26	2	no	—
251	C₂³	—	26	1	no	—
252	C₂³	C₂×C₄×#261	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	13	4	no	—
257	C₂×C₄	—	13	4	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	26	1	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	—	52	4	no	—
268	C₂×C₄	—	52	4	no	—
269	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂×#6118	1	1	yes	characteristic
272	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
273	C₂×C₄	#88237	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	13	4	no	—
277	C₈	—	26	8	no	—
278	C₈	—	26	2	no	—
279	C₈	—	52	4	no	—
280	C₈	#88111	1	4	yes	complemented
281	C₂²	—	26	4	no	—
282	C₂²	—	26	2	no	—
283	C₂²	—	52	1	no	—
284	C₂²	—	52	1	no	—
285	C₂²	—	52	2	no	—
286	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
287	C₂²	C₄×#6118	1	1	yes	characteristic
288	C₂²	M₁₆×#261	1	2	yes	—
289	C₂²	#11719099	1	2	yes	—
290	C₂²	C₄×C₂²⋊#261	1	1	yes	characteristic
291	C₄	—	4	2	no	—
292	C₄	—	13	4	no	—
293	C₄	—	13	4	no	—
294	C₄	—	26	2	no	—
295	C₄	#11719049	1	1	yes	characteristic
296	C₄	#11719051	1	1	yes	characteristic
297	C₄	C₂×#88111	1	1	yes	characteristic
298	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
299	C₂	—	52	2	no	—
300	C₂	#13130034	1	2	yes	—
301	C₂	#13130056	1	1	yes	characteristic
302	C₂	#13130060	1	1	yes	characteristic
303	C₂	C₄×#88111	1	1	yes	characteristic
304	C₂	#13130576	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#97	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₄²
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#8146
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#11718251	C₂	semidirect
3. C₂×C₈×#1057	C₂	semidirect
4. #13129658	C₂	semidirect
36. C₂×#6094	C₈	semidirect
37. C₂×#6094	C₈	semidirect
73. C₂×#1057	M₁₆	semidirect
131. C₈×C₁₃	C₂²⋊C₈	semidirect
133. #1055	C₂²⋊C₈	semidirect
246. C₁₃	#8146	semidirect
280. C₈	#88111	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88111	2
C₂×#11718251	2
C₂×C₈×#1057	2
#13129658	2
C₂×#11718368	2
#8146	13