Subgroups — #16789049

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88098	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13129875	C₂	1	2	yes	maximal, complemented
4	#13129874	C₂	1	2	yes	maximal, complemented
5	C₂×#11718369	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	C₂×#88098	—	2	2	no	—
12	C₂×#88098	—	2	2	no	—
13	C₂×#88098	C₄	1	4	yes	complemented
14	C₂×D₁₃⋊C₁₆	—	2	2	no	—
15	C₂×D₁₃⋊C₁₆	—	2	2	no	—
16	C₂×D₁₃⋊C₁₆	C₄	1	4	yes	complemented
17	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
18	C₂×#88055	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	#11718369	—	2	2	no	—
22	#11718369	—	2	2	no	—
23	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#6103	C₈	1	2	yes	—
28	C₂×#6101	C₈	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	C₂×#6094	—	2	2	no	—
33	C₂×#6094	—	2	2	no	—
34	C₂×#6094	C₈	1	4	yes	—
35	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	D₄	1	2	yes	—
38	C₂×C₈×D₁₃	D₄	1	2	yes	—
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	Q₈	1	2	yes	—
41	#88126	—	2	2	no	—
42	#88126	—	2	2	no	—
43	C₂×#6092	—	2	2	no	—
44	C₂×#6092	—	2	2	no	—
45	C₂×#6092	—	2	2	no	—
46	C₂×#6092	—	2	2	no	—
47	C₂×#6092	C₂×C₄	1	4	yes	—
48	C₂×#6092	C₂×C₄	1	4	yes	—
49	#88098	—	2	8	no	—
50	D₁₃⋊C₁₆	—	2	8	no	—
51	C₁₃×#97	—	2	2	no	—
52	#88055	—	2	2	no	—
53	C₈×Dic₁₃	—	2	1	no	—
54	C₈×Dic₁₃	—	2	1	no	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	#12545	—	13	1	no	maximal, Sylow
58	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
62	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₄²	1	2	yes	—
64	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
68	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	C₄²	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	—	2	1	no	—
77	#6103	—	2	4	no	—
78	#6101	—	2	4	no	—
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	C₂×D₄	1	1	yes	characteristic
89	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
90	#6094	—	2	8	no	—
91	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
92	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
93	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
94	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
95	#6092	—	2	4	no	—
96	#6092	—	2	4	no	—
97	C₂²×M₃₂	—	13	1	no	—
98	C₂³×C₁₆	—	13	1	no	—
99	C₂×#405	—	13	1	no	—
100	#1374	—	13	2	no	—
101	#1374	—	13	2	no	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₁₃	—	2	4	no	—
104	C₂²×D₁₃	C₄×C₈	1	2	yes	—
105	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
106	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	2	no	—
116	C₂×Dic₁₃	—	2	4	no	—
117	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
118	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
119	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
120	C₄×D₁₃	#95	1	4	yes	—
121	C₄×D₁₃	#97	1	4	yes	—
122	C₄×D₁₃	C₂²⋊C₈	1	4	yes	—
123	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	#95	1	4	yes	—
126	#1055	—	2	2	no	—
127	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
130	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×M₁₆	—	13	1	no	—
132	C₂³×C₈	—	13	1	no	—
133	C₂×#117	—	13	1	no	—
134	C₂×M₃₂	—	13	4	no	—
135	C₂×M₃₂	—	26	2	no	—
136	C₂×M₃₂	—	26	2	no	—
137	C₂²×C₁₆	—	13	2	no	—
138	C₂²×C₁₆	—	13	4	no	—
139	C₂²×C₁₆	—	13	2	no	—
140	C₂²×C₁₆	—	26	2	no	—
141	C₂²×C₁₆	—	26	2	no	—
142	#405	—	26	2	no	—
143	#405	—	26	2	no	—
144	C₂×#97	—	13	1	no	—
145	C₂×#97	#261	1	1	yes	characteristic
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₁₃	#504	1	4	yes	—
152	C₂×D₁₃	—	2	4	no	—
153	C₂×D₁₃	—	2	4	no	—
154	C₂×D₁₃	—	2	4	no	—
155	C₂×D₁₃	#336	1	4	yes	—
156	C₄×C₁₃	—	4	2	no	—
157	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
158	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
159	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
161	Dic₁₃	—	4	2	no	—
162	Dic₁₃	#336	1	4	yes	—
163	C₂³×C₄	—	13	1	no	—
164	C₂×M₁₆	—	13	4	no	—
165	C₂×M₁₆	—	26	2	no	—
166	C₂×M₁₆	—	26	2	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	—	13	2	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂²×C₈	—	13	2	no	—
171	C₂²×C₈	—	13	2	no	—
172	C₂²×C₈	—	13	2	no	—
173	C₂²×C₈	—	13	2	no	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	C₂×#261	1	1	yes	characteristic
176	C₂²×C₈	C₂×#261	1	1	yes	characteristic
177	#117	—	26	2	no	—
178	#117	—	26	2	no	—
179	C₂×C₄⋊C₄	—	13	2	no	—
180	C₂×C₂²⋊C₄	—	13	2	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	C₂×#261	1	1	yes	characteristic
183	M₃₂	—	26	8	no	—
184	C₂×C₁₆	—	13	4	no	—
185	C₂×C₁₆	—	13	4	no	—
186	C₂×C₁₆	—	26	8	no	—
187	C₂×C₁₆	—	26	2	no	—
188	C₂×C₁₆	—	26	2	no	—
189	C₂×C₁₆	—	26	2	no	—
190	C₂×C₁₆	—	26	2	no	—
191	#97	—	2	2	no	—
192	#97	—	26	2	no	—
193	C₄×C₈	—	26	1	no	—
194	C₄×C₈	—	26	1	no	—
195	C₄×C₈	—	26	1	no	—
196	C₄×C₈	—	26	1	no	—
197	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
198	C₂×C₁₃	#2108	1	2	yes	—
199	C₂×C₁₃	#2113	1	2	yes	—
200	C₂×C₁₃	#2151	1	2	yes	—
201	D₁₃	—	2	4	no	—
202	C₂⁴	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	2	no	—
205	C₂²×C₄	—	13	2	no	—
206	C₂²×C₄	—	13	2	no	—
207	C₂²×C₄	—	13	2	no	—
208	C₂²×C₄	—	13	2	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
212	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
213	M₁₆	—	26	8	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	13	1	no	—
219	C₂×C₈	—	13	1	no	—
220	C₂×C₈	—	13	1	no	—
221	C₂×C₈	—	13	1	no	—
222	C₂×C₈	—	13	4	no	—
223	C₂×C₈	—	13	4	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	13	4	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	26	1	no	—
228	C₂×C₈	—	26	1	no	—
229	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
230	C₂×C₈	C₄×#261	1	2	yes	—
231	C₂×C₈	C₄⋊#261	1	2	yes	—
232	C₂×C₈	C₂²⋊#261	1	2	yes	—
233	C₄⋊C₄	—	26	4	no	—
234	C₂²⋊C₄	—	26	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	26	1	no	—
238	C₄²	—	26	1	no	—
239	C₁₆	—	26	4	no	—
240	C₁₆	—	26	4	no	—
241	C₁₃	#12545	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	13	2	no	—
243	C₂³	—	13	2	no	—
244	C₂³	—	13	2	no	—
245	C₂³	—	26	4	no	—
246	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	13	4	no	—
253	C₂×C₄	—	13	4	no	—
254	C₂×C₄	—	13	4	no	—
255	C₂×C₄	—	13	4	no	—
256	C₂×C₄	—	26	4	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
259	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
260	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	13	1	no	—
266	C₈	—	13	1	no	—
267	C₈	—	13	1	no	—
268	C₈	—	13	1	no	—
269	C₈	—	26	2	no	—
270	C₈	(C₂×#261)⋊C₄	1	4	yes	—
271	C₂²	—	13	4	no	—
272	C₂²	—	26	4	no	—
273	C₂²	—	26	4	no	—
274	C₂²	—	26	4	no	—
275	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
276	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
277	C₂²	#11719096	1	4	yes	—
278	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	13	4	no	—
281	C₄	—	52	2	no	—
282	C₄	#11719047	1	1	yes	characteristic
283	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
284	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
285	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
286	C₂	—	26	4	no	—
287	C₂	#13130497	1	1	yes	characteristic
288	C₂	#13131046	1	2	yes	—
289	C₂	#13131067	1	2	yes	—
290	C₂	#13131172	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	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#12545
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
3. #13129875	C₂	semidirect
4. #13129874	C₂	semidirect
13. C₂×#88098	C₄	semidirect
16. C₂×D₁₃⋊C₁₆	C₄	semidirect
241. C₁₃	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88098	2
C₂²×D₁₃⋊C₁₆	2
#13129875	2
#13129874	2
C₂×#11718369	2
#12545	13