Subgroups — #16789126

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719092	C₂	1	2	yes	maximal
2	#13129904	C₂	1	4	yes	maximal, complemented
3	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×#88237	—	2	2	no	—
5	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
6	#11719092	—	2	2	no	—
7	#11719092	—	2	2	no	—
8	#11719092	C₄	1	4	yes	complemented
9	#11719092	C₂²	1	4	yes	—
10	C₂×#88102	—	2	2	no	—
11	C₂×#88101	C₄	1	4	yes	complemented
12	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
13	C₂×#88052	C₄	1	1	yes	characteristic, complemented
14	D₁₃×M₃₂	—	2	2	no	—
15	D₁₃×M₃₂	—	2	2	no	—
16	C₂×#88038	C₂²	1	2	yes	—
17	C₂×C₁₆×D₁₃	C₄	1	2	yes	complemented
18	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊#261	—	4	2	no	—
20	#88237	—	2	4	no	—
21	#88237	—	2	4	no	—
22	#88237	—	4	2	no	—
23	#88237	—	4	2	no	—
24	C₂×C₄⋊#261	—	2	4	no	—
25	C₂×C₄×#261	—	4	2	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
32	C₂×C₈×D₁₃	D₄	1	2	yes	—
33	C₂×C₈×D₁₃	D₄	1	2	yes	—
34	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
36	#88102	—	2	4	no	—
37	#88102	—	2	4	no	—
38	#88101	—	2	4	no	—
39	#88101	C₂×C₄	1	8	yes	complemented
40	C₁₃×M₃₂	—	2	2	no	—
41	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
42	#88052	—	2	2	no	—
43	C₂×#6090	C₂×C₄	1	2	yes	complemented
44	#88038	—	2	4	no	—
45	C₁₆×D₁₃	—	2	2	no	—
46	C₁₆×D₁₃	—	2	2	no	—
47	C₂×#1384	—	13	1	no	maximal, Sylow
48	C₂³×D₁₃	D₈	1	1	yes	characteristic
49	C₂²×#261	—	4	4	no	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	Q₁₆	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
56	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
57	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
58	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
59	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊#261	—	4	4	no	—
61	C₂²⋊#261	—	4	4	no	—
62	C₄⋊#261	—	2	8	no	—
63	C₄⋊#261	—	4	4	no	—
64	C₄×#261	—	4	4	no	—
65	C₄×#261	—	4	4	no	—
66	C₂×C₈×C₁₃	C₄²	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
71	C₂×#1055	C₄⋊C₄	1	2	yes	—
72	C₂×#1055	C₄⋊C₄	1	2	yes	—
73	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
74	C₈×D₁₃	C₄²	1	1	yes	characteristic
75	C₈×D₁₃	C₄²	1	1	yes	characteristic
76	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
77	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
78	C₈×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₈×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
81	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
82	C₈×D₁₃	C₂²×C₄	1	2	yes	—
83	C₈×D₁₃	C₂×D₄	1	2	yes	—
84	C₁₃×C₁₆	—	2	2	no	—
85	#6090	—	2	2	no	—
86	C₂²×M₃₂	—	13	1	no	—
87	C₂×#427	—	13	2	no	—
88	#1384	—	13	4	no	—
89	C₂³×C₁₃	#107	1	1	yes	characteristic
90	C₂²×D₁₃	—	4	2	no	—
91	C₂²×D₁₃	#102	1	2	yes	—
92	C₂²×D₁₃	#102	1	2	yes	—
93	C₂²×D₁₃	#107	1	1	yes	characteristic
94	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×#261	—	4	8	no	—
96	C₂×#261	—	8	8	no	—
97	C₂×C₄×C₁₃	#95	1	2	yes	—
98	C₂×C₄×C₁₃	#106	1	2	yes	—
99	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×Dic₁₃	#103	1	2	yes	—
102	C₂×Dic₁₃	#103	1	2	yes	—
103	C₂×Dic₁₃	#107	1	1	yes	characteristic
104	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
105	C₄×D₁₃	#95	1	1	yes	characteristic
106	C₄×D₁₃	#95	1	1	yes	characteristic
107	C₄×D₁₃	#102	1	2	yes	—
108	C₄×D₁₃	#102	1	2	yes	—
109	C₄×D₁₃	#103	1	2	yes	—
110	C₄×D₁₃	#103	1	2	yes	—
111	C₄×D₁₃	#106	1	1	yes	characteristic
112	C₄×D₁₃	#106	1	1	yes	characteristic
113	C₄×D₁₃	C₂×C₂²⋊C₄	1	2	yes	—
114	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
115	C₈×C₁₃	#95	1	2	yes	—
116	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	#1055	#95	1	1	yes	characteristic
119	#1055	#95	1	1	yes	characteristic
120	#1055	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³×C₈	—	13	1	no	—
122	C₂×#117	—	26	2	no	—
123	C₂×M₃₂	—	13	1	no	—
124	C₂×M₃₂	—	13	2	no	—
125	C₂×M₃₂	—	26	2	no	—
126	C₂×M₃₂	—	26	2	no	—
127	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
128	C₂²×C₁₆	—	13	2	no	—
129	#427	—	13	4	no	—
130	#427	—	13	4	no	—
131	#427	—	26	2	no	—
132	#427	—	26	2	no	—
133	C₂×#107	—	26	2	no	—
134	C₂×#106	—	13	4	no	—
135	C₂²×C₁₃	—	4	1	no	—
136	C₂²×C₁₃	#340	1	2	yes	—
137	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
138	C₂×D₁₃	—	2	4	no	—
139	C₂×D₁₃	—	2	2	no	—
140	C₂×D₁₃	—	4	2	no	—
141	C₂×D₁₃	—	4	1	no	—
142	C₂×D₁₃	#340	1	1	yes	characteristic
143	C₂×D₁₃	#340	1	1	yes	characteristic
144	C₂×D₁₃	C₂×#102	1	2	yes	—
145	#261	—	8	8	no	—
146	C₄×C₁₃	#340	1	2	yes	—
147	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
148	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
149	Dic₁₃	#340	1	1	yes	characteristic
150	Dic₁₃	#340	1	1	yes	characteristic
151	Dic₁₃	C₂×#103	1	2	yes	—
152	C₂³×C₄	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	—	13	2	no	—
155	C₂²×C₈	—	13	2	no	—
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	2	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₂²×C₈	—	13	1	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	C₂×#261	1	1	yes	characteristic
163	#117	—	26	4	no	—
164	#117	—	26	4	no	—
165	#117	—	52	2	no	—
166	#117	—	52	2	no	—
167	C₂×C₄⋊C₄	—	26	4	no	—
168	C₂×C₂²⋊C₄	—	52	2	no	—
169	C₂×C₄²	—	52	2	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	26	4	no	—
172	M₃₂	—	26	2	no	—
173	C₂×C₁₆	—	13	2	no	—
174	C₂×C₁₆	—	26	2	no	—
175	C₂×C₁₆	—	26	2	no	—
176	C₂×C₁₆	C₂×#261	1	2	yes	complemented
177	#107	—	26	4	no	—
178	#107	—	26	4	no	—
179	#106	—	13	8	no	—
180	#106	—	26	4	no	—
181	C₂×C₁₃	—	2	2	no	—
182	C₂×C₁₃	—	2	1	no	—
183	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
184	D₁₃	—	2	2	no	—
185	D₁₃	—	2	1	no	—
186	D₁₃	#1384	1	2	yes	complemented
187	C₂⁴	—	13	1	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	52	4	no	—
198	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	1	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	1	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	C₄×#261	1	2	yes	—
214	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊#261	1	2	yes	—
216	C₂×C₈	C₂²×#261	1	1	yes	characteristic
217	C₄⋊C₄	—	26	8	no	—
218	C₄⋊C₄	—	52	4	no	—
219	C₂²⋊C₄	—	52	4	no	—
220	C₂²⋊C₄	—	52	4	no	—
221	C₄²	—	52	4	no	—
222	C₄²	—	52	4	no	—
223	C₁₆	—	2	2	no	—
224	C₁₆	—	26	2	no	—
225	C₁₃	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	13	2	no	—
227	C₂³	—	13	1	no	—
228	C₂³	—	13	2	no	—
229	C₂³	—	13	1	no	—
230	C₂³	—	52	2	no	—
231	C₂³	#88102	1	1	yes	characteristic
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	1	no	—
235	C₂×C₄	—	13	1	no	—
236	C₂×C₄	—	13	2	no	—
237	C₂×C₄	—	13	2	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	52	8	no	—
247	C₂×C₄	—	104	8	no	—
248	C₂×C₄	#88101	1	2	yes	—
249	C₂×C₄	#88102	1	1	yes	characteristic
250	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
251	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
252	C₈	—	13	1	no	—
253	C₈	—	13	1	no	—
254	C₈	—	13	2	no	—
255	C₈	(C₂×#261)⋊C₄	1	2	yes	—
256	C₈	C₂×C₄×#261	1	1	yes	characteristic
257	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
258	C₂²	—	4	1	no	—
259	C₂²	—	13	1	no	—
260	C₂²	—	13	1	no	—
261	C₂²	—	13	2	no	—
262	C₂²	—	26	4	no	—
263	C₂²	—	26	2	no	—
264	C₂²	—	52	2	no	—
265	C₂²	—	52	1	no	—
266	C₂²	#11718265	1	2	yes	—
267	C₂²	C₂×#88102	1	1	yes	characteristic
268	C₄	—	13	1	no	—
269	C₄	—	13	1	no	—
270	C₄	—	13	2	no	—
271	C₄	—	104	8	no	—
272	C₄	#11718265	1	2	yes	—
273	C₄	C₂×#88101	1	1	yes	characteristic
274	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
275	C₂	—	2	2	no	—
276	C₂	—	2	1	no	—
277	C₂	—	13	2	no	—
278	C₂	—	26	2	no	—
279	C₂	—	26	1	no	—
280	C₂	C₂×#11718265	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88101
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂×C₁₃	C₂×#340
prime radical	C₂³×D₁₃	D₈
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	C₂×#1384
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13129904	C₂	semidirect
8. #11719092	C₄	semidirect
11. C₂×#88101	C₄	semidirect
12. C₂×C₁₃×M₃₂	C₄	semidirect
13. C₂×#88052	C₄	semidirect
17. C₂×C₁₆×D₁₃	C₄	semidirect
39. #88101	C₂×C₄	semidirect
41. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
43. C₂×#6090	C₂×C₄	semidirect
127. C₂×M₃₂	#261	semidirect
176. C₂×C₁₆	C₂×#261	semidirect
186. D₁₃	#1384	semidirect
225. C₁₃	C₂×#1384	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719092	2
#13129904	2
C₂×D₁₃×M₃₂	2
C₂×#1384	13