Subgroups — #16016001

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	4	yes	maximal, complemented
2	#13116635	C₂	1	4	yes	maximal, complemented
3	#13116592	C₂	1	2	yes	maximal, complemented
4	#13116447	C₂	1	2	yes	maximal
5	C₄×#88052	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
18	C₂×#88052	C₂²	1	4	yes	—
19	C₂²⋊#6090	C₂²	1	8	yes	—
20	C₄⋊#6090	C₂²	1	8	yes	complemented
21	#11718053	C₂²	1	4	yes	complemented
22	C₄×#6090	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₁₃	C₈	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	4	no	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₈	1	2	yes	—
36	C₂×C₄²×C₁₃	C₈	1	2	yes	—
37	C₂×C₄²×C₁₃	C₈	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#88052	—	2	8	no	—
53	C₂×#6090	C₂³	1	8	yes	—
54	#33418	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	4	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	C₂×C₈	1	2	yes	—
85	C₄²×C₁₃	C₂×C₈	1	2	yes	—
86	C₄²×C₁₃	C₂×C₈	1	2	yes	—
87	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×C₈	1	2	yes	—
89	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	#6090	—	2	8	no	—
94	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
95	#2163	—	13	4	no	—
96	#2162	—	13	4	no	—
97	#2150	—	13	2	no	—
98	#2112	—	13	2	no	—
99	C₄×M₃₂	—	13	2	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	M₃₂	1	8	yes	complemented
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	#117	1	2	yes	—
126	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
127	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
128	C₈×C₁₃	C₂×#41	1	2	yes	—
129	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
130	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
131	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
132	C₂×M₃₂	—	13	4	no	—
133	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
134	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
135	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
136	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
141	C₄⋊C₁₆	—	13	8	no	—
142	C₂²⋊C₁₆	—	13	8	no	—
143	#346	—	13	4	no	—
144	C₄×C₁₆	—	13	4	no	—
145	C₂²×C₁₃	—	2	8	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	#431	1	2	yes	—
150	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
151	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	#431	1	2	yes	—
156	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
157	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
158	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
159	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
160	C₂³×C₄	#1055	1	2	yes	—
161	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
166	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
167	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
169	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
171	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×C₄²	#1055	1	2	yes	—
173	C₂×C₄²	#1055	1	2	yes	—
174	C₂×C₄²	#1055	1	2	yes	—
175	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
176	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
179	M₃₂	—	26	8	no	—
180	C₂×C₁₆	—	13	8	no	—
181	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
182	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
183	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
184	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
185	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
187	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₃	1	2	yes	—
190	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	2	4	no	—
193	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
194	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
195	C₂⁴	C₂×#1055	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
200	C₂²×C₄	C₂×#1055	1	2	yes	—
201	C₂²×C₄	C₂×#1055	1	2	yes	—
202	C₂²×C₄	C₂×#1055	1	2	yes	—
203	C₂²×C₄	C₂×#1055	1	2	yes	—
204	C₂²×C₄	C₂×#1055	1	2	yes	—
205	C₂²×C₄	C₂×#1055	1	2	yes	—
206	C₂²×C₄	C₂×#1055	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	#6099	1	4	yes	—
216	C₂×C₈	#6127	1	4	yes	—
217	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
218	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
219	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
223	C₄²	C₂×#1055	1	2	yes	—
224	C₄²	C₂×#1055	1	2	yes	—
225	C₄²	C₂×#1055	1	2	yes	—
226	C₄²	C₂×#1055	1	2	yes	—
227	C₄²	C₂×#1055	1	1	yes	characteristic
228	C₄²	C₂×#1055	1	2	yes	—
229	C₄²	C₂×#1055	1	1	yes	characteristic
230	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
231	C₄²	C₂²×Dic₁₃	1	2	yes	—
232	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
233	C₁₆	—	26	8	no	—
234	C₁₃	#33418	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	4	no	—
236	C₂³	—	2	2	no	—
237	C₂³	C₂²×#1055	1	2	yes	—
238	C₂³	C₂²×#1055	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	#88052	1	8	yes	complemented
249	C₂×C₄	C₂²×#1055	1	2	yes	—
250	C₂×C₄	C₂²×#1055	1	2	yes	—
251	C₂×C₄	C₂²×#1055	1	2	yes	—
252	C₂×C₄	C₂²×#1055	1	2	yes	—
253	C₂×C₄	C₂²×#1055	1	2	yes	—
254	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
255	C₂×C₄	C₂×#6099	1	2	yes	—
256	C₂×C₄	#88180	1	2	yes	—
257	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	C₂×#6099	1	2	yes	—
261	C₈	C₂×#6099	1	2	yes	—
262	C₈	#88180	1	2	yes	—
263	C₈	C₂×#6127	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#11718586	1	2	yes	—
269	C₂²	C₂×#88052	1	4	yes	—
270	C₂²	C₂³×#1055	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	#11718586	1	2	yes	—
275	C₄	C₂×#88052	1	4	yes	—
276	C₄	C₂×#88052	1	4	yes	—
277	C₄	C₂²×#6099	1	1	yes	characteristic
278	C₄	C₂×#88180	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	—	2	2	no	—
281	C₂	C₂×#11718586	1	1	yes	characteristic
282	C₂	C₂²×#88052	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#33418
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #13116639	C₂	semidirect
2. #13116635	C₂	semidirect
3. #13116592	C₂	semidirect
5. C₄×#88052	C₂	semidirect
20. C₄⋊#6090	C₂²	semidirect
21. #11718053	C₂²	semidirect
22. C₄×#6090	C₂²	semidirect
113. C₂×C₄×C₁₃	M₃₂	semidirect
234. C₁₃	#33418	semidirect
248. C₂×C₄	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
#13116635	2
#13116592	2
#13116447	2
C₄×#88052	2
C₂²×C₄×C₈×C₁₃	2
#33418	13