Subgroups — #15772143

Order 3328.

454 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	C₂×C₂²⋊#6090	C₂	1	2	yes	maximal, complemented
2	#13112776	C₂	1	2	yes	maximal, complemented
3	#13112760	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
8	C₈×D₄×C₁₃	—	2	2	no	—
9	C₈×D₄×C₁₃	—	2	2	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂²×#6090	—	2	2	no	—
15	C₂²⋊#6090	—	2	2	no	—
16	C₂²⋊#6090	C₂²	1	4	yes	—
17	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	4	2	no	—
23	C₂²×C₈×C₁₃	—	4	2	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
33	C₂×C₁₃×C₂²⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₁₃×C₂²⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₂²⋊C₈	—	2	2	no	—
38	C₁₃×C₂²⋊C₈	—	2	2	no	—
39	C₄×C₈×C₁₃	—	2	1	no	—
40	C₄×C₈×C₁₃	—	2	1	no	—
41	C₂×#6090	—	2	4	no	—
42	C₂×#6090	—	4	4	no	—
43	#11465	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	C₁₆	1	4	yes	complemented
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	1	no	—
54	C₂²×C₄×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	1	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	1	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	—	2	2	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	#6090	—	4	4	no	—
81	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₁₆	—	13	2	no	—
83	#1322	—	13	2	no	—
84	#1314	—	13	2	no	—
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	—	2	1	no	—
87	C₂³×C₁₃	—	4	1	no	—
88	C₂³×C₁₃	—	4	1	no	—
89	C₂³×C₁₃	—	4	1	no	—
90	C₂³×C₁₃	—	4	1	no	—
91	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
92	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
93	C₂³×C₁₃	C₂×C₁₆	1	2	yes	complemented
94	C₂³×C₁₃	M₃₂	1	2	yes	—
95	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	D₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	complemented
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	4	1	no	—
119	C₈×C₁₃	—	4	1	no	—
120	C₈×C₁₃	#99	1	2	yes	—
121	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂×C₄×D₄	Dic₁₃	1	1	yes	characteristic
124	C₂²×C₁₆	—	26	2	no	—
125	C₈×D₄	—	2	2	no	—
126	C₈×D₄	—	2	2	no	—
127	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₈	Dic₁₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
131	C₂²⋊C₁₆	—	13	4	no	—
132	C₂²⋊C₁₆	—	26	2	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	4	2	no	—
139	C₂²×C₁₃	—	4	2	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	C₂²⋊C₁₆	1	4	yes	—
145	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
154	C₄×C₁₃	#413	1	1	yes	characteristic
155	C₂²×D₄	#1055	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	4	2	no	—
161	C₂²×C₈	—	4	2	no	—
162	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄⋊C₄	#1055	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	#1055	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	#1055	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂×C₁₆	—	26	4	no	—
175	C₂×C₁₆	—	52	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	1	no	—
183	C₂×C₁₃	—	2	4	no	—
184	C₂×C₁₃	—	4	2	no	—
185	C₂×C₁₃	#1459	1	1	yes	characteristic
186	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
187	C₂×C₁₃	#2115	1	1	yes	characteristic
188	C₂⁴	C₂×#1055	1	1	yes	characteristic
189	C₂⁴	C₂×#1055	1	1	yes	characteristic
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	#6090	1	4	yes	complemented
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
200	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
201	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
202	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
203	C₂²×C₄	#6099	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₁₆	—	52	4	no	—
225	C₁₃	#11465	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	C₂×#6090	1	2	yes	—
233	C₂³	C₂×#6090	1	2	yes	complemented
234	C₂³	#88052	1	2	yes	—
235	C₂³	C₂²⋊#1055	1	2	yes	—
236	C₂³	C₂²×#1055	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂×#6090	1	2	yes	complemented
254	C₂×C₄	C₂²⋊#1055	1	2	yes	—
255	C₂×C₄	C₂×#6099	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	#88074	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₂²⋊#6090	1	4	yes	—
274	C₂²	C₂²×#6090	1	1	yes	characteristic
275	C₂²	C₂×#88052	1	1	yes	characteristic
276	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂×#88074	1	1	yes	characteristic
283	C₄	#11718898	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	4	no	—
287	C₂	—	4	2	no	—
288	C₂	#13113154	1	1	yes	characteristic
289	C₂	C₂×C₂²⋊#6090	1	1	yes	characteristic
290	C₂	#13116457	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₁₃
commutator	C₂²×C₁₃	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₁₃	C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	#11465
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊#6090	C₂	semidirect
2. #13112776	C₂	semidirect
3. #13112760	C₂	semidirect
48. C₂×D₄×C₁₃	C₁₆	semidirect
93. C₂³×C₁₃	C₂×C₁₆	semidirect
113. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
192. C₂×D₄	#6090	semidirect
225. C₁₃	#11465	semidirect
233. C₂³	C₂×#6090	semidirect
253. C₂×C₄	C₂×#6090	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#6090	2
#13112776	2
#13112760	2
C₂×C₈×D₄×C₁₃	2
#11465	13