Subgroups — #15753392

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718046	C₂	1	2	yes	maximal, complemented
4	#13112663	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	—	2	4	no	—
9	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊#1055	C₄	1	2	yes	—
13	C₂×C₂²⋊#1055	C₄	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	#11718116	C₄	1	2	yes	—
17	#11718116	C₄	1	2	yes	—
18	#11718046	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	—	2	4	no	—
24	C₂²×C₈×C₁₃	—	2	4	no	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	4	yes	—
34	C₂²×#1055	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₁₃	—	2	4	no	—
36	C₄×C₈×C₁₃	—	2	2	no	—
37	C₄×C₈×C₁₃	D₄	1	4	yes	—
38	C₂²⋊#1055	—	2	4	no	—
39	C₂²⋊#1055	C₈	1	8	yes	complemented
40	C₄×#1055	—	2	2	no	—
41	C₄×#1055	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	C₄×#1055	—	2	2	no	—
44	#8145	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
57	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
58	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	—	2	16	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
68	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	C₂×D₄	1	2	yes	—
74	C₄²×C₁₃	#41	1	2	yes	—
75	C₂×#1055	—	2	4	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	2	4	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	2	4	no	—
80	C₂×#1055	C₂×C₈	1	8	yes	complemented
81	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
82	C₄×C₂²⋊C₈	—	13	1	no	—
83	C₄×C₂²⋊C₈	—	13	1	no	—
84	C₂×#322	—	13	2	no	—
85	#1273	—	13	2	no	—
86	C₂³×C₁₃	—	2	1	no	—
87	C₂³×C₁₃	—	2	1	no	—
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	2	1	no	—
90	C₂³×C₁₃	C₄×C₈	1	2	yes	—
91	C₂³×C₁₃	#97	1	2	yes	—
92	C₂³×C₁₃	#97	1	2	yes	—
93	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	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	4	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	8	no	—
104	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
105	C₂×C₄×C₁₃	#97	1	2	yes	—
106	C₂×C₄×C₁₃	#97	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	#131	1	2	yes	—
116	C₂×C₄×C₁₃	#131	1	2	yes	—
117	C₈×C₁₃	—	2	4	no	—
118	C₈×C₁₃	—	2	4	no	—
119	#1055	—	2	8	no	—
120	#1055	—	4	4	no	—
121	C₂³×C₈	Dic₁₃	1	2	yes	—
122	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
123	C₂×C₂²⋊C₈	—	13	2	no	—
124	C₂×C₂²⋊C₈	—	13	2	no	—
125	C₂×C₄×C₈	—	2	4	no	—
126	C₂×C₄×C₈	—	13	2	no	—
127	C₂×C₄×C₈	—	13	2	no	—
128	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	#336	—	13	2	no	—
132	#336	—	13	2	no	—
133	#322	—	26	4	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	#322	1	4	yes	complemented
141	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
145	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
146	C₂²×C₁₃	#405	1	1	yes	characteristic
147	C₂²×C₁₃	#405	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	#407	1	2	yes	—
152	C₄×C₁₃	#432	1	2	yes	—
153	C₄×C₁₃	C₈×D₄	1	4	yes	—
154	C₄×C₁₃	#435	1	4	yes	—
155	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
156	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	—	2	4	no	—
161	C₂²×C₈	—	13	4	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
164	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	C₂²⋊C₈	—	13	8	no	—
172	C₂²⋊C₈	—	26	4	no	—
173	C₄×C₈	—	2	4	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	26	2	no	—
176	C₄×C₈	—	26	2	no	—
177	C₄×C₈	—	26	2	no	—
178	C₄×C₈	—	26	2	no	—
179	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
180	C₂×C₁₃	—	2	4	no	—
181	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
182	C₂×C₁₃	C₈×M₁₆	1	1	yes	characteristic
183	C₂×C₁₃	#1451	1	1	yes	characteristic
184	C₂×C₁₃	#1452	1	1	yes	characteristic
185	C₂×C₁₃	#1749	1	1	yes	characteristic
186	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₁₃	#1752	1	1	yes	characteristic
188	C₂⁴	C₄×Dic₁₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₈×D₁₃	1	4	yes	—
197	C₂²×C₄	#6094	1	4	yes	—
198	C₂²×C₄	#6099	1	4	yes	—
199	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
205	C₂×C₈	—	2	16	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	13	8	no	—
211	C₂×C₈	—	26	4	no	—
212	C₂×C₈	—	26	4	no	—
213	C₂×C₈	—	26	4	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	#6099	1	4	yes	—
217	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	#6127	1	2	yes	—
223	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
224	C₁₃	#8145	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	#88043	1	2	yes	—
230	C₂³	C₈×Dic₁₃	1	2	yes	—
231	C₂³	#88055	1	2	yes	—
232	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	#88043	1	2	yes	—
244	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
245	C₂×C₄	#88055	1	2	yes	—
246	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
247	C₂×C₄	C₂×#6094	1	2	yes	—
248	C₂×C₄	#88155	1	2	yes	—
249	C₂×C₄	#88155	1	2	yes	—
250	C₂×C₄	C₂×#6099	1	2	yes	—
251	C₂×C₄	C₂×#6099	1	2	yes	—
252	C₂×C₄	#88180	1	1	yes	characteristic
253	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
254	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	2	4	no	—
258	C₈	—	26	8	no	—
259	C₈	—	52	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	#11718046	1	4	yes	complemented
267	C₂²	C₂×#88043	1	1	yes	characteristic
268	C₂²	C₄×#6099	1	1	yes	characteristic
269	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
270	C₂²	C₂×#88055	1	1	yes	characteristic
271	C₂²	#11718762	1	1	yes	characteristic
272	C₂²	#11718762	1	1	yes	characteristic
273	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	#11718587	1	2	yes	—
278	C₄	C₈×C₂²⋊D₁₃	1	4	yes	—
279	C₄	#11718772	1	4	yes	—
280	C₄	#11718868	1	2	yes	—
281	C₂	—	2	4	no	—
282	C₂	C₂×#11718046	1	1	yes	characteristic
283	C₂	C₈×#6099	1	1	yes	characteristic
284	C₂	#13113125	1	1	yes	characteristic
285	C₂	#13113128	1	1	yes	characteristic
286	C₂	#13114598	1	1	yes	characteristic
287	C₂	C₈×C₂²⋊Dic₁₃	1	1	yes	characteristic
288	C₂	#13114612	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#8145
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718046	C₂	semidirect
4. #13112663	C₂	semidirect
39. C₂²⋊#1055	C₈	semidirect
80. C₂×#1055	C₂×C₈	semidirect
140. C₂²×C₁₃	#322	semidirect
224. C₁₃	#8145	semidirect
266. C₂²	#11718046	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#1055	2
C₂×#11718046	2
#13112663	2
C₂²×C₄×C₈×C₁₃	2
#8145	13