Subgroups — #15761856

Order 3328.

498 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	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113227	C₂	1	2	yes	maximal, complemented
3	#13113157	C₂	1	2	yes	maximal, complemented
4	#13113128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×D₈	—	2	1	no	—
9	C₄×C₁₃×D₈	—	2	1	no	—
10	C₄×C₁₃×D₈	—	2	1	no	—
11	C₄×C₁₃×D₈	—	2	1	no	—
12	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
13	C₂×C₁₃×#102	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718586	C₂²	1	2	yes	—
16	#11718586	C₂²	1	1	yes	characteristic
17	#11718586	C₂²	1	1	yes	characteristic
18	D₄⋊#1055	—	2	4	no	—
19	#11718117	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	#11718046	—	2	1	no	—
23	#11718046	—	2	1	no	—
24	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	—	2	2	no	—
26	C₂×C₁₃×D₈	—	2	4	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₄×D₄×C₁₃	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#107	—	2	1	no	—
39	C₁₃×#107	—	2	1	no	—
40	C₁₃×#102	—	2	2	no	—
41	C₁₃×#102	—	2	2	no	—
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²⋊#1055	—	2	1	no	—
47	C₂²⋊#1055	—	2	2	no	—
48	C₂²⋊#1055	D₄	1	2	yes	—
49	C₄⋊#1055	—	2	1	no	—
50	C₄⋊#1055	—	2	2	no	—
51	C₄⋊#1055	D₄	1	2	yes	—
52	C₄×#1055	—	2	1	no	—
53	C₄×#1055	—	2	1	no	—
54	C₄×#1055	—	2	2	no	—
55	#9670	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	M₁₆	1	4	yes	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	8	4	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₁₃×D₈	—	2	4	no	—
71	C₁₃×D₈	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₂²⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃	#41	1	1	yes	characteristic
86	C₂×#1055	—	2	2	no	—
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
90	#1564	—	13	1	no	—
91	#1478	—	13	2	no	—
92	#1460	—	13	2	no	—
93	#1452	—	13	1	no	—
94	C₂³×C₁₃	—	2	4	no	—
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	—	8	2	no	—
98	C₂³×C₁₃	—	8	2	no	—
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	D₄×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	2	4	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	8	4	no	—
108	C₂×C₄×C₁₃	—	8	4	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	#131	1	2	yes	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	—	2	1	no	—
122	#1055	—	4	2	no	—
123	#1055	—	4	2	no	—
124	#1055	—	8	2	no	—
125	C₂²×D₈	Dic₁₃	1	1	yes	characteristic
126	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
127	C₄×D₈	—	2	1	no	—
128	C₄×D₈	—	2	1	no	—
129	C₄×D₈	—	2	1	no	—
130	C₄×D₈	—	2	1	no	—
131	#431	—	13	1	no	—
132	#431	—	13	1	no	—
133	#431	—	13	2	no	—
134	C₂×#107	Dic₁₃	1	1	yes	characteristic
135	C₂×#102	Dic₁₃	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
137	#335	—	26	1	no	—
138	#334	—	26	1	no	—
139	D₄⋊C₈	—	26	4	no	—
140	#323	—	26	4	no	—
141	#322	—	26	1	no	—
142	#322	—	26	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	—	8	4	no	—
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	—	8	2	no	—
149	C₂²×C₁₃	—	8	2	no	—
150	C₂²×C₁₃	#390	1	1	yes	characteristic
151	C₂²×C₁₃	#435	1	2	yes	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	8	2	no	—
156	C₄×C₁₃	#407	1	1	yes	characteristic
157	C₄×C₁₃	#408	1	1	yes	characteristic
158	C₄×C₁₃	#435	1	2	yes	—
159	C₄×C₁₃	#472	1	2	yes	—
160	C₂²×D₄	C₂×Dic₁₃	1	2	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	C₄⋊C₈	—	13	2	no	—
177	C₄⋊C₈	—	26	1	no	—
178	C₄⋊C₈	—	26	2	no	—
179	#102	—	2	2	no	—
180	#102	—	2	2	no	—
181	C₂²⋊C₈	—	13	2	no	—
182	C₂²⋊C₈	—	26	1	no	—
183	C₂²⋊C₈	—	26	2	no	—
184	C₄×C₈	—	26	2	no	—
185	C₄×C₈	—	26	1	no	—
186	C₄×C₈	—	26	1	no	—
187	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	8	4	no	—
193	C₂×C₁₃	#1955	1	1	yes	characteristic
194	C₂×C₁₃	#1976	1	1	yes	characteristic
195	C₂×C₁₃	#1978	1	1	yes	characteristic
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	#6099	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	8	4	no	—
207	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	#6129	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
210	D₈	—	2	4	no	—
211	D₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	26	2	no	—
215	C₂×C₈	—	26	2	no	—
216	C₂×C₈	—	52	2	no	—
217	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	D₄×D₁₃	1	1	yes	characteristic
226	C₄²	#6129	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₁₃	#9670	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	8	2	no	—
234	C₂³	—	8	2	no	—
235	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₂×#6099	1	2	yes	—
247	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
248	C₂×C₄	#88190	1	1	yes	characteristic
249	C₂×C₄	#88192	1	1	yes	characteristic
250	C₂×C₄	#88192	1	1	yes	characteristic
251	C₂×C₄	#88194	1	1	yes	characteristic
252	C₂×C₄	#88202	1	2	yes	—
253	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	52	2	no	—
259	C₈	—	52	2	no	—
260	C₈	—	104	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	#11718672	1	2	yes	—
269	C₂²	#11718882	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	8	2	no	—
274	C₄	#11718672	1	2	yes	—
275	C₄	#11718821	1	2	yes	—
276	C₄	#11718868	1	1	yes	characteristic
277	C₄	#11718895	1	1	yes	characteristic
278	C₂	—	2	2	no	—
279	C₂	—	8	4	no	—
280	C₂	#13115635	1	1	yes	characteristic
281	C₂	#13115735	1	1	yes	characteristic
282	C₂	#13115747	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂²×C₁₃	#390
prime radical	C₂²×C₁₃×D₈	C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	#9670
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #13113584	C₂	semidirect
2. #13113227	C₂	semidirect
3. #13113157	C₂	semidirect
4. #13113128	C₂	semidirect
229. C₁₃	#9670	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
#13113227	2
#13113157	2
#13113128	2
C₂×C₄×C₁₃×D₈	2
#9670	13