Subgroups — #15763077

Order 3328.

438 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	#13123199	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	2	yes	maximal
4	#13112720	C₂	1	2	yes	maximal
5	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#88124	C₄	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718597	C₄	1	2	yes	—
12	#11718596	C₂²	1	1	yes	characteristic
13	#11718596	C₂²	1	1	yes	characteristic
14	#11718589	—	2	1	no	—
15	#11718589	—	2	1	no	—
16	#11718589	—	2	1	no	—
17	#11718589	—	2	1	no	—
18	C₂×C₄⋊#1055	C₂²	1	2	yes	—
19	C₂×#88043	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂×#6101	—	2	2	no	—
36	C₂×#6101	—	2	2	no	—
37	C₂²×#1055	—	2	2	no	—
38	#88124	—	4	4	no	—
39	C₁₃×C₄⋊C₈	—	2	4	no	—
40	C₁₃×C₄⋊C₈	—	4	2	no	—
41	C₁₃×C₄⋊C₈	—	4	2	no	—
42	#88071	—	2	2	no	—
43	#88071	—	2	2	no	—
44	C₄⋊#1055	—	2	2	no	—
45	C₄⋊#1055	Q₈	1	4	yes	—
46	#88043	—	4	2	no	—
47	#9906	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	—	4	2	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	—	2	1	no	—
67	C₄²×C₁₃	—	2	1	no	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
70	C₄²×C₁₃	#41	1	2	yes	—
71	C₄⋊Dic₁₃	—	4	2	no	—
72	C₄⋊Dic₁₃	—	4	1	no	—
73	C₄⋊Dic₁₃	—	4	1	no	—
74	C₄⋊Dic₁₃	—	4	1	no	—
75	C₄⋊Dic₁₃	—	4	1	no	—
76	C₄⋊Dic₁₃	—	4	1	no	—
77	C₄⋊Dic₁₃	—	4	1	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	C₄⋊Dic₁₃	—	4	2	no	—
80	#6101	—	4	4	no	—
81	#6101	—	4	4	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	2	no	—
85	#2859	—	13	1	no	—
86	#1842	D₁₃	1	1	yes	characteristic
87	#1767	—	13	1	no	—
88	#1296	—	13	2	no	—
89	#1296	—	13	2	no	—
90	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
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	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	#100	1	2	yes	—
104	C₂×C₄×C₁₃	#101	1	2	yes	—
105	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#135	1	2	yes	—
110	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
116	C₂×Dic₁₃	—	4	2	no	—
117	C₂×Dic₁₃	—	4	2	no	—
118	C₂×Dic₁₃	—	4	2	no	—
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	—	4	2	no	—
121	C₂×Dic₁₃	—	4	2	no	—
122	C₈×C₁₃	—	8	2	no	—
123	#1055	—	4	4	no	—
124	C₂×#125	—	13	2	no	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄⋊C₈	—	13	2	no	—
128	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
129	C₂×#97	—	26	1	no	—
130	#389	—	26	1	no	—
131	#389	—	26	1	no	—
132	#389	—	26	1	no	—
133	#389	—	26	1	no	—
134	#384	—	13	1	no	—
135	#384	—	13	1	no	—
136	#382	—	13	2	no	—
137	C₄³	C₂×D₁₃	1	1	yes	characteristic
138	#336	—	26	1	no	—
139	C₂²×C₁₃	#391	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
142	C₂²×C₁₃	#416	1	1	yes	characteristic
143	C₂²×C₁₃	#417	1	1	yes	characteristic
144	C₂²×C₁₃	#417	1	1	yes	characteristic
145	C₂²×C₁₃	#417	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	—	4	1	no	—
151	C₄×C₁₃	#478	1	2	yes	—
152	C₄×C₁₃	#479	1	2	yes	—
153	C₄×C₁₃	#490	1	1	yes	characteristic
154	C₄×C₁₃	#491	1	1	yes	characteristic
155	C₄×C₁₃	#501	1	1	yes	characteristic
156	C₄×C₁₃	#501	1	1	yes	characteristic
157	Dic₁₃	—	8	2	no	—
158	Dic₁₃	—	8	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	26	2	no	—
161	#125	—	52	4	no	—
162	C₂×C₄⋊C₄	—	13	2	no	—
163	C₂×C₄⋊C₄	—	13	2	no	—
164	C₂×C₄⋊C₄	—	26	2	no	—
165	C₂×C₄⋊C₄	—	26	2	no	—
166	C₂×C₄⋊C₄	—	26	1	no	—
167	C₂×C₄⋊C₄	—	26	1	no	—
168	C₂×C₄⋊C₄	—	26	1	no	—
169	C₂×C₄⋊C₄	—	26	1	no	—
170	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
172	C₂×C₄²	D₅₂	1	1	yes	characteristic
173	C₂×C₄²	D₅₂	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	—	4	2	no	—
180	C₄⋊C₈	—	13	4	no	—
181	C₄⋊C₈	—	26	2	no	—
182	#97	—	52	2	no	—
183	#95	—	26	2	no	—
184	#95	—	26	2	no	—
185	C₂×C₁₃	#1551	1	1	yes	characteristic
186	C₂×C₁₃	#1552	1	1	yes	characteristic
187	C₂×C₁₃	#1560	1	1	yes	characteristic
188	C₂×C₁₃	#1561	1	1	yes	characteristic
189	C₂×C₁₃	#1989	1	1	yes	characteristic
190	C₂×C₁₃	#1992	1	1	yes	characteristic
191	C₂×C₁₃	#1997	1	1	yes	characteristic
192	C₂²×C₄	—	26	2	no	—
193	C₂²×C₄	—	26	2	no	—
194	C₂²×C₄	#6103	1	1	yes	characteristic
195	C₂²×C₄	#6103	1	1	yes	characteristic
196	C₂²×C₄	#6103	1	1	yes	characteristic
197	C₂²×C₄	#6103	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	—	52	2	no	—
206	C₂×C₈	—	52	2	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₄⋊C₄	—	52	4	no	—
210	C₄⋊C₄	—	52	2	no	—
211	C₄⋊C₄	—	52	1	no	—
212	C₄⋊C₄	—	52	1	no	—
213	C₄⋊C₄	—	52	1	no	—
214	C₄⋊C₄	—	52	1	no	—
215	C₄⋊C₄	—	52	1	no	—
216	C₄⋊C₄	—	52	1	no	—
217	C₄⋊C₄	—	52	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	#6129	1	2	yes	—
228	C₄²	Q₈×D₁₃	1	2	yes	—
229	C₁₃	#9906	1	1	yes	characteristic, complemented, Sylow
230	C₂³	C₂×#6103	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
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	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	52	2	no	—
244	C₂×C₄	—	52	2	no	—
245	C₂×C₄	—	52	2	no	—
246	C₂×C₄	—	52	2	no	—
247	C₂×C₄	—	52	2	no	—
248	C₂×C₄	—	52	2	no	—
249	C₂×C₄	#88063	1	2	yes	—
250	C₂×C₄	#88064	1	2	yes	—
251	C₂×C₄	#88141	1	2	yes	—
252	C₂×C₄	#88151	1	2	yes	—
253	C₂×C₄	#88158	1	2	yes	—
254	C₂×C₄	#88162	1	1	yes	characteristic
255	C₂×C₄	#88163	1	1	yes	characteristic
256	C₂×C₄	#88186	1	1	yes	characteristic
257	C₂×C₄	#88186	1	1	yes	characteristic
258	C₂×C₄	#88190	1	1	yes	characteristic
259	C₂×C₄	#88196	1	1	yes	characteristic
260	C₂×C₄	#88196	1	1	yes	characteristic
261	C₂×C₄	#88198	1	1	yes	characteristic
262	C₈	—	8	2	no	—
263	C₈	—	52	4	no	—
264	C₂²	#11718627	1	1	yes	characteristic
265	C₂²	#11718627	1	1	yes	characteristic
266	C₂²	#11718635	1	1	yes	characteristic
267	C₂²	#11718774	1	1	yes	characteristic
268	C₂²	C₂×#88063	1	1	yes	characteristic
269	C₂²	#11718794	1	1	yes	characteristic
270	C₂²	C₂×#88064	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	104	2	no	—
277	C₄	—	104	2	no	—
278	C₄	#11718490	1	2	yes	—
279	C₄	#11718508	1	2	yes	—
280	C₄	#11718732	1	1	yes	characteristic
281	C₄	#11718741	1	1	yes	characteristic
282	C₄	#11718741	1	1	yes	characteristic
283	C₄	#11718743	1	1	yes	characteristic
284	C₂	#13113529	1	1	yes	characteristic
285	C₂	#13113534	1	1	yes	characteristic
286	C₂	#13113569	1	1	yes	characteristic
287	C₂	#13113573	1	1	yes	characteristic
288	C₂	#13115791	1	1	yes	characteristic
289	C₂	#13115804	1	1	yes	characteristic
290	C₂	#13115824	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1842	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#9906
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
229. C₁₃	#9906	semidirect

Maximal subgroups

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

subgroup	index
#13123199	2
#13114689	2
#13112720	2
C₁₃×#1842	2
#9906	13