Subgroups — #15767179

Order 3328.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115760	C₂	1	2	yes	maximal
2	#13115554	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718139	C₂	1	2	yes	maximal
4	#13112671	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#500	—	2	4	no	—
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₁₃×#107	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88075	—	2	2	no	—
12	C₂×#88075	—	2	2	no	—
13	C₂×#88054	—	2	1	no	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
17	#11718142	—	2	1	no	—
18	#11718139	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	—	2	1	no	—
32	C₂²×#1055	—	2	1	no	—
33	C₂²×#1055	—	2	2	no	—
34	C₁₃×#107	—	2	4	no	—
35	C₁₃×#107	—	2	4	no	—
36	C₄×C₈×C₁₃	—	2	2	no	—
37	#88075	—	4	4	no	—
38	#88075	—	4	4	no	—
39	#88071	—	2	1	no	—
40	#88054	—	4	2	no	—
41	C₄⋊#1055	—	2	2	no	—
42	C₄⋊#1055	—	2	2	no	—
43	C₄⋊#1055	—	2	2	no	—
44	#10545	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	2	1	no	—
56	C₂²×Dic₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
62	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
63	C₁₃×C₄⋊C₄	—	2	4	no	—
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₁₃	C₂×D₄	1	2	yes	—
68	C₄²×C₁₃	#41	1	2	yes	—
69	C₄⋊Dic₁₃	—	2	1	no	—
70	C₄⋊Dic₁₃	—	2	2	no	—
71	C₄⋊Dic₁₃	D₈	1	2	yes	—
72	C₄×Dic₁₃	—	4	1	no	—
73	C₄×Dic₁₃	—	4	1	no	—
74	C₂×#1055	—	4	1	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	1	no	—
78	C₂×#1055	—	4	2	no	—
79	C₂×#1055	—	4	1	no	—
80	C₂×#500	D₁₃	1	1	yes	characteristic
81	#1981	—	13	2	no	—
82	#1939	—	13	1	no	—
83	C₂×#332	—	13	2	no	—
84	#1277	—	13	1	no	—
85	C₂³×C₁₃	#121	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	4	no	—
87	Q₈×C₁₃	—	4	4	no	—
88	Q₈×C₁₃	—	8	2	no	—
89	Q₈×C₁₃	—	8	2	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	8	2	no	—
94	C₂×C₄×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#121	1	2	yes	—
97	C₂×C₄×C₁₃	#123	1	2	yes	—
98	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
103	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	4	1	no	—
108	C₂×Dic₁₃	—	4	1	no	—
109	C₂×Dic₁₃	—	4	1	no	—
110	C₂×Dic₁₃	—	4	1	no	—
111	C₂×Dic₁₃	—	4	1	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	—	2	2	no	—
114	#1055	—	8	1	no	—
115	#1055	—	8	2	no	—
116	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
117	#500	—	2	4	no	—
118	C₂×#107	—	2	1	no	—
119	C₂×#107	—	2	1	no	—
120	C₂×C₄⋊C₈	—	13	1	no	—
121	C₂×C₄⋊C₈	—	13	2	no	—
122	C₂×C₄⋊C₈	—	26	1	no	—
123	C₂×#103	—	26	2	no	—
124	C₂×#103	—	26	2	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	13	1	no	—
127	#336	—	26	1	no	—
128	#332	—	26	4	no	—
129	C₂²×C₁₃	#394	1	1	yes	characteristic
130	C₂²×C₁₃	#461	1	1	yes	characteristic
131	C₂²×C₁₃	#462	1	1	yes	characteristic
132	C₂²×C₁₃	#466	1	1	yes	characteristic
133	C₂²×C₁₃	#469	1	1	yes	characteristic
134	C₂²×C₁₃	#473	1	1	yes	characteristic
135	C₂²×C₁₃	#473	1	1	yes	characteristic
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	8	4	no	—
138	C₄×C₁₃	#448	1	1	yes	characteristic
139	C₄×C₁₃	#451	1	1	yes	characteristic
140	C₄×C₁₃	#459	1	2	yes	—
141	C₄×C₁₃	#463	1	2	yes	—
142	C₄×C₁₃	#487	1	1	yes	characteristic
143	C₄×C₁₃	#491	1	1	yes	characteristic
144	Dic₁₃	—	4	2	no	—
145	Dic₁₃	—	8	1	no	—
146	C₂²×Q₈	—	2	2	no	—
147	C₂²×C₈	—	26	2	no	—
148	C₂²×C₈	—	26	1	no	—
149	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	C₂²⋊D₁₃	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	13	1	no	—
158	C₂×C₄⋊C₄	—	13	1	no	—
159	C₂×C₄²	—	26	1	no	—
160	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	#107	—	2	4	no	—
162	#107	—	2	4	no	—
163	C₄⋊C₈	—	26	2	no	—
164	C₄⋊C₈	—	26	2	no	—
165	C₄⋊C₈	—	26	2	no	—
166	C₄⋊C₈	—	52	2	no	—
167	#103	—	52	4	no	—
168	#103	—	52	4	no	—
169	C₄×C₈	—	2	2	no	—
170	#95	—	26	1	no	—
171	C₂×C₁₃	#1673	1	1	yes	characteristic
172	C₂×C₁₃	#1674	1	1	yes	characteristic
173	C₂×C₁₃	#1693	1	1	yes	characteristic
174	C₂×C₁₃	#1694	1	1	yes	characteristic
175	C₂×C₁₃	#2043	1	1	yes	characteristic
176	C₂×C₁₃	#2047	1	1	yes	characteristic
177	C₂×C₁₃	#2054	1	1	yes	characteristic
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	26	1	no	—
186	C₂²×C₄	—	26	1	no	—
187	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
188	C₂²×C₄	#6129	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	52	2	no	—
195	C₂×C₈	—	52	2	no	—
196	C₂×C₈	—	52	2	no	—
197	C₂×C₈	—	52	1	no	—
198	C₂×C₈	—	52	1	no	—
199	C₂×C₈	—	52	1	no	—
200	C₂×C₈	#6106	1	2	yes	—
201	C₂×C₈	#6107	1	2	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	13	2	no	—
207	C₄⋊C₄	—	26	1	no	—
208	C₄⋊C₄	—	26	2	no	—
209	C₄²	—	52	1	no	—
210	C₄²	—	52	1	no	—
211	C₄²	#6129	1	2	yes	—
212	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
213	C₁₃	#10545	1	1	yes	characteristic, complemented, Sylow
214	C₂³	#88192	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	52	1	no	—
226	C₂×C₄	—	52	1	no	—
227	C₂×C₄	—	52	1	no	—
228	C₂×C₄	—	52	1	no	—
229	C₂×C₄	—	52	1	no	—
230	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
231	C₂×C₄	#88165	1	2	yes	—
232	C₂×C₄	#88166	1	1	yes	characteristic
233	C₂×C₄	#88189	1	2	yes	—
234	C₂×C₄	#88190	1	1	yes	characteristic
235	C₂×C₄	#88193	1	2	yes	—
236	C₂×C₄	C₂×#6106	1	1	yes	characteristic
237	C₂×C₄	#88196	1	1	yes	characteristic
238	C₂×C₄	#88196	1	1	yes	characteristic
239	C₂×C₄	C₂×#6107	1	1	yes	characteristic
240	C₂×C₄	#88207	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	104	2	no	—
244	C₈	—	104	1	no	—
245	C₂²	#11718744	1	1	yes	characteristic
246	C₂²	#11718751	1	1	yes	characteristic
247	C₂²	#11718818	1	1	yes	characteristic
248	C₂²	#11718820	1	1	yes	characteristic
249	C₂²	#11718853	1	1	yes	characteristic
250	C₂²	#11718853	1	1	yes	characteristic
251	C₂²	#11718883	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	—	52	2	no	—
255	C₄	—	104	1	no	—
256	C₄	#11718742	1	1	yes	characteristic
257	C₄	#11718743	1	1	yes	characteristic
258	C₄	#11718811	1	2	yes	—
259	C₄	#11718814	1	2	yes	—
260	C₄	#11718888	1	1	yes	characteristic
261	C₄	#11718889	1	1	yes	characteristic
262	C₂	#13114222	1	1	yes	characteristic
263	C₂	#13114227	1	1	yes	characteristic
264	C₂	#13114346	1	1	yes	characteristic
265	C₂	#13114351	1	1	yes	characteristic
266	C₂	#13116079	1	1	yes	characteristic
267	C₂	#13116101	1	1	yes	characteristic
268	C₂	#13116144	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#10545
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
213. C₁₃	#10545	semidirect

Maximal subgroups

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

subgroup	index
#13115760	2
#13115554	2
C₂×#11718139	2
#13112671	2
C₂×C₁₃×#500	2
#10545	13