Subgroups — #15871723

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×Dic₁₃	C₂	1	2	yes	maximal
2	#13115802	C₂	1	2	yes	maximal
3	#13115580	C₂	1	2	yes	maximal
4	#13115575	C₂	1	2	yes	maximal
5	#13115000	C₂	1	1	yes	maximal, characteristic
6	#13114736	C₂	1	2	yes	maximal
7	#13114735	C₂	1	2	yes	maximal
8	C₄×#88055	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₁₃×#499	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88058	C₂²	1	2	yes	—
16	C₂×#88057	C₂²	1	2	yes	—
17	C₂×#88055	C₂²	1	1	yes	characteristic
18	C₂×#88055	C₂²	1	1	yes	characteristic
19	C₂×#88055	C₂²	1	1	yes	characteristic
20	C₂×#88055	C₂²	1	1	yes	characteristic
21	#11718646	C₂²	1	4	yes	—
22	#11718642	—	2	4	no	—
23	C₄⋊C₄×Dic₁₃	—	2	4	no	—
24	C₄⋊C₄×Dic₁₃	C₂²	1	4	yes	—
25	#11718597	C₂²	1	2	yes	—
26	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718121	C₂²	1	4	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#125	—	2	4	no	—
33	C₁₃×#125	C₂×C₄	1	8	yes	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	—	2	4	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	C₂×C₄×Dic₁₃	—	2	4	no	—
40	C₂×C₄×Dic₁₃	D₄	1	4	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	C₂²×#1055	C₂³	1	1	yes	characteristic
45	C₁₃×#107	C₂×C₄	1	8	yes	—
46	C₁₃×#106	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
48	#88071	—	2	4	no	—
49	#88071	—	2	4	no	—
50	#88071	—	2	4	no	—
51	#88058	—	2	4	no	—
52	#88057	—	2	4	no	—
53	#88055	—	2	2	no	—
54	#88055	—	2	2	no	—
55	#88055	Q₈	1	4	yes	—
56	#88055	Q₈	1	4	yes	—
57	C₄×#1055	—	2	2	no	—
58	#21728	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	4	no	—
63	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
64	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₁₃	#41	1	2	yes	—
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₁₃×C₄⋊C₄	—	2	16	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₃	#41	1	2	yes	—
76	C₄⋊Dic₁₃	—	2	4	no	—
77	C₄⋊Dic₁₃	—	2	4	no	—
78	C₄⋊Dic₁₃	—	4	8	no	—
79	C₄×Dic₁₃	—	2	1	no	—
80	C₄×Dic₁₃	—	2	1	no	—
81	C₄×Dic₁₃	—	4	4	no	—
82	C₄×Dic₁₃	—	4	8	no	—
83	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
84	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
85	C₂×#1055	—	2	1	no	—
86	C₂×#1055	—	2	1	no	—
87	C₂×#1055	—	2	1	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	C₂×Q₈	1	2	yes	—
90	C₂×#1055	C₂×Q₈	1	2	yes	—
91	C₂×#499	D₁₃	1	1	yes	characteristic
92	C₄×#125	—	13	2	no	—
93	#1991	—	13	2	no	—
94	#1944	—	13	2	no	—
95	#1943	—	13	2	no	—
96	#1846	—	13	1	no	—
97	#1776	—	13	2	no	—
98	#1776	—	13	2	no	—
99	C₄×#97	—	13	1	no	—
100	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	2	2	no	—
114	C₂×Dic₁₃	—	2	4	no	—
115	C₂×Dic₁₃	—	4	4	no	—
116	C₂×Dic₁₃	—	4	4	no	—
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
119	C₈×C₁₃	C₄×Q₈	1	4	yes	—
120	C₈×C₁₃	C₄×Q₈	1	4	yes	—
121	#1055	—	2	2	no	—
122	#1055	—	2	2	no	—
123	C₂×#125	C₂×D₁₃	1	2	yes	—
124	#499	Dic₁₃	1	8	yes	complemented
125	C₂×#107	—	13	2	no	—
126	C₂×#107	C₂×D₁₃	1	2	yes	—
127	C₂×#106	—	13	2	no	—
128	C₂×#106	C₂×D₁₃	1	2	yes	—
129	C₂×#97	—	13	1	no	—
130	C₂×#97	—	13	1	no	—
131	C₂×#97	—	13	1	no	—
132	C₂×#97	—	13	1	no	—
133	C₂×C₄×C₈	—	13	1	no	—
134	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
135	#389	—	13	4	no	—
136	#387	—	26	4	no	—
137	#382	—	13	2	no	—
138	C₄×C₄⋊C₄	—	13	4	no	—
139	C₄×C₄⋊C₄	—	26	4	no	—
140	C₄³	—	13	1	no	—
141	#340	—	13	4	no	—
142	#340	—	13	4	no	—
143	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₁₃	#532	1	1	yes	characteristic
146	C₂²×C₁₃	#575	1	4	yes	—
147	C₄×C₁₃	—	2	4	no	—
148	C₄×C₁₃	—	4	8	no	—
149	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₄×C₁₃	#533	1	1	yes	characteristic
152	C₄×C₁₃	#533	1	1	yes	characteristic
153	Dic₁₃	—	2	4	no	—
154	Dic₁₃	—	4	2	no	—
155	Dic₁₃	—	8	4	no	—
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
160	#125	—	2	4	no	—
161	#125	C₂×Dic₁₃	1	8	yes	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	13	4	no	—
164	C₂×C₄⋊C₄	—	26	4	no	—
165	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	4	no	—
169	C₂×C₄²	—	26	4	no	—
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	#107	—	26	4	no	—
172	#107	C₂×Dic₁₃	1	8	yes	—
173	#106	—	26	4	no	—
174	#106	C₂×Dic₁₃	1	8	yes	—
175	#97	—	13	4	no	—
176	#97	—	13	4	no	—
177	#97	—	26	2	no	—
178	#97	—	26	2	no	—
179	C₄×C₈	—	26	2	no	—
180	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
181	#95	—	26	4	no	—
182	#95	—	26	4	no	—
183	#95	—	26	4	no	—
184	C₂×C₁₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
185	C₂×C₁₃	#2943	1	2	yes	—
186	C₂×C₁₃	#3166	1	2	yes	—
187	C₂×C₁₃	#3183	1	2	yes	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	26	4	no	—
192	C₂²×C₄	D₄×D₁₃	1	2	yes	—
193	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	26	1	no	—
197	C₂×C₈	—	26	1	no	—
198	C₂×C₈	—	26	1	no	—
199	C₂×C₈	—	26	1	no	—
200	C₂×C₈	Q₈×D₁₃	1	2	yes	—
201	C₂×C₈	Q₈×D₁₃	1	2	yes	—
202	C₂×C₈	#6131	1	2	yes	—
203	C₂×C₈	#6131	1	2	yes	—
204	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
205	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
206	C₄⋊C₄	—	2	16	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	26	4	no	—
209	C₄⋊C₄	—	26	4	no	—
210	C₄⋊C₄	—	52	8	no	—
211	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
212	C₄²	—	13	2	no	—
213	C₄²	—	13	2	no	—
214	C₄²	—	26	1	no	—
215	C₄²	—	26	1	no	—
216	C₄²	—	52	4	no	—
217	C₄²	—	52	8	no	—
218	C₄²	#6129	1	2	yes	—
219	C₄²	C₂²×Dic₁₃	1	2	yes	—
220	C₁₃	#21728	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	13	4	no	—
227	C₂×C₄	—	26	4	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	4	no	—
230	C₂×C₄	—	52	4	no	—
231	C₂×C₄	—	52	4	no	—
232	C₂×C₄	—	52	4	no	—
233	C₂×C₄	D₄×Dic₁₃	1	4	yes	—
234	C₂×C₄	C₂×#6129	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×#6131	1	1	yes	characteristic
238	C₂×C₄	C₂×#6131	1	1	yes	characteristic
239	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
240	C₈	—	26	2	no	—
241	C₈	—	26	2	no	—
242	C₈	Q₈×Dic₁₃	1	4	yes	—
243	C₈	Q₈×Dic₁₃	1	4	yes	—
244	C₂²	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
245	C₂²	#11719430	1	1	yes	characteristic
246	C₂²	#11719473	1	4	yes	—
247	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	4	8	no	—
250	C₄	—	26	4	no	—
251	C₄	—	52	2	no	—
252	C₄	—	104	4	no	—
253	C₄	#11719433	1	1	yes	characteristic
254	C₄	#11719433	1	1	yes	characteristic
255	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
256	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
257	C₂	C₄⋊Q₈×Dic₁₃	1	1	yes	characteristic
258	C₂	#13123637	1	2	yes	—
259	C₂	#13125384	1	2	yes	—
260	C₂	#13125567	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#499	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#499	D₁₃
hypocenter	C₁₃	#21728
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
11. C₁₃×#499	C₄	semidirect
124. #499	Dic₁₃	semidirect
220. C₁₃	#21728	semidirect

Maximal subgroups

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

subgroup	index
#125×Dic₁₃	2
#13115802	2
#13115580	2
#13115575	2
#13115000	2
#13114736	2
#13114735	2
C₄×#88055	2
C₂×C₁₃×#499	2
#21728	13