Subgroups — #15767840

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115599	C₂	1	2	yes	maximal
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	2	yes	maximal
4	#13112671	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×#88058	—	2	1	no	—
11	C₂×#88058	—	2	1	no	—
12	C₂×#88048	—	2	2	no	—
13	#11718596	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	1	no	—
16	#11718121	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	—	2	1	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂²×#1055	—	2	2	no	—
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	C₁₃×C₄⋊C₈	—	4	2	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₁₃×#95	—	2	1	no	—
32	#88058	—	4	2	no	—
33	#88058	—	4	2	no	—
34	#88048	—	4	4	no	—
35	C₄⋊#1055	—	2	2	no	—
36	C₄⋊#1055	Q₈	1	4	yes	—
37	#10652	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	—	2	1	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	4	1	no	—
49	C₂×C₈×C₁₃	—	4	1	no	—
50	C₂×C₈×C₁₃	—	4	1	no	—
51	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
52	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	Q₁₆	1	2	yes	—
56	C₄²×C₁₃	—	4	1	no	—
57	C₄²×C₁₃	—	4	1	no	—
58	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
59	C₄²×C₁₃	#41	1	2	yes	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	8	2	no	—
65	C₂×#1055	—	2	4	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	4	2	no	—
68	#1948	—	13	2	no	—
69	#1939	D₁₃	1	1	yes	characteristic
70	#1845	—	13	1	no	—
71	#1296	—	13	2	no	—
72	#1277	—	13	1	no	—
73	C₂³×C₁₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	1	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	—	4	1	no	—
80	C₂×C₄×C₁₃	—	4	1	no	—
81	C₂×C₄×C₁₃	—	4	1	no	—
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#125	1	2	yes	—
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
90	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
93	C₂×Dic₁₃	—	8	2	no	—
94	C₂×Dic₁₃	—	8	2	no	—
95	C₂×Dic₁₃	—	8	2	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	—	8	1	no	—
99	#1055	—	4	4	no	—
100	C₂×#107	—	26	1	no	—
101	C₂×#107	—	26	1	no	—
102	C₂×#106	—	26	2	no	—
103	C₂×C₄⋊C₈	—	2	1	no	—
104	C₂×C₄⋊C₈	—	13	2	no	—
105	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
107	#384	—	13	2	no	—
108	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
109	#340	—	26	2	no	—
110	#336	—	2	1	no	—
111	C₂²×C₁₃	#398	1	1	yes	characteristic
112	C₂²×C₁₃	#460	1	1	yes	characteristic
113	C₂²×C₁₃	#461	1	1	yes	characteristic
114	C₂²×C₁₃	#467	1	1	yes	characteristic
115	C₂²×C₁₃	#469	1	1	yes	characteristic
116	C₂²×C₁₃	#472	1	1	yes	characteristic
117	C₂²×C₁₃	#473	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	—	8	1	no	—
121	C₄×C₁₃	#477	1	2	yes	—
122	C₄×C₁₃	#478	1	2	yes	—
123	C₄×C₁₃	#481	1	1	yes	characteristic
124	C₄×C₁₃	#483	1	1	yes	characteristic
125	C₄×C₁₃	#498	1	1	yes	characteristic
126	C₄×C₁₃	#501	1	1	yes	characteristic
127	Dic₁₃	—	16	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	26	2	no	—
130	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	26	2	no	—
133	C₂×C₄⋊C₄	—	26	2	no	—
134	C₂×C₄⋊C₄	—	26	2	no	—
135	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
136	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
137	C₂×C₄²	—	2	1	no	—
138	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
139	#107	—	52	2	no	—
140	#107	—	52	2	no	—
141	#106	—	52	4	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	4	2	no	—
144	C₄⋊C₈	—	13	4	no	—
145	C₄⋊C₈	—	26	2	no	—
146	C₄×C₈	—	2	2	no	—
147	#95	—	2	1	no	—
148	C₂×C₁₃	#1670	1	1	yes	characteristic
149	C₂×C₁₃	#1675	1	1	yes	characteristic
150	C₂×C₁₃	#1688	1	1	yes	characteristic
151	C₂×C₁₃	#1693	1	1	yes	characteristic
152	C₂×C₁₃	#2084	1	1	yes	characteristic
153	C₂×C₁₃	#2087	1	1	yes	characteristic
154	C₂×C₁₃	#2089	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	52	2	no	—
158	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
159	C₂²×C₄	#6127	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	26	4	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	—	52	2	no	—
171	C₂×C₈	#6105	1	2	yes	—
172	C₂×C₈	#6106	1	2	yes	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	52	2	no	—
176	C₄⋊C₄	—	52	2	no	—
177	C₄⋊C₄	—	52	2	no	—
178	C₄⋊C₄	—	52	2	no	—
179	C₄⋊C₄	—	104	2	no	—
180	C₄⋊C₄	Dic₅₂	1	2	yes	—
181	C₄²	—	4	1	no	—
182	C₄²	—	4	1	no	—
183	C₄²	#6129	1	2	yes	—
184	C₄²	Q₈×D₁₃	1	2	yes	—
185	C₁₃	#10652	1	1	yes	characteristic, complemented, Sylow
186	C₂³	#88184	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	104	2	no	—
196	C₂×C₄	—	104	2	no	—
197	C₂×C₄	—	104	2	no	—
198	C₂×C₄	#88143	1	2	yes	—
199	C₂×C₄	#88151	1	2	yes	—
200	C₂×C₄	#88158	1	1	yes	characteristic
201	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
202	C₂×C₄	#88162	1	2	yes	—
203	C₂×C₄	C₂×#6105	1	1	yes	characteristic
204	C₂×C₄	#88189	1	1	yes	characteristic
205	C₂×C₄	C₂×#6106	1	1	yes	characteristic
206	C₂×C₄	#88198	1	1	yes	characteristic
207	C₂×C₄	#88203	1	1	yes	characteristic
208	C₂×C₄	#88205	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	8	1	no	—
212	C₈	—	52	4	no	—
213	C₂²	#11718644	1	1	yes	characteristic
214	C₂²	#11718680	1	1	yes	characteristic
215	C₂²	#11718688	1	1	yes	characteristic
216	C₂²	#11718777	1	1	yes	characteristic
217	C₂²	#11718796	1	1	yes	characteristic
218	C₂²	#11718803	1	1	yes	characteristic
219	C₂²	#11718804	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	208	2	no	—
224	C₄	#11718508	1	2	yes	—
225	C₄	#11718511	1	2	yes	—
226	C₄	#11718695	1	1	yes	characteristic
227	C₄	#11718704	1	1	yes	characteristic
228	C₄	#11718726	1	1	yes	characteristic
229	C₄	#11718740	1	1	yes	characteristic
230	C₂	#13114206	1	1	yes	characteristic
231	C₂	#13114235	1	1	yes	characteristic
232	C₂	#13114320	1	1	yes	characteristic
233	C₂	#13114349	1	1	yes	characteristic
234	C₂	#13116312	1	1	yes	characteristic
235	C₂	#13116329	1	1	yes	characteristic
236	C₂	#13116341	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1939	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	#1939	D₁₃
hypocenter	C₁₃	#10652
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
185. C₁₃	#10652	semidirect

Maximal subgroups

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

subgroup	index
#13115599	2
#13114996	2
#13112720	2
#13112671	2
C₁₃×#1939	2
#10652	13