Subgroups — #15781552

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	#13115008	C₂	1	1	yes	maximal, characteristic
3	#13112945	C₂	1	4	yes	maximal
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	—	2	1	no	—
9	C₂×#88059	C₂²	1	2	yes	—
10	C₂×#88058	—	2	1	no	—
11	C₂×#88058	C₂²	1	2	yes	—
12	#11718585	—	2	1	no	—
13	#11718343	—	2	1	no	—
14	#11718343	—	2	1	no	—
15	#11718343	C₄	1	2	yes	—
16	#11718343	C₄	1	2	yes	—
17	C₁₃×#346	—	2	1	no	—
18	C₄×C₁₃×C₁₆	—	2	1	no	—
19	C₂²×Dic₂₆	—	4	1	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	4	1	no	—
25	C₂×#6099	—	2	2	no	—
26	#88123	—	2	2	no	—
27	#88123	—	2	2	no	—
28	#88123	—	4	1	no	—
29	#88123	—	4	1	no	—
30	C₁₃×M₃₂	—	2	2	no	—
31	C₁₃×M₃₂	Q₈	1	4	yes	complemented
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
37	#88059	—	2	4	no	—
38	#88059	—	4	2	no	—
39	#88058	—	2	2	no	—
40	#88058	—	2	2	no	—
41	#88058	—	2	2	no	—
42	#88058	C₂×C₄	1	4	yes	—
43	C₄⋊#1055	—	4	1	no	—
44	C₄⋊#1055	—	4	1	no	—
45	#13290	—	13	1	no	maximal, Sylow
46	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	—	4	2	no	—
50	C₂×Dic₂₆	—	4	2	no	—
51	C₂×Dic₂₆	—	4	2	no	—
52	C₂×Dic₂₆	—	8	4	no	—
53	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₃	D₈	1	1	yes	characteristic
64	C₄²×C₁₃	D₈	1	1	yes	characteristic
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	—	4	2	no	—
67	C₄⋊Dic₁₃	—	4	2	no	—
68	C₄⋊Dic₁₃	—	4	2	no	—
69	#6099	—	4	2	no	—
70	#6099	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	C₁₃×C₁₆	—	2	4	no	—
73	C₂×#500	—	13	1	no	—
74	C₄×M₃₂	D₁₃	1	1	yes	characteristic
75	#1848	—	13	1	no	—
76	#1387	—	13	4	no	—
77	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×Dic₁₃	—	4	4	no	—
91	C₂×Dic₁₃	—	8	4	no	—
92	Dic₂₆	—	8	4	no	—
93	Dic₂₆	—	8	4	no	—
94	C₈×C₁₃	#103	1	2	yes	—
95	C₈×C₁₃	#103	1	2	yes	—
96	C₈×C₁₃	C₄×Q₈	1	2	yes	—
97	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
98	#1055	—	8	2	no	—
99	C₂×C₄⋊Q₈	—	26	1	no	—
100	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
101	#500	—	13	2	no	—
102	#500	—	13	2	no	—
103	#500	—	26	1	no	—
104	#500	—	26	1	no	—
105	C₂×#108	—	13	2	no	—
106	C₂×#108	—	26	1	no	—
107	C₂×#107	—	13	2	no	—
108	C₂×#107	—	26	1	no	—
109	#423	—	26	1	no	—
110	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
111	#346	—	2	1	no	—
112	C₄×C₁₆	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₁₃	#487	1	1	yes	characteristic
117	C₂²×C₁₃	#498	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	#361	1	2	yes	—
121	C₄×C₁₃	#362	1	2	yes	—
122	C₄×C₁₃	#391	1	1	yes	characteristic
123	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
124	C₄×C₁₃	#486	1	1	yes	characteristic
125	C₄×C₁₃	#500	1	1	yes	characteristic
126	Dic₁₃	—	8	4	no	—
127	C₂²×Q₈	—	52	1	no	—
128	C₂×M₁₆	—	26	2	no	—
129	C₂²×C₈	D₅₂	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	26	2	no	—
132	C₄⋊Q₈	—	26	2	no	—
133	C₄⋊Q₈	—	52	1	no	—
134	C₄⋊Q₈	—	52	1	no	—
135	C₂×C₄⋊C₄	—	26	2	no	—
136	C₂×C₄⋊C₄	—	52	1	no	—
137	C₂×C₄²	D₅₂	1	1	yes	characteristic
138	M₃₂	—	2	2	no	—
139	M₃₂	Dic₂₆	1	4	yes	complemented
140	C₂×C₁₆	—	2	2	no	—
141	#108	—	26	4	no	—
142	#108	—	52	2	no	—
143	#107	—	13	4	no	—
144	#107	—	26	2	no	—
145	#107	—	26	2	no	—
146	#107	—	26	2	no	—
147	C₄⋊C₈	—	52	1	no	—
148	C₄⋊C₈	—	52	1	no	—
149	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
150	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	2	1	no	—
154	C₂×C₁₃	—	2	1	no	—
155	C₂×C₁₃	#1990	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
158	C₂×Q₈	—	52	2	no	—
159	C₂×Q₈	—	52	2	no	—
160	C₂×Q₈	—	104	4	no	—
161	C₂²×C₄	—	52	2	no	—
162	C₂²×C₄	#6095	1	1	yes	characteristic
163	C₂²×C₄	#6095	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
165	M₁₆	—	52	2	no	—
166	M₁₆	—	52	2	no	—
167	C₂×C₈	—	52	2	no	—
168	C₂×C₈	#6095	1	2	yes	—
169	C₂×C₈	Dic₅₂	1	2	yes	—
170	C₂×C₈	#6103	1	1	yes	characteristic
171	C₂×C₈	#6103	1	1	yes	characteristic
172	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
173	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
174	C₂×C₈	#6127	1	2	yes	—
175	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
176	C₄⋊C₄	—	26	4	no	—
177	C₄⋊C₄	—	52	2	no	—
178	C₄⋊C₄	—	52	2	no	—
179	C₄⋊C₄	—	52	2	no	—
180	C₄²	D₁₀₄	1	1	yes	characteristic
181	C₄²	D₁₀₄	1	1	yes	characteristic
182	C₄²	#6103	1	1	yes	characteristic
183	C₄²	#6103	1	1	yes	characteristic
184	C₁₆	—	2	4	no	—
185	C₁₃	#13290	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂×#6095	1	1	yes	characteristic
187	Q₈	—	104	4	no	—
188	Q₈	—	104	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	52	4	no	—
192	C₂×C₄	—	104	4	no	—
193	C₂×C₄	#88061	1	1	yes	characteristic
194	C₂×C₄	#88061	1	1	yes	characteristic
195	C₂×C₄	#88061	1	1	yes	characteristic
196	C₂×C₄	#88061	1	1	yes	characteristic
197	C₂×C₄	#88123	1	1	yes	characteristic
198	C₂×C₄	#88129	1	1	yes	characteristic
199	C₂×C₄	C₂×#6095	1	1	yes	characteristic
200	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
201	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
202	C₂×C₄	C₂×#6103	1	1	yes	characteristic
203	C₈	—	104	2	no	—
204	C₈	#88056	1	2	yes	—
205	C₈	#88056	1	2	yes	—
206	C₈	C₄×Dic₂₆	1	2	yes	—
207	C₈	#88178	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#11718341	1	1	yes	characteristic
211	C₂²	#11718342	1	1	yes	characteristic
212	C₂²	C₂×#88061	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	104	4	no	—
216	C₄	#11718108	1	2	yes	—
217	C₄	#11718109	1	2	yes	—
218	C₄	#11718343	1	1	yes	characteristic
219	C₄	#11718355	1	1	yes	characteristic
220	C₄	#11718590	1	1	yes	characteristic
221	C₄	C₂×#88056	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#13115797	1	1	yes	characteristic
225	C₂	C₂×#11718108	1	1	yes	characteristic
226	C₂	C₂×#11718109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88061
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#13290
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
31. C₁₃×M₃₂	Q₈	semidirect
139. M₃₂	Dic₂₆	semidirect
185. C₁₃	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718343	2
#13115008	2
#13112945	2
C₄×C₁₃×M₃₂	2
#13290	13