Subgroups — #15781520

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13112928	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	1	no	—
9	C₂×#88058	—	2	1	no	—
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88058	C₂²	1	2	yes	—
12	#11718596	—	2	1	no	—
13	#11718343	—	2	2	no	—
14	#11718343	C₄	1	4	yes	complemented
15	C₄×C₁₃×C₁₆	—	2	2	no	—
16	C₂²×Dic₂₆	—	4	1	no	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×C₄⋊Dic₁₃	—	2	2	no	—
22	C₂×C₄⋊Dic₁₃	—	4	1	no	—
23	C₂×C₄⋊Dic₁₃	—	4	1	no	—
24	C₂×C₄⋊Dic₁₃	—	4	1	no	—
25	#88123	—	2	4	no	—
26	#88123	—	4	2	no	—
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃	D₄	1	2	yes	—
32	#88058	—	2	4	no	—
33	#88058	—	2	2	no	—
34	#88058	—	2	4	no	—
35	#88058	—	4	2	no	—
36	#88058	C₂×C₄	1	4	yes	—
37	#13282	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	4	2	no	—
42	C₂²×Dic₁₃	—	4	2	no	—
43	C₂×Dic₂₆	—	4	4	no	—
44	C₂×Dic₂₆	—	8	2	no	—
45	C₂×Dic₂₆	—	8	2	no	—
46	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₁₃	#41	1	2	yes	—
53	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₁₃	SD₁₆	1	2	yes	—
55	C₄⋊Dic₁₃	—	2	4	no	—
56	C₄⋊Dic₁₃	—	4	4	no	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	C₄⋊Dic₁₃	—	8	2	no	—
61	C₄⋊Dic₁₃	—	8	2	no	—
62	C₁₃×C₁₆	—	2	4	no	—
63	C₂×#500	—	13	1	no	—
64	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
65	#1845	—	13	1	no	—
66	#1380	—	13	4	no	—
67	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#102	1	2	yes	—
69	C₂×C₄×C₁₃	#102	1	2	yes	—
70	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
71	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
72	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
73	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×Dic₁₃	—	4	4	no	—
81	C₂×Dic₁₃	—	8	2	no	—
82	C₂×Dic₁₃	—	8	2	no	—
83	C₂×Dic₁₃	—	8	2	no	—
84	C₂×Dic₁₃	—	8	2	no	—
85	C₂×Dic₁₃	—	8	2	no	—
86	Dic₂₆	—	8	8	no	—
87	C₈×C₁₃	#103	1	4	yes	—
88	C₈×C₁₃	C₄×Q₈	1	2	yes	—
89	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄⋊Q₈	—	26	1	no	—
91	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
92	#500	—	13	4	no	—
93	#500	—	26	2	no	—
94	C₂×#107	—	13	2	no	—
95	C₂×#107	—	13	2	no	—
96	C₂×#107	—	26	1	no	—
97	C₂×#107	—	26	1	no	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	#384	—	26	1	no	—
100	C₄×C₁₆	—	2	2	no	—
101	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
102	C₂²×C₁₃	#486	1	1	yes	characteristic
103	C₂²×C₁₃	#500	1	1	yes	characteristic
104	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
108	C₄×C₁₃	#357	1	4	yes	—
109	C₄×C₁₃	#358	1	4	yes	—
110	C₄×C₁₃	#391	1	1	yes	characteristic
111	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
112	C₄×C₁₃	#487	1	1	yes	characteristic
113	C₄×C₁₃	#498	1	1	yes	characteristic
114	Dic₁₃	—	8	4	no	—
115	Dic₁₃	—	16	2	no	—
116	C₂²×Q₈	—	52	1	no	—
117	C₂²×C₈	D₅₂	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	26	4	no	—
120	C₄⋊Q₈	—	52	2	no	—
121	C₂×C₄⋊C₄	—	26	2	no	—
122	C₂×C₄⋊C₄	—	26	2	no	—
123	C₂×C₄⋊C₄	—	52	1	no	—
124	C₂×C₄⋊C₄	—	52	1	no	—
125	C₂×C₄⋊C₄	—	52	1	no	—
126	C₂×C₄²	D₅₂	1	1	yes	characteristic
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
130	#107	—	13	4	no	—
131	#107	—	26	4	no	—
132	#107	—	26	2	no	—
133	#107	—	26	4	no	—
134	#107	—	52	2	no	—
135	C₄×C₈	C₄×D₁₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₁₃	#1990	1	1	yes	characteristic
138	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
139	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
140	C₂×C₁₃	#2240	1	1	yes	characteristic
141	C₂×C₁₃	#2241	1	1	yes	characteristic
142	C₂×C₁₃	#2252	1	1	yes	characteristic
143	C₂×C₁₃	#2254	1	1	yes	characteristic
144	C₂×Q₈	—	52	4	no	—
145	C₂×Q₈	—	104	2	no	—
146	C₂×Q₈	—	104	2	no	—
147	C₂²×C₄	—	52	2	no	—
148	C₂²×C₄	—	52	2	no	—
149	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
150	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
152	C₂×C₈	#6095	1	2	yes	—
153	C₂×C₈	Dic₅₂	1	2	yes	—
154	C₂×C₈	#6103	1	2	yes	—
155	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
157	C₂×C₈	#6127	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄⋊C₄	—	26	4	no	—
160	C₄⋊C₄	—	52	2	no	—
161	C₄⋊C₄	—	52	4	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₄⋊C₄	—	52	2	no	—
164	C₄⋊C₄	—	104	2	no	—
165	C₄⋊C₄	—	104	2	no	—
166	C₄²	#6095	1	2	yes	—
167	C₄²	#6103	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₁₃	#13282	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
171	Q₈	—	104	8	no	—
172	C₂×C₄	—	52	4	no	—
173	C₂×C₄	—	104	2	no	—
174	C₂×C₄	—	104	2	no	—
175	C₂×C₄	—	104	2	no	—
176	C₂×C₄	—	104	2	no	—
177	C₂×C₄	—	104	2	no	—
178	C₂×C₄	D₂₀₈	1	2	yes	—
179	C₂×C₄	#88040	1	2	yes	—
180	C₂×C₄	#88040	1	2	yes	—
181	C₂×C₄	Dic₁₀₄	1	2	yes	—
182	C₂×C₄	#88061	1	2	yes	—
183	C₂×C₄	#88061	1	2	yes	—
184	C₂×C₄	#88123	1	1	yes	characteristic
185	C₂×C₄	#88129	1	1	yes	characteristic
186	C₂×C₄	C₂×#6095	1	1	yes	characteristic
187	C₂×C₄	C₂×#6095	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
189	C₂×C₄	C₂×#6103	1	1	yes	characteristic
190	C₈	#88056	1	4	yes	—
191	C₈	C₄×Dic₂₆	1	2	yes	—
192	C₈	#88178	1	2	yes	—
193	C₂²	#11718343	1	1	yes	characteristic
194	C₂²	#11718355	1	1	yes	characteristic
195	C₂²	C₂×D₂₀₈	1	1	yes	characteristic
196	C₂²	C₂×#88040	1	1	yes	characteristic
197	C₂²	C₂×#88040	1	1	yes	characteristic
198	C₂²	C₂×Dic₁₀₄	1	1	yes	characteristic
199	C₂²	C₂×#88061	1	1	yes	characteristic
200	C₄	—	104	4	no	—
201	C₄	—	208	2	no	—
202	C₄	#11718095	1	4	yes	—
203	C₄	#11718101	1	4	yes	—
204	C₄	#11718341	1	1	yes	characteristic
205	C₄	#11718342	1	1	yes	characteristic
206	C₄	#11718590	1	1	yes	characteristic
207	C₄	C₂×#88056	1	1	yes	characteristic
208	C₂	#13115797	1	1	yes	characteristic
209	C₂	C₂×#11718101	1	1	yes	characteristic
210	C₂	C₂×#11718095	1	1	yes	characteristic
211	C₂	#13117060	1	1	yes	characteristic
212	C₂	#13117064	1	1	yes	characteristic
213	C₂	#13117111	1	1	yes	characteristic
214	C₂	#13117118	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#13282
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
14. #11718343	C₄	semidirect
29. C₂×C₁₃×C₁₆	Q₈	semidirect
129. C₂×C₁₆	Dic₂₆	semidirect
169. C₁₃	#13282	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
#13114996	2
#13112928	2
C₂×C₄×C₁₃×C₁₆	2
#13282	13