Subgroups — #16018409

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125348	C₂	1	2	yes	maximal
2	#13117117	C₂	1	2	yes	maximal, complemented
3	#13117114	C₂	1	2	yes	maximal, complemented
4	#13117113	C₂	1	2	yes	maximal, complemented
5	#13117110	C₂	1	2	yes	maximal, complemented
6	#13116620	C₂	1	1	yes	maximal, characteristic
7	#13116618	C₂	1	2	yes	maximal
8	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic
10	#11719222	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#427	C₂²	1	2	yes	—
13	#11718767	C₂²	1	2	yes	—
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	#11718659	—	2	2	no	—
17	#11718539	C₂²	1	4	yes	—
18	#11718535	C₂²	1	4	yes	—
19	#11718519	—	2	2	no	—
20	#11718516	—	2	2	no	—
21	C₁₃×#367	C₂²	1	2	yes	—
22	C₁₃×#366	C₂²	1	2	yes	—
23	#11718097	C₂²	1	2	yes	—
24	#11718096	C₂²	1	2	yes	—
25	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
28	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
29	#11718082	C₂²	1	4	yes	—
30	#11718081	C₂²	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#117	—	2	2	no	—
33	#88180	—	2	2	no	—
34	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
35	C₂²×#1055	D₄	1	1	yes	characteristic
36	#88144	—	2	4	no	—
37	#88142	—	2	4	no	—
38	#88132	—	4	4	no	—
39	C₄×Dic₂₆	—	4	4	no	—
40	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×#107	C₂³	1	4	yes	—
44	C₁₃×#106	—	2	2	no	—
45	#88058	C₂³	1	4	yes	—
46	#88057	—	2	2	no	—
47	C₈×Dic₁₃	D₄	1	1	yes	characteristic
48	C₈×Dic₁₃	D₄	1	1	yes	characteristic
49	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
50	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×#6090	Q₈	1	4	yes	complemented
52	C₂×#6090	C₂³	1	1	yes	characteristic
53	C₂×#6090	C₂³	1	1	yes	characteristic
54	#88048	—	2	4	no	—
55	#88047	—	2	4	no	—
56	#33726	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	D₈	1	2	yes	—
59	C₂×Dic₂₆	—	4	4	no	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₂²⋊C₄	—	4	2	no	—
66	C₄²×C₁₃	—	4	2	no	—
67	C₂²⋊Dic₁₃	—	4	2	no	—
68	C₄⋊Dic₁₃	—	2	4	no	—
69	C₄⋊Dic₁₃	—	4	4	no	—
70	#6101	—	4	4	no	—
71	#6101	—	4	4	no	—
72	C₄×Dic₁₃	—	4	2	no	—
73	C₄×Dic₁₃	D₈	1	2	yes	—
74	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×#1055	Q₁₆	1	4	yes	—
77	C₂×#1055	C₂×D₄	1	1	yes	characteristic
78	C₂×#1055	C₂×D₄	1	1	yes	characteristic
79	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
80	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
81	#6090	C₂×Q₈	1	2	yes	—
82	#6090	C₂×Q₈	1	2	yes	—
83	#3160	—	13	2	no	—
84	#2254	—	13	2	no	—
85	#2253	—	13	2	no	—
86	#2253	—	13	2	no	—
87	#2252	—	13	2	no	—
88	#2158	—	13	1	no	—
89	#2158	—	13	2	no	—
90	#2158	D₁₃	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	13	1	no	—
92	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₁₃	—	2	4	no	—
99	C₂×Dic₁₃	—	4	4	no	—
100	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
101	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
102	Dic₂₆	—	8	8	no	—
103	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
104	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
105	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
106	#1055	C₂×Q₁₆	1	2	yes	—
107	#1055	C₂×Q₁₆	1	2	yes	—
108	#533	—	26	2	no	—
109	C₂²×C₁₆	—	13	1	no	—
110	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
111	#500	—	13	4	no	—
112	#499	—	13	4	no	—
113	#499	—	26	2	no	—
114	#498	—	26	2	no	—
115	#427	—	13	2	no	—
116	#427	—	26	2	no	—
117	#427	C₂×D₁₃	1	2	yes	—
118	C₂×C₄×C₈	—	13	1	no	—
119	#367	—	13	2	no	—
120	#367	—	13	4	no	—
121	#367	C₂×D₁₃	1	2	yes	—
122	#366	—	13	2	no	—
123	#366	—	13	4	no	—
124	#366	C₂×D₁₃	1	2	yes	—
125	C₄×C₁₆	—	13	1	no	—
126	C₄×C₁₆	—	13	1	no	—
127	C₄×C₁₆	—	13	1	no	—
128	C₄×C₁₆	—	13	1	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	#500	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	8	4	no	—
133	C₄×C₁₃	#500	1	2	yes	—
134	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
136	Dic₁₃	—	2	2	no	—
137	Dic₁₃	—	8	4	no	—
138	Dic₁₃	#508	1	4	yes	—
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	26	4	no	—
142	#125	—	26	4	no	—
143	C₂²⋊Q₈	—	52	4	no	—
144	C₄×Q₈	—	52	4	no	—
145	#117	—	2	2	no	—
146	#117	—	26	2	no	—
147	C₂×C₄²	—	13	1	no	—
148	C₂×C₁₆	—	13	4	no	—
149	C₂×C₁₆	—	13	1	no	—
150	C₂×C₁₆	—	13	1	no	—
151	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
152	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
153	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
154	#107	—	13	4	no	—
155	#107	—	26	4	no	—
156	#107	C₂²×D₁₃	1	4	yes	—
157	#106	—	2	2	no	—
158	#106	—	26	4	no	—
159	#106	—	26	2	no	—
160	C₄×C₈	—	13	1	no	—
161	C₄×C₈	—	13	1	no	—
162	C₄×C₈	—	13	1	no	—
163	C₄×C₈	—	13	1	no	—
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
166	C₂×C₁₃	C₂×#508	1	2	yes	—
167	C₂×Q₈	—	52	4	no	—
168	C₂²×C₄	—	13	2	no	—
169	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	1	no	—
173	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
174	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
175	C₂×C₈	Q₈×D₁₃	1	2	yes	—
176	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	26	4	no	—
179	C₄⋊C₄	—	52	4	no	—
180	C₄⋊C₄	—	52	4	no	—
181	C₄⋊C₄	—	52	4	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	52	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	13	2	no	—
186	C₄²	—	13	1	no	—
187	C₄²	—	13	1	no	—
188	C₄²	—	52	2	no	—
189	C₁₆	—	13	2	no	—
190	C₁₆	—	13	2	no	—
191	C₁₆	C₂×Dic₂₆	1	2	yes	—
192	C₁₆	C₂×Dic₂₆	1	2	yes	—
193	C₁₃	#33726	1	1	yes	characteristic, complemented, Sylow
194	C₂³	D₈×D₁₃	1	1	yes	characteristic
195	Q₈	—	104	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	2	no	—
199	C₂×C₄	—	26	4	no	—
200	C₂×C₄	—	52	4	no	—
201	C₂×C₄	#88142	1	2	yes	—
202	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
205	C₈	—	13	2	no	—
206	C₈	—	13	2	no	—
207	C₈	#88142	1	2	yes	—
208	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
209	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	#11718535	1	2	yes	—
212	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	13	4	no	—
215	C₄	—	26	2	no	—
216	C₄	—	104	4	no	—
217	C₄	#11718535	1	2	yes	—
218	C₄	C₂×#88142	1	1	yes	characteristic
219	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#11718535	1	1	yes	characteristic
222	C₂	D₁₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33726
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #13117117	C₂	semidirect
3. #13117114	C₂	semidirect
4. #13117113	C₂	semidirect
5. #13117110	C₂	semidirect
40. C₂×C₁₃×C₁₆	Q₈	semidirect
51. C₂×#6090	Q₈	semidirect
151. C₂×C₁₆	Dic₂₆	semidirect
193. C₁₃	#33726	semidirect

Maximal subgroups

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

subgroup	index
#13125348	2
#13117117	2
#13117114	2
#13117113	2
#13117110	2
#13116620	2
#13116618	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2158	2
#33726	13