Subgroups — #16016281

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123688	C₂	1	2	yes	maximal, complemented
2	#13116712	C₂	1	4	yes	maximal, complemented
3	D₁₆×Dic₁₃	C₂	1	4	yes	maximal, complemented
4	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116552	C₂	1	2	yes	maximal, complemented
6	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	#11719503	—	2	2	no	—
10	C₂×C₁₃×D₁₆	C₄	1	8	yes	complemented
11	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718876	—	2	2	no	—
15	#11718813	—	2	4	no	—
16	D₈×Dic₁₃	C₂²	1	8	yes	complemented
17	#11718767	C₂²	1	2	yes	—
18	#11718762	C₂²	1	1	yes	characteristic
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	#11718154	C₂²	1	8	yes	complemented
21	#11718097	C₂²	1	2	yes	complemented
22	#11718096	C₂²	1	2	yes	complemented
23	#11718094	C₂²	1	2	yes	complemented
24	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
25	C₂²×D₄×C₁₃	—	2	2	no	—
26	C₂×C₁₃×D₈	—	2	2	no	—
27	C₂×C₁₃×D₈	—	2	4	no	—
28	C₂×C₁₃×D₈	C₂×C₄	1	8	yes	complemented
29	C₂×C₁₃×D₈	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₁₃	—	4	4	no	—
32	D₄×Dic₁₃	—	2	8	no	—
33	D₄×Dic₁₃	—	4	4	no	—
34	#88180	—	2	2	no	—
35	#88180	D₄	1	1	yes	characteristic
36	C₂×#6099	D₄	1	1	yes	characteristic
37	C₁₃×D₁₆	C₂×C₄	1	8	yes	complemented
38	C₁₃×D₁₆	C₂×C₄	1	8	yes	complemented
39	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	#88072	—	2	8	no	—
43	#88058	C₂³	1	4	yes	—
44	#88057	—	2	2	no	—
45	#88055	D₄	1	2	yes	—
46	C₈×Dic₁₃	C₂³	1	2	yes	—
47	#88052	—	2	2	no	—
48	C₂×#6090	C₂³	1	2	yes	—
49	#33456	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	—	2	8	no	—
52	C₂×D₄×C₁₃	—	2	4	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	4	8	no	—
57	C₁₃×D₈	—	2	8	no	—
58	C₁₃×D₈	—	2	4	no	—
59	C₁₃×D₈	C₂²×C₄	1	8	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	#41	1	2	yes	—
63	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²⋊Dic₁₃	—	4	2	no	—
65	C₂²⋊Dic₁₃	—	4	8	no	—
66	C₂²⋊Dic₁₃	D₈	1	2	yes	—
67	C₄⋊Dic₁₃	—	2	4	no	—
68	C₄⋊Dic₁₃	D₈	1	2	yes	—
69	C₄×Dic₁₃	—	4	2	no	—
70	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
71	#6099	—	2	2	no	—
72	C₂×#1055	C₂×D₄	1	2	yes	—
73	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
74	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
75	#6090	—	2	2	no	—
76	C₂²×D₁₆	D₁₃	1	1	yes	characteristic
77	#2948	—	13	2	no	—
78	#2177	—	13	4	no	—
79	C₄×D₁₆	—	13	4	no	—
80	#2158	—	13	1	no	—
81	#2140	—	13	2	no	—
82	#2108	—	13	1	no	—
83	C₂³×C₁₃	—	4	4	no	—
84	C₂³×C₁₃	—	4	8	no	—
85	C₂³×C₁₃	—	16	4	no	—
86	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
87	D₄×C₁₃	—	2	8	no	—
88	D₄×C₁₃	—	4	4	no	—
89	D₄×C₁₃	—	4	8	no	—
90	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#135	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₁₃	—	4	4	no	—
95	C₂×Dic₁₃	—	16	8	no	—
96	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
97	C₈×C₁₃	C₄×D₄	1	2	yes	—
98	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
100	#1055	—	2	2	no	—
101	C₂²×D₈	C₂×D₁₃	1	2	yes	—
102	#518	—	26	2	no	—
103	C₂×D₁₆	Dic₁₃	1	8	yes	complemented
104	C₂×D₁₆	C₂×D₁₃	1	2	yes	—
105	C₂×D₁₆	C₂×D₁₃	1	2	yes	—
106	C₂×M₃₂	—	13	1	no	—
107	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
108	#442	—	26	4	no	—
109	C₄×D₈	—	13	8	no	—
110	#427	—	13	2	no	—
111	#418	—	26	2	no	—
112	#405	—	13	1	no	—
113	#367	—	13	2	no	—
114	#366	—	13	2	no	—
115	#357	—	13	8	no	—
116	#346	—	13	2	no	—
117	C₄×C₁₆	—	13	2	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	4	8	no	—
120	C₂²×C₁₃	—	8	8	no	—
121	C₂²×C₁₃	—	16	8	no	—
122	C₂²×C₁₃	C₄×D₈	1	2	yes	—
123	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₁₃	C₄×D₈	1	2	yes	—
125	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
127	Dic₁₃	—	2	4	no	—
128	Dic₁₃	—	8	4	no	—
129	C₂²×D₄	—	2	2	no	—
130	C₂×D₈	—	2	2	no	—
131	C₂×D₈	—	2	4	no	—
132	C₂×D₈	C₂×Dic₁₃	1	8	yes	complemented
133	C₂×D₈	C₂²×D₁₃	1	4	yes	—
134	C₂×M₁₆	—	13	1	no	—
135	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₄×D₄	—	26	8	no	—
137	C₄×D₄	—	52	4	no	—
138	#117	—	13	1	no	—
139	#117	—	26	2	no	—
140	C₂×C₂²⋊C₄	—	52	4	no	—
141	D₁₆	C₂×Dic₁₃	1	8	yes	complemented
142	D₁₆	C₂×Dic₁₃	1	8	yes	complemented
143	M₃₂	—	26	2	no	—
144	C₂×C₁₆	—	13	2	no	—
145	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	complemented
146	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
148	#107	—	13	4	no	—
149	#106	—	26	2	no	—
150	#102	—	26	8	no	—
151	#97	—	13	2	no	—
152	C₄×C₈	—	13	2	no	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	—	8	8	no	—
155	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
156	C₂×C₁₃	#3415	1	2	yes	—
157	C₂⁴	—	4	2	no	—
158	C₂×D₄	—	2	8	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	4	no	—
162	C₂²×C₄	—	52	8	no	—
163	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
164	D₈	—	2	8	no	—
165	D₈	—	2	4	no	—
166	D₈	C₂²×Dic₁₃	1	8	yes	—
167	M₁₆	—	26	2	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
170	C₂×C₈	#6129	1	2	yes	—
171	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
172	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
173	C₄⋊C₄	—	13	2	no	—
174	C₄⋊C₄	—	26	4	no	—
175	C₂²⋊C₄	—	13	2	no	—
176	C₂²⋊C₄	—	52	8	no	—
177	C₂²⋊C₄	—	52	2	no	—
178	C₄²	—	13	2	no	—
179	C₄²	—	52	2	no	—
180	C₁₆	—	26	2	no	—
181	C₁₆	C₂²×Dic₁₃	1	2	yes	—
182	C₁₆	C₂²×Dic₁₃	1	2	yes	—
183	C₁₃	#33456	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	D₈×D₁₃	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	4	no	—
190	D₄	—	4	8	no	—
191	C₂×C₄	—	13	4	no	—
192	C₂×C₄	—	52	4	no	—
193	C₂×C₄	—	208	8	no	—
194	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
195	C₂×C₄	#88166	1	2	yes	—
196	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
198	C₈	—	26	2	no	—
199	C₈	D₄×Dic₁₃	1	2	yes	—
200	C₈	C₂×#6129	1	1	yes	characteristic
201	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	D₈×Dic₁₃	1	2	yes	—
207	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
208	C₄	—	26	4	no	—
209	C₄	—	104	4	no	—
210	C₄	D₈×Dic₁₃	1	2	yes	—
211	C₄	C₂×#88166	1	1	yes	characteristic
212	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic
216	C₂	#13127973	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₂²×C₁₃×D₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×D₁₆	D₁₃
hypocenter	C₁₃	#33456
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. #13123688	C₂	semidirect
2. #13116712	C₂	semidirect
3. D₁₆×Dic₁₃	C₂	semidirect
4. #13116620	C₂	semidirect
5. #13116552	C₂	semidirect
6. #13116432	C₂	semidirect
10. C₂×C₁₃×D₁₆	C₄	semidirect
16. D₈×Dic₁₃	C₂²	semidirect
20. #11718154	C₂²	semidirect
21. #11718097	C₂²	semidirect
22. #11718096	C₂²	semidirect
23. #11718094	C₂²	semidirect
24. C₁₆×Dic₁₃	C₂²	semidirect
28. C₂×C₁₃×D₈	C₂×C₄	semidirect
37. C₁₃×D₁₆	C₂×C₄	semidirect
38. C₁₃×D₁₆	C₂×C₄	semidirect
39. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
103. C₂×D₁₆	Dic₁₃	semidirect
132. C₂×D₈	C₂×Dic₁₃	semidirect
141. D₁₆	C₂×Dic₁₃	semidirect
142. D₁₆	C₂×Dic₁₃	semidirect
145. C₂×C₁₆	C₂×Dic₁₃	semidirect
183. C₁₃	#33456	semidirect

Maximal subgroups

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

subgroup	index
#13123688	2
#13116712	2
D₁₆×Dic₁₃	2
#13116620	2
#13116552	2
#13116432	2
C₂²×C₁₃×D₁₆	2
#33456	13