Subgroups — #16016109

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718562	C₂	1	2	yes	maximal, complemented
2	#13123549	C₂	1	1	yes	maximal, characteristic
3	#13116669	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₅₂	C₂	1	4	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116449	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	C₂²	1	2	yes	—
9	#11719182	C₄	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718771	C₄	1	2	yes	—
14	C₂×#88052	C₂²	1	2	yes	—
15	#11718585	C₄	1	1	yes	characteristic
16	#11718562	—	2	8	no	—
17	C₂×#88038	C₂²	1	4	yes	—
18	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
19	#11718349	C₄	1	4	yes	—
20	C₈×D₅₂	C₂²	1	4	yes	—
21	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
22	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
23	#11718099	C₂²	1	8	yes	—
24	C₄⋊#6090	C₂²	1	4	yes	—
25	C₂×#6127	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#88184	C₈	1	4	yes	—
30	C₂×#6099	C₂×C₄	1	2	yes	—
31	#88155	—	2	4	no	—
32	#88155	—	2	4	no	—
33	C₂×#6094	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₃	C₂³	1	4	yes	—
35	#88129	C₈	1	2	yes	—
36	#88128	C₈	1	1	yes	characteristic
37	C₄×D₅₂	C₂×C₄	1	4	yes	—
38	#88123	C₈	1	1	yes	characteristic
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
41	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₃	C₂³	1	2	yes	—
46	#88060	C₂×C₄	1	8	yes	—
47	#88052	—	2	4	no	—
48	C₂×#6090	C₂³	1	4	yes	—
49	C₄⋊#1055	C₂×C₄	1	4	yes	—
50	#88038	—	2	8	no	—
51	C₁₆×D₁₃	—	2	8	no	—
52	#33434	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
56	#6127	—	2	4	no	—
57	#6127	—	2	4	no	—
58	C₂×D₅₂	C₂×C₈	1	4	yes	—
59	C₂×D₅₂	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
61	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₃	#41	1	2	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
71	#6103	C₂×C₈	1	8	yes	—
72	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
73	#6099	—	2	4	no	—
74	C₂×#1055	C₂²×C₄	1	4	yes	—
75	#6094	—	2	8	no	—
76	C₈×D₁₃	—	2	4	no	—
77	C₈×D₁₃	—	2	4	no	—
78	C₁₃×C₁₆	—	2	2	no	—
79	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
80	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
81	#6090	—	2	4	no	—
82	C₂×#504	—	13	2	no	—
83	#2931	—	13	1	no	—
84	#2169	—	13	4	no	—
85	D₄×C₁₆	—	13	4	no	—
86	#2150	—	13	1	no	—
87	#2113	—	13	2	no	—
88	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
89	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₁₃	#131	1	2	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂²⋊D₁₃	—	2	8	no	—
99	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
100	D₅₂	—	2	4	no	—
101	D₅₂	—	2	8	no	—
102	C₄×D₁₃	—	2	4	no	—
103	C₄×D₁₃	—	2	4	no	—
104	Dic₂₆	—	2	4	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	C₄×D₄	1	2	yes	—
107	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
109	#1055	—	2	4	no	—
110	C₂×#131	—	13	2	no	—
111	#532	—	13	1	no	—
112	#504	—	26	8	no	—
113	C₂×M₃₂	—	13	2	no	—
114	C₂×M₃₂	—	13	4	no	—
115	C₂²×C₁₆	—	13	4	no	—
116	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
117	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
118	#436	—	13	4	no	—
119	C₈×D₄	—	13	4	no	—
120	#423	—	13	1	no	—
121	#408	—	13	2	no	—
122	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₄⋊C₁₆	—	13	4	no	—
124	C₂²⋊C₁₆	—	13	8	no	—
125	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
126	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	C₈×D₄	1	2	yes	—
129	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
130	C₂×D₁₃	—	2	8	no	—
131	C₂×D₁₃	—	2	4	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	C₈×D₄	1	2	yes	—
134	C₄×C₁₃	#504	1	4	yes	complemented
135	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
137	Dic₁₃	—	2	4	no	—
138	C₂×#41	—	13	2	no	—
139	#131	—	26	4	no	—
140	#131	—	26	4	no	—
141	C₂×M₁₆	—	13	2	no	—
142	C₂×M₁₆	—	13	4	no	—
143	C₂²×C₈	—	13	4	no	—
144	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
146	C₄⋊Q₈	—	13	1	no	—
147	#127	—	13	1	no	—
148	#124	—	13	2	no	—
149	#121	—	13	4	no	—
150	C₄×D₄	—	13	4	no	—
151	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
152	M₃₂	—	26	4	no	—
153	M₃₂	—	26	8	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	13	4	no	—
156	C₂×C₁₆	—	26	8	no	—
157	C₂×C₁₆	D₅₂	1	4	yes	complemented
158	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
159	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
160	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
161	C₄⋊C₈	—	13	4	no	—
162	C₂²⋊C₈	—	13	8	no	—
163	C₄×C₈	C₄×D₁₃	1	2	yes	—
164	C₄×C₈	C₂²×D₁₃	1	2	yes	—
165	C₂×C₁₃	—	2	2	no	—
166	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#504	1	2	yes	—
168	D₁₃	—	4	4	no	—
169	#41	—	26	4	no	—
170	#41	—	26	4	no	—
171	C₂×Q₈	—	13	2	no	—
172	C₂×D₄	—	13	4	no	—
173	C₂×D₄	—	13	4	no	—
174	C₂×D₄	—	13	2	no	—
175	C₂²×C₄	—	13	4	no	—
176	C₂²×C₄	C₈×D₁₃	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
178	M₁₆	—	26	4	no	—
179	M₁₆	—	26	8	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	13	4	no	—
182	C₂×C₈	—	26	4	no	—
183	C₂×C₈	—	26	4	no	—
184	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂×D₅₂	1	2	yes	—
187	C₂×C₈	#6127	1	2	yes	—
188	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
189	C₄⋊C₄	—	13	4	no	—
190	C₂²⋊C₄	—	13	8	no	—
191	C₄²	C₈×D₁₃	1	1	yes	characteristic
192	C₄²	C₈×D₁₃	1	1	yes	characteristic
193	C₄²	C₂×C₄×D₁₃	1	2	yes	—
194	C₁₆	—	2	2	no	—
195	C₁₆	—	26	4	no	—
196	C₁₆	C₂×D₅₂	1	2	yes	—
197	C₁₆	C₂×D₅₂	1	2	yes	—
198	C₁₃	#33434	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	13	4	no	—
200	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
201	Q₈	—	26	4	no	—
202	D₄	—	26	8	no	—
203	D₄	—	26	8	no	—
204	D₄	—	26	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	13	4	no	—
207	C₂×C₄	—	26	4	no	—
208	C₂×C₄	—	26	4	no	—
209	C₂×C₄	C₄×D₅₂	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
213	C₂×C₄	#88155	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	26	4	no	—
217	C₈	C₄×D₅₂	1	2	yes	—
218	C₈	C₂²×D₅₂	1	1	yes	characteristic
219	C₈	C₂×#6127	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	26	4	no	—
222	C₂²	—	26	8	no	—
223	C₂²	C₈×D₅₂	1	2	yes	—
224	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	26	4	no	—
227	C₄	C₈×D₅₂	1	2	yes	—
228	C₄	#11718562	1	4	yes	complemented
229	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
230	C₄	C₂×#88155	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	52	4	no	—
233	C₂	C₂×C₈×D₅₂	1	1	yes	characteristic
234	C₂	C₂×#11718562	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
prime radical	#11719182	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#33434
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11718562	C₂	semidirect
3. #13116669	C₂	semidirect
4. C₁₆×D₅₂	C₂	semidirect
5. #13116592	C₂	semidirect
6. #13116449	C₂	semidirect
7. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
40. C₂×C₁₃×C₁₆	D₄	semidirect
88. C₂×C₄×C₁₆	D₁₃	semidirect
134. C₄×C₁₃	#504	semidirect
157. C₂×C₁₆	D₅₂	semidirect
198. C₁₃	#33434	semidirect
228. C₄	#11718562	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718562	2
#13123549	2
#13116669	2
C₁₆×D₅₂	2
#13116592	2
#13116449	2
C₂×C₄×C₁₃×C₁₆	2
#33434	13