Subgroups — #16789603

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#261	C₂	1	2	yes	maximal
3	C₈×C₄⋊#261	C₂	1	1	yes	maximal, characteristic
4	(C₂×#261)⋊C₁₆	C₂	1	2	yes	maximal
5	D₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#261	C₂²	1	2	yes	—
7	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊#261	C₄	1	1	yes	characteristic
10	D₁₃⋊(C₄⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
12	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	(C₂×#261)⋊C₈	C₄	1	2	yes	—
14	#11718257	C₄	1	2	yes	complemented
15	C₄×#6092	C₄	1	2	yes	complemented
16	C₁₆×#261	—	2	4	no	—
17	C₁₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
18	#11718093	C₄	1	2	yes	complemented
19	C₄⋊#6090	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄⋊#261	C₈	1	2	yes	—
21	C₂×C₄×#261	C₂×C₄	1	2	yes	—
22	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
25	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
26	(C₂×#261)⋊C₄	C₈	1	2	yes	—
27	#88112	C₈	1	2	yes	—
28	C₄⋊#1057	C₈	1	2	yes	—
29	C₂×#6092	C₂×C₄	1	2	yes	—
30	C₂×#6092	C₂×C₄	1	2	yes	—
31	C₈×#261	—	2	2	no	—
32	C₈×#261	—	2	2	no	—
33	D₁₃⋊C₁₆	—	2	1	no	—
34	D₁₃⋊C₁₆	—	2	1	no	—
35	D₁₃⋊C₁₆	D₄	1	2	yes	—
36	D₁₃⋊C₁₆	Q₈	1	2	yes	—
37	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#6090	C₂×C₄	1	2	yes	—
42	C₄×#1055	C₂×C₄	1	1	yes	characteristic
43	C₁₆×D₁₃	—	2	2	no	—
44	C₁₆×D₁₃	—	2	2	no	—
45	C₁₆×C₄⋊C₄	—	13	1	no	maximal, Sylow
46	C₂²×#261	C₂×C₈	1	4	yes	—
47	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×#1057	C₂×C₈	1	4	yes	—
51	C₄⋊#261	C₁₆	1	8	yes	complemented
52	C₄×#261	—	2	2	no	—
53	C₄×#261	—	2	2	no	—
54	D₁₃⋊C₈	—	2	2	no	—
55	D₁₃⋊C₈	—	2	2	no	—
56	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₄²	1	1	yes	characteristic
59	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
60	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
61	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂×#1055	C₄²	1	1	yes	characteristic
63	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
64	C₈×D₁₃	—	2	1	no	—
65	C₈×D₁₃	—	2	1	no	—
66	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
67	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₈×D₁₃	#41	1	1	yes	characteristic
69	C₈×D₁₃	#41	1	1	yes	characteristic
70	#6092	—	2	2	no	—
71	#6092	C₄⋊C₄	1	4	yes	complemented
72	C₁₃×C₁₆	—	2	2	no	—
73	#6090	—	2	2	no	—
74	C₂×C₄⋊C₁₆	—	13	1	no	—
75	C₂×C₄×C₁₆	—	13	1	no	—
76	C₂×C₄×C₁₆	—	13	2	no	—
77	C₈×C₄⋊C₄	—	13	1	no	—
78	#1374	—	13	2	no	—
79	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×#261	—	2	4	no	—
81	C₂×#261	—	2	4	no	—
82	C₂×#261	C₂×C₁₆	1	8	yes	—
83	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
87	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
88	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	C₂×C₁₆	1	4	yes	—
91	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
92	C₄×D₁₃	C₄×Q₈	1	1	yes	characteristic
93	C₄×D₁₃	#131	1	1	yes	characteristic
94	C₄×D₁₃	#131	1	1	yes	characteristic
95	#1057	—	2	4	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
99	#1055	—	2	1	no	—
100	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
101	#1055	#117	1	1	yes	characteristic
102	C₂²×C₁₆	—	13	1	no	—
103	C₂²×C₁₆	—	13	1	no	—
104	C₂²×C₁₆	—	13	2	no	—
105	C₂×C₄⋊C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	2	no	—
108	C₄×C₄⋊C₄	—	13	1	no	—
109	C₄⋊C₁₆	—	13	2	no	—
110	C₄⋊C₁₆	—	13	1	no	—
111	C₄⋊C₁₆	#261	1	1	yes	characteristic, complemented
112	C₄×C₁₆	—	13	2	no	—
113	C₄×C₁₆	—	13	2	no	—
114	C₄×C₁₆	—	26	4	no	—
115	#336	—	13	2	no	—
116	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
118	C₂×D₁₃	C₈×Q₈	1	1	yes	characteristic
119	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
120	C₂×D₁₃	#504	1	2	yes	—
121	#261	—	2	8	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	C₄×C₁₆	1	2	yes	complemented
124	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₁₃	#405	1	1	yes	characteristic
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	C₄×C₁₆	1	2	yes	complemented
128	Dic₁₃	C₈×D₄	1	1	yes	characteristic
129	Dic₁₃	C₈×Q₈	1	1	yes	characteristic
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	2	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂×C₄⋊C₄	—	13	2	no	—
134	C₂×C₄²	—	13	1	no	—
135	C₂×C₄²	—	13	2	no	—
136	C₂×C₁₆	—	13	2	no	—
137	C₂×C₁₆	—	13	2	no	—
138	C₂×C₁₆	—	13	2	no	—
139	C₂×C₁₆	—	13	2	no	—
140	C₂×C₁₆	—	13	2	no	—
141	C₂×C₁₆	—	26	2	no	—
142	C₂×C₁₆	—	26	2	no	—
143	C₂×C₁₆	—	26	1	no	—
144	C₂×C₁₆	—	26	1	no	—
145	C₂×C₁₆	C₂×#261	1	2	yes	—
146	C₄⋊C₈	—	13	2	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₄×C₈	—	13	1	no	—
149	C₄×C₈	—	13	1	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	—	26	2	no	—
152	C₄×C₈	—	26	2	no	—
153	C₄×C₈	C₂×#261	1	1	yes	characteristic
154	#95	—	13	2	no	—
155	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
157	C₂×C₁₃	#2108	1	1	yes	characteristic
158	D₁₃	D₄×C₁₆	1	2	yes	—
159	D₁₃	Q₈×C₁₆	1	2	yes	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	4	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂×C₈	—	13	1	no	—
165	C₂×C₈	—	13	1	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	4	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	26	1	no	—
172	C₂×C₈	—	26	1	no	—
173	C₂×C₈	—	26	2	no	—
174	C₂×C₈	—	26	2	no	—
175	C₂×C₈	C₄×#261	1	1	yes	characteristic
176	C₂×C₈	C₂²×#261	1	1	yes	characteristic
177	C₄⋊C₄	—	13	8	no	—
178	C₄²	—	13	1	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	26	2	no	—
182	C₄²	—	26	2	no	—
183	C₄²	C₄×#261	1	1	yes	characteristic
184	C₁₆	—	2	2	no	—
185	C₁₆	—	13	4	no	—
186	C₁₆	—	26	2	no	—
187	C₁₆	—	26	2	no	—
188	C₁₃	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	13	1	no	—
190	C₂×C₄	—	13	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	4	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	8	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	26	4	no	—
201	C₂×C₄	—	26	4	no	—
202	C₂×C₄	C₈×#261	1	1	yes	characteristic
203	C₂×C₄	C₈×#261	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	13	1	no	—
207	C₈	—	13	1	no	—
208	C₈	—	26	4	no	—
209	C₈	—	26	1	no	—
210	C₈	D₄×#261	1	1	yes	characteristic
211	C₈	Q₈×#261	1	1	yes	characteristic
212	C₂²	—	13	2	no	—
213	C₂²	—	13	2	no	—
214	C₂²	—	13	1	no	—
215	C₂²	—	13	1	no	—
216	C₂²	C₂×C₈×#261	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	13	2	no	—
219	C₄	—	13	1	no	—
220	C₄	—	13	1	no	—
221	C₄	—	26	8	no	—
222	C₄	—	26	1	no	—
223	C₄	C₁₆×#261	1	2	yes	complemented
224	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
225	C₄	#11719099	1	1	yes	characteristic
226	C₂	—	13	2	no	—
227	C₂	—	13	2	no	—
228	C₂	#13130629	1	1	yes	characteristic
229	C₂	C₂×C₁₆×#261	1	1	yes	characteristic
230	C₂	#13131048	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₁₃×C₄⋊C₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₄⋊C₁₆	#261
hypocenter	C₁₃	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
14. #11718257	C₄	semidirect
15. C₄×#6092	C₄	semidirect
17. C₁₃×C₄⋊C₁₆	C₄	semidirect
18. #11718093	C₄	semidirect
19. C₄⋊#6090	C₄	semidirect
51. C₄⋊#261	C₁₆	semidirect
71. #6092	C₄⋊C₄	semidirect
111. C₄⋊C₁₆	#261	semidirect
123. C₄×C₁₃	C₄×C₁₆	semidirect
127. Dic₁₃	C₄×C₁₆	semidirect
188. C₁₃	C₁₆×C₄⋊C₄	semidirect
223. C₄	C₁₆×#261	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₃⋊C₁₆	2
C₂×C₁₆×#261	2
C₈×C₄⋊#261	2
(C₂×#261)⋊C₁₆	2
D₁₃×C₄⋊C₁₆	2
C₁₆×C₄⋊C₄	13