Subgroups — #16789041

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₃⋊C₁₆	C₂	1	2	yes	maximal
2	#13129875	C₂	1	2	yes	maximal, complemented
3	#13129874	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	4	yes	—
9	C₂×D₁₃⋊C₁₆	—	2	4	no	—
10	C₂×D₁₃⋊C₁₆	—	2	4	no	—
11	C₂×D₁₃⋊C₁₆	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₃	—	2	4	no	—
17	C₂²×C₄×D₁₃	C₈	1	2	yes	—
18	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂³	1	1	yes	characteristic
26	C₂×C₈×D₁₃	—	2	2	no	—
27	C₂×C₈×D₁₃	—	2	2	no	—
28	C₂×C₈×D₁₃	C₈	1	4	yes	—
29	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₁₃	D₄	1	2	yes	—
31	C₂×C₈×D₁₃	D₄	1	4	yes	—
32	C₂×C₈×D₁₃	Q₈	1	2	yes	—
33	C₄²×D₁₃	—	2	2	no	—
34	C₄²×D₁₃	—	2	2	no	—
35	C₂×#6092	—	2	4	no	—
36	C₂×#6092	—	2	4	no	—
37	C₂×#6092	C₂×C₄	1	8	yes	complemented
38	D₁₃⋊C₁₆	—	2	16	no	—
39	C₄×C₈×C₁₃	—	2	2	no	—
40	C₈×Dic₁₃	—	2	2	no	—
41	C₈×Dic₁₃	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	C₂×#1374	—	13	1	no	maximal, Sylow
44	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
48	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×D₁₃	—	2	4	no	—
50	C₂×C₄×D₁₃	—	2	4	no	—
51	C₂×C₄×D₁₃	C₁₆	1	8	yes	complemented
52	C₂×C₄×D₁₃	C₄²	1	2	yes	—
53	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
57	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	C₄²	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄×Dic₁₃	—	2	1	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	—	2	4	no	—
69	C₂×#1055	—	2	1	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	C₂×C₈	1	2	yes	—
72	C₂×#1055	C₂²×C₄	1	2	yes	—
73	C₂×#1055	C₂×D₄	1	1	yes	characteristic
74	C₂×#1055	C₂×D₄	1	2	yes	—
75	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
76	C₈×D₁₃	—	2	8	no	—
77	C₈×D₁₃	C₂²⋊C₄	1	8	yes	—
78	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
79	#6092	—	2	8	no	—
80	C₂³×C₁₆	—	13	2	no	—
81	C₂²×C₄×C₈	—	13	1	no	—
82	#1374	—	13	2	no	—
83	#1374	—	13	2	no	—
84	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂²×D₁₃	C₄×C₈	1	2	yes	—
86	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
87	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
88	C₂²×D₁₃	C₂×C₁₆	1	4	yes	—
89	C₂²×D₁₃	M₃₂	1	4	yes	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
93	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
102	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
103	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
104	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	complemented
105	C₄×D₁₃	—	2	16	no	—
106	C₄×D₁₃	#95	1	4	yes	—
107	C₄×D₁₃	#97	1	4	yes	—
108	C₄×D₁₃	C₂²⋊C₈	1	4	yes	—
109	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	#95	1	4	yes	—
112	#1055	—	2	2	no	—
113	#1055	C₂×C₂²⋊C₄	1	2	yes	—
114	#1055	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₈	—	13	1	no	—
116	C₂³×C₈	—	13	1	no	—
117	C₂²×C₄²	—	13	1	no	—
118	C₂²×C₁₆	—	13	4	no	—
119	C₂²×C₁₆	—	13	8	no	—
120	C₂²×C₁₆	—	26	4	no	—
121	C₂²×C₁₆	—	26	4	no	—
122	C₂×C₄×C₈	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂×C₄×C₈	—	13	1	no	—
125	C₂×C₄×C₈	—	26	4	no	—
126	C₂×C₄×C₈	#261	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
131	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
132	C₂×D₁₃	#336	1	4	yes	—
133	C₂×D₁₃	C₄×C₁₆	1	4	yes	—
134	C₂×D₁₃	#346	1	4	yes	—
135	C₂×D₁₃	C₂²⋊C₁₆	1	8	yes	—
136	C₂×D₁₃	C₄⋊C₁₆	1	8	yes	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
140	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
142	Dic₁₃	—	2	4	no	—
143	Dic₁₃	#336	1	4	yes	—
144	C₂³×C₄	—	13	1	no	—
145	C₂³×C₄	—	13	2	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	4	no	—
148	C₂²×C₈	—	13	2	no	—
149	C₂²×C₈	—	13	2	no	—
150	C₂²×C₈	—	13	4	no	—
151	C₂²×C₈	—	13	1	no	—
152	C₂²×C₈	—	13	4	no	—
153	C₂²×C₈	—	26	2	no	—
154	C₂²×C₈	—	26	2	no	—
155	C₂²×C₈	C₂×#261	1	1	yes	characteristic
156	C₂²×C₈	C₂×#261	1	1	yes	characteristic
157	C₂×C₄²	—	13	2	no	—
158	C₂×C₄²	—	13	1	no	—
159	C₂×C₄²	—	26	2	no	—
160	C₂×C₄²	—	26	2	no	—
161	C₂×C₄²	C₂×#261	1	1	yes	characteristic
162	C₂×C₁₆	—	13	8	no	—
163	C₂×C₁₆	—	26	16	no	—
164	C₂×C₁₆	—	26	4	no	—
165	C₂×C₁₆	—	26	4	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	26	2	no	—
168	C₄×C₈	—	26	2	no	—
169	C₄×C₈	—	26	2	no	—
170	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
171	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
172	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
173	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	2	yes	—
174	C₂×C₁₃	C₂×C₄⋊C₁₆	1	2	yes	—
175	D₁₃	#1374	1	8	yes	complemented
176	C₂⁴	—	13	1	no	—
177	C₂²×C₄	—	13	1	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	2	no	—
181	C₂²×C₄	—	13	2	no	—
182	C₂²×C₄	—	13	2	no	—
183	C₂²×C₄	—	13	2	no	—
184	C₂²×C₄	—	13	2	no	—
185	C₂²×C₄	—	13	8	no	—
186	C₂²×C₄	—	26	4	no	—
187	C₂²×C₄	—	26	4	no	—
188	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
189	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	1	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	8	no	—
197	C₂×C₈	—	13	8	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	26	8	no	—
200	C₂×C₈	—	26	1	no	—
201	C₂×C₈	—	26	1	no	—
202	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
203	C₂×C₈	C₄×#261	1	2	yes	—
204	C₂×C₈	C₄⋊#261	1	2	yes	—
205	C₂×C₈	C₂²⋊#261	1	2	yes	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	26	4	no	—
209	C₄²	—	26	1	no	—
210	C₄²	—	26	1	no	—
211	C₁₆	—	26	8	no	—
212	C₁₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	13	2	no	—
214	C₂³	—	13	2	no	—
215	C₂³	—	13	4	no	—
216	C₂³	—	13	4	no	—
217	C₂³	—	13	2	no	—
218	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	13	2	no	—
222	C₂×C₄	—	13	2	no	—
223	C₂×C₄	—	13	2	no	—
224	C₂×C₄	—	13	4	no	—
225	C₂×C₄	—	13	4	no	—
226	C₂×C₄	—	13	4	no	—
227	C₂×C₄	—	13	4	no	—
228	C₂×C₄	—	13	4	no	—
229	C₂×C₄	—	26	16	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	D₁₃⋊C₁₆	1	4	yes	complemented
233	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
234	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	13	2	no	—
241	C₈	—	13	2	no	—
242	C₈	—	26	2	no	—
243	C₈	(C₂×#261)⋊C₄	1	4	yes	—
244	C₂²	—	13	4	no	—
245	C₂²	—	13	4	no	—
246	C₂²	—	13	4	no	—
247	C₂²	—	13	8	no	—
248	C₂²	—	13	8	no	—
249	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
250	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
251	C₂²	C₂×D₁₃⋊C₁₆	1	2	yes	—
252	C₂²	D₁₃⋊M₃₂	1	2	yes	—
253	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	13	4	no	—
256	C₄	—	26	4	no	—
257	C₄	#11719047	1	1	yes	characteristic
258	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
259	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
260	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
261	C₂	—	13	8	no	—
262	C₂	#13130497	1	1	yes	characteristic
263	C₂	C₄×D₁₃⋊C₁₆	1	1	yes	characteristic
264	C₂	#13131037	1	1	yes	characteristic
265	C₂	D₁₃⋊(C₂²⋊C₁₆)	1	2	yes	—
266	C₂	D₁₃⋊(C₄⋊C₁₆)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1374
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129875	C₂	semidirect
3. #13129874	C₂	semidirect
11. C₂×D₁₃⋊C₁₆	C₄	semidirect
37. C₂×#6092	C₂×C₄	semidirect
51. C₂×C₄×D₁₃	C₁₆	semidirect
92. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
104. C₂×Dic₁₃	C₂×C₁₆	semidirect
175. D₁₃	#1374	semidirect
212. C₁₃	C₂×#1374	semidirect
232. C₂×C₄	D₁₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₃⋊C₁₆	2
#13129875	2
#13129874	2
C₂×C₄×C₈×D₁₃	2
C₂×#1374	13