Subgroups — C₂×#41×#1057 #16808451

Order 3328.

2566 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#1057	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#1057	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#1057	C₂	1	6	yes	maximal, complemented
4	C₂×#11719119	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×#1057	C₂	1	3	yes	maximal, complemented
6	C₂×#41×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×#41	C₄	1	1	yes	characteristic
8	C₂³×#1057	C₂²	1	6	yes	—
9	#41×Dic₁₃	C₄	1	8	yes	—
10	#41×Dic₁₃	C₂²	1	8	yes	—
11	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×Dic₁₃	C₄	1	3	yes	—
14	C₂×D₄×Dic₁₃	C₂²	1	3	yes	—
15	C₂×#88180	C₄	1	3	yes	—
16	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
17	C₂×C₂²⋊#1057	C₂²	1	6	yes	—
18	Q₈×#1057	C₂²	1	16	yes	complemented
19	D₄×#1057	C₂²	1	48	yes	complemented
20	#11719119	C₂²	1	24	yes	complemented
21	C₂×C₄⋊#1057	C₂²	1	6	yes	—
22	C₂×C₄×#1057	C₂²	1	6	yes	—
23	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
24	C₂×C₄×#1057	C₂²	1	3	yes	—
25	C₂×C₄×#1057	C₂²	1	24	yes	complemented
26	C₂×C₁₃×#41	C₈	1	16	yes	complemented
27	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₃	C₂×C₄	1	3	yes	—
30	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
31	C₂³×Dic₁₃	C₂×C₄	1	3	yes	—
32	C₂³×Dic₁₃	C₂³	1	3	yes	—
33	C₂²×#1057	—	2	24	no	—
34	C₂²×#1057	C₂³	1	6	yes	—
35	C₂²×#1057	C₂³	1	24	yes	—
36	C₂²×#1057	C₂³	1	2	yes	—
37	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	6	yes	—
38	Q₈×Dic₁₃	C₂×C₄	1	8	yes	—
39	Q₈×Dic₁₃	C₂×C₄	1	4	yes	—
40	Q₈×Dic₁₃	C₂³	1	4	yes	—
41	D₄×Dic₁₃	C₂×C₄	1	24	yes	—
42	D₄×Dic₁₃	C₂×C₄	1	12	yes	—
43	D₄×Dic₁₃	C₂³	1	12	yes	—
44	#88180	C₂×C₄	1	24	yes	—
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
50	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
53	C₂²⋊#1057	C₂³	1	24	yes	—
54	C₄⋊#1057	C₂³	1	24	yes	—
55	C₄×#1057	C₂³	1	4	yes	—
56	C₄×#1057	C₂³	1	12	yes	—
57	C₄×#1057	C₂³	1	24	yes	—
58	C₂×C₈×#41	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	C₂²×C₄	1	3	yes	—
60	C₁₃×#41	C₂×C₈	1	64	yes	complemented
61	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	complemented
62	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂×D₄×C₁₃	C₂×C₈	1	24	yes	complemented
64	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₁₃	C₂×C₈	1	24	yes	complemented
66	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	12	no	—
70	C₂²×Dic₁₃	—	2	12	no	—
71	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
72	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
73	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
74	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
75	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂⁴	1	6	yes	—
77	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×#1057	—	2	12	no	—
79	C₂×#1057	—	2	48	no	—
80	C₂×#1057	#41	1	8	yes	direct product factor
81	C₂×#1057	C₂⁴	1	12	yes	—
82	C₂×#1057	C₂⁴	1	4	yes	—
83	C₂²⋊Dic₁₃	C₂²×C₄	1	24	yes	—
84	C₄⋊Dic₁₃	C₂²×C₄	1	12	yes	—
85	C₄⋊Dic₁₃	C₂²×C₄	1	12	yes	—
86	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
88	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	6	yes	—
90	C₄×Dic₁₃	C₂⁴	1	2	yes	—
91	C₄×Dic₁₃	C₂⁴	1	6	yes	—
92	C₂×C₄×#41	—	13	1	no	—
93	C₈×#41	—	13	16	no	—
94	C₂×C₈×Q₈	—	13	2	no	—
95	C₂×C₈×D₄	—	13	6	no	—
96	C₂×#431	—	13	3	no	—
97	C₂²×C₄×C₈	—	13	3	no	—
98	C₂³×C₁₃	—	2	6	no	—
99	C₂³×C₁₃	—	2	6	no	—
100	C₂³×C₁₃	C₂²×C₈	1	12	yes	—
101	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
102	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
103	Q₈×C₁₃	C₂²×C₈	1	16	yes	complemented
104	D₄×C₁₃	C₂²×C₈	1	48	yes	complemented
105	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
107	C₂×C₄×C₁₃	C₂²×C₈	1	48	yes	complemented
108	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
110	C₂×Dic₁₃	—	2	6	no	—
111	C₂×Dic₁₃	—	2	6	no	—
112	C₂×Dic₁₃	—	2	12	no	—
113	C₂×Dic₁₃	—	2	24	no	—
114	C₂×Dic₁₃	—	2	12	no	—
115	C₂×Dic₁₃	#131	1	4	yes	—
116	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
117	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
118	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
120	C₂×Dic₁₃	C₂×#41	1	4	yes	—
121	C₂×Dic₁₃	C₂⁵	1	1	yes	characteristic
122	#1057	—	2	12	no	—
123	#1057	C₂×#41	1	8	yes	direct product factor
124	C₂²×#41	#261	1	1	yes	characteristic
125	C₂³×C₈	—	13	6	no	—
126	C₄×#41	—	13	8	no	—
127	C₄×#41	—	13	8	no	—
128	C₂×C₄×Q₈	—	13	1	no	—
129	C₂×C₄×Q₈	—	13	1	no	—
130	C₂×C₄×D₄	—	13	3	no	—
131	C₂×C₄×D₄	—	13	3	no	—
132	C₂×#117	—	13	3	no	—
133	C₂²×C₄²	—	13	3	no	—
134	C₈×Q₈	—	13	16	no	—
135	C₈×D₄	—	13	48	no	—
136	#431	—	13	24	no	—
137	C₂×C₄⋊C₈	—	13	6	no	—
138	C₂×C₂²⋊C₈	—	13	6	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	C₂×C₄×C₈	—	13	3	no	—
141	C₂×C₄×C₈	—	13	6	no	—
142	C₂×C₄×C₈	—	13	24	no	—
143	C₂²×C₁₃	—	2	6	no	—
144	C₂²×C₁₃	—	2	24	no	—
145	C₂²×C₁₃	—	2	6	no	—
146	C₂²×C₁₃	C₄×#41	1	2	yes	—
147	C₂²×C₁₃	C₂³×C₈	1	12	yes	—
148	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
149	C₂²×C₁₃	C₂×#131	1	2	yes	—
150	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
151	C₄×C₁₃	C₂³×C₈	1	4	yes	—
152	C₄×C₁₃	C₂³×C₈	1	12	yes	—
153	Dic₁₃	—	2	6	no	—
154	Dic₁₃	—	2	6	no	—
155	Dic₁₃	C₄×#41	1	2	yes	—
156	Dic₁₃	C₂×#131	1	4	yes	—
157	Dic₁₃	C₂²×#41	1	2	yes	—
158	C₂×#41	#1057	1	16	yes	direct product factor
159	C₂×#41	C₂×#261	1	8	yes	—
160	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
161	C₂²×D₄	C₂×#261	1	3	yes	—
162	C₂³×C₄	—	13	3	no	—
163	C₂³×C₄	—	13	3	no	—
164	C₂³×C₄	C₂×#261	1	3	yes	—
165	C₂²×C₈	—	13	2	no	—
166	C₂²×C₈	—	13	6	no	—
167	C₂²×C₈	—	13	24	no	—
168	C₂²×C₈	—	26	24	no	—
169	C₄×Q₈	—	13	8	no	—
170	C₄×Q₈	—	13	4	no	—
171	C₄×Q₈	—	13	4	no	—
172	C₄×D₄	—	13	24	no	—
173	C₄×D₄	—	13	12	no	—
174	C₄×D₄	—	13	12	no	—
175	#117	—	13	24	no	—
176	C₂×C₄⋊C₄	—	13	3	no	—
177	C₂×C₄⋊C₄	—	13	3	no	—
178	C₂×C₂²⋊C₄	—	13	6	no	—
179	C₂×C₄²	—	13	3	no	—
180	C₂×C₄²	—	13	3	no	—
181	C₂×C₄²	—	13	12	no	—
182	C₂×C₄²	—	13	12	no	—
183	C₂×C₄²	—	13	3	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₄⋊C₈	—	13	24	no	—
186	C₂²⋊C₈	—	13	24	no	—
187	C₄×C₈	—	13	4	no	—
188	C₄×C₈	—	13	12	no	—
189	C₄×C₈	—	13	24	no	—
190	C₂×C₁₃	—	2	12	no	—
191	C₂×C₁₃	C₈×#41	1	4	yes	complemented
192	C₂×C₁₃	C₂×C₄×#41	1	1	yes	characteristic
193	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
194	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
195	C₂⁴	C₂²×#261	1	3	yes	—
196	#41	C₂×#1057	1	64	yes	direct product factor
197	C₂×Q₈	C₂×#1057	1	8	yes	complemented
198	C₂×Q₈	C₂²×#261	1	4	yes	—
199	C₂×D₄	C₂×#1057	1	24	yes	complemented
200	C₂×D₄	C₂²×#261	1	12	yes	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	6	no	—
204	C₂²×C₄	—	13	3	no	—
205	C₂²×C₄	—	13	3	no	—
206	C₂²×C₄	—	13	6	no	—
207	C₂²×C₄	—	13	12	no	—
208	C₂²×C₄	—	26	12	no	—
209	C₂²×C₄	—	26	12	no	—
210	C₂²×C₄	C₂×#1057	1	24	yes	complemented
211	C₂²×C₄	C₂²×#261	1	3	yes	—
212	C₂²×C₄	C₂²×#261	1	12	yes	—
213	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
214	C₂×C₈	—	13	4	no	—
215	C₂×C₈	—	13	8	no	—
216	C₂×C₈	—	13	12	no	—
217	C₂×C₈	—	26	48	no	—
218	C₂×C₈	—	26	12	no	—
219	C₄⋊C₄	—	13	12	no	—
220	C₄⋊C₄	—	13	12	no	—
221	C₂²⋊C₄	—	13	24	no	—
222	C₄²	—	13	2	no	—
223	C₄²	—	13	6	no	—
224	C₄²	—	13	2	no	—
225	C₄²	—	13	12	no	—
226	C₄²	—	13	12	no	—
227	C₄²	—	13	6	no	—
228	C₁₃	C₂×C₈×#41	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	2	6	no	—
230	C₂³	—	2	6	no	—
231	C₂³	C₂²×#1057	1	12	yes	—
232	C₂³	C₂³×#261	1	6	yes	—
233	C₂³	C₂³×#261	1	1	yes	characteristic
234	Q₈	C₂²×#1057	1	16	yes	complemented
235	D₄	C₂²×#1057	1	48	yes	complemented
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	1	no	—
239	C₂×C₄	—	13	1	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	6	no	—
242	C₂×C₄	—	13	6	no	—
243	C₂×C₄	—	26	12	no	—
244	C₂×C₄	—	26	12	no	—
245	C₂×C₄	—	26	24	no	—
246	C₂×C₄	—	26	6	no	—
247	C₂×C₄	—	26	6	no	—
248	C₂×C₄	C₂²×#1057	1	4	yes	—
249	C₂×C₄	C₂²×#1057	1	12	yes	—
250	C₂×C₄	C₂²×#1057	1	48	yes	complemented
251	C₂×C₄	C₂³×#261	1	2	yes	—
252	C₂×C₄	C₂³×#261	1	6	yes	—
253	C₈	—	13	8	no	—
254	C₈	—	26	12	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	2	6	no	—
257	C₂²	—	2	24	no	—
258	C₂²	D₁₃⋊#131	1	2	yes	—
259	C₂²	#41×#261	1	2	yes	—
260	C₂²	C₂³×#1057	1	12	yes	—
261	C₂²	C₂³×#1057	1	2	yes	—
262	C₂²	C₂⁴×#261	1	1	yes	characteristic
263	C₄	—	13	2	no	—
264	C₄	—	13	2	no	—
265	C₄	—	13	4	no	—
266	C₄	—	26	6	no	—
267	C₄	—	26	6	no	—
268	C₄	C₂³×#1057	1	4	yes	—
269	C₄	C₂³×#1057	1	12	yes	—
270	C₂	—	2	12	no	—
271	C₂	#41×#1057	1	4	yes	direct product factor
272	C₂	C₂×#41×#261	1	1	yes	characteristic
273	C₂	C₂⁴×#1057	1	1	yes	characteristic
274	C₂	C₂×D₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#261
Fitting, prime radical	C₂²×C₁₃×#41	C₄
socle	C₂³×C₁₃	C₂³×C₄
cosocle	C₂×Dic₁₃	C₂⁵
hypercenter	C₂²×#41	#261
hypocenter	C₁₃	C₂×C₈×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #41×#1057	C₂	direct (N × H)
2. C₂×Q₈×#1057	C₂	semidirect
3. C₂×D₄×#1057	C₂	semidirect
4. C₂×#11719119	C₂	semidirect
5. C₂²×C₄×#1057	C₂	semidirect
18. Q₈×#1057	C₂²	semidirect
19. D₄×#1057	C₂²	semidirect
20. #11719119	C₂²	semidirect
25. C₂×C₄×#1057	C₂²	semidirect
26. C₂×C₁₃×#41	C₈	semidirect
60. C₁₃×#41	C₂×C₈	semidirect
61. C₂×Q₈×C₁₃	C₂×C₈	semidirect
63. C₂×D₄×C₁₃	C₂×C₈	semidirect
65. C₂²×C₄×C₁₃	C₂×C₈	semidirect
80. C₂×#1057	#41	direct (N × H)
103. Q₈×C₁₃	C₂²×C₈	semidirect
104. D₄×C₁₃	C₂²×C₈	semidirect
107. C₂×C₄×C₁₃	C₂²×C₈	semidirect
123. #1057	C₂×#41	direct (N × H)
158. C₂×#41	#1057	direct (N × H)
191. C₂×C₁₃	C₈×#41	semidirect
196. #41	C₂×#1057	direct (N × H)
197. C₂×Q₈	C₂×#1057	semidirect
199. C₂×D₄	C₂×#1057	semidirect
210. C₂²×C₄	C₂×#1057	semidirect
228. C₁₃	C₂×C₈×#41	semidirect
234. Q₈	C₂²×#1057	semidirect
235. D₄	C₂²×#1057	semidirect
250. C₂×C₄	C₂²×#1057	semidirect
271. C₂	#41×#1057	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×#1057	2
C₂×Q₈×#1057	2
C₂×D₄×#1057	2
C₂×#11719119	2
C₂²×C₄×#1057	2
C₂×#41×Dic₁₃	2
C₂×C₈×#41	13

Subgroups — C2×#41×#1057 #16808451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×#1057 #16808451