Subgroups — C₂²×#41×Dic₁₃ #16769495

Order 3328.

12878 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₁₃	C₂	1	48	yes	maximal, direct product factor
2	C₂²×Q₈×Dic₁₃	C₂	1	2	yes	maximal, complemented
3	C₂²×D₄×Dic₁₃	C₂	1	6	yes	maximal, complemented
4	C₂²×#88180	C₂	1	3	yes	maximal, complemented
5	C₂³×C₄×Dic₁₃	C₂	1	3	yes	maximal, complemented
6	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×#41	C₄	1	32	yes	complemented
8	C₂²×C₁₃×#41	C₂²	1	24	yes	—
9	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
11	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
12	C₂⁴×Dic₁₃	C₂²	1	6	yes	—
13	C₂²×C₂²⋊Dic₁₃	C₂²	1	6	yes	—
14	#41×Dic₁₃	C₂²	1	256	yes	direct product factor
15	C₂×Q₈×Dic₁₃	C₂²	1	48	yes	complemented
16	C₂×D₄×Dic₁₃	C₂²	1	144	yes	complemented
17	C₂×#88180	C₂²	1	72	yes	complemented
18	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
19	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
21	C₂²×C₄×Dic₁₃	C₂²	1	6	yes	—
22	C₂²×C₄×Dic₁₃	C₂²	1	72	yes	complemented
23	C₂⁵×C₁₃	C₂³	1	3	yes	—
24	C₂×C₁₃×#41	C₂×C₄	1	384	yes	complemented
25	C₂×C₁₃×#41	C₂³	1	64	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	16	yes	complemented
27	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₃	C₂×C₄	1	48	yes	complemented
29	C₂²×D₄×C₁₃	C₂³	1	36	yes	—
30	C₂³×C₄×C₁₃	C₂×C₄	1	48	yes	complemented
31	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	36	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×Dic₁₃	—	2	48	no	—
35	C₂³×Dic₁₃	C₂³	1	6	yes	—
36	C₂³×Dic₁₃	C₂³	1	72	yes	—
37	C₂³×Dic₁₃	C₂³	1	2	yes	—
38	C₂×C₂²⋊Dic₁₃	C₂³	1	72	yes	—
39	Q₈×Dic₁₃	C₂³	1	128	yes	complemented
40	D₄×Dic₁₃	C₂³	1	384	yes	complemented
41	#88180	C₂³	1	192	yes	complemented
42	C₂×C₄⋊Dic₁₃	C₂³	1	72	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	36	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	72	yes	—
46	C₂×C₄×Dic₁₃	C₂³	1	192	yes	complemented
47	C₂²×C₄×#41	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	—	2	24	no	—
49	C₂⁴×C₁₃	C₂²×C₄	1	24	yes	—
50	C₂⁴×C₁₃	C₂⁴	1	18	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₁₃×#41	C₂²×C₄	1	512	yes	complemented
53	C₂×Q₈×C₁₃	C₂²×C₄	1	96	yes	complemented
54	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	—
55	C₂×D₄×C₁₃	C₂²×C₄	1	288	yes	complemented
56	C₂×D₄×C₁₃	C₂⁴	1	48	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	288	yes	complemented
60	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	48	yes	—
63	C₂²×Dic₁₃	—	2	24	no	—
64	C₂²×Dic₁₃	—	2	288	no	—
65	C₂²×Dic₁₃	#41	1	16	yes	direct product factor
66	C₂²×Dic₁₃	C₂⁴	1	36	yes	—
67	C₂²×Dic₁₃	C₂⁴	1	96	yes	—
68	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
69	C₂²⋊Dic₁₃	C₂⁴	1	96	yes	—
70	C₄⋊Dic₁₃	C₂⁴	1	96	yes	—
71	C₄×Dic₁₃	C₂⁴	1	16	yes	—
72	C₄×Dic₁₃	C₂⁴	1	48	yes	—
73	C₄×Dic₁₃	C₂⁴	1	96	yes	—
74	C₂³×#41	D₁₃	1	1	yes	characteristic
75	C₂×C₄×#41	—	13	48	no	—
76	C₂²×C₄×Q₈	—	13	2	no	—
77	C₂²×C₄×D₄	—	13	6	no	—
78	C₂²×#117	—	13	3	no	—
79	C₂³×C₄²	—	13	3	no	—
80	C₂³×C₁₃	—	2	96	no	—
81	C₂³×C₁₃	—	2	72	no	—
82	C₂³×C₁₃	C₂³×C₄	1	72	yes	—
83	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
84	C₂³×C₁₃	C₂×#41	1	8	yes	—
85	C₂³×C₁₃	C₂⁵	1	12	yes	—
86	C₂³×C₁₃	C₂⁵	1	3	yes	—
87	Q₈×C₁₃	C₂³×C₄	1	64	yes	complemented
88	D₄×C₁₃	C₂³×C₄	1	192	yes	complemented
89	C₂×C₄×C₁₃	C₂³×C₄	1	24	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	72	yes	—
91	C₂×C₄×C₁₃	C₂³×C₄	1	192	yes	complemented
92	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
93	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
94	C₂×Dic₁₃	—	2	72	no	—
95	C₂×Dic₁₃	—	2	192	no	—
96	C₂×Dic₁₃	C₂×#41	1	48	yes	direct product factor
97	C₂×Dic₁₃	C₂⁵	1	24	yes	—
98	C₂×Dic₁₃	C₂⁵	1	8	yes	—
99	C₂²×#41	Dic₁₃	1	32	yes	direct product factor
100	C₂²×#41	C₂×D₁₃	1	24	yes	—
101	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
102	C₂³×D₄	C₂×D₁₃	1	3	yes	—
103	C₂⁴×C₄	—	13	6	no	—
104	C₂⁴×C₄	C₂×D₁₃	1	3	yes	—
105	C₄×#41	—	13	256	no	—
106	C₂×C₄×Q₈	—	13	48	no	—
107	C₂×C₄×D₄	—	13	144	no	—
108	C₂×#117	—	13	72	no	—
109	C₂²×C₄⋊C₄	—	13	6	no	—
110	C₂²×C₂²⋊C₄	—	13	6	no	—
111	C₂²×C₄²	—	13	6	no	—
112	C₂²×C₄²	—	13	72	no	—
113	C₂²×C₄²	—	13	1	no	—
114	C₂²×C₄²	—	13	3	no	—
115	C₂²×C₁₃	—	2	24	no	—
116	C₂²×C₁₃	—	2	144	no	—
117	C₂²×C₁₃	C₄×#41	1	16	yes	complemented
118	C₂²×C₁₃	C₂⁴×C₄	1	24	yes	—
119	C₂²×C₁₃	C₂⁴×C₄	1	6	yes	—
120	C₂²×C₁₃	C₂²×#41	1	12	yes	—
121	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
122	C₄×C₁₃	C₂⁴×C₄	1	8	yes	—
123	C₄×C₁₃	C₂⁴×C₄	1	24	yes	—
124	Dic₁₃	—	2	24	no	—
125	Dic₁₃	C₂²×#41	1	16	yes	direct product factor
126	C₂⁵	C₂²×D₁₃	1	3	yes	—
127	C₂×#41	C₂×Dic₁₃	1	384	yes	direct product factor
128	C₂×#41	C₂²×D₁₃	1	64	yes	—
129	C₂²×Q₈	C₂×Dic₁₃	1	16	yes	complemented
130	C₂²×Q₈	C₂²×D₁₃	1	12	yes	—
131	C₂²×D₄	C₂×Dic₁₃	1	48	yes	complemented
132	C₂²×D₄	C₂²×D₁₃	1	36	yes	—
133	C₂³×C₄	—	13	2	no	—
134	C₂³×C₄	—	13	6	no	—
135	C₂³×C₄	—	13	72	no	—
136	C₂³×C₄	—	26	48	no	—
137	C₂³×C₄	C₂×Dic₁₃	1	48	yes	complemented
138	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
139	C₂³×C₄	C₂²×D₁₃	1	36	yes	—
140	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
141	C₄×Q₈	—	13	128	no	—
142	C₄×D₄	—	13	384	no	—
143	#117	—	13	192	no	—
144	C₂×C₄⋊C₄	—	13	72	no	—
145	C₂×C₂²⋊C₄	—	13	72	no	—
146	C₂×C₄²	—	13	36	no	—
147	C₂×C₄²	—	13	72	no	—
148	C₂×C₄²	—	13	192	no	—
149	C₂×C₄²	—	13	12	no	—
150	C₂×C₁₃	—	2	24	no	—
151	C₂×C₁₃	C₂×C₄×#41	1	12	yes	complemented
152	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
153	C₂×C₁₃	C₂³×#41	1	2	yes	—
154	C₂⁴	—	2	24	no	—
155	C₂⁴	C₂²×Dic₁₃	1	24	yes	—
156	C₂⁴	C₂³×D₁₃	1	18	yes	—
157	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
158	#41	C₂²×Dic₁₃	1	512	yes	direct product factor
159	C₂×Q₈	C₂²×Dic₁₃	1	96	yes	complemented
160	C₂×Q₈	C₂³×D₁₃	1	16	yes	—
161	C₂×D₄	C₂²×Dic₁₃	1	288	yes	complemented
162	C₂×D₄	C₂³×D₁₃	1	48	yes	—
163	C₂²×C₄	—	13	16	no	—
164	C₂²×C₄	—	13	12	no	—
165	C₂²×C₄	—	13	36	no	—
166	C₂²×C₄	—	13	96	no	—
167	C₂²×C₄	—	26	288	no	—
168	C₂²×C₄	—	26	24	no	—
169	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
170	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
171	C₂²×C₄	C₂²×Dic₁₃	1	288	yes	complemented
172	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
173	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
174	C₂²×C₄	C₂³×D₁₃	1	48	yes	—
175	C₄⋊C₄	—	13	96	no	—
176	C₂²⋊C₄	—	13	96	no	—
177	C₄²	—	13	16	no	—
178	C₄²	—	13	48	no	—
179	C₄²	—	13	96	no	—
180	C₁₃	C₂²×C₄×#41	1	1	yes	characteristic, complemented, Sylow
181	C₂³	—	2	96	no	—
182	C₂³	—	2	72	no	—
183	C₂³	#41×D₁₃	1	8	yes	—
184	C₂³	C₂³×Dic₁₃	1	72	yes	—
185	C₂³	C₂³×Dic₁₃	1	4	yes	—
186	C₂³	C₂⁴×D₁₃	1	12	yes	—
187	C₂³	C₂⁴×D₁₃	1	3	yes	—
188	Q₈	C₂³×Dic₁₃	1	64	yes	complemented
189	D₄	C₂³×Dic₁₃	1	192	yes	complemented
190	C₂×C₄	—	13	48	no	—
191	C₂×C₄	—	13	8	no	—
192	C₂×C₄	—	13	24	no	—
193	C₂×C₄	—	26	192	no	—
194	C₂×C₄	—	26	72	no	—
195	C₂×C₄	C₂³×Dic₁₃	1	24	yes	—
196	C₂×C₄	C₂³×Dic₁₃	1	72	yes	—
197	C₂×C₄	C₂³×Dic₁₃	1	192	yes	complemented
198	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
199	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
200	C₂²	—	2	144	no	—
201	C₂²	—	2	24	no	—
202	C₂²	#41×Dic₁₃	1	16	yes	direct product factor
203	C₂²	C₂×#41×D₁₃	1	12	yes	—
204	C₂²	C₂⁴×Dic₁₃	1	24	yes	—
205	C₂²	C₂⁴×Dic₁₃	1	6	yes	—
206	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
207	C₄	—	13	16	no	—
208	C₄	—	26	24	no	—
209	C₄	C₂⁴×Dic₁₃	1	8	yes	—
210	C₄	C₂⁴×Dic₁₃	1	24	yes	—
211	C₂	—	2	24	no	—
212	C₂	C₂×#41×Dic₁₃	1	12	yes	direct product factor
213	C₂	C₂⁵×Dic₁₃	1	1	yes	characteristic
214	C₂	C₂²×#41×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting, prime radical	C₂³×C₁₃×#41	C₂
socle	C₂⁴×C₁₃	C₂⁴
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	C₂²×C₄×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×Dic₁₃	C₂	direct (N × H)
2. C₂²×Q₈×Dic₁₃	C₂	semidirect
3. C₂²×D₄×Dic₁₃	C₂	semidirect
4. C₂²×#88180	C₂	semidirect
5. C₂³×C₄×Dic₁₃	C₂	semidirect
7. C₂²×C₁₃×#41	C₄	semidirect
14. #41×Dic₁₃	C₂²	direct (N × H)
15. C₂×Q₈×Dic₁₃	C₂²	semidirect
16. C₂×D₄×Dic₁₃	C₂²	semidirect
17. C₂×#88180	C₂²	semidirect
22. C₂²×C₄×Dic₁₃	C₂²	semidirect
24. C₂×C₁₃×#41	C₂×C₄	semidirect
26. C₂²×Q₈×C₁₃	C₂×C₄	semidirect
28. C₂²×D₄×C₁₃	C₂×C₄	semidirect
30. C₂³×C₄×C₁₃	C₂×C₄	semidirect
39. Q₈×Dic₁₃	C₂³	semidirect
40. D₄×Dic₁₃	C₂³	semidirect
41. #88180	C₂³	semidirect
46. C₂×C₄×Dic₁₃	C₂³	semidirect
52. C₁₃×#41	C₂²×C₄	semidirect
53. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
55. C₂×D₄×C₁₃	C₂²×C₄	semidirect
59. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
65. C₂²×Dic₁₃	#41	direct (N × H)
87. Q₈×C₁₃	C₂³×C₄	semidirect
88. D₄×C₁₃	C₂³×C₄	semidirect
91. C₂×C₄×C₁₃	C₂³×C₄	semidirect
96. C₂×Dic₁₃	C₂×#41	direct (N × H)
99. C₂²×#41	Dic₁₃	direct (N × H)
117. C₂²×C₁₃	C₄×#41	semidirect
125. Dic₁₃	C₂²×#41	direct (N × H)
127. C₂×#41	C₂×Dic₁₃	direct (N × H)
129. C₂²×Q₈	C₂×Dic₁₃	semidirect
131. C₂²×D₄	C₂×Dic₁₃	semidirect
137. C₂³×C₄	C₂×Dic₁₃	semidirect
151. C₂×C₁₃	C₂×C₄×#41	semidirect
158. #41	C₂²×Dic₁₃	direct (N × H)
159. C₂×Q₈	C₂²×Dic₁₃	semidirect
161. C₂×D₄	C₂²×Dic₁₃	semidirect
171. C₂²×C₄	C₂²×Dic₁₃	semidirect
180. C₁₃	C₂²×C₄×#41	semidirect
188. Q₈	C₂³×Dic₁₃	semidirect
189. D₄	C₂³×Dic₁₃	semidirect
197. C₂×C₄	C₂³×Dic₁₃	semidirect
202. C₂²	#41×Dic₁₃	direct (N × H)
212. C₂	C₂×#41×Dic₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×Dic₁₃	2
C₂²×Q₈×Dic₁₃	2
C₂²×D₄×Dic₁₃	2
C₂²×#88180	2
C₂³×C₄×Dic₁₃	2
C₂³×C₁₃×#41	2
C₂²×C₄×#41	13

Subgroups — C22×#41×Dic13 #16769495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×Dic₁₃ #16769495