Subgroups — C₂²×C₁₃×#492 #15746967

Order 3328.

3950 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	C₂³×C₁₃×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#492	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₂³×D₄×C₁₃	—	2	2	no	—
8	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₁₃×SD₁₆	—	2	32	no	—
10	C₂²×C₁₃×SD₁₆	C₂²	1	48	yes	complemented
11	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₁₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
14	C₂×C₁₃×#127	—	2	6	no	—
15	C₂×C₁₃×#127	C₂²	1	12	yes	—
16	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#492	C₂²	1	64	yes	direct product factor
19	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
20	C₂⁵×C₁₃	—	2	2	no	—
21	C₂²×Q₈×C₁₃	—	2	6	no	—
22	C₂²×Q₈×C₁₃	—	2	16	no	—
23	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
24	C₂²×D₄×C₁₃	—	2	16	no	—
25	C₂²×D₄×C₁₃	—	2	6	no	—
26	C₂²×D₄×C₁₃	—	2	24	no	—
27	C₂²×D₄×C₁₃	—	4	16	no	—
28	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
29	C₂³×C₄×C₁₃	—	2	2	no	—
30	C₂³×C₄×C₁₃	D₄	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×SD₁₆	—	2	192	no	—
33	C₂×C₁₃×SD₁₆	C₂³	1	64	yes	complemented
34	C₂²×C₈×C₁₃	D₄	1	16	yes	complemented
35	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
36	C₁₃×C₄⋊Q₈	—	2	24	no	—
37	C₁₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
38	C₁₃×#127	—	2	24	no	—
39	C₁₃×#127	C₂³	1	16	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
41	C₂×C₄²×C₁₃	D₄	1	6	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
43	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
44	C₂²×#492	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁴×C₁₃	—	2	12	no	—
46	C₂⁴×C₁₃	—	4	8	no	—
47	C₂⁴×C₁₃	—	4	16	no	—
48	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	—	2	12	no	—
50	C₂×Q₈×C₁₃	—	2	48	no	—
51	C₂×Q₈×C₁₃	—	4	24	no	—
52	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
53	C₂×D₄×C₁₃	—	2	48	no	—
54	C₂×D₄×C₁₃	—	2	12	no	—
55	C₂×D₄×C₁₃	—	2	32	no	—
56	C₂×D₄×C₁₃	—	4	24	no	—
57	C₂×D₄×C₁₃	—	4	96	no	—
58	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₁₃	—	2	12	no	—
60	C₂²×C₄×C₁₃	—	4	8	no	—
61	C₂²×C₄×C₁₃	SD₁₆	1	16	yes	complemented
62	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
66	C₁₃×SD₁₆	—	2	128	no	—
67	C₂×C₈×C₁₃	C₂×D₄	1	48	yes	complemented
68	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
69	C₁₃×C₄⋊C₄	—	2	32	no	—
70	C₄²×C₁₃	C₂×D₄	1	12	yes	—
71	C₄²×C₁₃	C₂⁴	1	4	yes	—
72	C₂³×SD₁₆	C₂×C₁₃	1	4	yes	complemented
73	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
74	C₂²×#127	C₂×C₁₃	1	1	yes	characteristic
75	C₂×#492	C₂×C₁₃	1	24	yes	direct product factor
76	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
77	C₂³×C₁₃	—	2	8	no	—
78	C₂³×C₁₃	—	4	24	no	—
79	C₂³×C₁₃	—	4	48	no	—
80	C₂³×C₁₃	—	8	32	no	—
81	C₂³×C₁₃	#127	1	4	yes	—
82	C₂³×C₁₃	C₂×SD₁₆	1	8	yes	—
83	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
84	Q₈×C₁₃	—	2	16	no	—
85	Q₈×C₁₃	—	4	24	no	—
86	D₄×C₁₃	—	2	16	no	—
87	D₄×C₁₃	—	4	24	no	—
88	D₄×C₁₃	—	4	64	no	—
89	C₂×C₄×C₁₃	—	2	8	no	—
90	C₂×C₄×C₁₃	—	4	24	no	—
91	C₂×C₄×C₁₃	#127	1	12	yes	—
92	C₂×C₄×C₁₃	C₂×SD₁₆	1	48	yes	complemented
93	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
94	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
95	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
96	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
97	C₈×C₁₃	C₂²×D₄	1	16	yes	complemented
98	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
99	C₂³×D₄	—	2	2	no	—
100	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
101	C₂²×SD₁₆	—	2	32	no	—
102	C₂²×SD₁₆	C₂²×C₁₃	1	48	yes	complemented
103	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
104	C₂×C₄⋊Q₈	—	2	6	no	—
105	C₂×C₄⋊Q₈	C₂²×C₁₃	1	12	yes	complemented
106	C₂×#127	—	2	6	no	—
107	C₂×#127	C₂²×C₁₃	1	12	yes	—
108	C₂²×C₄⋊C₄	—	2	2	no	—
109	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
110	#492	C₂²×C₁₃	1	64	yes	direct product factor
111	C₂×C₄×C₈	C₂²×C₁₃	1	12	yes	complemented
112	C₂²×C₁₃	—	4	8	no	—
113	C₂²×C₁₃	—	4	16	no	—
114	C₂²×C₁₃	—	8	48	no	—
115	C₂²×C₁₃	#492	1	16	yes	direct product factor
116	C₂²×C₁₃	C₂×#127	1	6	yes	—
117	C₂²×C₁₃	C₂²×SD₁₆	1	12	yes	—
118	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₁₃	—	4	8	no	—
120	C₄×C₁₃	C₂×#127	1	6	yes	—
121	C₄×C₁₃	C₂²×SD₁₆	1	16	yes	complemented
122	C₄×C₁₃	C₂³×D₄	1	2	yes	—
123	C₂⁵	—	2	2	no	—
124	C₂²×Q₈	—	2	6	no	—
125	C₂²×Q₈	—	2	16	no	—
126	C₂²×Q₈	C₂³×C₁₃	1	12	yes	—
127	C₂²×D₄	—	2	16	no	—
128	C₂²×D₄	—	2	6	no	—
129	C₂²×D₄	—	2	24	no	—
130	C₂²×D₄	—	4	16	no	—
131	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	D₄×C₁₃	1	2	yes	—
134	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
135	C₂×SD₁₆	—	2	192	no	—
136	C₂×SD₁₆	C₂³×C₁₃	1	64	yes	complemented
137	C₂²×C₈	D₄×C₁₃	1	16	yes	complemented
138	C₂²×C₈	C₂³×C₁₃	1	12	yes	—
139	C₄⋊Q₈	—	2	24	no	—
140	C₄⋊Q₈	C₂³×C₁₃	1	16	yes	complemented
141	#127	—	2	24	no	—
142	#127	C₂³×C₁₃	1	16	yes	—
143	C₂×C₄⋊C₄	—	2	24	no	—
144	C₂×C₄²	D₄×C₁₃	1	6	yes	—
145	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
146	C₄×C₈	C₂³×C₁₃	1	16	yes	complemented
147	C₂×C₁₃	—	8	8	no	—
148	C₂×C₁₃	C₂×#492	1	12	yes	direct product factor
149	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
150	C₂×C₁₃	C₂³×SD₁₆	1	2	yes	—
151	C₂⁴	—	2	12	no	—
152	C₂⁴	—	4	8	no	—
153	C₂⁴	—	4	16	no	—
154	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
155	C₂×Q₈	—	2	12	no	—
156	C₂×Q₈	—	2	48	no	—
157	C₂×Q₈	—	4	24	no	—
158	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
159	C₂×D₄	—	2	48	no	—
160	C₂×D₄	—	2	12	no	—
161	C₂×D₄	—	2	32	no	—
162	C₂×D₄	—	4	24	no	—
163	C₂×D₄	—	4	96	no	—
164	C₂×D₄	C₂⁴×C₁₃	1	8	yes	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	4	8	no	—
167	C₂²×C₄	C₁₃×SD₁₆	1	16	yes	complemented
168	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
169	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
170	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
171	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
172	SD₁₆	—	2	128	no	—
173	C₂×C₈	C₂×D₄×C₁₃	1	48	yes	complemented
174	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
175	C₄⋊C₄	—	2	32	no	—
176	C₄²	C₂×D₄×C₁₃	1	12	yes	—
177	C₄²	C₂⁴×C₁₃	1	4	yes	—
178	C₁₃	C₂²×#492	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³	—	2	8	no	—
180	C₂³	—	4	24	no	—
181	C₂³	—	4	48	no	—
182	C₂³	—	8	32	no	—
183	C₂³	C₁₃×#127	1	4	yes	—
184	C₂³	C₂×C₁₃×SD₁₆	1	8	yes	—
185	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
186	Q₈	—	2	16	no	—
187	Q₈	—	4	24	no	—
188	D₄	—	2	16	no	—
189	D₄	—	4	24	no	—
190	D₄	—	4	64	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	24	no	—
193	C₂×C₄	C₁₃×#127	1	12	yes	—
194	C₂×C₄	C₂×C₁₃×SD₁₆	1	48	yes	complemented
195	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
196	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
197	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
198	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
199	C₈	C₂²×D₄×C₁₃	1	16	yes	complemented
200	C₂²	—	4	16	no	—
201	C₂²	—	4	8	no	—
202	C₂²	—	8	48	no	—
203	C₂²	C₁₃×#492	1	16	yes	direct product factor
204	C₂²	C₂×C₁₃×#127	1	6	yes	—
205	C₂²	C₂²×C₁₃×SD₁₆	1	12	yes	—
206	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
207	C₄	—	4	8	no	—
208	C₄	C₂×C₁₃×#127	1	6	yes	—
209	C₄	C₂²×C₁₃×SD₁₆	1	16	yes	complemented
210	C₄	C₂³×D₄×C₁₃	1	2	yes	—
211	C₂	—	8	8	no	—
212	C₂	C₂×C₁₃×#492	1	12	yes	direct product factor
213	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
214	C₂	C₂³×C₁₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂²×C₁₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₃×SD₁₆	C₂	semidirect
2. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₁₃×#492	C₂	direct (N × H)
5. C₂²×C₄×C₈×C₁₃	C₂	semidirect
10. C₂²×C₁₃×SD₁₆	C₂²	semidirect
13. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
18. C₁₃×#492	C₂²	direct (N × H)
19. C₂×C₄×C₈×C₁₃	C₂²	semidirect
33. C₂×C₁₃×SD₁₆	C₂³	semidirect
34. C₂²×C₈×C₁₃	D₄	semidirect
37. C₁₃×C₄⋊Q₈	C₂³	semidirect
43. C₄×C₈×C₁₃	C₂³	semidirect
44. C₂²×#492	C₁₃	direct (N × H)
61. C₂²×C₄×C₁₃	SD₁₆	semidirect
67. C₂×C₈×C₁₃	C₂×D₄	semidirect
72. C₂³×SD₁₆	C₂×C₁₃	semidirect
73. C₂²×C₄⋊Q₈	C₂×C₁₃	semidirect
75. C₂×#492	C₂×C₁₃	direct (N × H)
76. C₂²×C₄×C₈	C₂×C₁₃	semidirect
92. C₂×C₄×C₁₃	C₂×SD₁₆	semidirect
97. C₈×C₁₃	C₂²×D₄	semidirect
102. C₂²×SD₁₆	C₂²×C₁₃	semidirect
105. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
110. #492	C₂²×C₁₃	direct (N × H)
111. C₂×C₄×C₈	C₂²×C₁₃	semidirect
115. C₂²×C₁₃	#492	direct (N × H)
121. C₄×C₁₃	C₂²×SD₁₆	semidirect
136. C₂×SD₁₆	C₂³×C₁₃	semidirect
137. C₂²×C₈	D₄×C₁₃	semidirect
140. C₄⋊Q₈	C₂³×C₁₃	semidirect
146. C₄×C₈	C₂³×C₁₃	semidirect
148. C₂×C₁₃	C₂×#492	direct (N × H)
167. C₂²×C₄	C₁₃×SD₁₆	semidirect
173. C₂×C₈	C₂×D₄×C₁₃	semidirect
178. C₁₃	C₂²×#492	direct (N × H)
194. C₂×C₄	C₂×C₁₃×SD₁₆	semidirect
199. C₈	C₂²×D₄×C₁₃	semidirect
203. C₂²	C₁₃×#492	direct (N × H)
209. C₄	C₂²×C₁₃×SD₁₆	semidirect
212. C₂	C₂×C₁₃×#492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×SD₁₆	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₁₃×#127	2
C₂×C₁₃×#492	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#492	13

Subgroups — C22×C13×#492 #15746967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#492 #15746967