Subgroups — C₂²×C₁₃×#496 #15746972

Order 3328.

4038 conjugacy classes of subgroups.

The subgroup lattice has 278 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	2	yes	maximal, complemented
2	C₂³×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#496	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₁₃	—	2	2	no	—
9	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₁₃×SD₁₆	—	2	16	no	—
12	C₂²×C₁₃×SD₁₆	C₂²	1	24	yes	complemented
13	C₂²×C₁₃×D₈	—	2	16	no	—
14	C₂²×C₁₃×D₈	C₂²	1	24	yes	complemented
15	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
16	C₂×C₁₃×#127	—	2	6	no	—
17	C₂×C₁₃×#127	C₂²	1	12	yes	complemented
18	C₂×C₁₃×#124	—	2	6	no	—
19	C₂×C₁₃×#124	C₂²	1	12	yes	complemented
20	C₂²×C₁₃×C₂²⋊C₄	—	2	2	no	—
21	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#496	C₂²	1	64	yes	direct product factor
23	C₂×C₁₃×#97	C₂²	1	12	yes	complemented
24	C₂⁵×C₁₃	—	2	1	no	—
25	C₂⁵×C₁₃	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	3	no	—
27	C₂²×Q₈×C₁₃	—	2	8	no	—
28	C₂²×Q₈×C₁₃	C₂³	1	6	yes	—
29	C₂²×D₄×C₁₃	—	2	8	no	—
30	C₂²×D₄×C₁₃	—	2	8	no	—
31	C₂²×D₄×C₁₃	—	2	8	no	—
32	C₂²×D₄×C₁₃	—	2	3	no	—
33	C₂²×D₄×C₁₃	—	2	6	no	—
34	C₂²×D₄×C₁₃	—	2	24	no	—
35	C₂²×D₄×C₁₃	—	4	16	no	—
36	C₂²×D₄×C₁₃	C₂³	1	6	yes	—
37	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
38	C₂³×C₄×C₁₃	—	2	1	no	—
39	C₂³×C₄×C₁₃	D₄	1	2	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×SD₁₆	—	2	96	no	—
42	C₂×C₁₃×SD₁₆	C₂³	1	32	yes	complemented
43	C₂×C₁₃×D₈	—	2	96	no	—
44	C₂×C₁₃×D₈	C₂³	1	32	yes	complemented
45	C₂²×C₈×C₁₃	D₄	1	16	yes	complemented
46	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
47	C₁₃×#127	—	2	24	no	—
48	C₁₃×#127	C₂³	1	16	yes	complemented
49	C₁₃×#124	—	2	24	no	—
50	C₁₃×#124	C₂³	1	16	yes	complemented
51	C₂×C₁₃×C₂²⋊C₄	—	2	24	no	—
52	C₂×C₄²×C₁₃	D₄	1	6	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
54	C₁₃×#97	C₂³	1	16	yes	complemented
55	C₂²×#496	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	—	2	6	no	—
57	C₂⁴×C₁₃	—	2	12	no	—
58	C₂⁴×C₁₃	—	4	4	no	—
59	C₂⁴×C₁₃	—	4	8	no	—
60	C₂⁴×C₁₃	—	4	16	no	—
61	C₂⁴×C₁₃	—	4	8	no	—
62	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Q₈×C₁₃	—	2	6	no	—
64	C₂×Q₈×C₁₃	—	2	24	no	—
65	C₂×Q₈×C₁₃	—	4	12	no	—
66	C₂×Q₈×C₁₃	C₂⁴	1	4	yes	—
67	C₂×D₄×C₁₃	—	2	24	no	—
68	C₂×D₄×C₁₃	—	2	24	no	—
69	C₂×D₄×C₁₃	—	2	24	no	—
70	C₂×D₄×C₁₃	—	2	6	no	—
71	C₂×D₄×C₁₃	—	2	12	no	—
72	C₂×D₄×C₁₃	—	2	32	no	—
73	C₂×D₄×C₁₃	—	4	96	no	—
74	C₂×D₄×C₁₃	—	4	12	no	—
75	C₂×D₄×C₁₃	—	4	24	no	—
76	C₂×D₄×C₁₃	C₂⁴	1	4	yes	—
77	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₁₃	—	2	8	no	—
79	C₂²×C₄×C₁₃	—	2	6	no	—
80	C₂²×C₄×C₁₃	—	4	4	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
85	C₁₃×SD₁₆	—	2	64	no	—
86	C₁₃×D₈	—	2	64	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	48	yes	complemented
88	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
89	C₁₃×C₂²⋊C₄	—	2	32	no	—
90	C₄²×C₁₃	C₂×D₄	1	12	yes	—
91	C₄²×C₁₃	C₂⁴	1	4	yes	—
92	C₂³×SD₁₆	C₂×C₁₃	1	2	yes	complemented
93	C₂³×D₈	C₂×C₁₃	1	2	yes	complemented
94	C₂²×#127	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂²×#124	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂×#496	C₂×C₁₃	1	24	yes	direct product factor
97	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂³×C₁₃	—	2	4	no	—
99	C₂³×C₁₃	—	2	8	no	—
100	C₂³×C₁₃	—	4	12	no	—
101	C₂³×C₁₃	—	4	24	no	—
102	C₂³×C₁₃	—	4	48	no	—
103	C₂³×C₁₃	—	4	24	no	—
104	C₂³×C₁₃	—	8	32	no	—
105	C₂³×C₁₃	—	8	16	no	—
106	C₂³×C₁₃	#127	1	4	yes	—
107	C₂³×C₁₃	AGL(1,ℤ/8)	1	8	yes	—
108	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
109	Q₈×C₁₃	—	2	8	no	—
110	Q₈×C₁₃	—	4	12	no	—
111	D₄×C₁₃	—	2	8	no	—
112	D₄×C₁₃	—	2	8	no	—
113	D₄×C₁₃	—	2	8	no	—
114	D₄×C₁₃	—	4	64	no	—
115	D₄×C₁₃	—	4	12	no	—
116	D₄×C₁₃	—	4	24	no	—
117	C₂×C₄×C₁₃	—	2	24	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	12	no	—
120	C₂×C₄×C₁₃	#127	1	12	yes	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
124	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
125	C₈×C₁₃	C₂²×D₄	1	16	yes	complemented
126	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂³×D₄	—	2	2	no	—
128	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
129	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
130	C₂²×SD₁₆	—	2	16	no	—
131	C₂²×SD₁₆	C₂²×C₁₃	1	24	yes	complemented
132	C₂²×D₈	—	2	16	no	—
133	C₂²×D₈	C₂²×C₁₃	1	24	yes	complemented
134	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
135	C₂×#127	—	2	6	no	—
136	C₂×#127	C₂²×C₁₃	1	12	yes	complemented
137	C₂×#124	—	2	6	no	—
138	C₂×#124	C₂²×C₁₃	1	12	yes	complemented
139	C₂²×C₂²⋊C₄	—	2	2	no	—
140	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
141	#496	C₂²×C₁₃	1	64	yes	direct product factor
142	C₂×#97	C₂²×C₁₃	1	12	yes	complemented
143	C₂²×C₁₃	—	4	4	no	—
144	C₂²×C₁₃	—	4	8	no	—
145	C₂²×C₁₃	—	4	16	no	—
146	C₂²×C₁₃	—	4	8	no	—
147	C₂²×C₁₃	—	8	48	no	—
148	C₂²×C₁₃	—	8	24	no	—
149	C₂²×C₁₃	#496	1	16	yes	direct product factor
150	C₂²×C₁₃	C₂×#127	1	6	yes	—
151	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
152	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
153	C₄×C₁₃	—	2	8	no	—
154	C₄×C₁₃	—	4	4	no	—
155	C₄×C₁₃	C₂×#127	1	6	yes	—
156	C₄×C₁₃	C₂³×D₄	1	2	yes	—
157	C₂⁵	—	2	1	no	—
158	C₂⁵	—	2	2	no	—
159	C₂²×Q₈	—	2	3	no	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	C₂³×C₁₃	1	6	yes	—
162	C₂²×D₄	—	2	8	no	—
163	C₂²×D₄	—	2	8	no	—
164	C₂²×D₄	—	2	8	no	—
165	C₂²×D₄	—	2	3	no	—
166	C₂²×D₄	—	2	6	no	—
167	C₂²×D₄	—	2	24	no	—
168	C₂²×D₄	—	4	16	no	—
169	C₂²×D₄	C₂³×C₁₃	1	6	yes	—
170	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	D₄×C₁₃	1	2	yes	—
173	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×SD₁₆	—	2	96	no	—
175	C₂×SD₁₆	C₂³×C₁₃	1	32	yes	complemented
176	C₂×D₈	—	2	96	no	—
177	C₂×D₈	C₂³×C₁₃	1	32	yes	complemented
178	C₂²×C₈	D₄×C₁₃	1	16	yes	complemented
179	C₂²×C₈	C₂³×C₁₃	1	12	yes	—
180	#127	—	2	24	no	—
181	#127	C₂³×C₁₃	1	16	yes	complemented
182	#124	—	2	24	no	—
183	#124	C₂³×C₁₃	1	16	yes	complemented
184	C₂×C₂²⋊C₄	—	2	24	no	—
185	C₂×C₄²	D₄×C₁₃	1	6	yes	—
186	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
187	#97	C₂³×C₁₃	1	16	yes	complemented
188	C₂×C₁₃	—	8	8	no	—
189	C₂×C₁₃	—	8	4	no	—
190	C₂×C₁₃	C₂×#496	1	12	yes	direct product factor
191	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
192	C₂×C₁₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
193	C₂⁴	—	2	6	no	—
194	C₂⁴	—	2	12	no	—
195	C₂⁴	—	4	4	no	—
196	C₂⁴	—	4	8	no	—
197	C₂⁴	—	4	16	no	—
198	C₂⁴	—	4	8	no	—
199	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×Q₈	—	2	6	no	—
201	C₂×Q₈	—	2	24	no	—
202	C₂×Q₈	—	4	12	no	—
203	C₂×Q₈	C₂⁴×C₁₃	1	4	yes	—
204	C₂×D₄	—	2	24	no	—
205	C₂×D₄	—	2	24	no	—
206	C₂×D₄	—	2	24	no	—
207	C₂×D₄	—	2	6	no	—
208	C₂×D₄	—	2	12	no	—
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	4	96	no	—
211	C₂×D₄	—	4	12	no	—
212	C₂×D₄	—	4	24	no	—
213	C₂×D₄	C₂⁴×C₁₃	1	4	yes	—
214	C₂×D₄	C₂⁴×C₁₃	1	8	yes	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	6	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
219	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
221	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
222	SD₁₆	—	2	64	no	—
223	D₈	—	2	64	no	—
224	C₂×C₈	C₂×D₄×C₁₃	1	48	yes	complemented
225	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
226	C₂²⋊C₄	—	2	32	no	—
227	C₄²	C₂×D₄×C₁₃	1	12	yes	—
228	C₄²	C₂⁴×C₁₃	1	4	yes	—
229	C₁₃	C₂²×#496	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	2	4	no	—
231	C₂³	—	2	8	no	—
232	C₂³	—	4	12	no	—
233	C₂³	—	4	24	no	—
234	C₂³	—	4	48	no	—
235	C₂³	—	4	24	no	—
236	C₂³	—	8	32	no	—
237	C₂³	—	8	16	no	—
238	C₂³	C₁₃×#127	1	4	yes	—
239	C₂³	C₁₃×AGL(1,ℤ/8)	1	8	yes	—
240	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
241	Q₈	—	2	8	no	—
242	Q₈	—	4	12	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	8	no	—
245	D₄	—	2	8	no	—
246	D₄	—	4	64	no	—
247	D₄	—	4	12	no	—
248	D₄	—	4	24	no	—
249	C₂×C₄	—	2	24	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	12	no	—
252	C₂×C₄	C₁₃×#127	1	12	yes	—
253	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
254	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
255	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
256	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
257	C₈	C₂²×D₄×C₁₃	1	16	yes	complemented
258	C₂²	—	4	16	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	48	no	—
263	C₂²	—	8	24	no	—
264	C₂²	C₁₃×#496	1	16	yes	direct product factor
265	C₂²	C₂×C₁₃×#127	1	6	yes	—
266	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	12	yes	—
267	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	4	4	no	—
270	C₄	C₂×C₁₃×#127	1	6	yes	—
271	C₄	C₂³×D₄×C₁₃	1	2	yes	—
272	C₂	—	8	8	no	—
273	C₂	—	8	4	no	—
274	C₂	C₂×C₁₃×#496	1	12	yes	direct product factor
275	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
276	C₂	C₂²×C₁₃×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
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. C₂³×C₁₃×D₈	C₂	semidirect
3. C₂²×C₁₃×#127	C₂	semidirect
4. C₂²×C₁₃×#124	C₂	semidirect
5. C₂×C₁₃×#496	C₂	direct (N × H)
6. C₂²×C₁₃×#97	C₂	semidirect
12. C₂²×C₁₃×SD₁₆	C₂²	semidirect
14. C₂²×C₁₃×D₈	C₂²	semidirect
17. C₂×C₁₃×#127	C₂²	semidirect
19. C₂×C₁₃×#124	C₂²	semidirect
22. C₁₃×#496	C₂²	direct (N × H)
23. C₂×C₁₃×#97	C₂²	semidirect
42. C₂×C₁₃×SD₁₆	C₂³	semidirect
44. C₂×C₁₃×D₈	C₂³	semidirect
45. C₂²×C₈×C₁₃	D₄	semidirect
48. C₁₃×#127	C₂³	semidirect
50. C₁₃×#124	C₂³	semidirect
54. C₁₃×#97	C₂³	semidirect
55. C₂²×#496	C₁₃	direct (N × H)
87. C₂×C₈×C₁₃	C₂×D₄	semidirect
92. C₂³×SD₁₆	C₂×C₁₃	semidirect
93. C₂³×D₈	C₂×C₁₃	semidirect
94. C₂²×#127	C₂×C₁₃	semidirect
95. C₂²×#124	C₂×C₁₃	semidirect
96. C₂×#496	C₂×C₁₃	direct (N × H)
97. C₂²×#97	C₂×C₁₃	semidirect
125. C₈×C₁₃	C₂²×D₄	semidirect
131. C₂²×SD₁₆	C₂²×C₁₃	semidirect
133. C₂²×D₈	C₂²×C₁₃	semidirect
136. C₂×#127	C₂²×C₁₃	semidirect
138. C₂×#124	C₂²×C₁₃	semidirect
141. #496	C₂²×C₁₃	direct (N × H)
142. C₂×#97	C₂²×C₁₃	semidirect
149. C₂²×C₁₃	#496	direct (N × H)
175. C₂×SD₁₆	C₂³×C₁₃	semidirect
177. C₂×D₈	C₂³×C₁₃	semidirect
178. C₂²×C₈	D₄×C₁₃	semidirect
181. #127	C₂³×C₁₃	semidirect
183. #124	C₂³×C₁₃	semidirect
187. #97	C₂³×C₁₃	semidirect
190. C₂×C₁₃	C₂×#496	direct (N × H)
224. C₂×C₈	C₂×D₄×C₁₃	semidirect
229. C₁₃	C₂²×#496	direct (N × H)
257. C₈	C₂²×D₄×C₁₃	semidirect
264. C₂²	C₁₃×#496	direct (N × H)
274. C₂	C₂×C₁₃×#496	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
C₂³×C₁₃×D₈	2
C₂²×C₁₃×#127	2
C₂²×C₁₃×#124	2
C₂×C₁₃×#496	2
C₂²×C₁₃×#97	2
C₂²×#496	13

Subgroups — C22×C13×#496 #15746972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#496 #15746972