Subgroups — C₂²×C₁₃×#578 #15749425

Order 3328.

10774 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#578	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₁₃×#137	C₂	1	6	yes	maximal, complemented
4	C₂²×C₁₃×#135	C₂	1	3	yes	maximal, complemented
5	C₂³×C₁₃×Q₁₆	C₂	1	3	yes	maximal, complemented
6	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×2₋¹⁺⁴	—	2	24	no	—
8	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	48	yes	complemented
9	C₂²×C₁₃×#41	—	2	6	no	—
10	C₂²×C₁₃×#41	C₂²	1	6	yes	—
11	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₁₃	—	2	2	no	—
13	C₂³×Q₈×C₁₃	C₂²	1	6	yes	—
14	C₁₃×#578	C₂²	1	256	yes	direct product factor
15	C₂×C₁₃×#137	C₂²	1	144	yes	complemented
16	C₂×C₁₃×#135	C₂²	1	72	yes	complemented
17	C₂²×C₁₃×Q₁₆	C₂²	1	3	yes	—
18	C₂²×C₁₃×Q₁₆	C₂²	1	6	yes	—
19	C₂²×C₁₃×Q₁₆	C₂²	1	72	yes	complemented
20	C₂²×C₁₃×SD₁₆	C₂²	1	6	yes	—
21	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#131	C₂²	1	24	yes	complemented
23	C₂²×C₁₃×M₁₆	C₂²	1	3	yes	—
24	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
25	C₁₃×2₋¹⁺⁴	—	2	192	no	—
26	C₁₃×2₋¹⁺⁴	C₂³	1	128	yes	complemented
27	C₂×C₁₃×#41	—	2	144	no	—
28	C₂×C₁₃×#41	—	2	36	no	—
29	C₂×C₁₃×#41	D₄	1	12	yes	—
30	C₂×C₁₃×#41	C₂³	1	72	yes	—
31	C₂×C₁₃×#41	C₂³	1	12	yes	—
32	C₂²×Q₈×C₁₃	—	2	48	no	—
33	C₂²×Q₈×C₁₃	—	2	6	no	—
34	C₂²×Q₈×C₁₃	—	2	36	no	—
35	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃	C₂³	1	6	yes	—
37	C₂²×Q₈×C₁₃	C₂³	1	72	yes	—
38	C₂²×D₄×C₁₃	—	2	6	no	—
39	C₂²×D₄×C₁₃	D₄	1	3	yes	—
40	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₁₃	—	2	6	no	—
42	C₂³×C₄×C₁₃	—	2	6	no	—
43	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
44	C₁₃×#137	C₂³	1	384	yes	complemented
45	C₁₃×#135	C₂³	1	192	yes	complemented
46	C₂×C₁₃×Q₁₆	C₂³	1	36	yes	—
47	C₂×C₁₃×Q₁₆	C₂³	1	72	yes	—
48	C₂×C₁₃×Q₁₆	C₂³	1	192	yes	complemented
49	C₂×C₁₃×SD₁₆	C₂³	1	72	yes	—
50	C₂×C₁₃×D₈	C₂³	1	12	yes	—
51	C₁₃×#131	C₂³	1	64	yes	complemented
52	C₂×C₁₃×M₁₆	C₂³	1	36	yes	—
53	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
55	C₂²×C₈×C₁₃	C₂³	1	36	yes	—
56	C₂²×#578	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	2	2	no	—
58	C₂⁴×C₁₃	C₂×D₄	1	3	yes	—
59	C₁₃×#41	—	2	384	no	—
60	C₁₃×#41	—	2	144	no	—
61	C₁₃×#41	C₂×D₄	1	48	yes	—
62	C₁₃×#41	C₂⁴	1	96	yes	—
63	C₁₃×#41	C₂⁴	1	16	yes	—
64	C₂×Q₈×C₁₃	—	2	128	no	—
65	C₂×Q₈×C₁₃	—	2	72	no	—
66	C₂×Q₈×C₁₃	—	2	18	no	—
67	C₂×Q₈×C₁₃	—	2	144	no	—
68	C₂×Q₈×C₁₃	C₂×D₄	1	12	yes	—
69	C₂×Q₈×C₁₃	C₂⁴	1	36	yes	—
70	C₂×Q₈×C₁₃	C₂⁴	1	96	yes	—
71	C₂×D₄×C₁₃	—	2	72	no	—
72	C₂×D₄×C₁₃	—	2	6	no	—
73	C₂×D₄×C₁₃	C₂×D₄	1	36	yes	—
74	C₂×D₄×C₁₃	C₂⁴	1	12	yes	—
75	C₂²×C₄×C₁₃	—	2	72	no	—
76	C₂²×C₄×C₁₃	—	2	6	no	—
77	C₂²×C₄×C₁₃	—	2	72	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	18	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×Q₁₆	C₂⁴	1	48	yes	—
83	C₁₃×Q₁₆	C₂⁴	1	96	yes	—
84	C₁₃×SD₁₆	C₂⁴	1	96	yes	—
85	C₁₃×D₈	C₂⁴	1	16	yes	—
86	C₁₃×M₁₆	C₂⁴	1	48	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	18	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	48	yes	—
90	C₂²×2₋¹⁺⁴	C₂×C₁₃	1	2	yes	complemented
91	C₂×#578	C₂×C₁₃	1	48	yes	direct product factor
92	C₂²×#137	C₂×C₁₃	1	6	yes	complemented
93	C₂²×#135	C₂×C₁₃	1	3	yes	complemented
94	C₂³×Q₁₆	C₂×C₁₃	1	3	yes	complemented
95	C₂²×#131	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₁₃	—	2	12	no	—
97	C₂³×C₁₃	—	2	12	no	—
98	C₂³×C₁₃	—	4	8	no	—
99	C₂³×C₁₃	C₂²×D₄	1	18	yes	—
100	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	96	no	—
102	Q₈×C₁₃	—	2	36	no	—
103	Q₈×C₁₃	C₂²×D₄	1	16	yes	—
104	Q₈×C₁₃	C₂⁵	1	24	yes	—
105	D₄×C₁₃	—	2	96	no	—
106	D₄×C₁₃	—	2	12	no	—
107	D₄×C₁₃	C₂²×D₄	1	48	yes	—
108	D₄×C₁₃	C₂⁵	1	8	yes	—
109	C₂×C₄×C₁₃	—	2	36	no	—
110	C₂×C₄×C₁₃	—	2	96	no	—
111	C₂×C₄×C₁₃	—	2	96	no	—
112	C₂×C₄×C₁₃	C₂²×D₄	1	18	yes	—
113	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	36	yes	—
115	C₂×C₄×C₁₃	C₂⁵	1	3	yes	—
116	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
117	C₈×C₁₃	C₂⁵	1	4	yes	—
118	C₈×C₁₃	C₂⁵	1	12	yes	—
119	C₂×2₋¹⁺⁴	—	2	24	no	—
120	C₂×2₋¹⁺⁴	C₂²×C₁₃	1	48	yes	complemented
121	C₂²×#41	—	2	6	no	—
122	C₂²×#41	C₂²×C₁₃	1	6	yes	—
123	C₂²×#41	C₂²×C₁₃	1	1	yes	characteristic
124	C₂³×Q₈	—	2	2	no	—
125	C₂³×Q₈	C₂²×C₁₃	1	6	yes	—
126	#578	C₂²×C₁₃	1	256	yes	direct product factor
127	C₂×#137	C₂²×C₁₃	1	144	yes	complemented
128	C₂×#135	C₂²×C₁₃	1	72	yes	complemented
129	C₂²×Q₁₆	C₂²×C₁₃	1	3	yes	—
130	C₂²×Q₁₆	C₂²×C₁₃	1	6	yes	—
131	C₂²×Q₁₆	C₂²×C₁₃	1	72	yes	complemented
132	C₂²×SD₁₆	C₂²×C₁₃	1	6	yes	—
133	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#131	C₂²×C₁₃	1	24	yes	complemented
135	C₂²×M₁₆	C₂²×C₁₃	1	3	yes	—
136	C₂³×C₈	C₂²×C₁₃	1	3	yes	—
137	C₂²×C₁₃	—	2	36	no	—
138	C₂²×C₁₃	—	2	8	no	—
139	C₂²×C₁₃	—	4	24	no	—
140	C₂²×C₁₃	#578	1	4	yes	direct product factor
141	C₂²×C₁₃	C₂³×D₄	1	12	yes	—
142	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
143	C₄×C₁₃	—	2	24	no	—
144	C₄×C₁₃	C₂³×D₄	1	12	yes	—
145	C₄×C₁₃	C₂³×D₄	1	3	yes	—
146	C₄×C₁₃	C₂⁶	1	1	yes	characteristic
147	2₋¹⁺⁴	—	2	192	no	—
148	2₋¹⁺⁴	C₂³×C₁₃	1	128	yes	complemented
149	C₂×#41	—	2	144	no	—
150	C₂×#41	—	2	36	no	—
151	C₂×#41	D₄×C₁₃	1	12	yes	—
152	C₂×#41	C₂³×C₁₃	1	72	yes	—
153	C₂×#41	C₂³×C₁₃	1	12	yes	—
154	C₂²×Q₈	—	2	48	no	—
155	C₂²×Q₈	—	2	6	no	—
156	C₂²×Q₈	—	2	36	no	—
157	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
158	C₂²×Q₈	C₂³×C₁₃	1	6	yes	—
159	C₂²×Q₈	C₂³×C₁₃	1	72	yes	—
160	C₂²×D₄	—	2	6	no	—
161	C₂²×D₄	D₄×C₁₃	1	3	yes	—
162	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
163	C₂³×C₄	—	2	6	no	—
164	C₂³×C₄	—	2	6	no	—
165	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
166	#137	C₂³×C₁₃	1	384	yes	complemented
167	#135	C₂³×C₁₃	1	192	yes	complemented
168	C₂×Q₁₆	C₂³×C₁₃	1	36	yes	—
169	C₂×Q₁₆	C₂³×C₁₃	1	72	yes	—
170	C₂×Q₁₆	C₂³×C₁₃	1	192	yes	complemented
171	C₂×SD₁₆	C₂³×C₁₃	1	72	yes	—
172	C₂×D₈	C₂³×C₁₃	1	12	yes	—
173	#131	C₂³×C₁₃	1	64	yes	complemented
174	C₂×M₁₆	C₂³×C₁₃	1	36	yes	—
175	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
177	C₂²×C₈	C₂³×C₁₃	1	36	yes	—
178	C₂×C₁₃	—	2	12	no	—
179	C₂×C₁₃	—	4	8	no	—
180	C₂×C₁₃	C₂×#578	1	6	yes	direct product factor
181	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
182	C₂⁴	—	2	2	no	—
183	C₂⁴	C₂×D₄×C₁₃	1	3	yes	—
184	#41	—	2	384	no	—
185	#41	—	2	144	no	—
186	#41	C₂×D₄×C₁₃	1	48	yes	—
187	#41	C₂⁴×C₁₃	1	96	yes	—
188	#41	C₂⁴×C₁₃	1	16	yes	—
189	C₂×Q₈	—	2	128	no	—
190	C₂×Q₈	—	2	72	no	—
191	C₂×Q₈	—	2	18	no	—
192	C₂×Q₈	—	2	144	no	—
193	C₂×Q₈	C₂×D₄×C₁₃	1	12	yes	—
194	C₂×Q₈	C₂⁴×C₁₃	1	36	yes	—
195	C₂×Q₈	C₂⁴×C₁₃	1	96	yes	—
196	C₂×D₄	—	2	72	no	—
197	C₂×D₄	—	2	6	no	—
198	C₂×D₄	C₂×D₄×C₁₃	1	36	yes	—
199	C₂×D₄	C₂⁴×C₁₃	1	12	yes	—
200	C₂²×C₄	—	2	6	no	—
201	C₂²×C₄	—	2	72	no	—
202	C₂²×C₄	—	2	72	no	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	18	yes	—
205	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂⁴×C₁₃	1	18	yes	—
207	Q₁₆	C₂⁴×C₁₃	1	48	yes	—
208	Q₁₆	C₂⁴×C₁₃	1	96	yes	—
209	SD₁₆	C₂⁴×C₁₃	1	96	yes	—
210	D₈	C₂⁴×C₁₃	1	16	yes	—
211	M₁₆	C₂⁴×C₁₃	1	48	yes	—
212	C₂×C₈	C₂⁴×C₁₃	1	6	yes	—
213	C₂×C₈	C₂⁴×C₁₃	1	18	yes	—
214	C₂×C₈	C₂⁴×C₁₃	1	48	yes	—
215	C₁₃	C₂²×#578	1	1	yes	characteristic, direct product factor, Sylow
216	C₂³	—	2	12	no	—
217	C₂³	—	2	12	no	—
218	C₂³	—	4	8	no	—
219	C₂³	C₂²×D₄×C₁₃	1	18	yes	—
220	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
221	Q₈	—	2	96	no	—
222	Q₈	—	2	36	no	—
223	Q₈	C₂²×D₄×C₁₃	1	16	yes	—
224	Q₈	C₂⁵×C₁₃	1	24	yes	—
225	D₄	—	2	96	no	—
226	D₄	—	2	12	no	—
227	D₄	C₂²×D₄×C₁₃	1	48	yes	—
228	D₄	C₂⁵×C₁₃	1	8	yes	—
229	C₂×C₄	—	2	36	no	—
230	C₂×C₄	—	2	96	no	—
231	C₂×C₄	—	2	96	no	—
232	C₂×C₄	C₂²×D₄×C₁₃	1	18	yes	—
233	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
234	C₂×C₄	C₂²×D₄×C₁₃	1	36	yes	—
235	C₂×C₄	C₂⁵×C₁₃	1	3	yes	—
236	C₂×C₄	C₂⁵×C₁₃	1	12	yes	—
237	C₈	C₂⁵×C₁₃	1	4	yes	—
238	C₈	C₂⁵×C₁₃	1	12	yes	—
239	C₂²	—	2	36	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	4	24	no	—
242	C₂²	C₁₃×#578	1	4	yes	direct product factor
243	C₂²	C₂³×D₄×C₁₃	1	12	yes	—
244	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
245	C₄	—	2	24	no	—
246	C₄	C₂³×D₄×C₁₃	1	12	yes	—
247	C₄	C₂³×D₄×C₁₃	1	3	yes	—
248	C₄	C₂⁶×C₁₃	1	1	yes	characteristic
249	C₂	—	2	12	no	—
250	C₂	—	4	8	no	—
251	C₂	C₂×C₁₃×#578	1	6	yes	direct product factor
252	C₂	C₂⁴×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₁₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₁₃×#578	C₂	direct (N × H)
3. C₂²×C₁₃×#137	C₂	semidirect
4. C₂²×C₁₃×#135	C₂	semidirect
5. C₂³×C₁₃×Q₁₆	C₂	semidirect
6. C₂²×C₁₃×#131	C₂	semidirect
8. C₂×C₁₃×2₋¹⁺⁴	C₂²	semidirect
14. C₁₃×#578	C₂²	direct (N × H)
15. C₂×C₁₃×#137	C₂²	semidirect
16. C₂×C₁₃×#135	C₂²	semidirect
19. C₂²×C₁₃×Q₁₆	C₂²	semidirect
22. C₂×C₁₃×#131	C₂²	semidirect
26. C₁₃×2₋¹⁺⁴	C₂³	semidirect
44. C₁₃×#137	C₂³	semidirect
45. C₁₃×#135	C₂³	semidirect
48. C₂×C₁₃×Q₁₆	C₂³	semidirect
51. C₁₃×#131	C₂³	semidirect
56. C₂²×#578	C₁₃	direct (N × H)
90. C₂²×2₋¹⁺⁴	C₂×C₁₃	semidirect
91. C₂×#578	C₂×C₁₃	direct (N × H)
92. C₂²×#137	C₂×C₁₃	semidirect
93. C₂²×#135	C₂×C₁₃	semidirect
94. C₂³×Q₁₆	C₂×C₁₃	semidirect
95. C₂²×#131	C₂×C₁₃	semidirect
120. C₂×2₋¹⁺⁴	C₂²×C₁₃	semidirect
126. #578	C₂²×C₁₃	direct (N × H)
127. C₂×#137	C₂²×C₁₃	semidirect
128. C₂×#135	C₂²×C₁₃	semidirect
131. C₂²×Q₁₆	C₂²×C₁₃	semidirect
134. C₂×#131	C₂²×C₁₃	semidirect
140. C₂²×C₁₃	#578	direct (N × H)
148. 2₋¹⁺⁴	C₂³×C₁₃	semidirect
166. #137	C₂³×C₁₃	semidirect
167. #135	C₂³×C₁₃	semidirect
170. C₂×Q₁₆	C₂³×C₁₃	semidirect
173. #131	C₂³×C₁₃	semidirect
180. C₂×C₁₃	C₂×#578	direct (N × H)
215. C₁₃	C₂²×#578	direct (N × H)
242. C₂²	C₁₃×#578	direct (N × H)
251. C₂	C₂×C₁₃×#578	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×2₋¹⁺⁴	2
C₂×C₁₃×#578	2
C₂²×C₁₃×#137	2
C₂²×C₁₃×#135	2
C₂³×C₁₃×Q₁₆	2
C₂²×C₁₃×#131	2
C₂²×#578	13

Subgroups — C22×C13×#578 #15749425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#578 #15749425