Subgroups — C₂²×C₁₃×#541 #15749020

Order 3328.

7110 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#541	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#126	C₂	1	4	yes	maximal
4	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₃×#125	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
7	C₂²×C₄×Q₈×C₁₃	C₂	1	2	yes	maximal, complemented
8	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#541	C₂²	1	256	yes	direct product factor
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
13	C₂×C₁₃×#126	C₂²	1	96	yes	—
14	C₂×C₁₃×#125	C₂²	1	24	yes	complemented
15	C₂×C₁₃×#125	C₂²	1	96	yes	complemented
16	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	48	yes	—
17	C₂×C₄×Q₈×C₁₃	C₂²	1	48	yes	complemented
18	C₂×C₁₃×#117	C₂²	1	24	yes	—
19	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
23	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
24	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃	—	2	16	no	—
32	C₂²×Q₈×C₁₃	C₂³	1	24	yes	—
33	C₂³×C₄×C₁₃	—	2	8	no	—
34	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
42	C₁₃×#126	C₂³	1	256	yes	—
43	C₁₃×#125	C₂³	1	64	yes	complemented
44	C₁₃×#125	C₂³	1	256	yes	complemented
45	C₁₃×C₂²⋊Q₈	C₂³	1	128	yes	—
46	C₄×Q₈×C₁₃	C₂³	1	128	yes	complemented
47	C₁₃×#117	C₂³	1	64	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
54	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
55	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
57	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	48	yes	—
59	C₂²×#541	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₃	—	2	4	no	—
61	C₂⁴×C₁₃	—	2	8	no	—
62	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
63	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂×Q₈×C₁₃	—	2	96	no	—
65	C₂×Q₈×C₁₃	C₂⁴	1	32	yes	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	16	no	—
68	C₂²×C₄×C₁₃	—	2	8	no	—
69	C₂²×C₄×C₁₃	—	2	8	no	—
70	C₂²×C₄×C₁₃	—	2	16	no	—
71	C₂²×C₄×C₁₃	—	2	48	no	—
72	C₂²×C₄×C₁₃	#41	1	8	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
80	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
81	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
82	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
83	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
84	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
85	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
86	C₁₃×C₂²⋊C₄	C₂⁴	1	32	yes	—
87	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	—
88	C₄²×C₁₃	C₂⁴	1	16	yes	—
89	C₄²×C₁₃	C₂⁴	1	16	yes	—
90	C₄²×C₁₃	C₂⁴	1	64	yes	—
91	C₂×#541	C₂×C₁₃	1	48	yes	direct product factor
92	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂²×#126	C₂×C₁₃	1	4	yes	—
94	C₂²×#125	C₂×C₁₃	1	4	yes	complemented
95	C₂²×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂²×C₂²⋊Q₈	C₂×C₁₃	1	2	yes	—
97	C₂²×C₄×Q₈	C₂×C₁₃	1	2	yes	complemented
98	C₂²×#117	C₂×C₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	—	2	12	no	—
100	C₂³×C₁₃	—	2	24	no	—
101	C₂³×C₁₃	—	4	16	no	—
102	C₂³×C₁₃	C₂×#41	1	4	yes	—
103	C₂³×C₁₃	2₋¹⁺⁴	1	8	yes	—
104	C₂³×C₁₃	C₂⁵	1	4	yes	—
105	C₂³×C₁₃	C₂⁵	1	3	yes	—
106	Q₈×C₁₃	—	2	64	no	—
107	C₂×C₄×C₁₃	—	2	12	no	—
108	C₂×C₄×C₁₃	—	2	48	no	—
109	C₂×C₄×C₁₃	—	2	24	no	—
110	C₂×C₄×C₁₃	—	2	24	no	—
111	C₂×C₄×C₁₃	—	2	48	no	—
112	C₂×C₄×C₁₃	—	2	32	no	—
113	C₂×C₄×C₁₃	C₂×#41	1	24	yes	—
114	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
115	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
116	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
117	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
118	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
120	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
121	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
122	#541	C₂²×C₁₃	1	256	yes	direct product factor
123	C₂×C₄⋊Q₈	C₂²×C₁₃	1	24	yes	complemented
124	C₂×#126	C₂²×C₁₃	1	96	yes	—
125	C₂×#125	C₂²×C₁₃	1	24	yes	complemented
126	C₂×#125	C₂²×C₁₃	1	96	yes	complemented
127	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	48	yes	—
128	C₂×C₄×Q₈	C₂²×C₁₃	1	48	yes	complemented
129	C₂×#117	C₂²×C₁₃	1	24	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
134	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
136	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
137	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
138	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
139	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₄²	C₂²×C₁₃	1	4	yes	—
141	C₂²×C₁₃	—	2	4	no	—
142	C₂²×C₁₃	—	2	8	no	—
143	C₂²×C₁₃	—	4	24	no	—
144	C₂²×C₁₃	#541	1	16	yes	direct product factor
145	C₂²×C₁₃	C₂²×#41	1	6	yes	—
146	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	12	yes	—
147	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	16	no	—
150	C₄×C₁₃	—	2	8	no	—
151	C₄×C₁₃	—	2	8	no	—
152	C₄×C₁₃	—	2	16	no	—
153	C₄×C₁₃	C₂²×#41	1	8	yes	—
154	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
155	C₂²×Q₈	—	2	16	no	—
156	C₂²×Q₈	C₂³×C₁₃	1	24	yes	—
157	C₂³×C₄	—	2	8	no	—
158	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
159	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
160	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
161	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
162	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
164	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
165	C₄⋊Q₈	C₂³×C₁₃	1	64	yes	complemented
166	#126	C₂³×C₁₃	1	256	yes	—
167	#125	C₂³×C₁₃	1	64	yes	complemented
168	#125	C₂³×C₁₃	1	256	yes	complemented
169	C₂²⋊Q₈	C₂³×C₁₃	1	128	yes	—
170	C₄×Q₈	C₂³×C₁₃	1	128	yes	complemented
171	#117	C₂³×C₁₃	1	64	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
173	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
174	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
178	C₂×C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
179	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
180	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
181	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
182	C₂×C₄²	C₂³×C₁₃	1	48	yes	—
183	C₂×C₁₃	—	4	4	no	—
184	C₂×C₁₃	C₂×#541	1	12	yes	direct product factor
185	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
186	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	2	yes	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	8	no	—
189	C₂⁴	C₂⁴×C₁₃	1	6	yes	—
190	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
191	C₂×Q₈	—	2	96	no	—
192	C₂×Q₈	C₂⁴×C₁₃	1	32	yes	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	16	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	48	no	—
199	C₂²×C₄	C₁₃×#41	1	8	yes	—
200	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
201	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
202	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
203	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
204	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
205	C₂²×C₄	C₂⁴×C₁₃	1	16	yes	—
206	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
207	C₄⋊C₄	C₂⁴×C₁₃	1	32	yes	—
208	C₄⋊C₄	C₂⁴×C₁₃	1	32	yes	—
209	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
210	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
211	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
212	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
213	C₂²⋊C₄	C₂⁴×C₁₃	1	32	yes	—
214	C₂²⋊C₄	C₂⁴×C₁₃	1	64	yes	—
215	C₄²	C₂⁴×C₁₃	1	16	yes	—
216	C₄²	C₂⁴×C₁₃	1	16	yes	—
217	C₄²	C₂⁴×C₁₃	1	64	yes	—
218	C₁₃	C₂²×#541	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	2	12	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	4	16	no	—
222	C₂³	C₂×C₁₃×#41	1	4	yes	—
223	C₂³	C₁₃×2₋¹⁺⁴	1	8	yes	—
224	C₂³	C₂⁵×C₁₃	1	4	yes	—
225	C₂³	C₂⁵×C₁₃	1	3	yes	—
226	Q₈	—	2	64	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	2	24	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	2	48	no	—
232	C₂×C₄	—	2	32	no	—
233	C₂×C₄	C₂×C₁₃×#41	1	24	yes	—
234	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
235	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
236	C₂×C₄	C₂⁵×C₁₃	1	16	yes	—
237	C₂×C₄	C₂⁵×C₁₃	1	16	yes	—
238	C₂×C₄	C₂⁵×C₁₃	1	4	yes	—
239	C₂×C₄	C₂⁵×C₁₃	1	4	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	24	no	—
243	C₂²	C₁₃×#541	1	16	yes	direct product factor
244	C₂²	C₂²×C₁₃×#41	1	6	yes	—
245	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	12	yes	—
246	C₂²	C₂⁶×C₁₃	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	16	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	8	no	—
251	C₄	—	2	16	no	—
252	C₄	C₂²×C₁₃×#41	1	8	yes	—
253	C₂	—	4	4	no	—
254	C₂	C₂×C₁₃×#541	1	12	yes	direct product factor
255	C₂	C₂³×C₁₃×#41	1	1	yes	characteristic
256	C₂	C₂²×C₁₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₃
prime radical	C₂⁵×C₁₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#541	C₂	direct (N × H)
2. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
4. C₂²×C₁₃×#125	C₂	semidirect
5. C₂²×C₁₃×#125	C₂	semidirect
7. C₂²×C₄×Q₈×C₁₃	C₂	semidirect
11. C₁₃×#541	C₂²	direct (N × H)
12. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
14. C₂×C₁₃×#125	C₂²	semidirect
15. C₂×C₁₃×#125	C₂²	semidirect
17. C₂×C₄×Q₈×C₁₃	C₂²	semidirect
41. C₁₃×C₄⋊Q₈	C₂³	semidirect
43. C₁₃×#125	C₂³	semidirect
44. C₁₃×#125	C₂³	semidirect
46. C₄×Q₈×C₁₃	C₂³	semidirect
59. C₂²×#541	C₁₃	direct (N × H)
91. C₂×#541	C₂×C₁₃	direct (N × H)
92. C₂²×C₄⋊Q₈	C₂×C₁₃	semidirect
94. C₂²×#125	C₂×C₁₃	semidirect
95. C₂²×#125	C₂×C₁₃	semidirect
97. C₂²×C₄×Q₈	C₂×C₁₃	semidirect
122. #541	C₂²×C₁₃	direct (N × H)
123. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
125. C₂×#125	C₂²×C₁₃	semidirect
126. C₂×#125	C₂²×C₁₃	semidirect
128. C₂×C₄×Q₈	C₂²×C₁₃	semidirect
144. C₂²×C₁₃	#541	direct (N × H)
165. C₄⋊Q₈	C₂³×C₁₃	semidirect
167. #125	C₂³×C₁₃	semidirect
168. #125	C₂³×C₁₃	semidirect
170. C₄×Q₈	C₂³×C₁₃	semidirect
184. C₂×C₁₃	C₂×#541	direct (N × H)
218. C₁₃	C₂²×#541	direct (N × H)
243. C₂²	C₁₃×#541	direct (N × H)
254. C₂	C₂×C₁₃×#541	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×C₁₃×#541	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₁₃×#126	2
C₂²×C₁₃×#125	2
C₂²×C₁₃×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×C₁₃×#117	2
C₂²×#541	13

Subgroups — C22×C13×#541 #15749020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#541 #15749020