Subgroups — C₂²×C₁₃×#495 #15746970

Order 3328.

3518 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×#124	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#495	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₁₃×Q₁₆	—	2	8	no	—
10	C₂²×C₁₃×Q₁₆	C₂²	1	12	yes	complemented
11	C₂²×C₁₃×SD₁₆	—	2	16	no	—
12	C₂²×C₁₃×SD₁₆	C₂²	1	24	yes	complemented
13	C₂²×C₁₃×D₈	—	2	8	no	—
14	C₂²×C₁₃×D₈	C₂²	1	12	yes	—
15	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#124	—	2	12	no	—
18	C₂×C₁₃×#124	C₂²	1	24	yes	complemented
19	C₂²×C₁₃×C₂²⋊C₄	—	2	4	no	—
20	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#495	C₂²	1	64	yes	direct product factor
22	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
23	C₂⁵×C₁₃	—	2	2	no	—
24	C₂²×Q₈×C₁₃	—	2	6	no	—
25	C₂²×Q₈×C₁₃	—	2	8	no	—
26	C₂²×Q₈×C₁₃	—	2	8	no	—
27	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₃	—	2	8	no	—
29	C₂²×D₄×C₁₃	—	2	8	no	—
30	C₂²×D₄×C₁₃	—	2	6	no	—
31	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
32	C₂³×C₄×C₁₃	—	2	2	no	—
33	C₂³×C₄×C₁₃	D₄	1	2	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₁₃×Q₁₆	—	2	48	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	16	yes	complemented
37	C₂×C₁₃×SD₁₆	—	2	96	no	—
38	C₂×C₁₃×SD₁₆	C₂³	1	32	yes	complemented
39	C₂×C₁₃×D₈	—	2	48	no	—
40	C₂×C₁₃×D₈	C₂³	1	16	yes	—
41	C₂²×C₈×C₁₃	D₄	1	8	yes	—
42	C₂²×C₈×C₁₃	D₄	1	8	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
45	C₁₃×#124	—	2	48	no	—
46	C₁₃×#124	C₂³	1	32	yes	complemented
47	C₂×C₁₃×C₂²⋊C₄	—	2	48	no	—
48	C₂×C₄²×C₁₃	D₄	1	6	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
50	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
51	C₂²×#495	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	12	no	—
53	C₂⁴×C₁₃	—	4	8	no	—
54	C₂⁴×C₁₃	—	4	16	no	—
55	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₃	—	2	12	no	—
57	C₂×Q₈×C₁₃	—	2	24	no	—
58	C₂×Q₈×C₁₃	—	2	24	no	—
59	C₂×Q₈×C₁₃	—	4	24	no	—
60	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
61	C₂×D₄×C₁₃	—	2	24	no	—
62	C₂×D₄×C₁₃	—	2	24	no	—
63	C₂×D₄×C₁₃	—	2	12	no	—
64	C₂×D₄×C₁₃	—	4	24	no	—
65	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₁₃	—	2	8	no	—
67	C₂²×C₄×C₁₃	—	2	12	no	—
68	C₂²×C₄×C₁₃	—	4	8	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
74	C₁₃×Q₁₆	—	2	32	no	—
75	C₁₃×SD₁₆	—	2	64	no	—
76	C₁₃×D₈	—	2	32	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
81	C₁₃×C₂²⋊C₄	—	2	64	no	—
82	C₄²×C₁₃	C₂×D₄	1	12	yes	—
83	C₄²×C₁₃	C₂⁴	1	4	yes	—
84	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂³×SD₁₆	C₂×C₁₃	1	2	yes	complemented
86	C₂³×D₈	C₂×C₁₃	1	1	yes	characteristic
87	C₂²×#124	C₂×C₁₃	1	2	yes	complemented
88	C₂×#495	C₂×C₁₃	1	24	yes	direct product factor
89	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₁₃	—	2	8	no	—
91	C₂³×C₁₃	—	4	24	no	—
92	C₂³×C₁₃	—	4	48	no	—
93	C₂³×C₁₃	—	8	32	no	—
94	C₂³×C₁₃	#127	1	4	yes	—
95	C₂³×C₁₃	#135	1	8	yes	—
96	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
97	Q₈×C₁₃	—	2	8	no	—
98	Q₈×C₁₃	—	2	8	no	—
99	Q₈×C₁₃	—	4	24	no	—
100	D₄×C₁₃	—	2	8	no	—
101	D₄×C₁₃	—	2	8	no	—
102	D₄×C₁₃	—	4	24	no	—
103	C₂×C₄×C₁₃	—	2	24	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	4	24	no	—
106	C₂×C₄×C₁₃	#127	1	12	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
110	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
112	C₈×C₁₃	C₂²×D₄	1	8	yes	—
113	C₈×C₁₃	C₂²×D₄	1	8	yes	—
114	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
115	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
116	C₂²×Q₁₆	—	2	8	no	—
117	C₂²×Q₁₆	C₂²×C₁₃	1	12	yes	complemented
118	C₂²×SD₁₆	—	2	16	no	—
119	C₂²×SD₁₆	C₂²×C₁₃	1	24	yes	complemented
120	C₂²×D₈	—	2	8	no	—
121	C₂²×D₈	C₂²×C₁₃	1	12	yes	—
122	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#124	—	2	12	no	—
125	C₂×#124	C₂²×C₁₃	1	24	yes	complemented
126	C₂²×C₂²⋊C₄	—	2	4	no	—
127	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
128	#495	C₂²×C₁₃	1	64	yes	direct product factor
129	C₂×C₄×C₈	C₂²×C₁₃	1	12	yes	complemented
130	C₂²×C₁₃	—	4	8	no	—
131	C₂²×C₁₃	—	4	16	no	—
132	C₂²×C₁₃	—	8	48	no	—
133	C₂²×C₁₃	#495	1	16	yes	direct product factor
134	C₂²×C₁₃	C₂×#127	1	6	yes	—
135	C₂²×C₁₃	C₂×#135	1	12	yes	—
136	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
137	C₄×C₁₃	—	2	8	no	—
138	C₄×C₁₃	—	4	8	no	—
139	C₄×C₁₃	C₂×#127	1	6	yes	—
140	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
142	C₂⁵	—	2	2	no	—
143	C₂²×Q₈	—	2	6	no	—
144	C₂²×Q₈	—	2	8	no	—
145	C₂²×Q₈	—	2	8	no	—
146	C₂²×Q₈	C₂³×C₁₃	1	12	yes	—
147	C₂²×D₄	—	2	8	no	—
148	C₂²×D₄	—	2	8	no	—
149	C₂²×D₄	—	2	6	no	—
150	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	D₄×C₁₃	1	2	yes	—
153	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
154	C₂×Q₁₆	—	2	48	no	—
155	C₂×Q₁₆	C₂³×C₁₃	1	16	yes	complemented
156	C₂×SD₁₆	—	2	96	no	—
157	C₂×SD₁₆	C₂³×C₁₃	1	32	yes	complemented
158	C₂×D₈	—	2	48	no	—
159	C₂×D₈	C₂³×C₁₃	1	16	yes	—
160	C₂²×C₈	D₄×C₁₃	1	8	yes	—
161	C₂²×C₈	D₄×C₁₃	1	8	yes	—
162	C₂²×C₈	C₂³×C₁₃	1	6	yes	—
163	C₂²×C₈	C₂³×C₁₃	1	6	yes	—
164	#124	—	2	48	no	—
165	#124	C₂³×C₁₃	1	32	yes	complemented
166	C₂×C₂²⋊C₄	—	2	48	no	—
167	C₂×C₄²	D₄×C₁₃	1	6	yes	—
168	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
169	C₄×C₈	C₂³×C₁₃	1	16	yes	complemented
170	C₂×C₁₃	—	8	8	no	—
171	C₂×C₁₃	C₂×#495	1	12	yes	direct product factor
172	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
173	C₂×C₁₃	C₂²×#135	1	2	yes	—
174	C₂⁴	—	2	12	no	—
175	C₂⁴	—	4	8	no	—
176	C₂⁴	—	4	16	no	—
177	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
178	C₂×Q₈	—	2	12	no	—
179	C₂×Q₈	—	2	24	no	—
180	C₂×Q₈	—	2	24	no	—
181	C₂×Q₈	—	4	24	no	—
182	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
183	C₂×D₄	—	2	24	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	—	2	12	no	—
186	C₂×D₄	—	4	24	no	—
187	C₂×D₄	C₂⁴×C₁₃	1	8	yes	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	12	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
192	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
193	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
194	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
196	Q₁₆	—	2	32	no	—
197	SD₁₆	—	2	64	no	—
198	D₈	—	2	32	no	—
199	C₂×C₈	C₂×D₄×C₁₃	1	24	yes	—
200	C₂×C₈	C₂×D₄×C₁₃	1	24	yes	—
201	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
202	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
203	C₂²⋊C₄	—	2	64	no	—
204	C₄²	C₂×D₄×C₁₃	1	12	yes	—
205	C₄²	C₂⁴×C₁₃	1	4	yes	—
206	C₁₃	C₂²×#495	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³	—	2	8	no	—
208	C₂³	—	4	24	no	—
209	C₂³	—	4	48	no	—
210	C₂³	—	8	32	no	—
211	C₂³	C₁₃×#127	1	4	yes	—
212	C₂³	C₁₃×#135	1	8	yes	—
213	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
214	Q₈	—	2	8	no	—
215	Q₈	—	2	8	no	—
216	Q₈	—	4	24	no	—
217	D₄	—	2	8	no	—
218	D₄	—	2	8	no	—
219	D₄	—	4	24	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	24	no	—
223	C₂×C₄	C₁₃×#127	1	12	yes	—
224	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
225	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
226	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
227	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
228	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
229	C₈	C₂²×D₄×C₁₃	1	8	yes	—
230	C₈	C₂²×D₄×C₁₃	1	8	yes	—
231	C₂²	—	4	16	no	—
232	C₂²	—	4	8	no	—
233	C₂²	—	8	48	no	—
234	C₂²	C₁₃×#495	1	16	yes	direct product factor
235	C₂²	C₂×C₁₃×#127	1	6	yes	—
236	C₂²	C₂×C₁₃×#135	1	12	yes	—
237	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
238	C₄	—	2	8	no	—
239	C₄	—	4	8	no	—
240	C₄	C₂×C₁₃×#127	1	6	yes	—
241	C₄	C₂³×D₄×C₁₃	1	1	yes	characteristic
242	C₄	C₂³×D₄×C₁₃	1	1	yes	characteristic
243	C₂	—	8	8	no	—
244	C₂	C₂×C₁₃×#495	1	12	yes	direct product factor
245	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
246	C₂	C₂²×C₁₃×#135	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂³×C₁₃×D₈	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×Q₁₆	C₂	semidirect
2. C₂³×C₁₃×SD₁₆	C₂	semidirect
4. C₂²×C₁₃×#124	C₂	semidirect
5. C₂×C₁₃×#495	C₂	direct (N × H)
6. C₂²×C₄×C₈×C₁₃	C₂	semidirect
10. C₂²×C₁₃×Q₁₆	C₂²	semidirect
12. C₂²×C₁₃×SD₁₆	C₂²	semidirect
18. C₂×C₁₃×#124	C₂²	semidirect
21. C₁₃×#495	C₂²	direct (N × H)
22. C₂×C₄×C₈×C₁₃	C₂²	semidirect
36. C₂×C₁₃×Q₁₆	C₂³	semidirect
38. C₂×C₁₃×SD₁₆	C₂³	semidirect
46. C₁₃×#124	C₂³	semidirect
50. C₄×C₈×C₁₃	C₂³	semidirect
51. C₂²×#495	C₁₃	direct (N × H)
84. C₂³×Q₁₆	C₂×C₁₃	semidirect
85. C₂³×SD₁₆	C₂×C₁₃	semidirect
87. C₂²×#124	C₂×C₁₃	semidirect
88. C₂×#495	C₂×C₁₃	direct (N × H)
89. C₂²×C₄×C₈	C₂×C₁₃	semidirect
117. C₂²×Q₁₆	C₂²×C₁₃	semidirect
119. C₂²×SD₁₆	C₂²×C₁₃	semidirect
125. C₂×#124	C₂²×C₁₃	semidirect
128. #495	C₂²×C₁₃	direct (N × H)
129. C₂×C₄×C₈	C₂²×C₁₃	semidirect
133. C₂²×C₁₃	#495	direct (N × H)
155. C₂×Q₁₆	C₂³×C₁₃	semidirect
157. C₂×SD₁₆	C₂³×C₁₃	semidirect
165. #124	C₂³×C₁₃	semidirect
169. C₄×C₈	C₂³×C₁₃	semidirect
171. C₂×C₁₃	C₂×#495	direct (N × H)
206. C₁₃	C₂²×#495	direct (N × H)
234. C₂²	C₁₃×#495	direct (N × H)
244. C₂	C₂×C₁₃×#495	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×Q₁₆	2
C₂³×C₁₃×SD₁₆	2
C₂³×C₁₃×D₈	2
C₂²×C₁₃×#124	2
C₂×C₁₃×#495	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#495	13

Subgroups — C22×C13×#495 #15746970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#495 #15746970