Subgroups — C₁₃×#15253 #15701592

Order 3328.

1338 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#391	C₂	1	2	yes	maximal
2	C₁₃×#1446	C₂	1	4	yes	maximal
3	C₁₃×#1445	C₂	1	4	yes	maximal
4	C₁₃×#1434	C₂	1	4	yes	maximal
5	C₄×C₁₃×#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
10	C₁₃×#391	—	2	8	no	—
11	C₂×C₁₃×#95	C₂²	1	4	yes	—
12	C₂×C₁₃×#95	C₂²	1	4	yes	—
13	C₂×C₁₃×#95	C₂²	1	4	yes	—
14	C₂×C₁₃×#95	C₂²	1	8	yes	—
15	C₂×C₁₃×#95	C₂²	1	4	yes	—
16	C₂×C₁₃×#95	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₃	—	2	4	no	—
18	C₂²×Q₈×C₁₃	—	2	8	no	—
19	C₂²×Q₈×C₁₃	—	2	8	no	—
20	C₂²×Q₈×C₁₃	—	4	4	no	—
21	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
26	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
28	C₂×C₄²×C₁₃	—	4	2	no	—
29	C₂×C₄²×C₁₃	—	4	2	no	—
30	C₂×C₄²×C₁₃	—	4	2	no	—
31	C₂×C₄²×C₁₃	D₄	1	4	yes	—
32	C₂×C₄²×C₁₃	Q₈	1	8	yes	complemented
33	C₁₃×#95	—	2	8	no	—
34	C₁₃×#95	—	2	8	no	—
35	C₁₃×#95	—	2	8	no	—
36	C₁₃×#95	—	2	16	no	—
37	C₁₃×#95	—	4	4	no	—
38	C₁₃×#95	—	4	4	no	—
39	#15253	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	16	no	—
42	C₂×Q₈×C₁₃	—	4	32	no	—
43	C₂×Q₈×C₁₃	—	4	8	no	—
44	C₂×Q₈×C₁₃	—	4	8	no	—
45	C₂²×C₄×C₁₃	—	2	4	no	—
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	8	no	—
52	C₂²×C₄×C₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	8	no	—
60	C₂²×C₄×C₁₃	—	2	8	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₃	#41	1	8	yes	—
68	C₂²×C₄×C₁₃	#41	1	8	yes	—
69	C₁₃×C₄⋊C₄	—	4	8	no	—
70	C₄²×C₁₃	—	4	4	no	—
71	C₄²×C₁₃	—	4	8	no	—
72	C₂×#391	C₂×C₁₃	1	2	yes	—
73	#1446	C₂×C₁₃	1	4	yes	—
74	#1445	C₂×C₁₃	1	4	yes	—
75	#1434	C₂×C₁₃	1	4	yes	—
76	C₄×#95	C₂×C₁₃	1	1	yes	characteristic
77	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
78	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
79	C₂³×C₁₃	C₂×#41	1	4	yes	—
80	C₂³×C₁₃	C₂×#41	1	4	yes	—
81	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
82	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂³×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
84	Q₈×C₁₃	—	4	32	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	16	no	—
92	C₂×C₄×C₁₃	—	2	8	no	—
93	C₂×C₄×C₁₃	—	2	8	no	—
94	C₂×C₄×C₁₃	—	2	8	no	—
95	C₂×C₄×C₁₃	—	2	8	no	—
96	C₂×C₄×C₁₃	—	2	16	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	4	8	no	—
100	C₂×C₄×C₁₃	—	4	8	no	—
101	C₂×C₄×C₁₃	—	4	8	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	—	4	8	no	—
107	C₂×C₄×C₁₃	#124	1	8	yes	—
108	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
109	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
110	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
111	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
112	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
113	#391	—	2	8	no	—
114	C₂×#95	C₂²×C₁₃	1	4	yes	—
115	C₂×#95	C₂²×C₁₃	1	4	yes	—
116	C₂×#95	C₂²×C₁₃	1	8	yes	—
117	C₂×#95	C₂²×C₁₃	1	4	yes	—
118	C₂×#95	C₂²×C₁₃	1	4	yes	—
119	C₂×#95	C₂²×C₁₃	1	4	yes	—
120	C₂²×C₁₃	#525	1	2	yes	—
121	C₂²×C₁₃	C₂×#124	1	2	yes	—
122	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₁₃	#533	1	4	yes	—
124	C₂²×C₁₃	#533	1	4	yes	—
125	C₂²×C₁₃	#534	1	1	yes	characteristic
126	C₂²×C₁₃	#537	1	1	yes	characteristic
127	C₂²×C₁₃	#543	1	2	yes	—
128	C₂²×C₁₃	#544	1	2	yes	—
129	C₂²×C₁₃	#551	1	8	yes	—
130	C₂²×C₁₃	#556	1	4	yes	—
131	C₂²×C₁₃	#559	1	4	yes	—
132	C₄×C₁₃	—	2	8	no	—
133	C₄×C₁₃	—	2	4	no	—
134	C₄×C₁₃	—	4	8	no	—
135	C₄×C₁₃	—	4	8	no	—
136	C₄×C₁₃	—	8	4	no	—
137	C₂²×Q₈	—	2	4	no	—
138	C₂²×Q₈	—	2	8	no	—
139	C₂²×Q₈	—	2	8	no	—
140	C₂²×Q₈	—	4	4	no	—
141	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
142	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
143	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
144	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
145	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	8	no	—
148	C₂×C₄²	—	4	2	no	—
149	C₂×C₄²	—	4	2	no	—
150	C₂×C₄²	—	4	2	no	—
151	C₂×C₄²	D₄×C₁₃	1	4	yes	—
152	C₂×C₄²	Q₈×C₁₃	1	8	yes	complemented
153	#95	—	2	8	no	—
154	#95	—	2	8	no	—
155	#95	—	2	8	no	—
156	#95	—	2	16	no	—
157	#95	—	4	4	no	—
158	#95	—	4	4	no	—
159	C₂×C₁₃	#2399	1	1	yes	characteristic
160	C₂×C₁₃	#2561	1	2	yes	—
161	C₂×C₁₃	#2592	1	2	yes	—
162	C₂×C₁₃	#2605	1	4	yes	—
163	C₂×C₁₃	#2607	1	4	yes	—
164	C₂×C₁₃	#2659	1	1	yes	characteristic
165	C₂×C₁₃	#2862	1	1	yes	characteristic
166	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
167	C₂×Q₈	—	2	16	no	—
168	C₂×Q₈	—	4	32	no	—
169	C₂×Q₈	—	4	8	no	—
170	C₂×Q₈	—	4	8	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
191	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
192	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
193	C₂²×C₄	C₁₃×#41	1	8	yes	—
194	C₂²×C₄	C₁₃×#41	1	8	yes	—
195	C₄⋊C₄	—	4	8	no	—
196	C₄²	—	4	4	no	—
197	C₄²	—	4	8	no	—
198	C₁₃	#15253	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
200	C₂³	C₂²×Q₈×C₁₃	1	2	yes	—
201	C₂³	C₂×C₁₃×#41	1	4	yes	—
202	C₂³	C₂×C₁₃×#41	1	4	yes	—
203	C₂³	C₁₃×H(5,𝔽₂)	1	2	yes	—
204	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
205	C₂³	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
206	Q₈	—	4	32	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	16	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	C₁₃×#124	1	8	yes	—
230	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
231	C₂²	C₁₃×#525	1	2	yes	—
232	C₂²	C₂×C₁₃×#124	1	2	yes	—
233	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
234	C₂²	C₁₃×#533	1	4	yes	—
235	C₂²	C₁₃×#533	1	4	yes	—
236	C₂²	C₁₃×#534	1	1	yes	characteristic
237	C₂²	C₁₃×#537	1	1	yes	characteristic
238	C₂²	C₁₃×#543	1	2	yes	—
239	C₂²	C₁₃×#544	1	2	yes	—
240	C₂²	C₁₃×#551	1	8	yes	—
241	C₂²	C₁₃×#556	1	4	yes	—
242	C₂²	C₁₃×#559	1	4	yes	—
243	C₄	—	2	8	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	—	4	8	no	—
247	C₄	—	8	4	no	—
248	C₂	C₁₃×#2399	1	1	yes	characteristic
249	C₂	C₁₃×#2561	1	2	yes	—
250	C₂	C₁₃×#2592	1	2	yes	—
251	C₂	C₁₃×#2605	1	4	yes	—
252	C₂	C₁₃×#2607	1	4	yes	—
253	C₂	C₁₃×#2659	1	1	yes	characteristic
254	C₂	C₁₃×#2862	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
32. C₂×C₄²×C₁₃	Q₈	semidirect
39. #15253	C₁₃	direct (N × H)
152. C₂×C₄²	Q₈×C₁₃	semidirect
198. C₁₃	#15253	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×#391	2
C₁₃×#1446	2
C₁₃×#1445	2
C₁₃×#1434	2
C₄×C₁₃×#95	2
#15253	13

Subgroups — C13×#15253 #15701592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#15253 #15701592