Subgroups — C₁₃×#10265 #15696604

Order 3328.

574 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×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#462	C₂	1	2	yes	maximal
3	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#332	C₂	1	2	yes	maximal
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	—	2	2	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#500	—	2	4	no	—
10	C₁₃×#462	—	2	4	no	—
11	C₁₃×#462	—	2	4	no	—
12	C₂×C₁₃×#107	—	2	2	no	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#103	—	2	2	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#336	—	2	2	no	—
18	C₁₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₁₃×Q₁₆	—	2	8	no	—
22	C₂×C₁₃×Q₁₆	—	4	4	no	—
23	C₂×C₁₃×Q₁₆	—	4	4	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
31	C₄×Q₈×C₁₃	—	2	2	no	—
32	C₄×Q₈×C₁₃	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#107	—	2	8	no	—
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
43	C₁₃×#103	—	2	8	no	—
44	C₄×C₈×C₁₃	—	2	2	no	—
45	#10265	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×Q₈×C₁₃	—	2	1	no	—
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂×Q₈×C₁₃	D₈	1	2	yes	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₁₃×Q₁₆	—	4	16	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	—	4	1	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	2	4	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₄²×C₁₃	—	4	1	no	—
78	C₄²×C₁₃	—	4	1	no	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₄²×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×#500	C₂×C₁₃	1	1	yes	characteristic
82	C₂×#462	C₂×C₁₃	1	2	yes	—
83	#1984	C₂×C₁₃	1	1	yes	characteristic
84	C₂×#332	C₂×C₁₃	1	2	yes	—
85	#1277	C₂×C₁₃	1	1	yes	characteristic
86	C₂³×C₁₃	#121	1	1	yes	characteristic
87	Q₈×C₁₃	—	2	4	no	—
88	Q₈×C₁₃	—	4	2	no	—
89	Q₈×C₁₃	—	4	4	no	—
90	Q₈×C₁₃	—	4	4	no	—
91	Q₈×C₁₃	—	8	2	no	—
92	Q₈×C₁₃	—	8	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	4	1	no	—
96	C₂×C₄×C₁₃	—	4	1	no	—
97	C₂×C₄×C₁₃	—	4	1	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	8	2	no	—
101	C₂×C₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	#120	1	2	yes	—
103	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#121	1	2	yes	—
105	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
109	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₁₃	#137	1	2	yes	—
111	C₈×C₁₃	—	2	4	no	—
112	C₈×C₁₃	—	4	4	no	—
113	C₈×C₁₃	—	8	1	no	—
114	C₂²×Q₁₆	—	2	2	no	—
115	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
116	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
117	#500	—	2	4	no	—
118	#462	—	2	4	no	—
119	#462	—	2	4	no	—
120	C₂×#107	—	2	2	no	—
121	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
123	C₂×#103	—	2	2	no	—
124	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	#336	—	2	2	no	—
126	#332	—	2	4	no	—
127	C₂²×C₁₃	#392	1	1	yes	characteristic
128	C₂²×C₁₃	#462	1	2	yes	—
129	C₂²×C₁₃	#466	1	1	yes	characteristic
130	C₂²×C₁₃	#469	1	1	yes	characteristic
131	C₂²×C₁₃	#473	1	2	yes	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	—	8	4	no	—
135	C₄×C₁₃	—	8	1	no	—
136	C₄×C₁₃	#447	1	2	yes	—
137	C₄×C₁₃	#449	1	2	yes	—
138	C₄×C₁₃	#451	1	2	yes	—
139	C₄×C₁₃	#494	1	1	yes	characteristic
140	C₄×C₁₃	#497	1	1	yes	characteristic
141	C₂²×Q₈	—	2	2	no	—
142	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
143	C₂×Q₁₆	—	2	8	no	—
144	C₂×Q₁₆	—	4	4	no	—
145	C₂×Q₁₆	—	4	4	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	D₄×C₁₃	1	2	yes	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	D₄×C₁₃	1	4	yes	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
161	#107	—	2	8	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
165	#103	—	2	8	no	—
166	C₄×C₈	—	2	2	no	—
167	C₂×C₁₃	#1674	1	2	yes	—
168	C₂×C₁₃	#1694	1	2	yes	—
169	C₂×C₁₃	#1999	1	1	yes	characteristic
170	C₂×C₁₃	#2016	1	1	yes	characteristic
171	C₂×C₁₃	#2020	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	C₁₃×D₈	1	2	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
188	Q₁₆	—	4	16	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₁₃×Q₁₆	1	4	yes	complemented
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	C₂×D₄×C₁₃	1	2	yes	—
206	C₄²	C₂×D₄×C₁₃	1	2	yes	—
207	C₁₃	#10265	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	C₁₃×#121	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	Q₈	—	8	2	no	—
214	Q₈	—	8	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	C₁₃×#120	1	2	yes	—
225	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
226	C₂×C₄	C₁₃×#121	1	2	yes	—
227	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
228	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
229	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
230	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
231	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₂×C₄	C₁₃×#137	1	2	yes	—
233	C₈	—	2	4	no	—
234	C₈	—	4	4	no	—
235	C₈	—	8	1	no	—
236	C₂²	C₁₃×#392	1	1	yes	characteristic
237	C₂²	C₁₃×#462	1	2	yes	—
238	C₂²	C₁₃×#466	1	1	yes	characteristic
239	C₂²	C₁₃×#469	1	1	yes	characteristic
240	C₂²	C₁₃×#473	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	4	no	—
244	C₄	—	8	1	no	—
245	C₄	C₁₃×#447	1	2	yes	—
246	C₄	C₁₃×#449	1	2	yes	—
247	C₄	C₁₃×#451	1	2	yes	—
248	C₄	C₁₃×#494	1	1	yes	characteristic
249	C₄	C₁₃×#497	1	1	yes	characteristic
250	C₂	C₁₃×#1674	1	2	yes	—
251	C₂	C₁₃×#1694	1	2	yes	—
252	C₂	C₁₃×#1999	1	1	yes	characteristic
253	C₂	C₁₃×#2016	1	1	yes	characteristic
254	C₂	C₁₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
45. #10265	C₁₃	direct (N × H)
70. C₂×C₈×C₁₃	Q₁₆	semidirect
196. C₂×C₈	C₁₃×Q₁₆	semidirect
207. C₁₃	#10265	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×#500	2
C₂×C₁₃×#462	2
C₁₃×#1984	2
C₂×C₁₃×#332	2
C₁₃×#1277	2
#10265	13

Subgroups — C13×#10265 #15696604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10265 #15696604