Subgroups — C₁₃×#10821 #15697160

Order 3328.

854 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#340	C₂	1	2	yes	maximal
3	C₂×C₁₃×#340	C₂	1	2	yes	maximal
4	C₁₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#97	—	2	4	no	—
10	C₂×C₁₃×#97	C₄	1	4	yes	complemented
11	C₂×C₁₃×#95	—	2	4	no	—
12	C₁₃×#340	—	2	4	no	—
13	C₁₃×#340	C₄	1	8	yes	complemented
14	C₂³×C₄×C₁₃	—	2	4	no	—
15	C₂³×C₄×C₁₃	D₄	1	2	yes	—
16	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	D₄	1	4	yes	—
25	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
35	C₁₃×#97	—	2	8	no	—
36	C₁₃×#95	—	4	8	no	—
37	#10821	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	8	no	—
44	C₂²×C₄×C₁₃	—	4	4	no	—
45	C₂²×C₄×C₁₃	—	4	4	no	—
46	C₂²×C₄×C₁₃	—	4	4	no	—
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	4	no	—
49	C₂²×C₄×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	2	yes	—
56	C₂²×C₄×C₁₃	#41	1	2	yes	—
57	C₂²×C₄×C₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	8	no	—
64	C₂×C₈×C₁₃	—	2	8	no	—
65	C₂×C₈×C₁₃	C₄²	1	4	yes	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
67	C₁₃×C₄⋊C₄	—	2	8	no	—
68	C₁₃×C₄⋊C₄	—	2	8	no	—
69	C₁₃×C₄⋊C₄	—	4	4	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	4	no	—
74	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
75	C₂×#340	C₂×C₁₃	1	2	yes	—
76	C₂×#340	C₂×C₁₃	1	2	yes	—
77	C₄⋊#95	C₂×C₁₃	1	2	yes	—
78	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₁₃	C₄×D₄	1	2	yes	—
80	C₂³×C₁₃	#121	1	2	yes	—
81	C₂³×C₁₃	#124	1	2	yes	—
82	C₂³×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
83	C₂³×C₁₃	#137	1	4	yes	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	8	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	8	no	—
90	C₂×C₄×C₁₃	—	4	8	no	—
91	C₂×C₄×C₁₃	—	4	8	no	—
92	C₂×C₄×C₁₃	—	8	4	no	—
93	C₂×C₄×C₁₃	—	8	4	no	—
94	C₂×C₄×C₁₃	—	8	4	no	—
95	C₂×C₄×C₁₃	—	8	4	no	—
96	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#117	1	4	yes	—
101	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₁₃	#123	1	4	yes	—
107	C₂×C₄×C₁₃	#125	1	2	yes	—
108	C₂×C₄×C₁₃	#126	1	4	yes	—
109	C₈×C₁₃	—	2	4	no	—
110	C₈×C₁₃	—	2	4	no	—
111	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
114	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
115	C₂×#97	—	2	4	no	—
116	C₂×#97	C₄×C₁₃	1	4	yes	complemented
117	C₂×#95	—	2	4	no	—
118	#340	—	2	4	no	—
119	#340	C₄×C₁₃	1	8	yes	complemented
120	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₃	#380	1	1	yes	characteristic
122	C₂²×C₁₃	#382	1	1	yes	characteristic
123	C₂²×C₁₃	#388	1	4	yes	—
124	C₂²×C₁₃	#428	1	4	yes	—
125	C₂²×C₁₃	#440	1	4	yes	—
126	C₂²×C₁₃	#441	1	2	yes	—
127	C₂²×C₁₃	#442	1	2	yes	—
128	C₂²×C₁₃	#468	1	4	yes	—
129	C₂²×C₁₃	#469	1	2	yes	—
130	C₂²×C₁₃	#470	1	2	yes	—
131	C₂²×C₁₃	#489	1	4	yes	—
132	C₂²×C₁₃	#490	1	2	yes	—
133	C₂²×C₁₃	#491	1	2	yes	—
134	C₄×C₁₃	—	4	4	no	—
135	C₄×C₁₃	—	8	8	no	—
136	C₄×C₁₃	#376	1	1	yes	characteristic
137	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₁₃	#381	1	1	yes	characteristic
139	C₄×C₁₃	#383	1	1	yes	characteristic
140	C₄×C₁₃	#387	1	4	yes	—
141	C₂³×C₄	—	2	4	no	—
142	C₂³×C₄	D₄×C₁₃	1	2	yes	—
143	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
149	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
151	C₂²×C₈	D₄×C₁₃	1	4	yes	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
161	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
162	#97	—	2	8	no	—
163	#95	—	4	8	no	—
164	C₂×C₁₃	#1434	1	1	yes	characteristic
165	C₂×C₁₃	#1776	1	2	yes	—
166	C₂×C₁₃	#1809	1	2	yes	—
167	C₂×C₁₃	#1836	1	2	yes	—
168	C₂×C₁₃	#1935	1	4	yes	—
169	C₂×C₁₃	#1936	1	4	yes	—
170	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
186	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₁₃×#41	1	2	yes	—
188	C₂²×C₄	C₁₃×#41	1	2	yes	—
189	C₂²×C₄	C₁₃×#41	1	2	yes	—
190	C₂²×C₄	C₁₃×#41	1	2	yes	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	C₄²×C₁₃	1	4	yes	—
198	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	4	no	—
206	C₁₃	#10821	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
208	C₂³	C₄×D₄×C₁₃	1	2	yes	—
209	C₂³	C₁₃×#121	1	2	yes	—
210	C₂³	C₁₃×#124	1	2	yes	—
211	C₂³	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
212	C₂³	C₁₃×#137	1	4	yes	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
228	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
229	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
230	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
231	C₂×C₄	C₁₃×#117	1	4	yes	—
232	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
233	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
234	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
235	C₂×C₄	C₁₃×#123	1	4	yes	—
236	C₂×C₄	C₁₃×#125	1	2	yes	—
237	C₂×C₄	C₁₃×#126	1	4	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
241	C₂²	C₁₃×#380	1	1	yes	characteristic
242	C₂²	C₁₃×#382	1	1	yes	characteristic
243	C₂²	C₁₃×#388	1	4	yes	—
244	C₂²	C₁₃×#428	1	4	yes	—
245	C₂²	C₁₃×#440	1	4	yes	—
246	C₂²	C₁₃×#441	1	2	yes	—
247	C₂²	C₁₃×#442	1	2	yes	—
248	C₂²	C₁₃×#468	1	4	yes	—
249	C₂²	C₁₃×#469	1	2	yes	—
250	C₂²	C₁₃×#470	1	2	yes	—
251	C₂²	C₁₃×#489	1	4	yes	—
252	C₂²	C₁₃×#490	1	2	yes	—
253	C₂²	C₁₃×#491	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	—	8	8	no	—
256	C₄	C₁₃×#376	1	1	yes	characteristic
257	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
258	C₄	C₁₃×#381	1	1	yes	characteristic
259	C₄	C₁₃×#383	1	1	yes	characteristic
260	C₄	C₁₃×#387	1	4	yes	—
261	C₂	C₁₃×#1434	1	1	yes	characteristic
262	C₂	C₁₃×#1776	1	2	yes	—
263	C₂	C₁₃×#1809	1	2	yes	—
264	C₂	C₁₃×#1836	1	2	yes	—
265	C₂	C₁₃×#1935	1	4	yes	—
266	C₂	C₁₃×#1936	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
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
10. C₂×C₁₃×#97	C₄	semidirect
13. C₁₃×#340	C₄	semidirect
37. #10821	C₁₃	direct (N × H)
116. C₂×#97	C₄×C₁₃	semidirect
119. #340	C₄×C₁₃	semidirect
206. C₁₃	#10821	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×#97	2
C₂×C₁₃×#340	2
C₁₃×C₄⋊#95	2
#10821	13

Subgroups — C13×#10821 #15697160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10821 #15697160