Subgroups — C₁₃×#7880 #15694219

Order 3328.

618 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	—	2	1	no	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#431	—	2	2	no	—
8	C₁₃×#431	—	2	2	no	—
9	C₁₃×#431	—	2	2	no	—
10	C₁₃×#431	C₄	1	4	yes	—
11	C₁₃×#423	—	2	2	no	—
12	C₁₃×#423	C₄	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
14	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	—	4	4	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊C₈	—	2	4	no	—
38	C₁₃×C₄⋊C₈	—	2	4	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₁₃×C₂²⋊C₈	—	2	4	no	—
44	C₁₃×C₂²⋊C₈	—	2	4	no	—
45	C₄×C₈×C₁₃	—	2	4	no	—
46	C₄×C₈×C₁₃	—	2	2	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	#7880	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₁₃×M₁₆	—	4	8	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	8	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	4	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₄²×C₁₃	C₄²	1	2	yes	—
75	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
85	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
86	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
87	#1277	C₂×C₁₃	1	4	yes	complemented
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	#95	1	2	yes	—
92	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	#95	1	2	yes	—
104	C₂×C₄×C₁₃	#95	1	2	yes	—
105	C₂×C₄×C₁₃	#95	1	2	yes	—
106	C₂×C₄×C₁₃	#108	1	4	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#131	1	4	yes	—
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₁₃	#137	1	2	yes	—
116	C₈×C₁₃	—	4	4	no	—
117	C₈×C₁₃	—	4	4	no	—
118	C₈×C₁₃	—	4	4	no	—
119	C₂²×M₁₆	—	2	1	no	—
120	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
121	#431	—	2	2	no	—
122	#431	—	2	2	no	—
123	#431	—	2	2	no	—
124	#431	C₄×C₁₃	1	4	yes	—
125	#423	—	2	2	no	—
126	#423	C₄×C₁₃	1	4	yes	—
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	C₄×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×C₂²⋊C₈	—	2	1	no	—
132	C₂×C₂²⋊C₈	C₄×C₁₃	1	2	yes	—
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	4	4	no	—
139	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₃	#424	1	2	yes	—
141	C₂²×C₁₃	#428	1	2	yes	—
142	C₂²×C₁₃	C₂×#108	1	2	yes	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	#341	1	4	yes	—
148	C₄×C₁₃	#405	1	2	yes	—
149	C₄×C₁₃	#408	1	2	yes	—
150	C₄×C₁₃	#417	1	2	yes	—
151	C₄×C₁₃	#418	1	1	yes	characteristic
152	C₄×C₁₃	#419	1	1	yes	characteristic
153	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
154	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
155	C₂³×C₄	Q₈×C₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	4	4	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
161	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
163	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
164	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₁₃	1	2	yes	—
166	C₂×C₄²	D₄×C₁₃	1	2	yes	—
167	C₂×C₄²	Q₈×C₁₃	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
174	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
175	C₂²⋊C₈	—	2	4	no	—
176	C₂²⋊C₈	—	2	4	no	—
177	C₄×C₈	—	2	4	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₂×C₁₃	—	4	2	no	—
181	C₂×C₁₃	#1574	1	4	yes	—
182	C₂×C₁₃	#1729	1	1	yes	characteristic
183	C₂×C₁₃	#1736	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#341	1	1	yes	characteristic
185	C₂⁴	C₁₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
191	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
192	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
193	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
194	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
197	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
198	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
202	M₁₆	—	4	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₄²	C₄²×C₁₃	1	2	yes	—
211	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
212	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
213	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
214	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
215	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
216	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₁₃	#7880	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₁₃×#95	1	2	yes	—
225	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	C₁₃×#95	1	2	yes	—
237	C₂×C₄	C₁₃×#95	1	2	yes	—
238	C₂×C₄	C₁₃×#95	1	2	yes	—
239	C₂×C₄	C₁₃×#108	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₄	C₁₃×#131	1	4	yes	—
247	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
248	C₂×C₄	C₁₃×#137	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	4	4	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
257	C₂²	C₁₃×#424	1	2	yes	—
258	C₂²	C₁₃×#428	1	2	yes	—
259	C₂²	C₂×C₁₃×#108	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	C₁₃×#341	1	4	yes	—
265	C₄	C₁₃×#405	1	2	yes	—
266	C₄	C₁₃×#408	1	2	yes	—
267	C₄	C₁₃×#417	1	2	yes	—
268	C₄	C₁₃×#418	1	1	yes	characteristic
269	C₄	C₁₃×#419	1	1	yes	characteristic
270	C₂	—	4	2	no	—
271	C₂	C₁₃×#1574	1	4	yes	—
272	C₂	C₁₃×#1729	1	1	yes	characteristic
273	C₂	C₁₃×#1736	1	1	yes	characteristic
274	C₂	C₂×C₁₃×#341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1277	C₂	semidirect
48. #7880	C₁₃	direct (N × H)
87. #1277	C₂×C₁₃	semidirect
220. C₁₃	#7880	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×#431	2
C₂×C₁₃×#423	2
C₁₃×#1277	2
#7880	13

Subgroups — C13×#7880 #15694219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7880 #15694219