Subgroups — C₁₃×#13137 #15699476

Order 3328.

316 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2148	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2143	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1974	C₂	1	1	yes	maximal, characteristic
5	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#104	—	2	1	no	—
15	C₁₃×#408	—	2	1	no	—
16	C₄×C₁₃×M₁₆	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#368	—	2	2	no	—
19	C₁₃×#362	C₄	1	2	yes	complemented
20	C₁₃×#361	C₄	1	2	yes	complemented
21	C₁₃×#359	—	2	2	no	—
22	C₁₃×#341	—	2	1	no	—
23	C₂×C₁₃×#41	—	2	1	no	—
24	C₁₃×#135	—	2	1	no	—
25	C₁₃×#135	—	2	1	no	—
26	C₁₃×#135	—	2	1	no	—
27	C₁₃×#135	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×SD₁₆	—	2	1	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	—	2	1	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×M₃₂	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	—	2	1	no	—
40	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×C₁₃×C₁₆	—	2	1	no	—
42	C₂×C₁₃×C₁₆	—	2	1	no	—
43	C₁₃×#108	—	2	1	no	—
44	C₁₃×#108	—	2	1	no	—
45	C₁₃×#108	—	2	1	no	—
46	C₁₃×#108	—	2	1	no	—
47	C₁₃×#108	C₂×C₄	1	2	yes	—
48	C₁₃×#108	D₄	1	2	yes	—
49	C₁₃×#104	—	4	1	no	—
50	C₁₃×#104	—	4	1	no	—
51	C₁₃×C₂²⋊C₈	—	4	2	no	—
52	C₁₃×#97	—	4	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	C₄×C₈×C₁₃	—	4	1	no	—
56	#13137	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₁₃×#41	—	4	1	no	—
58	C₁₃×#41	—	4	1	no	—
59	C₁₃×#41	—	4	1	no	—
60	C₁₃×#41	—	4	1	no	—
61	C₂×Q₈×C₁₃	—	2	1	no	—
62	C₂×D₄×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×Q₁₆	—	2	1	no	—
68	C₁₃×Q₁₆	—	2	1	no	—
69	C₁₃×SD₁₆	—	4	1	no	—
70	C₁₃×SD₁₆	—	4	1	no	—
71	C₁₃×D₈	—	2	1	no	—
72	C₁₃×D₈	—	2	1	no	—
73	C₁₃×M₁₆	—	2	2	no	—
74	C₁₃×M₁₆	—	2	2	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₁₃×M₁₆	—	4	1	no	—
77	C₁₃×M₁₆	—	4	1	no	—
78	C₁₃×M₁₆	—	4	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	D₈	1	1	yes	characteristic
93	C₄²×C₁₃	D₈	1	1	yes	characteristic
94	C₁₃×C₁₆	—	2	2	no	—
95	C₁₃×C₁₆	—	4	1	no	—
96	#2161	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#2148	C₂×C₁₃	1	1	yes	characteristic
98	#2143	C₂×C₁₃	1	1	yes	characteristic
99	#1974	C₂×C₁₃	1	1	yes	characteristic
100	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#1388	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#1376	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂³×C₁₃	—	4	1	no	—
104	C₂³×C₁₃	#137	1	1	yes	characteristic
105	Q₈×C₁₃	—	4	1	no	—
106	Q₈×C₁₃	—	4	1	no	—
107	D₄×C₁₃	—	4	1	no	—
108	D₄×C₁₃	—	4	1	no	—
109	D₄×C₁₃	—	8	1	no	—
110	D₄×C₁₃	—	8	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	8	1	no	—
116	C₂×C₄×C₁₃	—	8	1	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₈×C₁₃	—	4	2	no	—
124	C₈×C₁₃	—	4	2	no	—
125	C₈×C₁₃	—	4	1	no	—
126	C₈×C₁₃	—	8	1	no	—
127	C₈×C₁₃	#117	1	1	yes	characteristic
128	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₈×C₁₃	#124	1	1	yes	characteristic
130	C₈×C₁₃	#126	1	1	yes	characteristic
131	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#104	—	2	1	no	—
138	#408	—	2	1	no	—
139	C₄×M₁₆	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
141	#368	—	2	2	no	—
142	#362	C₄×C₁₃	1	2	yes	complemented
143	#361	C₄×C₁₃	1	2	yes	complemented
144	#359	—	2	2	no	—
145	#341	—	2	1	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	8	2	no	—
148	C₂²×C₁₃	—	8	1	no	—
149	C₂²×C₁₃	#440	1	1	yes	characteristic
150	C₂²×C₁₃	#464	1	1	yes	characteristic
151	C₂²×C₁₃	#480	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	8	1	no	—
155	C₄×C₁₃	#388	1	1	yes	characteristic
156	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
157	C₄×C₁₃	#459	1	1	yes	characteristic
158	C₄×C₁₃	#482	1	1	yes	characteristic
159	C₂×#41	—	2	1	no	—
160	#135	—	2	1	no	—
161	#135	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂×SD₁₆	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
173	M₃₂	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	C₂×C₄×C₁₃	1	2	yes	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	#108	—	2	1	no	—
180	#108	—	2	1	no	—
181	#108	—	2	1	no	—
182	#108	—	2	1	no	—
183	#108	C₂×C₄×C₁₃	1	2	yes	—
184	#108	D₄×C₁₃	1	2	yes	—
185	#104	—	4	1	no	—
186	#104	—	4	1	no	—
187	C₂²⋊C₈	—	4	2	no	—
188	#97	—	4	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	4	1	no	—
192	C₂×C₁₃	—	2	1	no	—
193	C₂×C₁₃	—	2	1	no	—
194	C₂×C₁₃	—	2	1	no	—
195	C₂×C₁₃	—	16	1	no	—
196	C₂×C₁₃	#1925	1	1	yes	characteristic
197	#41	—	4	1	no	—
198	#41	—	4	1	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
207	Q₁₆	—	2	1	no	—
208	Q₁₆	—	2	1	no	—
209	SD₁₆	—	4	1	no	—
210	SD₁₆	—	4	1	no	—
211	D₈	—	2	1	no	—
212	D₈	—	2	1	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
228	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
229	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
230	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	C₁₃×D₈	1	1	yes	characteristic
233	C₄²	C₁₃×D₈	1	1	yes	characteristic
234	C₁₆	—	2	2	no	—
235	C₁₆	—	4	1	no	—
236	C₁₃	#13137	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	—	4	1	no	—
238	C₂³	C₁₃×#137	1	1	yes	characteristic
239	Q₈	—	4	1	no	—
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	1	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
253	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
254	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
255	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
256	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	1	no	—
260	C₈	—	8	1	no	—
261	C₈	C₁₃×#117	1	1	yes	characteristic
262	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
263	C₈	C₁₃×#124	1	1	yes	characteristic
264	C₈	C₁₃×#126	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	1	no	—
268	C₂²	C₁₃×#440	1	1	yes	characteristic
269	C₂²	C₁₃×#464	1	1	yes	characteristic
270	C₂²	C₁₃×#480	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₁₃×#388	1	1	yes	characteristic
275	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
276	C₄	C₁₃×#459	1	1	yes	characteristic
277	C₄	C₁₃×#482	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	16	1	no	—
282	C₂	C₁₃×#1925	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1925
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2161	C₂	semidirect
5. C₄×C₁₃×#108	C₂	semidirect
6. C₁₃×#1388	C₂	semidirect
7. C₁₃×#1376	C₂	semidirect
19. C₁₃×#362	C₄	semidirect
20. C₁₃×#361	C₄	semidirect
56. #13137	C₁₃	direct (N × H)
96. #2161	C₂×C₁₃	semidirect
100. C₄×#108	C₂×C₁₃	semidirect
101. #1388	C₂×C₁₃	semidirect
102. #1376	C₂×C₁₃	semidirect
142. #362	C₄×C₁₃	semidirect
143. #361	C₄×C₁₃	semidirect
236. C₁₃	#13137	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#2161	2
C₁₃×#2148	2
C₁₃×#2143	2
C₁₃×#1974	2
C₄×C₁₃×#108	2
C₁₃×#1388	2
C₁₃×#1376	2
#13137	13

Subgroups — C13×#13137 #15699476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13137 #15699476