Subgroups — C₁₃×#13156 #15699495

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2148	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#360	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1976	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1386	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	C₁₃×#428	—	2	1	no	—
13	C₁₃×#427	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₁₃×#421	—	2	1	no	—
16	C₁₃×#410	—	2	1	no	—
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	C₁₃×#362	C₄	1	2	yes	complemented
19	C₁₃×#361	C₄	1	2	yes	complemented
20	C₁₃×#360	—	2	2	no	—
21	C₁₃×#347	—	2	2	no	—
22	C₁₃×#343	—	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	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	—	2	1	no	—
35	C₁₃×#117	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₁₃×M₃₂	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
40	C₁₃×M₃₂	D₄	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	C₂×C₄	1	2	yes	—
45	C₁₃×#107	—	2	1	no	—
46	C₁₃×#107	—	4	1	no	—
47	C₁₃×#106	—	2	1	no	—
48	C₁₃×#106	—	2	1	no	—
49	C₁₃×#106	—	2	1	no	—
50	C₁₃×#106	—	4	1	no	—
51	C₁₃×#104	—	4	2	no	—
52	C₁₃×#99	—	4	2	no	—
53	C₁₃×#97	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	#13156	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₃×#41	—	4	1	no	—
57	C₁₃×#41	—	4	1	no	—
58	C₁₃×#41	—	4	1	no	—
59	C₁₃×#41	—	4	1	no	—
60	C₂×Q₈×C₁₃	—	2	1	no	—
61	C₂×D₄×C₁₃	—	2	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×Q₁₆	—	2	1	no	—
67	C₁₃×Q₁₆	—	2	1	no	—
68	C₁₃×SD₁₆	—	4	1	no	—
69	C₁₃×SD₁₆	—	4	1	no	—
70	C₁₃×D₈	—	2	1	no	—
71	C₁₃×D₈	—	2	1	no	—
72	C₁₃×M₁₆	—	2	2	no	—
73	C₁₃×M₁₆	—	2	1	no	—
74	C₁₃×M₁₆	—	2	1	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₄⋊C₄	—	4	1	no	—
87	C₁₃×C₄⋊C₄	—	4	1	no	—
88	C₁₃×C₂²⋊C₄	—	2	1	no	—
89	C₁₃×C₂²⋊C₄	—	4	1	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	4	1	no	—
92	C₁₃×C₁₆	—	2	2	no	—
93	C₁₃×C₁₆	—	2	2	no	—
94	#2148	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#360	C₂×C₁₃	1	1	yes	characteristic
96	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#1976	C₂×C₁₃	1	1	yes	characteristic
98	#1951	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#1389	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1386	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	—	4	1	no	—
102	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	Q₈×C₁₃	—	4	1	no	—
104	Q₈×C₁₃	—	4	1	no	—
105	D₄×C₁₃	—	4	1	no	—
106	D₄×C₁₃	—	4	1	no	—
107	D₄×C₁₃	—	8	1	no	—
108	D₄×C₁₃	—	8	1	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	8	1	no	—
113	C₂×C₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	—	8	1	no	—
115	C₂×C₄×C₁₃	—	16	1	no	—
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
122	C₈×C₁₃	—	4	2	no	—
123	C₈×C₁₃	—	4	1	no	—
124	C₈×C₁₃	#117	1	1	yes	characteristic
125	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
126	C₈×C₁₃	#123	1	1	yes	characteristic
127	C₈×C₁₃	#126	1	1	yes	characteristic
128	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
132	#428	—	2	1	no	—
133	#427	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
135	#421	—	2	1	no	—
136	#410	—	2	1	no	—
137	#405	C₂²×C₁₃	1	1	yes	characteristic
138	#362	C₄×C₁₃	1	2	yes	complemented
139	#361	C₄×C₁₃	1	2	yes	complemented
140	#360	—	2	2	no	—
141	#347	—	2	2	no	—
142	#343	—	2	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	8	1	no	—
145	C₂²×C₁₃	—	16	1	no	—
146	C₂²×C₁₃	#440	1	1	yes	characteristic
147	C₂²×C₁₃	#469	1	1	yes	characteristic
148	C₂²×C₁₃	#490	1	1	yes	characteristic
149	C₄×C₁₃	—	8	1	no	—
150	C₄×C₁₃	—	8	1	no	—
151	C₄×C₁₃	—	16	1	no	—
152	C₄×C₁₃	—	16	1	no	—
153	C₄×C₁₃	#388	1	1	yes	characteristic
154	C₄×C₁₃	#442	1	1	yes	characteristic
155	C₄×C₁₃	#468	1	1	yes	characteristic
156	C₄×C₁₃	#489	1	1	yes	characteristic
157	C₂×#41	—	2	1	no	—
158	#135	—	2	1	no	—
159	#135	—	2	1	no	—
160	#135	—	2	1	no	—
161	#135	—	2	1	no	—
162	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₂×SD₁₆	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	#117	—	2	1	no	—
169	#117	D₄×C₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	M₃₂	—	2	1	no	—
172	M₃₂	—	2	1	no	—
173	M₃₂	C₂×C₄×C₁₃	1	2	yes	—
174	M₃₂	D₄×C₁₃	1	2	yes	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	#108	—	2	1	no	—
178	#108	C₂×C₄×C₁₃	1	2	yes	—
179	#107	—	2	1	no	—
180	#107	—	4	1	no	—
181	#106	—	2	1	no	—
182	#106	—	2	1	no	—
183	#106	—	2	1	no	—
184	#106	—	4	1	no	—
185	#104	—	4	2	no	—
186	#99	—	4	2	no	—
187	#97	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	1	no	—
192	C₂×C₁₃	—	16	1	no	—
193	C₂×C₁₃	#1936	1	1	yes	characteristic
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	—	2	1	no	—
206	SD₁₆	—	4	1	no	—
207	SD₁₆	—	4	1	no	—
208	D₈	—	2	1	no	—
209	D₈	—	2	1	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	4	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
219	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
220	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
221	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	4	1	no	—
230	C₁₆	—	2	2	no	—
231	C₁₆	—	2	2	no	—
232	C₁₃	#13156	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	4	1	no	—
234	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	8	1	no	—
240	D₄	—	8	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	—	8	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	16	1	no	—
248	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
251	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
254	C₈	—	4	2	no	—
255	C₈	—	4	1	no	—
256	C₈	C₁₃×#117	1	1	yes	characteristic
257	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
258	C₈	C₁₃×#123	1	1	yes	characteristic
259	C₈	C₁₃×#126	1	1	yes	characteristic
260	C₂²	—	4	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	16	1	no	—
263	C₂²	C₁₃×#440	1	1	yes	characteristic
264	C₂²	C₁₃×#469	1	1	yes	characteristic
265	C₂²	C₁₃×#490	1	1	yes	characteristic
266	C₄	—	8	1	no	—
267	C₄	—	8	1	no	—
268	C₄	—	16	1	no	—
269	C₄	—	16	1	no	—
270	C₄	C₁₃×#388	1	1	yes	characteristic
271	C₄	C₁₃×#442	1	1	yes	characteristic
272	C₄	C₁₃×#468	1	1	yes	characteristic
273	C₄	C₁₃×#489	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	16	1	no	—
278	C₂	C₁₃×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1936
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2110	C₂	semidirect
5. C₁₃×#1951	C₂	semidirect
6. C₁₃×#1389	C₂	semidirect
7. C₁₃×#1386	C₂	semidirect
18. C₁₃×#362	C₄	semidirect
19. C₁₃×#361	C₄	semidirect
55. #13156	C₁₃	direct (N × H)
96. #2110	C₂×C₁₃	semidirect
98. #1951	C₂×C₁₃	semidirect
99. #1389	C₂×C₁₃	semidirect
100. #1386	C₂×C₁₃	semidirect
138. #362	C₄×C₁₃	semidirect
139. #361	C₄×C₁₃	semidirect
232. C₁₃	#13156	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×#2148	2
C₂×C₁₃×#360	2
C₁₃×#2110	2
C₁₃×#1976	2
C₁₃×#1951	2
C₁₃×#1389	2
C₁₃×#1386	2
#13156	13

Subgroups — C13×#13156 #15699495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13156 #15699495