Subgroups — C₁₃×#33659 #15719998

Order 3328.

546 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3060	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2239	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2238	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#365	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2148	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₁₃×#568	—	2	1	no	—
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₁₃×#537	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#470	C₂²	1	2	yes	—
14	C₁₃×#469	C₂²	1	2	yes	—
15	C₁₃×#465	—	2	2	no	—
16	C₁₃×#430	—	2	1	no	—
17	C₁₃×#430	C₂²	1	4	yes	—
18	C₂×C₁₃×#108	—	2	2	no	—
19	C₂×C₁₃×#108	C₂²	1	2	yes	—
20	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
21	C₁₃×#417	—	2	1	no	—
22	C₁₃×#365	C₂²	1	8	yes	—
23	C₁₃×#362	C₂²	1	4	yes	—
24	C₁₃×#361	C₂²	1	4	yes	—
25	C₁₃×#349	C₂²	1	4	yes	—
26	C₂×C₁₃×#41	—	2	1	no	—
27	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
28	C₁₃×#135	—	2	2	no	—
29	C₁₃×#135	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×SD₁₆	—	2	1	no	—
32	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
33	C₁₃×#131	—	2	2	no	—
34	C₁₃×#131	—	2	2	no	—
35	C₁₃×#131	—	4	4	no	—
36	C₂×C₁₃×M₁₆	—	2	4	no	—
37	C₂×C₁₃×M₁₆	—	2	2	no	—
38	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
39	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	D₄	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₂²⋊Q₈	—	2	2	no	—
43	C₁₃×C₂²⋊Q₈	—	4	1	no	—
44	C₁₃×#121	—	2	2	no	—
45	C₁₃×#121	—	4	1	no	—
46	C₁₃×#121	—	4	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₁₃×M₃₂	—	2	2	no	—
49	C₁₃×M₃₂	C₂³	1	4	yes	—
50	C₂×C₁₃×C₁₆	—	2	2	no	—
51	C₁₃×#108	—	2	2	no	—
52	C₁₃×#108	—	2	4	no	—
53	C₁₃×#108	C₂³	1	4	yes	—
54	C₁₃×#106	—	2	1	no	—
55	C₁₃×#106	C₂³	1	2	yes	—
56	C₁₃×#103	—	2	2	no	—
57	C₁₃×#102	—	2	2	no	—
58	#33659	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃×#41	—	2	2	no	—
60	C₁₃×#41	—	2	2	no	—
61	C₁₃×#41	—	4	2	no	—
62	C₁₃×#41	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	2	1	no	—
64	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
65	C₂×D₄×C₁₃	—	2	1	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
69	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×Q₁₆	—	2	2	no	—
75	C₁₃×SD₁₆	—	4	1	no	—
76	C₁₃×SD₁₆	—	4	1	no	—
77	C₁₃×D₈	—	2	2	no	—
78	C₁₃×M₁₆	—	2	2	no	—
79	C₁₃×M₁₆	—	2	4	no	—
80	C₁₃×M₁₆	—	2	4	no	—
81	C₁₃×M₁₆	—	4	4	no	—
82	C₁₃×M₁₆	—	4	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	4	4	no	—
86	C₂×C₈×C₁₃	—	4	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₁₃×C₁₆	—	2	4	no	—
98	#3060	C₂×C₁₃	1	1	yes	characteristic
99	#2916	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#2239	C₂×C₁₃	1	4	yes	complemented
101	#2238	C₂×C₁₃	1	4	yes	complemented
102	C₂×#365	C₂×C₁₃	1	2	yes	complemented
103	#2148	C₂×C₁₃	1	2	yes	complemented
104	#2115	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂³×C₁₃	—	4	1	no	—
106	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
107	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	Q₈×C₁₃	—	2	2	no	—
109	Q₈×C₁₃	—	4	1	no	—
110	Q₈×C₁₃	—	4	1	no	—
111	D₄×C₁₃	—	2	4	no	—
112	D₄×C₁₃	—	2	2	no	—
113	D₄×C₁₃	—	4	1	no	—
114	D₄×C₁₃	—	4	1	no	—
115	D₄×C₁₃	—	8	2	no	—
116	D₄×C₁₃	—	16	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	16	1	no	—
123	C₂×C₄×C₁₃	#123	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	4	4	no	—
130	C₈×C₁₃	#123	1	1	yes	characteristic
131	C₈×C₁₃	#123	1	1	yes	characteristic
132	C₈×C₁₃	C₂×#41	1	2	yes	—
133	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
134	#568	—	2	1	no	—
135	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
136	#537	—	2	1	no	—
137	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
138	#470	C₂²×C₁₃	1	2	yes	—
139	#469	C₂²×C₁₃	1	2	yes	—
140	#465	—	2	2	no	—
141	#430	—	2	1	no	—
142	#430	C₂²×C₁₃	1	4	yes	—
143	C₂×#108	—	2	2	no	—
144	C₂×#108	C₂²×C₁₃	1	2	yes	—
145	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
146	#417	—	2	1	no	—
147	#365	C₂²×C₁₃	1	8	yes	—
148	#362	C₂²×C₁₃	1	4	yes	—
149	#361	C₂²×C₁₃	1	4	yes	—
150	#349	C₂²×C₁₃	1	4	yes	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	1	no	—
156	C₂²×C₁₃	—	8	1	no	—
157	C₂²×C₁₃	#481	1	2	yes	—
158	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	8	2	no	—
161	C₄×C₁₃	—	8	1	no	—
162	C₄×C₁₃	#481	1	2	yes	—
163	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
164	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×#41	—	2	1	no	—
166	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
167	#135	—	2	2	no	—
168	#135	—	2	2	no	—
169	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
170	C₂×SD₁₆	—	2	1	no	—
171	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
172	#131	—	2	2	no	—
173	#131	—	2	2	no	—
174	#131	—	4	4	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
178	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₈	D₄×C₁₃	1	2	yes	—
180	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	—	4	1	no	—
183	#121	—	2	2	no	—
184	#121	—	4	1	no	—
185	#121	—	4	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	M₃₂	—	2	2	no	—
188	M₃₂	C₂³×C₁₃	1	4	yes	—
189	C₂×C₁₆	—	2	2	no	—
190	#108	—	2	2	no	—
191	#108	—	2	4	no	—
192	#108	C₂³×C₁₃	1	4	yes	—
193	#106	—	2	1	no	—
194	#106	C₂³×C₁₃	1	2	yes	—
195	#103	—	2	2	no	—
196	#102	—	2	2	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	4	2	no	—
200	C₂×C₁₃	—	16	1	no	—
201	C₂×C₁₃	C₂×#481	1	1	yes	characteristic
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	4	2	no	—
205	#41	—	4	2	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	C₁₃×SD₁₆	1	1	yes	characteristic
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	C₁₃×SD₁₆	1	2	yes	—
212	C₂×D₄	C₁₃×SD₁₆	1	1	yes	characteristic
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	4	1	no	—
219	SD₁₆	—	4	1	no	—
220	D₈	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	4	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂×C₈	C₁₃×#41	1	2	yes	—
233	C₂×C₈	C₁₃×#41	1	2	yes	—
234	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₁₆	—	2	4	no	—
241	C₁₃	#33659	1	1	yes	characteristic, direct product factor, Sylow
242	C₂³	—	4	1	no	—
243	C₂³	C₂×C₁₃×SD₁₆	1	2	yes	—
244	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	4	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	1	no	—
252	D₄	—	8	2	no	—
253	D₄	—	16	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	16	1	no	—
260	C₂×C₄	C₁₃×#123	1	2	yes	—
261	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
262	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
263	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
264	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	4	4	no	—
267	C₈	C₁₃×#123	1	1	yes	characteristic
268	C₈	C₁₃×#123	1	1	yes	characteristic
269	C₈	C₂×C₁₃×#41	1	2	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	8	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	1	no	—
276	C₂²	C₁₃×#481	1	2	yes	—
277	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	1	no	—
281	C₄	C₁₃×#481	1	2	yes	—
282	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
283	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	16	1	no	—
288	C₂	C₂×C₁₃×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2916	C₂	semidirect
3. C₁₃×#2239	C₂	semidirect
4. C₁₃×#2238	C₂	semidirect
5. C₂×C₁₃×#365	C₂	semidirect
6. C₁₃×#2148	C₂	semidirect
7. C₁₃×#2115	C₂	semidirect
58. #33659	C₁₃	direct (N × H)
99. #2916	C₂×C₁₃	semidirect
100. #2239	C₂×C₁₃	semidirect
101. #2238	C₂×C₁₃	semidirect
102. C₂×#365	C₂×C₁₃	semidirect
103. #2148	C₂×C₁₃	semidirect
104. #2115	C₂×C₁₃	semidirect
241. C₁₃	#33659	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×#3060	2
C₁₃×#2916	2
C₁₃×#2239	2
C₁₃×#2238	2
C₂×C₁₃×#365	2
C₁₃×#2148	2
C₁₃×#2115	2
#33659	13

Subgroups — C13×#33659 #15719998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33659 #15719998