Subgroups — C₁₃×#33858 #15720197

Order 3328.

668 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3415	C₂	1	2	yes	maximal, complemented
2	C₁₃×#3292	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2221	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2213	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2192	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#576	—	2	2	no	—
10	C₁₃×#576	C₂²	1	4	yes	complemented
11	C₁₃×#509	—	2	4	no	—
12	C₁₃×#509	C₂²	1	4	yes	complemented
13	C₁₃×#508	—	2	2	no	—
14	C₁₃×#508	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₁₆	—	2	2	no	—
16	C₂×C₁₃×D₁₆	C₂²	1	2	yes	complemented
17	C₂×C₁₃×D₁₆	C₂²	1	2	yes	complemented
18	C₁₃×#504	C₂²	1	2	yes	complemented
19	C₁₃×#493	C₂²	1	2	yes	complemented
20	C₁₃×#472	—	2	2	no	—
21	C₁₃×#472	C₂²	1	4	yes	complemented
22	C₁₃×#453	—	2	4	no	—
23	C₁₃×#443	C₂²	1	1	yes	characteristic
24	C₁₃×#443	C₂²	1	2	yes	complemented
25	C₁₃×#368	C₂²	1	1	yes	characteristic, complemented
26	C₁₃×#364	C₂²	1	1	yes	characteristic, complemented
27	C₁₃×#361	C₂²	1	4	yes	complemented
28	C₁₃×#359	C₂²	1	2	yes	complemented
29	C₁₃×#350	C₂²	1	2	yes	complemented
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×H(5,𝔽₂)	—	2	4	no	—
32	C₁₃×AGL(1,ℤ/8)	—	2	4	no	—
33	C₁₃×AGL(1,ℤ/8)	—	2	4	no	—
34	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
35	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
36	C₁₃×#135	—	2	4	no	—
37	C₁₃×#135	—	4	2	no	—
38	C₁₃×#135	D₄	1	1	yes	characteristic
39	C₁₃×#135	C₂³	1	2	yes	—
40	C₂×C₁₃×D₈	—	2	4	no	—
41	C₂×C₁₃×D₈	—	2	2	no	—
42	C₂×C₁₃×D₈	—	4	2	no	—
43	C₂×C₁₃×D₈	C₂³	1	4	yes	—
44	C₁₃×#131	—	2	2	no	—
45	C₁₃×#131	C₂³	1	2	yes	—
46	C₁₃×#127	—	2	2	no	—
47	C₁₃×Q₃₂	D₄	1	1	yes	characteristic, complemented
48	C₁₃×SD₃₂	—	2	4	no	—
49	C₁₃×SD₃₂	D₄	1	2	yes	complemented
50	C₁₃×D₁₆	—	2	4	no	—
51	C₁₃×D₁₆	—	2	2	no	—
52	C₁₃×D₁₆	—	2	2	no	—
53	C₁₃×D₁₆	—	4	2	no	—
54	C₁₃×D₁₆	D₄	1	1	yes	characteristic, complemented
55	C₁₃×M₃₂	—	2	2	no	—
56	C₁₃×M₃₂	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	—	2	2	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×#108	D₄	1	1	yes	characteristic
61	C₁₃×#108	C₂³	1	2	yes	—
62	C₁₃×#104	—	2	2	no	—
63	C₁₃×#104	D₄	1	2	yes	—
64	C₁₃×#100	—	2	4	no	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#33858	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₁₃×#41	—	4	4	no	—
70	C₁₃×#41	C₂×D₄	1	2	yes	—
71	C₂×D₄×C₁₃	—	2	4	no	—
72	C₂×D₄×C₁₃	—	4	4	no	—
73	C₂×D₄×C₁₃	—	4	4	no	—
74	C₂×D₄×C₁₃	—	8	2	no	—
75	C₂×D₄×C₁₃	—	8	4	no	—
76	C₁₃×Q₁₆	D₈	1	1	yes	characteristic
77	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
78	C₁₃×SD₁₆	—	2	2	no	—
79	C₁₃×SD₁₆	—	4	4	no	—
80	C₁₃×SD₁₆	—	4	4	no	—
81	C₁₃×SD₁₆	D₈	1	2	yes	—
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×D₈	—	2	4	no	—
84	C₁₃×D₈	—	4	4	no	—
85	C₁₃×D₈	—	4	4	no	—
86	C₁₃×D₈	—	4	2	no	—
87	C₁₃×D₈	—	4	2	no	—
88	C₁₃×D₈	D₈	1	1	yes	characteristic
89	C₁₃×D₈	C₂×D₄	1	2	yes	—
90	C₁₃×M₁₆	—	2	2	no	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	—	4	2	no	—
93	C₁₃×M₁₆	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	4	2	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₁₃×C₁₆	—	2	2	no	—
100	C₁₃×C₁₆	—	2	1	no	—
101	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
102	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
103	#3415	C₂×C₁₃	1	2	yes	complemented
104	#3292	C₂×C₁₃	1	2	yes	complemented
105	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#2221	C₂×C₁₃	1	2	yes	complemented
107	#2213	C₂×C₁₃	1	2	yes	complemented
108	#2192	C₂×C₁₃	1	4	yes	complemented
109	#2178	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#2170	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂³×C₁₃	—	4	4	no	—
112	C₂³×C₁₃	—	8	4	no	—
113	Q₈×C₁₃	—	2	2	no	—
114	Q₈×C₁₃	C₂×D₈	1	2	yes	—
115	D₄×C₁₃	—	2	2	no	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	4	no	—
119	D₄×C₁₃	—	4	4	no	—
120	D₄×C₁₃	—	8	4	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	16	4	no	—
123	D₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	—	4	2	no	—
131	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	#576	—	2	2	no	—
136	#576	C₂²×C₁₃	1	4	yes	complemented
137	#509	—	2	4	no	—
138	#509	C₂²×C₁₃	1	4	yes	complemented
139	#508	—	2	2	no	—
140	#508	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×D₁₆	—	2	2	no	—
142	C₂×D₁₆	C₂²×C₁₃	1	2	yes	complemented
143	C₂×D₁₆	C₂²×C₁₃	1	2	yes	complemented
144	#504	C₂²×C₁₃	1	2	yes	complemented
145	#493	C₂²×C₁₃	1	2	yes	complemented
146	#472	—	2	2	no	—
147	#472	C₂²×C₁₃	1	4	yes	complemented
148	#453	—	2	4	no	—
149	#443	C₂²×C₁₃	1	1	yes	characteristic
150	#443	C₂²×C₁₃	1	2	yes	complemented
151	#368	C₂²×C₁₃	1	1	yes	characteristic, complemented
152	#364	C₂²×C₁₃	1	1	yes	characteristic, complemented
153	#361	C₂²×C₁₃	1	4	yes	complemented
154	#359	C₂²×C₁₃	1	2	yes	complemented
155	#350	C₂²×C₁₃	1	2	yes	complemented
156	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	4	2	no	—
159	C₂²×C₁₃	—	8	4	no	—
160	C₂²×C₁₃	—	16	4	no	—
161	C₂²×C₁₃	—	32	4	no	—
162	C₂²×C₁₃	#576	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	4	2	no	—
166	C₄×C₁₃	D₄²	1	1	yes	characteristic
167	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
168	H(5,𝔽₂)	—	2	4	no	—
169	AGL(1,ℤ/8)	—	2	4	no	—
170	AGL(1,ℤ/8)	—	2	4	no	—
171	AGL(1,ℤ/8)	—	4	4	no	—
172	AGL(1,ℤ/8)	—	4	4	no	—
173	#135	—	2	4	no	—
174	#135	—	4	2	no	—
175	#135	D₄×C₁₃	1	1	yes	characteristic
176	#135	C₂³×C₁₃	1	2	yes	—
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	—	4	2	no	—
180	C₂×D₈	C₂³×C₁₃	1	4	yes	—
181	#131	—	2	2	no	—
182	#131	C₂³×C₁₃	1	2	yes	—
183	#127	—	2	2	no	—
184	Q₃₂	D₄×C₁₃	1	1	yes	characteristic, complemented
185	SD₃₂	—	2	4	no	—
186	SD₃₂	D₄×C₁₃	1	2	yes	complemented
187	D₁₆	—	2	4	no	—
188	D₁₆	—	2	2	no	—
189	D₁₆	—	2	2	no	—
190	D₁₆	—	4	2	no	—
191	D₁₆	D₄×C₁₃	1	1	yes	characteristic, complemented
192	M₃₂	—	2	2	no	—
193	M₃₂	C₂³×C₁₃	1	2	yes	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
197	#108	D₄×C₁₃	1	1	yes	characteristic
198	#108	C₂³×C₁₃	1	2	yes	—
199	#104	—	2	2	no	—
200	#104	D₄×C₁₃	1	2	yes	—
201	#100	—	2	4	no	—
202	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	4	2	no	—
205	C₂×C₁₃	—	8	2	no	—
206	C₂×C₁₃	—	16	4	no	—
207	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
208	#41	—	2	2	no	—
209	#41	—	4	4	no	—
210	#41	—	4	4	no	—
211	#41	C₂×D₄×C₁₃	1	2	yes	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	8	2	no	—
216	C₂×D₄	—	8	4	no	—
217	Q₁₆	C₁₃×D₈	1	1	yes	characteristic
218	Q₁₆	C₂×D₄×C₁₃	1	2	yes	—
219	SD₁₆	—	2	2	no	—
220	SD₁₆	—	4	4	no	—
221	SD₁₆	—	4	4	no	—
222	SD₁₆	C₁₃×D₈	1	2	yes	—
223	D₈	—	2	4	no	—
224	D₈	—	2	4	no	—
225	D₈	—	4	4	no	—
226	D₈	—	4	4	no	—
227	D₈	—	4	2	no	—
228	D₈	—	4	2	no	—
229	D₈	C₁₃×D₈	1	1	yes	characteristic
230	D₈	C₂×D₄×C₁₃	1	2	yes	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	C₂×D₄×C₁₃	1	2	yes	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	1	no	—
242	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₁₃	#33858	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	4	4	no	—
246	C₂³	—	8	4	no	—
247	Q₈	—	2	2	no	—
248	Q₈	C₂×C₁₃×D₈	1	2	yes	—
249	D₄	—	2	2	no	—
250	D₄	—	2	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	8	4	no	—
255	D₄	—	8	4	no	—
256	D₄	—	16	4	no	—
257	D₄	C₂×C₁₃×D₈	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
266	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
267	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
268	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	4	no	—
273	C₂²	—	32	4	no	—
274	C₂²	C₁₃×#576	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	D₄²×C₁₃	1	1	yes	characteristic
279	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	4	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	16	4	no	—
284	C₂	D₄×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3415	C₂	semidirect
2. C₁₃×#3292	C₂	semidirect
3. C₁₃×#2246	C₂	semidirect
4. C₁₃×#2221	C₂	semidirect
5. C₁₃×#2213	C₂	semidirect
6. C₁₃×#2192	C₂	semidirect
7. C₁₃×#2178	C₂	semidirect
8. C₁₃×#2170	C₂	semidirect
10. C₁₃×#576	C₂²	semidirect
12. C₁₃×#509	C₂²	semidirect
16. C₂×C₁₃×D₁₆	C₂²	semidirect
17. C₂×C₁₃×D₁₆	C₂²	semidirect
18. C₁₃×#504	C₂²	semidirect
19. C₁₃×#493	C₂²	semidirect
21. C₁₃×#472	C₂²	semidirect
24. C₁₃×#443	C₂²	semidirect
25. C₁₃×#368	C₂²	semidirect
26. C₁₃×#364	C₂²	semidirect
27. C₁₃×#361	C₂²	semidirect
28. C₁₃×#359	C₂²	semidirect
29. C₁₃×#350	C₂²	semidirect
30. C₄×C₁₃×C₁₆	C₂²	semidirect
47. C₁₃×Q₃₂	D₄	semidirect
49. C₁₃×SD₃₂	D₄	semidirect
54. C₁₃×D₁₆	D₄	semidirect
66. #33858	C₁₃	direct (N × H)
103. #3415	C₂×C₁₃	semidirect
104. #3292	C₂×C₁₃	semidirect
105. #2246	C₂×C₁₃	semidirect
106. #2221	C₂×C₁₃	semidirect
107. #2213	C₂×C₁₃	semidirect
108. #2192	C₂×C₁₃	semidirect
109. #2178	C₂×C₁₃	semidirect
110. #2170	C₂×C₁₃	semidirect
136. #576	C₂²×C₁₃	semidirect
138. #509	C₂²×C₁₃	semidirect
142. C₂×D₁₆	C₂²×C₁₃	semidirect
143. C₂×D₁₆	C₂²×C₁₃	semidirect
144. #504	C₂²×C₁₃	semidirect
145. #493	C₂²×C₁₃	semidirect
147. #472	C₂²×C₁₃	semidirect
150. #443	C₂²×C₁₃	semidirect
151. #368	C₂²×C₁₃	semidirect
152. #364	C₂²×C₁₃	semidirect
153. #361	C₂²×C₁₃	semidirect
154. #359	C₂²×C₁₃	semidirect
155. #350	C₂²×C₁₃	semidirect
156. C₄×C₁₆	C₂²×C₁₃	semidirect
184. Q₃₂	D₄×C₁₃	semidirect
186. SD₃₂	D₄×C₁₃	semidirect
191. D₁₆	D₄×C₁₃	semidirect
244. C₁₃	#33858	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×#3415	2
C₁₃×#3292	2
C₁₃×#2246	2
C₁₃×#2221	2
C₁₃×#2213	2
C₁₃×#2192	2
C₁₃×#2178	2
C₁₃×#2170	2
#33858	13

Subgroups — C13×#33858 #15720197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33858 #15720197