Subgroups — C₁₃×#3191 #13112249

Order 1664.

424 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#525	C₂	1	2	yes	maximal, complemented
2	C₁₃×#485	C₂	1	2	yes	maximal, complemented
3	C₁₃×#482	C₂	1	2	yes	maximal, complemented
4	C₁₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#465	C₂	1	2	yes	maximal, complemented
7	C₁₃×#452	C₂	1	2	yes	maximal, complemented
8	C₁₃×#449	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	C₂²	1	2	yes	—
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	complemented
14	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#123	—	2	2	no	—
18	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₁₃×#121	C₂²	1	2	yes	—
21	C₁₃×#121	C₂²	1	2	yes	—
22	C₁₃×#120	—	2	2	no	—
23	C₄×D₄×C₁₃	—	2	2	no	—
24	C₄×D₄×C₁₃	—	2	2	no	—
25	C₁₃×#117	C₂²	1	2	yes	—
26	C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
27	C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
28	C₁₃×#106	C₂²	1	2	yes	complemented
29	C₁₃×#103	C₂²	1	2	yes	complemented
30	C₁₃×#103	C₂²	1	2	yes	complemented
31	C₁₃×#102	C₂²	1	2	yes	complemented
32	C₁₃×#102	C₂²	1	2	yes	complemented
33	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
36	C₁₃×#41	—	4	2	no	—
37	C₁₃×#41	—	4	2	no	—
38	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	4	1	no	—
45	C₂²×C₄×C₁₃	—	4	1	no	—
46	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	D₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×Q₁₆	—	4	1	no	—
51	C₁₃×SD₁₆	—	4	2	no	—
52	C₁₃×D₈	—	4	1	no	—
53	C₂×C₈×C₁₃	—	4	1	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
61	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	#3191	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	—	2	2	no	—
71	C₂³×C₁₃	—	4	1	no	—
72	C₂³×C₁₃	—	4	1	no	—
73	C₂³×C₁₃	C₂×D₄	1	2	yes	—
74	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	2	no	—
76	Q₈×C₁₃	—	2	2	no	—
77	D₄×C₁₃	—	2	2	no	—
78	D₄×C₁₃	—	2	2	no	—
79	D₄×C₁₃	—	4	4	no	—
80	D₄×C₁₃	—	4	4	no	—
81	D₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	4	1	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	2	1	no	—
98	#525	C₂×C₁₃	1	2	yes	complemented
99	#485	C₂×C₁₃	1	2	yes	complemented
100	#482	C₂×C₁₃	1	2	yes	complemented
101	#467	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#466	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#465	C₂×C₁₃	1	2	yes	complemented
104	#452	C₂×C₁₃	1	2	yes	complemented
105	#449	C₂×C₁₃	1	2	yes	complemented
106	C₂×C₂²⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	4	4	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	#135	1	2	yes	—
117	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	—	4	2	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×#41	C₂²×C₁₃	1	2	yes	—
126	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
128	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	complemented
129	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
130	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
132	#123	—	2	2	no	—
133	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
134	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
135	#121	C₂²×C₁₃	1	2	yes	—
136	#121	C₂²×C₁₃	1	2	yes	—
137	#120	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	#117	C₂²×C₁₃	1	2	yes	—
141	#107	C₂²×C₁₃	1	1	yes	characteristic, complemented
142	#107	C₂²×C₁₃	1	1	yes	characteristic, complemented
143	#106	C₂²×C₁₃	1	2	yes	complemented
144	#103	C₂²×C₁₃	1	2	yes	complemented
145	#103	C₂²×C₁₃	1	2	yes	complemented
146	#102	C₂²×C₁₃	1	2	yes	complemented
147	#102	C₂²×C₁₃	1	2	yes	complemented
148	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
149	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
150	C₂×C₁₃	—	2	2	no	—
151	C₂×C₁₃	—	4	2	no	—
152	C₂×C₁₃	—	8	2	no	—
153	C₂×C₁₃	#534	1	1	yes	characteristic
154	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
155	C₂×C₁₃	#575	1	1	yes	characteristic
156	C₂⁴	D₄×C₁₃	1	1	yes	characteristic
157	#41	—	4	2	no	—
158	#41	—	4	2	no	—
159	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	C₂³×C₁₃	1	2	yes	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	4	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
168	C₂²×C₄	D₄×C₁₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
171	Q₁₆	—	4	1	no	—
172	SD₁₆	—	4	2	no	—
173	D₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₃	1	2	yes	—
178	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
182	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₁₃	#3191	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×D₄×C₁₃	1	2	yes	—
195	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₁₃×#135	1	2	yes	—
229	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
236	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	4	2	no	—
239	C₂	—	8	2	no	—
240	C₂	C₁₃×#534	1	1	yes	characteristic
241	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
242	C₂	C₁₃×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₁₃×#525	C₂	semidirect
2. C₁₃×#485	C₂	semidirect
3. C₁₃×#482	C₂	semidirect
4. C₁₃×#467	C₂	semidirect
5. C₁₃×#466	C₂	semidirect
6. C₁₃×#465	C₂	semidirect
7. C₁₃×#452	C₂	semidirect
8. C₁₃×#449	C₂	semidirect
9. C₂×C₁₃×C₂²⋊C₈	C₂	semidirect
12. C₂×C₁₃×Q₁₆	C₂²	semidirect
13. C₂×C₁₃×SD₁₆	C₂²	semidirect
14. C₂×C₁₃×D₈	C₂²	semidirect
26. C₁₃×#107	C₂²	semidirect
27. C₁₃×#107	C₂²	semidirect
28. C₁₃×#106	C₂²	semidirect
29. C₁₃×#103	C₂²	semidirect
30. C₁₃×#103	C₂²	semidirect
31. C₁₃×#102	C₂²	semidirect
32. C₁₃×#102	C₂²	semidirect
68. #3191	C₁₃	direct (N × H)
98. #525	C₂×C₁₃	semidirect
99. #485	C₂×C₁₃	semidirect
100. #482	C₂×C₁₃	semidirect
101. #467	C₂×C₁₃	semidirect
102. #466	C₂×C₁₃	semidirect
103. #465	C₂×C₁₃	semidirect
104. #452	C₂×C₁₃	semidirect
105. #449	C₂×C₁₃	semidirect
106. C₂×C₂²⋊C₈	C₂×C₁₃	semidirect
127. C₂×Q₁₆	C₂²×C₁₃	semidirect
128. C₂×SD₁₆	C₂²×C₁₃	semidirect
129. C₂×D₈	C₂²×C₁₃	semidirect
141. #107	C₂²×C₁₃	semidirect
142. #107	C₂²×C₁₃	semidirect
143. #106	C₂²×C₁₃	semidirect
144. #103	C₂²×C₁₃	semidirect
145. #103	C₂²×C₁₃	semidirect
146. #102	C₂²×C₁₃	semidirect
147. #102	C₂²×C₁₃	semidirect
189. C₁₃	#3191	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×#525	2
C₁₃×#485	2
C₁₃×#482	2
C₁₃×#467	2
C₁₃×#466	2
C₁₃×#465	2
C₁₃×#452	2
C₁₃×#449	2
C₂×C₁₃×C₂²⋊C₈	2
#3191	13

Subgroups — C13×#3191 #13112249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3191 #13112249