Subgroups — C₁₃×#3222 #13112280

Order 1664.

396 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#535	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#464	C₂	1	2	yes	maximal, complemented
8	C₁₃×#460	C₂	1	2	yes	maximal, complemented
9	C₁₃×#452	C₂	1	2	yes	maximal, complemented
10	C₁₃×#450	C₂	1	2	yes	maximal, complemented
11	C₁₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	complemented
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#127	C₂²	1	1	yes	characteristic
20	C₁₃×#127	C₂²	1	1	yes	characteristic
21	C₁₃×#126	—	2	2	no	—
22	C₁₃×#125	—	2	2	no	—
23	C₁₃×#125	C₂²	1	1	yes	characteristic, complemented
24	C₁₃×#124	C₂²	1	1	yes	characteristic
25	C₁₃×#124	C₂²	1	1	yes	characteristic
26	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₁₃×#121	—	2	2	no	—
28	C₁₃×#120	—	2	2	no	—
29	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
30	C₁₃×#117	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₁₃×#103	C₂²	1	2	yes	complemented
33	C₁₃×#102	C₂²	1	2	yes	—
34	C₁₃×#102	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	2	yes	complemented
36	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₁₃×#97	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₁₃	—	2	1	no	—
40	C₁₃×#41	—	4	2	no	—
41	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
43	C₂×D₄×C₁₃	—	2	1	no	—
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	4	2	no	—
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×Q₁₆	—	4	2	no	—
54	C₁₃×SD₁₆	—	4	2	no	—
55	C₁₃×SD₁₆	—	4	2	no	—
56	C₁₃×SD₁₆	—	4	2	no	—
57	C₂×C₈×C₁₃	C₂³	1	2	yes	—
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	#3222	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₃	—	2	1	no	—
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	4	2	no	—
77	C₂³×C₁₃	—	8	2	no	—
78	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
79	Q₈×C₁₃	—	2	2	no	—
80	Q₈×C₁₃	—	2	2	no	—
81	Q₈×C₁₃	—	2	2	no	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	—	2	2	no	—
84	D₄×C₁₃	—	2	2	no	—
85	D₄×C₁₃	—	4	2	no	—
86	D₄×C₁₃	—	4	2	no	—
87	D₄×C₁₃	—	8	2	no	—
88	D₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	2	2	no	—
100	#541	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#535	C₂×C₁₃	1	1	yes	characteristic
102	#497	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#490	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#488	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#464	C₂×C₁₃	1	2	yes	complemented
107	#460	C₂×C₁₃	1	2	yes	complemented
108	#452	C₂×C₁₃	1	2	yes	complemented
109	#450	C₂×C₁₃	1	2	yes	complemented
110	#433	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	1	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	2	no	—
118	C₂²×C₁₃	—	4	2	no	—
119	C₂²×C₁₃	—	8	4	no	—
120	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	4	1	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
129	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
131	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
132	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	complemented
134	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
135	#127	C₂²×C₁₃	1	1	yes	characteristic
136	#127	C₂²×C₁₃	1	1	yes	characteristic
137	#126	—	2	2	no	—
138	#125	—	2	2	no	—
139	#125	C₂²×C₁₃	1	1	yes	characteristic, complemented
140	#124	C₂²×C₁₃	1	1	yes	characteristic
141	#124	C₂²×C₁₃	1	1	yes	characteristic
142	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
143	#121	—	2	2	no	—
144	#120	—	2	2	no	—
145	C₄×Q₈	C₂²×C₁₃	1	2	yes	complemented
146	#117	C₂²×C₁₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂²×C₁₃	1	2	yes	complemented
148	#103	C₂²×C₁₃	1	2	yes	complemented
149	#102	C₂²×C₁₃	1	2	yes	—
150	#102	C₂²×C₁₃	1	2	yes	—
151	#102	C₂²×C₁₃	1	2	yes	complemented
152	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
153	#97	C₂²×C₁₃	1	1	yes	characteristic, complemented
154	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
155	C₂×C₁₃	—	4	1	no	—
156	C₂×C₁₃	—	8	1	no	—
157	C₂×C₁₃	—	8	1	no	—
158	C₂×C₁₃	—	8	1	no	—
159	C₂×C₁₃	#535	1	1	yes	characteristic
160	C₂×C₁₃	#577	1	2	yes	—
161	C₂⁴	—	2	1	no	—
162	#41	—	4	2	no	—
163	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
171	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
175	Q₁₆	—	4	2	no	—
176	SD₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	SD₁₆	—	4	2	no	—
179	C₂×C₈	C₂³×C₁₃	1	2	yes	—
180	C₂×C₈	C₂³×C₁₃	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
186	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₁₃	#3222	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	8	2	no	—
200	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
201	Q₈	—	2	2	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	2	no	—
210	D₄	—	8	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
239	C₂	—	4	1	no	—
240	C₂	—	8	1	no	—
241	C₂	—	8	1	no	—
242	C₂	—	8	1	no	—
243	C₂	C₁₃×#535	1	1	yes	characteristic
244	C₂	C₁₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#541	C₂	semidirect
3. C₁₃×#497	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#490	C₂	semidirect
6. C₁₃×#488	C₂	semidirect
7. C₁₃×#464	C₂	semidirect
8. C₁₃×#460	C₂	semidirect
9. C₁₃×#452	C₂	semidirect
10. C₁₃×#450	C₂	semidirect
11. C₁₃×#433	C₂	semidirect
14. C₂×C₁₃×Q₁₆	C₂²	semidirect
17. C₂×C₁₃×SD₁₆	C₂²	semidirect
18. C₁₃×C₄⋊Q₈	C₂²	semidirect
23. C₁₃×#125	C₂²	semidirect
29. C₄×Q₈×C₁₃	C₂²	semidirect
31. C₁₃×C₄⋊C₈	C₂²	semidirect
32. C₁₃×#103	C₂²	semidirect
35. C₁₃×#102	C₂²	semidirect
37. C₁₃×#97	C₂²	semidirect
38. C₄×C₈×C₁₃	C₂²	semidirect
72. #3222	C₁₃	direct (N × H)
100. #541	C₂×C₁₃	semidirect
102. #497	C₂×C₁₃	semidirect
103. #492	C₂×C₁₃	semidirect
104. #490	C₂×C₁₃	semidirect
105. #488	C₂×C₁₃	semidirect
106. #464	C₂×C₁₃	semidirect
107. #460	C₂×C₁₃	semidirect
108. #452	C₂×C₁₃	semidirect
109. #450	C₂×C₁₃	semidirect
110. #433	C₂×C₁₃	semidirect
130. C₂×Q₁₆	C₂²×C₁₃	semidirect
133. C₂×SD₁₆	C₂²×C₁₃	semidirect
134. C₄⋊Q₈	C₂²×C₁₃	semidirect
139. #125	C₂²×C₁₃	semidirect
145. C₄×Q₈	C₂²×C₁₃	semidirect
147. C₄⋊C₈	C₂²×C₁₃	semidirect
148. #103	C₂²×C₁₃	semidirect
151. #102	C₂²×C₁₃	semidirect
153. #97	C₂²×C₁₃	semidirect
154. C₄×C₈	C₂²×C₁₃	semidirect
194. C₁₃	#3222	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×#541	2
C₁₃×#535	2
C₁₃×#497	2
C₁₃×#492	2
C₁₃×#490	2
C₁₃×#488	2
C₁₃×#464	2
C₁₃×#460	2
C₁₃×#452	2
C₁₃×#450	2
C₁₃×#433	2
#3222	13

Subgroups — C13×#3222 #13112280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3222 #13112280