Subgroups — C₁₃×#3368 #13112426

Order 1664.

382 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#490	C₂	1	2	yes	maximal, complemented
4	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#478	C₂	1	2	yes	maximal, complemented
6	C₁₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#460	C₂	1	2	yes	maximal, complemented
8	C₁₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#440	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	—	2	1	no	—
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#127	C₂²	1	1	yes	characteristic
18	C₁₃×#127	C₂²	1	2	yes	—
19	C₁₃×#125	—	2	2	no	—
20	C₁₃×#125	C₂²	1	2	yes	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	—	2	2	no	—
23	C₄×Q₈×C₁₃	—	2	1	no	—
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₃×#103	C₂²	1	1	yes	characteristic
34	C₁₃×#102	C₂²	1	1	yes	characteristic
35	C₁₃×#102	C₂²	1	2	yes	—
36	C₁₃×#102	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₁₃×#41	—	4	4	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₁₃×SD₁₆	—	2	2	no	—
54	C₁₃×SD₁₆	—	2	4	no	—
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂³	1	2	yes	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	1	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	C₄²×C₁₃	C₂³	1	2	yes	—
74	#3368	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	Q₈×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	Q₈×C₁₃	#41	1	2	yes	complemented
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	—	2	1	no	—
84	D₄×C₁₃	—	2	2	no	—
85	D₄×C₁₃	—	4	4	no	—
86	D₄×C₁₃	—	4	2	no	—
87	D₄×C₁₃	—	8	2	no	—
88	D₄×C₁₃	—	8	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	—	2	1	no	—
103	#557	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#550	C₂×C₁₃	1	1	yes	characteristic
105	#490	C₂×C₁₃	1	2	yes	complemented
106	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#478	C₂×C₁₃	1	2	yes	complemented
108	#474	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#460	C₂×C₁₃	1	2	yes	complemented
110	#460	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#440	C₂×C₁₃	1	2	yes	complemented
113	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	—	4	2	no	—
119	C₂²×C₁₃	—	4	1	no	—
120	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×#41	—	2	1	no	—
132	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
134	C₄⋊Q₈	—	2	1	no	—
135	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	#127	C₂²×C₁₃	1	1	yes	characteristic
137	#127	C₂²×C₁₃	1	2	yes	—
138	#125	—	2	2	no	—
139	#125	C₂²×C₁₃	1	2	yes	—
140	#121	—	2	1	no	—
141	#121	—	2	2	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
145	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₁₃	1	2	yes	—
148	#107	C₂²×C₁₃	1	2	yes	—
149	#106	C₂²×C₁₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
152	#103	C₂²×C₁₃	1	1	yes	characteristic
153	#102	C₂²×C₁₃	1	1	yes	characteristic
154	#102	C₂²×C₁₃	1	2	yes	—
155	#102	C₂²×C₁₃	1	2	yes	—
156	#102	C₂²×C₁₃	1	2	yes	—
157	#102	C₂²×C₁₃	1	2	yes	—
158	#97	C₂²×C₁₃	1	2	yes	—
159	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×C₁₃	—	8	2	no	—
161	C₂×C₁₃	—	8	1	no	—
162	C₂×C₁₃	#548	1	1	yes	characteristic
163	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×C₁₃	#577	1	1	yes	characteristic
165	#41	—	4	4	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×D₄	C₂³×C₁₃	1	2	yes	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	1	no	—
177	SD₁₆	—	2	2	no	—
178	SD₁₆	—	2	4	no	—
179	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₂×C₈	C₂³×C₁₃	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
188	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
189	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
191	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₁₃	1	1	yes	characteristic
197	C₄²	C₂³×C₁₃	1	2	yes	—
198	C₁₃	#3368	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	4	2	no	—
204	Q₈	C₁₃×#41	1	2	yes	complemented
205	D₄	—	2	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	8	2	no	—
212	D₄	—	8	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
242	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
243	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
244	C₂	—	8	2	no	—
245	C₂	—	8	1	no	—
246	C₂	C₁₃×#548	1	1	yes	characteristic
247	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#557	C₂	semidirect
3. C₁₃×#490	C₂	semidirect
4. C₁₃×#486	C₂	semidirect
5. C₁₃×#478	C₂	semidirect
6. C₁₃×#474	C₂	semidirect
7. C₁₃×#460	C₂	semidirect
8. C₁₃×#460	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
10. C₁₃×#440	C₂	semidirect
11. C₄×C₁₃×SD₁₆	C₂	semidirect
74. #3368	C₁₃	direct (N × H)
80. Q₈×C₁₃	#41	semidirect
103. #557	C₂×C₁₃	semidirect
105. #490	C₂×C₁₃	semidirect
106. #486	C₂×C₁₃	semidirect
107. #478	C₂×C₁₃	semidirect
108. #474	C₂×C₁₃	semidirect
109. #460	C₂×C₁₃	semidirect
110. #460	C₂×C₁₃	semidirect
111. #446	C₂×C₁₃	semidirect
112. #440	C₂×C₁₃	semidirect
113. C₄×SD₁₆	C₂×C₁₃	semidirect
198. C₁₃	#3368	direct (N × H)
204. Q₈	C₁₃×#41	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#557	2
C₁₃×#550	2
C₁₃×#490	2
C₁₃×#486	2
C₁₃×#478	2
C₁₃×#474	2
C₁₃×#460	2
C₁₃×#446	2
C₁₃×#440	2
C₄×C₁₃×SD₁₆	2
#3368	13

Subgroups — C13×#3368 #13112426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3368 #13112426