Subgroups — C₁₃×#3366 #13112424

Order 1664.

396 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
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×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#459	C₂	1	2	yes	maximal, complemented
8	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#442	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×D₈	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#127	C₂²	1	1	yes	characteristic
17	C₁₃×#127	C₂²	1	2	yes	—
18	C₁₃×#125	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×C₂²⋊Q₈	—	2	2	no	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	—	2	2	no	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₄×D₄×C₁₃	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₃×#102	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×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₃	C₂³	1	2	yes	—
50	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₁₃×D₈	—	2	2	no	—
56	C₁₃×D₈	—	2	4	no	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂³	1	2	yes	—
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	2	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	C₄²×C₁₃	C₂³	1	2	yes	—
74	#3366	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	8	1	no	—
80	D₄×C₁₃	—	2	1	no	—
81	D₄×C₁₃	—	2	2	no	—
82	D₄×C₁₃	—	2	1	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	D₄×C₁₃	#41	1	2	yes	complemented
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	8	2	no	—
97	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	—	2	1	no	—
105	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#550	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#490	C₂×C₁₃	1	2	yes	complemented
108	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#478	C₂×C₁₃	1	2	yes	complemented
110	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#459	C₂×C₁₃	1	2	yes	complemented
112	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#442	C₂×C₁₃	1	2	yes	complemented
115	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	—	4	1	no	—
119	C₂²×C₁₃	—	4	2	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	2	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	2	no	—
125	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	4	1	no	—
130	C₄×C₁₃	—	4	1	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
133	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×#41	—	2	1	no	—
136	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×D₈	C₂²×C₁₃	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	#127	C₂²×C₁₃	1	1	yes	characteristic
140	#127	C₂²×C₁₃	1	2	yes	—
141	#125	C₂²×C₁₃	1	2	yes	—
142	C₂²⋊Q₈	—	2	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	#121	—	2	1	no	—
145	#121	—	2	2	no	—
146	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₁₃	1	2	yes	—
149	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×D₄	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	#107	C₂²×C₁₃	1	1	yes	characteristic
153	#106	C₂²×C₁₃	1	2	yes	—
154	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
156	#102	C₂²×C₁₃	1	1	yes	characteristic
157	#102	C₂²×C₁₃	1	1	yes	characteristic
158	#102	C₂²×C₁₃	1	2	yes	—
159	#102	C₂²×C₁₃	1	2	yes	—
160	#102	C₂²×C₁₃	1	2	yes	—
161	#102	C₂²×C₁₃	1	2	yes	—
162	#97	C₂²×C₁₃	1	2	yes	—
163	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₁₃	—	4	2	no	—
165	C₂×C₁₃	—	8	2	no	—
166	C₂×C₁₃	—	8	1	no	—
167	C₂×C₁₃	#548	1	1	yes	characteristic
168	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₁₃	#576	1	1	yes	characteristic
170	#41	—	4	4	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×D₄	C₂³×C₁₃	1	2	yes	—
179	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	2	no	—
184	D₈	—	2	2	no	—
185	D₈	—	2	4	no	—
186	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₁₃	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
193	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
194	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
195	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₄²	C₂³×C₁₃	1	2	yes	—
203	C₁₃	#3366	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	8	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	1	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	1	no	—
218	D₄	C₁₃×#41	1	2	yes	complemented
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
228	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
229	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
251	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
252	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	—	8	2	no	—
255	C₂	—	8	1	no	—
256	C₂	C₁₃×#548	1	1	yes	characteristic
257	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂	C₁₃×#576	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 ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#554	C₂	semidirect
2. C₁₃×#550	C₂	semidirect
3. C₁₃×#490	C₂	semidirect
4. C₁₃×#486	C₂	semidirect
5. C₁₃×#478	C₂	semidirect
6. C₁₃×#476	C₂	semidirect
7. C₁₃×#459	C₂	semidirect
8. C₁₃×#459	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
10. C₁₃×#442	C₂	semidirect
11. C₄×C₁₃×D₈	C₂	semidirect
74. #3366	C₁₃	direct (N × H)
89. D₄×C₁₃	#41	semidirect
105. #554	C₂×C₁₃	semidirect
106. #550	C₂×C₁₃	semidirect
107. #490	C₂×C₁₃	semidirect
108. #486	C₂×C₁₃	semidirect
109. #478	C₂×C₁₃	semidirect
110. #476	C₂×C₁₃	semidirect
111. #459	C₂×C₁₃	semidirect
112. #459	C₂×C₁₃	semidirect
113. #446	C₂×C₁₃	semidirect
114. #442	C₂×C₁₃	semidirect
115. C₄×D₈	C₂×C₁₃	semidirect
203. C₁₃	#3366	direct (N × H)
218. D₄	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×#554	2
C₁₃×#550	2
C₁₃×#490	2
C₁₃×#486	2
C₁₃×#478	2
C₁₃×#476	2
C₁₃×#459	2
C₁₃×#446	2
C₁₃×#442	2
C₄×C₁₃×D₈	2
#3366	13

Subgroups — C13×#3366 #13112424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3366 #13112424