Subgroups — C₁₃×#3077 #13112135

Order 1664.

440 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#478	C₂	1	4	yes	maximal, complemented
4	C₁₃×#476	C₂	1	2	yes	maximal, complemented
5	C₁₃×#474	C₂	1	2	yes	maximal, complemented
6	C₁₃×#428	C₂	1	2	yes	maximal, complemented
7	C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#418	C₂	1	2	yes	maximal
9	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	—	2	1	no	—
11	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#125	C₂²	1	2	yes	complemented
15	C₁₃×C₂²⋊Q₈	—	2	1	no	—
16	C₁₃×C₂²⋊Q₈	—	2	1	no	—
17	C₄×Q₈×C₁₃	—	2	1	no	—
18	C₄×Q₈×C₁₃	—	2	1	no	—
19	C₄×D₄×C₁₃	—	2	1	no	—
20	C₄×D₄×C₁₃	—	2	1	no	—
21	C₄×D₄×C₁₃	C₂²	1	2	yes	—
22	C₄×D₄×C₁₃	C₂²	1	2	yes	—
23	C₁₃×#117	C₂²	1	2	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
26	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#107	C₂²	1	4	yes	complemented
28	C₁₃×#106	C₂²	1	4	yes	complemented
29	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
31	C₁₃×#102	C₂²	1	4	yes	—
32	C₁₃×#102	C₂²	1	4	yes	—
33	C₂⁴×C₁₃	—	2	1	no	—
34	C₂×Q₈×C₁₃	—	2	1	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×D₄×C₁₃	—	2	1	no	—
37	C₂×D₄×C₁₃	—	2	2	no	—
38	C₂×D₄×C₁₃	Q₈	1	4	yes	complemented
39	C₂×D₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	—	2	1	no	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	4	4	no	—
44	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×M₁₆	—	2	4	no	—
48	C₂×C₈×C₁₃	C₂³	1	4	yes	—
49	C₁₃×C₄⋊C₄	—	2	1	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₁₃×C₂²⋊C₄	—	2	2	no	—
59	C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₄²×C₁₃	D₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₄²×C₁₃	C₂³	1	1	yes	characteristic
65	#3077	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₃	—	2	4	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	—	4	2	no	—
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₁₃	—	4	2	no	—
72	Q₈×C₁₃	—	4	2	no	—
73	D₄×C₁₃	—	2	4	no	—
74	D₄×C₁₃	—	2	2	no	—
75	D₄×C₁₃	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	4	1	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₁₃	#41	1	2	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₃	—	2	4	no	—
93	#533	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic
95	#478	C₂×C₁₃	1	4	yes	complemented
96	#476	C₂×C₁₃	1	2	yes	complemented
97	#474	C₂×C₁₃	1	2	yes	complemented
98	#428	C₂×C₁₃	1	2	yes	complemented
99	#423	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#418	C₂×C₁₃	1	2	yes	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	—	2	4	no	—
104	C₂²×C₁₃	—	4	8	no	—
105	C₂²×C₁₃	—	4	2	no	—
106	C₂²×C₁₃	—	4	2	no	—
107	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	—	4	2	no	—
114	C₄×C₁₃	—	4	2	no	—
115	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
116	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
120	C₂³×C₄	—	2	1	no	—
121	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
122	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
123	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
124	#125	C₂²×C₁₃	1	2	yes	complemented
125	C₂²⋊Q₈	—	2	1	no	—
126	C₂²⋊Q₈	—	2	1	no	—
127	C₄×Q₈	—	2	1	no	—
128	C₄×Q₈	—	2	1	no	—
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	—	2	1	no	—
131	C₄×D₄	C₂²×C₁₃	1	2	yes	—
132	C₄×D₄	C₂²×C₁₃	1	2	yes	—
133	#117	C₂²×C₁₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₂²⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
137	#107	C₂²×C₁₃	1	4	yes	complemented
138	#106	C₂²×C₁₃	1	4	yes	complemented
139	C₄⋊C₈	C₂²×C₁₃	1	2	yes	complemented
140	C₄⋊C₈	C₂²×C₁₃	1	2	yes	complemented
141	#102	C₂²×C₁₃	1	4	yes	—
142	#102	C₂²×C₁₃	1	4	yes	—
143	C₂×C₁₃	—	2	2	no	—
144	C₂×C₁₃	—	4	4	no	—
145	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₁₃	#575	1	1	yes	characteristic
148	C₂⁴	—	2	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	Q₈×C₁₃	1	4	yes	complemented
154	C₂×D₄	C₂³×C₁₃	1	2	yes	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
162	M₁₆	—	2	4	no	—
163	C₂×C₈	C₂³×C₁₃	1	4	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
170	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
171	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	D₄×C₁₃	1	1	yes	characteristic
177	C₄²	D₄×C₁₃	1	1	yes	characteristic
178	C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₁₃	#3077	1	1	yes	characteristic, direct product factor, Sylow
181	C₂³	—	2	4	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	2	2	no	—
190	D₄	C₂×Q₈×C₁₃	1	4	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
205	C₂×C₄	C₁₃×#41	1	2	yes	—
206	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₁₃×C₂²⋊Q₈	1	2	yes	—
215	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
223	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
224	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
225	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
229	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
3. C₁₃×#478	C₂	semidirect
4. C₁₃×#476	C₂	semidirect
5. C₁₃×#474	C₂	semidirect
6. C₁₃×#428	C₂	semidirect
7. C₁₃×#423	C₂	semidirect
12. C₁₃×C₄⋊Q₈	C₂²	semidirect
13. C₁₃×C₄⋊Q₈	C₂²	semidirect
14. C₁₃×#125	C₂²	semidirect
27. C₁₃×#107	C₂²	semidirect
28. C₁₃×#106	C₂²	semidirect
29. C₁₃×C₄⋊C₈	C₂²	semidirect
30. C₁₃×C₄⋊C₈	C₂²	semidirect
38. C₂×D₄×C₁₃	Q₈	semidirect
65. #3077	C₁₃	direct (N × H)
93. #533	C₂×C₁₃	semidirect
95. #478	C₂×C₁₃	semidirect
96. #476	C₂×C₁₃	semidirect
97. #474	C₂×C₁₃	semidirect
98. #428	C₂×C₁₃	semidirect
99. #423	C₂×C₁₃	semidirect
122. C₄⋊Q₈	C₂²×C₁₃	semidirect
123. C₄⋊Q₈	C₂²×C₁₃	semidirect
124. #125	C₂²×C₁₃	semidirect
137. #107	C₂²×C₁₃	semidirect
138. #106	C₂²×C₁₃	semidirect
139. C₄⋊C₈	C₂²×C₁₃	semidirect
140. C₄⋊C₈	C₂²×C₁₃	semidirect
153. C₂×D₄	Q₈×C₁₃	semidirect
180. C₁₃	#3077	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×#533	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#478	2
C₁₃×#476	2
C₁₃×#474	2
C₁₃×#428	2
C₁₃×#423	2
C₁₃×#418	2
#3077	13

Subgroups — C13×#3077 #13112135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3077 #13112135