Subgroups — C₁₃×#2970 #13112028

Order 1664.

406 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#446	C₂	1	2	yes	maximal, complemented
3	C₁₃×#435	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₁₃	C₂	1	2	yes	maximal, complemented
5	C₁₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#432	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#97	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#41	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₃	C₂²	1	4	yes	—
12	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₄	1	2	yes	—
15	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
16	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₃	C₂²	1	2	yes	—
18	C₁₃×#117	C₄	1	1	yes	characteristic
19	C₁₃×#117	C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
24	C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
25	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₁₃×#97	C₂²	1	2	yes	—
28	C₁₃×#97	C₂²	1	4	yes	—
29	C₁₃×#97	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂²	1	2	yes	—
31	C₁₃×#41	—	2	4	no	—
32	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×M₁₆	—	2	4	no	—
42	C₂×C₈×C₁₃	—	2	8	no	—
43	C₂×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₈×C₁₃	C₂³	1	4	yes	—
45	C₂×C₈×C₁₃	C₂³	1	2	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₄²×C₁₃	C₂×C₄	1	2	yes	—
54	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×C₄	1	2	yes	—
56	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃	C₂³	1	2	yes	—
60	#2970	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
63	Q₈×C₁₃	—	2	2	no	—
64	D₄×C₁₃	—	2	4	no	—
65	D₄×C₁₃	M₁₆	1	4	yes	complemented
66	C₂×C₄×C₁₃	—	2	4	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	1	no	—
72	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₃	—	2	4	no	—
82	C₈×C₁₃	—	2	2	no	—
83	C₈×C₁₃	#41	1	4	yes	complemented
84	C₄×#41	C₂×C₁₃	1	1	yes	characteristic
85	#446	C₂×C₁₃	1	2	yes	complemented
86	#435	C₂×C₁₃	1	4	yes	complemented
87	C₈×D₄	C₂×C₁₃	1	2	yes	complemented
88	#432	C₂×C₁₃	1	1	yes	characteristic, complemented
89	#432	C₂×C₁₃	1	2	yes	complemented
90	C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂×#97	C₂×C₁₃	1	2	yes	complemented
92	C₂²×C₁₃	—	2	2	no	—
93	C₂²×C₁₃	—	2	2	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	C₂×M₁₆	1	4	yes	—
98	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
107	C₄×C₁₃	C₂×#41	1	2	yes	—
108	C₂×#41	C₄×C₁₃	1	1	yes	characteristic
109	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
110	C₂²×C₈	C₂²×C₁₃	1	4	yes	—
111	C₄×Q₈	C₄×C₁₃	1	1	yes	characteristic
112	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
113	C₄×D₄	C₄×C₁₃	1	2	yes	—
114	C₄×D₄	C₄×C₁₃	1	1	yes	characteristic
115	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
116	C₄×D₄	C₂²×C₁₃	1	2	yes	—
117	#117	C₄×C₁₃	1	1	yes	characteristic
118	#117	C₄×C₁₃	1	2	yes	—
119	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
120	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
121	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
122	C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
123	C₂²⋊C₈	C₂²×C₁₃	1	4	yes	—
124	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
125	#97	C₂²×C₁₃	1	1	yes	characteristic
126	#97	C₂²×C₁₃	1	2	yes	—
127	#97	C₂²×C₁₃	1	4	yes	—
128	#97	C₂²×C₁₃	1	1	yes	characteristic
129	C₄×C₈	C₂²×C₁₃	1	2	yes	—
130	C₂×C₁₃	—	2	4	no	—
131	C₂×C₁₃	—	4	1	no	—
132	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
133	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₁₃	#568	1	1	yes	characteristic
135	#41	—	2	4	no	—
136	C₂×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
137	C₂×D₄	C₂×C₄×C₁₃	1	2	yes	—
138	C₂×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×C₁₃	1	2	yes	—
140	C₂²×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₄×C₁₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
145	M₁₆	—	2	4	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	C₂³×C₁₃	1	2	yes	—
148	C₂×C₈	C₂³×C₁₃	1	4	yes	—
149	C₂×C₈	C₂³×C₁₃	1	2	yes	—
150	C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
151	C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
152	C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
153	C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
154	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
155	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
156	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
157	C₄²	C₂×C₄×C₁₃	1	2	yes	—
158	C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
159	C₄²	C₂×C₄×C₁₃	1	2	yes	—
160	C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
161	C₄²	C₂³×C₁₃	1	1	yes	characteristic
162	C₄²	C₂³×C₁₃	1	1	yes	characteristic
163	C₄²	C₂³×C₁₃	1	2	yes	—
164	C₁₃	#2970	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³	C₂²×C₄×C₁₃	1	1	yes	characteristic
166	C₂³	C₂²×C₄×C₁₃	1	2	yes	—
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	C₁₃×M₁₆	1	4	yes	complemented
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
179	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
182	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
183	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	C₁₃×#41	1	4	yes	complemented
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₂×C₁₃×M₁₆	1	4	yes	—
194	C₂²	C₂³×C₄×C₁₃	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	C₂×C₁₃×M₁₆	1	2	yes	—
203	C₄	C₂×C₁₃×#41	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	4	1	no	—
206	C₂	C₄×C₁₃×#41	1	1	yes	characteristic
207	C₂	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
208	C₂	C₁₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₄
prime radical	C₂×C₁₃×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. C₁₃×#435	C₂	semidirect
4. C₈×D₄×C₁₃	C₂	semidirect
5. C₁₃×#432	C₂	semidirect
6. C₁₃×#432	C₂	semidirect
7. C₄×C₁₃×M₁₆	C₂	semidirect
8. C₂×C₁₃×#97	C₂	semidirect
60. #2970	C₁₃	direct (N × H)
65. D₄×C₁₃	M₁₆	semidirect
83. C₈×C₁₃	#41	semidirect
85. #446	C₂×C₁₃	semidirect
86. #435	C₂×C₁₃	semidirect
87. C₈×D₄	C₂×C₁₃	semidirect
88. #432	C₂×C₁₃	semidirect
89. #432	C₂×C₁₃	semidirect
90. C₄×M₁₆	C₂×C₁₃	semidirect
91. C₂×#97	C₂×C₁₃	semidirect
164. C₁₃	#2970	direct (N × H)
169. D₄	C₁₃×M₁₆	semidirect
187. C₈	C₁₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#41	2
C₁₃×#446	2
C₁₃×#435	2
C₈×D₄×C₁₃	2
C₁₃×#432	2
C₄×C₁₃×M₁₆	2
C₂×C₁₃×#97	2
#2970	13

Subgroups — C13×#2970 #13112028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2970 #13112028