Subgroups — C₁₃×#3235 #13112293

Order 1664.

366 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#501	C₂	1	2	yes	maximal, complemented
4	C₁₃×#490	C₂	1	2	yes	maximal, complemented
5	C₁₃×#482	C₂	1	4	yes	maximal, complemented
6	C₁₃×#476	C₂	1	2	yes	maximal, complemented
7	C₁₃×#474	C₂	1	2	yes	maximal, complemented
8	C₁₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	—	2	1	no	—
10	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₃×#127	C₂²	1	1	yes	characteristic
14	C₁₃×#125	C₂²	1	2	yes	—
15	C₁₃×#124	—	2	1	no	—
16	C₁₃×C₂²⋊Q₈	—	2	1	no	—
17	C₁₃×C₂²⋊Q₈	—	2	1	no	—
18	C₁₃×#121	—	2	1	no	—
19	C₁₃×#121	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₃	—	2	1	no	—
21	C₄×Q₈×C₁₃	—	2	1	no	—
22	C₄×D₄×C₁₃	—	2	1	no	—
23	C₄×D₄×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#117	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#107	C₂²	1	4	yes	—
27	C₁₃×#106	C₂²	1	4	yes	—
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₃×#102	C₂²	1	4	yes	—
30	C₁₃×#102	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₁₃×#97	C₂²	1	2	yes	—
33	C₁₃×#41	—	4	4	no	—
34	C₂×Q₈×C₁₃	—	2	1	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	—	2	1	no	—
37	C₂×D₄×C₁₃	—	2	2	no	—
38	C₂×D₄×C₁₃	—	2	1	no	—
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂³	1	4	yes	—
46	C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	2	1	no	—
50	C₁₃×C₄⋊C₄	—	2	1	no	—
51	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₁₃×C₂²⋊C₄	—	2	2	no	—
55	C₁₃×C₂²⋊C₄	—	2	2	no	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	D₄	1	1	yes	characteristic
59	C₄²×C₁₃	D₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂³	1	1	yes	characteristic
61	C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	#3235	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₁₃	—	4	2	no	—
66	Q₈×C₁₃	—	4	2	no	—
67	Q₈×C₁₃	—	4	2	no	—
68	D₄×C₁₃	—	2	4	no	—
69	D₄×C₁₃	—	4	2	no	—
70	D₄×C₁₃	—	4	2	no	—
71	D₄×C₁₃	—	4	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	1	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₈×C₁₃	—	2	4	no	—
86	#533	C₂×C₁₃	1	1	yes	characteristic, complemented
87	#532	C₂×C₁₃	1	1	yes	characteristic
88	#501	C₂×C₁₃	1	2	yes	complemented
89	#490	C₂×C₁₃	1	2	yes	complemented
90	#482	C₂×C₁₃	1	4	yes	complemented
91	#476	C₂×C₁₃	1	2	yes	complemented
92	#474	C₂×C₁₃	1	2	yes	complemented
93	#432	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	—	4	2	no	—
98	C₂²×C₁₃	—	4	2	no	—
99	C₂²×C₁₃	—	4	2	no	—
100	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	2	1	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	4	2	no	—
107	C₄×C₁₃	—	4	2	no	—
108	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
110	C₂×#41	—	2	1	no	—
111	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
113	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
114	#127	C₂²×C₁₃	1	1	yes	characteristic
115	#125	C₂²×C₁₃	1	2	yes	—
116	#124	—	2	1	no	—
117	C₂²⋊Q₈	—	2	1	no	—
118	C₂²⋊Q₈	—	2	1	no	—
119	#121	—	2	1	no	—
120	#121	C₂²×C₁₃	1	2	yes	—
121	C₄×Q₈	—	2	1	no	—
122	C₄×Q₈	—	2	1	no	—
123	C₄×D₄	—	2	1	no	—
124	C₄×D₄	C₂²×C₁₃	1	2	yes	—
125	#117	C₂²×C₁₃	1	2	yes	—
126	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
127	#107	C₂²×C₁₃	1	4	yes	—
128	#106	C₂²×C₁₃	1	4	yes	—
129	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	#102	C₂²×C₁₃	1	4	yes	—
131	#102	C₂²×C₁₃	1	4	yes	—
132	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
133	#97	C₂²×C₁₃	1	2	yes	—
134	C₂×C₁₃	—	4	1	no	—
135	C₂×C₁₃	—	8	2	no	—
136	C₂×C₁₃	#536	1	1	yes	characteristic
137	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₁₃	#575	1	1	yes	characteristic
139	#41	—	4	4	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	C₂³×C₁₃	1	2	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
149	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
151	C₂×C₈	C₂³×C₁₃	1	4	yes	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
158	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
159	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	D₄×C₁₃	1	1	yes	characteristic
165	C₄²	D₄×C₁₃	1	1	yes	characteristic
166	C₄²	C₂³×C₁₃	1	1	yes	characteristic
167	C₄²	C₂³×C₁₃	1	1	yes	characteristic
168	C₁₃	#3235	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³	—	2	2	no	—
170	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	4	2	no	—
173	Q₈	—	4	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
207	C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	—	8	2	no	—
210	C₂	C₁₃×#536	1	1	yes	characteristic
211	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
212	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	C₁₃×#532	C₂
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×#501	C₂	semidirect
4. C₁₃×#490	C₂	semidirect
5. C₁₃×#482	C₂	semidirect
6. C₁₃×#476	C₂	semidirect
7. C₁₃×#474	C₂	semidirect
8. C₁₃×#432	C₂	semidirect
62. #3235	C₁₃	direct (N × H)
86. #533	C₂×C₁₃	semidirect
88. #501	C₂×C₁₃	semidirect
89. #490	C₂×C₁₃	semidirect
90. #482	C₂×C₁₃	semidirect
91. #476	C₂×C₁₃	semidirect
92. #474	C₂×C₁₃	semidirect
93. #432	C₂×C₁₃	semidirect
168. C₁₃	#3235	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
C₁₃×#532	2
C₁₃×#501	2
C₁₃×#490	2
C₁₃×#482	2
C₁₃×#476	2
C₁₃×#474	2
C₁₃×#432	2
#3235	13

Subgroups — C13×#3235 #13112293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3235 #13112293