Subgroups — C₁₃×#3214 #13112272

Order 1664.

434 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#496	C₂	1	2	yes	maximal, complemented
4	C₁₃×#489	C₂	1	2	yes	maximal, complemented
5	C₁₃×#461	C₂	1	2	yes	maximal, complemented
6	C₁₃×#459	C₂	1	2	yes	maximal, complemented
7	C₁₃×#449	C₂	1	4	yes	maximal, complemented
8	C₁₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	—	2	1	no	—
10	C₂×C₁₃×#41	C₂²	1	2	yes	—
11	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	—
12	C₂×C₁₃×D₈	C₂²	1	4	yes	—
13	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×#127	C₂²	1	1	yes	characteristic
16	C₁₃×#127	C₂²	1	1	yes	characteristic
17	C₁₃×#124	—	2	1	no	—
18	C₁₃×#124	C₂²	1	2	yes	—
19	C₁₃×#121	—	2	1	no	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₁₃	—	2	1	no	—
24	C₄×D₄×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₃×#103	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	C₁₃×#41	—	4	4	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂×D₄×C₁₃	—	2	2	no	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	—	2	1	no	—
41	C₂×D₄×C₁₃	—	2	1	no	—
42	C₂×D₄×C₁₃	—	2	1	no	—
43	C₂×D₄×C₁₃	C₂³	1	2	yes	—
44	C₂×D₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×SD₁₆	—	4	4	no	—
51	C₁₃×D₈	—	4	4	no	—
52	C₂×C₈×C₁₃	C₂³	1	4	yes	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
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	C₁₃×C₂²⋊C₄	—	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	#3214	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₃	—	2	2	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₁₃	—	2	4	no	—
70	Q₈×C₁₃	—	4	2	no	—
71	D₄×C₁₃	—	2	2	no	—
72	D₄×C₁₃	—	2	4	no	—
73	D₄×C₁₃	—	2	2	no	—
74	D₄×C₁₃	—	4	2	no	—
75	D₄×C₁₃	—	4	2	no	—
76	D₄×C₁₃	—	4	2	no	—
77	D₄×C₁₃	—	4	4	no	—
78	D₄×C₁₃	—	4	4	no	—
79	D₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	4	1	no	—
85	C₂×C₄×C₁₃	—	4	4	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
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×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₃	—	2	4	no	—
94	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#496	C₂×C₁₃	1	2	yes	complemented
97	#489	C₂×C₁₃	1	2	yes	complemented
98	#461	C₂×C₁₃	1	2	yes	complemented
99	#459	C₂×C₁₃	1	2	yes	complemented
100	#449	C₂×C₁₃	1	4	yes	complemented
101	#432	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	4	2	no	—
106	C₂²×C₁₃	—	4	2	no	—
107	C₂²×C₁₃	—	4	2	no	—
108	C₂²×C₁₃	—	4	2	no	—
109	C₂²×C₁₃	—	4	2	no	—
110	C₂²×C₁₃	—	4	2	no	—
111	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	—	4	2	no	—
118	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
119	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
120	C₂×#41	—	2	1	no	—
121	C₂×#41	C₂²×C₁₃	1	2	yes	—
122	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	—
123	C₂×D₈	C₂²×C₁₃	1	4	yes	—
124	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
126	#127	C₂²×C₁₃	1	1	yes	characteristic
127	#127	C₂²×C₁₃	1	1	yes	characteristic
128	#124	—	2	1	no	—
129	#124	C₂²×C₁₃	1	2	yes	—
130	#121	—	2	1	no	—
131	#121	—	2	1	no	—
132	#121	—	2	1	no	—
133	#121	C₂²×C₁₃	1	2	yes	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	C₂²×C₁₃	1	2	yes	—
138	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
139	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
140	#103	C₂²×C₁₃	1	4	yes	—
141	#102	C₂²×C₁₃	1	4	yes	—
142	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
143	#97	C₂²×C₁₃	1	2	yes	—
144	C₂×C₁₃	—	4	1	no	—
145	C₂×C₁₃	—	8	2	no	—
146	C₂×C₁₃	—	8	2	no	—
147	C₂×C₁₃	#535	1	1	yes	characteristic
148	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₁₃	#575	1	1	yes	characteristic
150	#41	—	4	4	no	—
151	#41	—	4	4	no	—
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	C₂³×C₁₃	1	2	yes	—
161	C₂×D₄	C₂³×C₁₃	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
167	SD₁₆	—	4	4	no	—
168	D₈	—	4	4	no	—
169	C₂×C₈	C₂³×C₁₃	1	4	yes	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₄²	D₄×C₁₃	1	1	yes	characteristic
179	C₄²	D₄×C₁₃	1	1	yes	characteristic
180	C₄²	C₂³×C₁₃	1	1	yes	characteristic
181	C₄²	C₂³×C₁₃	1	1	yes	characteristic
182	C₁₃	#3214	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	4	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	4	2	no	—
192	D₄	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
228	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
229	C₂	—	4	1	no	—
230	C₂	—	8	2	no	—
231	C₂	—	8	2	no	—
232	C₂	C₁₃×#535	1	1	yes	characteristic
233	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂	C₁₃×#575	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 ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#532	C₂	semidirect
2. C₁₃×#532	C₂	semidirect
3. C₁₃×#496	C₂	semidirect
4. C₁₃×#489	C₂	semidirect
5. C₁₃×#461	C₂	semidirect
6. C₁₃×#459	C₂	semidirect
7. C₁₃×#449	C₂	semidirect
8. C₁₃×#432	C₂	semidirect
65. #3214	C₁₃	direct (N × H)
94. #532	C₂×C₁₃	semidirect
95. #532	C₂×C₁₃	semidirect
96. #496	C₂×C₁₃	semidirect
97. #489	C₂×C₁₃	semidirect
98. #461	C₂×C₁₃	semidirect
99. #459	C₂×C₁₃	semidirect
100. #449	C₂×C₁₃	semidirect
101. #432	C₂×C₁₃	semidirect
182. C₁₃	#3214	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×#532	2
C₁₃×#496	2
C₁₃×#489	2
C₁₃×#461	2
C₁₃×#459	2
C₁₃×#449	2
C₁₃×#432	2
#3214	13

Subgroups — C13×#3214 #13112272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3214 #13112272