Subgroups — C₁₃×#3213 #13112271

Order 1664.

444 conjugacy classes of subgroups.

The subgroup lattice has 242 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	C₂×C₁₃×#124	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×#463	C₂	1	4	yes	maximal, complemented
6	C₁₃×#448	C₂	1	2	yes	maximal, complemented
7	C₁₃×#447	C₂	1	2	yes	maximal, complemented
8	C₁₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	—	2	1	no	—
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	complemented
13	C₂×C₁₃×D₈	C₂²	1	4	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#127	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#124	—	2	1	no	—
17	C₁₃×#124	—	2	1	no	—
18	C₁₃×#124	—	2	1	no	—
19	C₁₃×#124	C₂²	1	2	yes	—
20	C₁₃×#124	C₂²	1	2	yes	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₁₃	—	2	1	no	—
24	C₄×D₄×C₁₃	C₂²	1	2	yes	complemented
25	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
27	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
29	C₁₃×#103	C₂²	1	4	yes	complemented
30	C₁₃×#102	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₁₃×#97	C₂²	1	2	yes	complemented
33	C₂⁴×C₁₃	—	2	1	no	—
34	C₁₃×#41	—	4	4	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	—	2	1	no	—
37	C₂×Q₈×C₁₃	—	4	2	no	—
38	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃	—	2	2	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	—	4	2	no	—
44	C₂×D₄×C₁₃	C₂³	1	2	yes	—
45	C₂×D₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×SD₁₆	—	4	4	no	—
52	C₁₃×D₈	—	4	4	no	—
53	C₂×C₈×C₁₃	C₂³	1	4	yes	—
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
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	C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₄²×C₁₃	D₄	1	1	yes	characteristic
63	C₄²×C₁₃	D₄	1	1	yes	characteristic
64	C₄²×C₁₃	C₂³	1	1	yes	characteristic
65	C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	#3213	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	—	2	2	no	—
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	—	4	2	no	—
71	C₂³×C₁₃	—	4	2	no	—
72	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
73	Q₈×C₁₃	—	2	4	no	—
74	Q₈×C₁₃	—	4	2	no	—
75	Q₈×C₁₃	—	4	2	no	—
76	D₄×C₁₃	—	2	4	no	—
77	D₄×C₁₃	—	2	4	no	—
78	D₄×C₁₃	—	4	2	no	—
79	D₄×C₁₃	—	4	2	no	—
80	D₄×C₁₃	—	4	2	no	—
81	D₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	4	1	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	4	no	—
96	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂×#124	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#496	C₂×C₁₃	1	2	yes	complemented
99	#489	C₂×C₁₃	1	2	yes	complemented
100	#463	C₂×C₁₃	1	4	yes	complemented
101	#448	C₂×C₁₃	1	2	yes	complemented
102	#447	C₂×C₁₃	1	2	yes	complemented
103	#432	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	1	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	—	4	2	no	—
112	C₂²×C₁₃	—	8	4	no	—
113	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	—	4	2	no	—
122	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
123	C₂×#41	—	2	1	no	—
124	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	complemented
127	C₂×D₈	C₂²×C₁₃	1	4	yes	—
128	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
129	#127	C₂²×C₁₃	1	1	yes	characteristic, complemented
130	#124	—	2	1	no	—
131	#124	—	2	1	no	—
132	#124	—	2	1	no	—
133	#124	C₂²×C₁₃	1	2	yes	—
134	#124	C₂²×C₁₃	1	2	yes	—
135	#121	—	2	1	no	—
136	#121	C₂²×C₁₃	1	2	yes	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	C₂²×C₁₃	1	2	yes	complemented
139	C₂×C₂²⋊C₄	—	2	1	no	—
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₃	1	2	yes	complemented
143	#103	C₂²×C₁₃	1	4	yes	complemented
144	#102	C₂²×C₁₃	1	4	yes	—
145	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
146	#97	C₂²×C₁₃	1	2	yes	complemented
147	C₂×C₁₃	—	2	2	no	—
148	C₂×C₁₃	—	8	2	no	—
149	C₂×C₁₃	—	8	2	no	—
150	C₂×C₁₃	#535	1	1	yes	characteristic
151	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₁₃	#575	1	1	yes	characteristic
153	C₂⁴	—	2	1	no	—
154	#41	—	4	4	no	—
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	C₂³×C₁₃	1	2	yes	—
165	C₂×D₄	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
169	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
171	SD₁₆	—	4	4	no	—
172	D₈	—	4	4	no	—
173	C₂×C₈	C₂³×C₁₃	1	4	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	D₄×C₁₃	1	1	yes	characteristic
183	C₄²	D₄×C₁₃	1	1	yes	characteristic
184	C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₁₃	#3213	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
226	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	8	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₁₃×#535	1	1	yes	characteristic
239	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	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. C₂×C₁₃×#124	C₂	semidirect
3. C₁₃×#496	C₂	semidirect
4. C₁₃×#489	C₂	semidirect
5. C₁₃×#463	C₂	semidirect
6. C₁₃×#448	C₂	semidirect
7. C₁₃×#447	C₂	semidirect
8. C₁₃×#432	C₂	semidirect
12. C₂×C₁₃×SD₁₆	C₂²	semidirect
14. C₁₃×C₄⋊Q₈	C₂²	semidirect
15. C₁₃×#127	C₂²	semidirect
24. C₄×D₄×C₁₃	C₂²	semidirect
28. C₁₃×C₄⋊C₈	C₂²	semidirect
29. C₁₃×#103	C₂²	semidirect
32. C₁₃×#97	C₂²	semidirect
66. #3213	C₁₃	direct (N × H)
96. #532	C₂×C₁₃	semidirect
97. C₂×#124	C₂×C₁₃	semidirect
98. #496	C₂×C₁₃	semidirect
99. #489	C₂×C₁₃	semidirect
100. #463	C₂×C₁₃	semidirect
101. #448	C₂×C₁₃	semidirect
102. #447	C₂×C₁₃	semidirect
103. #432	C₂×C₁₃	semidirect
126. C₂×SD₁₆	C₂²×C₁₃	semidirect
128. C₄⋊Q₈	C₂²×C₁₃	semidirect
129. #127	C₂²×C₁₃	semidirect
138. C₄×D₄	C₂²×C₁₃	semidirect
142. C₄⋊C₈	C₂²×C₁₃	semidirect
143. #103	C₂²×C₁₃	semidirect
146. #97	C₂²×C₁₃	semidirect
186. C₁₃	#3213	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
C₂×C₁₃×#124	2
C₁₃×#496	2
C₁₃×#489	2
C₁₃×#463	2
C₁₃×#448	2
C₁₃×#447	2
C₁₃×#432	2
#3213	13

Subgroups — C13×#3213 #13112271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3213 #13112271