Subgroups — C₁₃×#3223 #13112281

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#464	C₂	1	2	yes	maximal, complemented
8	C₁₃×#462	C₂	1	2	yes	maximal, complemented
9	C₁₃×#452	C₂	1	2	yes	maximal
10	C₁₃×#451	C₂	1	2	yes	maximal, complemented
11	C₁₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₃×#126	—	2	2	no	—
22	C₁₃×#125	—	2	2	no	—
23	C₁₃×#125	C₂²	1	1	yes	characteristic
24	C₁₃×#124	C₂²	1	1	yes	characteristic
25	C₁₃×#124	C₂²	1	1	yes	characteristic
26	C₁₃×#123	—	2	2	no	—
27	C₁₃×C₂²⋊Q₈	—	2	2	no	—
28	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
30	C₁₃×#117	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₁₃×#103	C₂²	1	2	yes	—
33	C₁₃×#103	C₂²	1	2	yes	—
34	C₁₃×#103	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₁₃×#97	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
39	C₁₃×#41	—	4	2	no	—
40	C₂×Q₈×C₁₃	—	2	1	no	—
41	C₂×Q₈×C₁₃	—	4	2	no	—
42	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×Q₁₆	—	4	2	no	—
52	C₁₃×Q₁₆	—	4	2	no	—
53	C₁₃×Q₁₆	—	4	2	no	—
54	C₁₃×SD₁₆	—	4	2	no	—
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	#3223	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₃	—	2	1	no	—
74	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	2	no	—
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	2	2	no	—
80	Q₈×C₁₃	—	4	2	no	—
81	D₄×C₁₃	—	2	2	no	—
82	D₄×C₁₃	—	4	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	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	2	no	—
97	#541	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#536	C₂×C₁₃	1	1	yes	characteristic
99	#497	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#494	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#491	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#488	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#464	C₂×C₁₃	1	2	yes	complemented
104	#462	C₂×C₁₃	1	2	yes	complemented
105	#452	C₂×C₁₃	1	2	yes	—
106	#451	C₂×C₁₃	1	2	yes	complemented
107	#433	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	4	1	no	—
111	C₂²×C₁₃	—	4	2	no	—
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	4	1	no	—
117	C₄×C₁₃	—	4	1	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
123	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
125	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
126	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
127	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
128	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
131	#126	—	2	2	no	—
132	#125	—	2	2	no	—
133	#125	C₂²×C₁₃	1	1	yes	characteristic
134	#124	C₂²×C₁₃	1	1	yes	characteristic
135	#124	C₂²×C₁₃	1	1	yes	characteristic
136	#123	—	2	2	no	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
139	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
140	#117	C₂²×C₁₃	1	1	yes	characteristic
141	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
142	#103	C₂²×C₁₃	1	2	yes	—
143	#103	C₂²×C₁₃	1	2	yes	—
144	#103	C₂²×C₁₃	1	2	yes	—
145	#102	C₂²×C₁₃	1	2	yes	—
146	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
147	#97	C₂²×C₁₃	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₁₃	—	4	1	no	—
150	C₂×C₁₃	—	8	1	no	—
151	C₂×C₁₃	#535	1	1	yes	characteristic
152	C₂×C₁₃	#578	1	2	yes	—
153	#41	—	4	2	no	—
154	C₂×Q₈	—	2	1	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
157	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
158	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
165	Q₁₆	—	4	2	no	—
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	SD₁₆	—	4	2	no	—
169	C₂×C₈	C₂³×C₁₃	1	2	yes	—
170	C₂×C₈	C₂³×C₁₃	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
183	C₄²	—	2	2	no	—
184	C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₁₃	#3223	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	—	2	1	no	—
188	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	2	no	—
195	D₄	—	2	2	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	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
225	C₂	—	4	1	no	—
226	C₂	—	8	1	no	—
227	C₂	C₁₃×#535	1	1	yes	characteristic
228	C₂	C₁₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#541	C₂	semidirect
3. C₁₃×#497	C₂	semidirect
4. C₁₃×#494	C₂	semidirect
5. C₁₃×#491	C₂	semidirect
6. C₁₃×#488	C₂	semidirect
7. C₁₃×#464	C₂	semidirect
8. C₁₃×#462	C₂	semidirect
10. C₁₃×#451	C₂	semidirect
11. C₁₃×#433	C₂	semidirect
72. #3223	C₁₃	direct (N × H)
97. #541	C₂×C₁₃	semidirect
99. #497	C₂×C₁₃	semidirect
100. #494	C₂×C₁₃	semidirect
101. #491	C₂×C₁₃	semidirect
102. #488	C₂×C₁₃	semidirect
103. #464	C₂×C₁₃	semidirect
104. #462	C₂×C₁₃	semidirect
106. #451	C₂×C₁₃	semidirect
107. #433	C₂×C₁₃	semidirect
186. C₁₃	#3223	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×#541	2
C₁₃×#536	2
C₁₃×#497	2
C₁₃×#494	2
C₁₃×#491	2
C₁₃×#488	2
C₁₃×#464	2
C₁₃×#462	2
C₁₃×#452	2
C₁₃×#451	2
C₁₃×#433	2
#3223	13

Subgroups — C13×#3223 #13112281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3223 #13112281