Subgroups — C₁₃×#3082 #13112140

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#544	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#519	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#479	C₂	1	2	yes	maximal, complemented
4	C₁₃×#479	C₂	1	2	yes	maximal, complemented
5	C₁₃×#477	C₂	1	2	yes	maximal, complemented
6	C₁₃×#475	C₂	1	2	yes	maximal, complemented
7	C₁₃×#428	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#424	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#419	C₂	1	1	yes	maximal, characteristic
11	C₂×C₁₃×#103	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	—	2	1	no	—
16	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×#125	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	—	2	2	no	—
20	C₄×Q₈×C₁₃	—	2	1	no	—
21	C₄×Q₈×C₁₃	—	2	1	no	—
22	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#117	—	2	2	no	—
25	C₁₃×#117	C₂²	1	1	yes	characteristic
26	C₁₃×#117	C₂²	1	1	yes	characteristic
27	C₁₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#107	C₂²	1	1	yes	characteristic
31	C₁₃×#107	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	1	yes	characteristic
33	C₁₃×#106	C₂²	1	1	yes	characteristic
34	C₁₃×#106	C₂²	1	2	yes	—
35	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#103	C₂²	1	2	yes	—
39	C₁₃×#103	C₂²	1	2	yes	—
40	C₁₃×#103	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	—	2	2	no	—
43	C₂×Q₈×C₁₃	—	2	2	no	—
44	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×M₁₆	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	2	yes	—
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
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	D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₂²⋊C₄	—	2	1	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	C₂³	1	2	yes	—
75	#3082	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₁₃	—	2	4	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	8	2	no	—
80	Q₈×C₁₃	C₂×Q₈	1	4	yes	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	8	1	no	—
89	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₄×C₁₃	#41	1	2	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	2	2	no	—
98	#544	C₂×C₁₃	1	1	yes	characteristic
99	#519	C₂×C₁₃	1	1	yes	characteristic
100	#479	C₂×C₁₃	1	2	yes	complemented
101	#479	C₂×C₁₃	1	2	yes	complemented
102	#477	C₂×C₁₃	1	2	yes	complemented
103	#475	C₂×C₁₃	1	2	yes	complemented
104	#428	C₂×C₁₃	1	1	yes	characteristic
105	#427	C₂×C₁₃	1	1	yes	characteristic
106	#424	C₂×C₁₃	1	1	yes	characteristic
107	#419	C₂×C₁₃	1	1	yes	characteristic
108	C₂×#103	C₂×C₁₃	1	1	yes	characteristic
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	4	1	no	—
117	C₄×C₁₃	—	4	2	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	1	no	—
120	C₄×C₁₃	—	4	1	no	—
121	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
122	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
129	#125	—	2	1	no	—
130	#125	C₂²×C₁₃	1	2	yes	—
131	C₂²⋊Q₈	—	2	2	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
135	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
136	#117	—	2	2	no	—
137	#117	C₂²×C₁₃	1	1	yes	characteristic
138	#117	C₂²×C₁₃	1	1	yes	characteristic
139	#117	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
141	#107	C₂²×C₁₃	1	1	yes	characteristic
142	#107	C₂²×C₁₃	1	1	yes	characteristic
143	#107	C₂²×C₁₃	1	2	yes	—
144	#106	C₂²×C₁₃	1	1	yes	characteristic
145	#106	C₂²×C₁₃	1	1	yes	characteristic
146	#106	C₂²×C₁₃	1	2	yes	—
147	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
148	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
149	#103	C₂²×C₁₃	1	2	yes	—
150	#103	C₂²×C₁₃	1	2	yes	—
151	#103	C₂²×C₁₃	1	2	yes	—
152	#103	C₂²×C₁₃	1	2	yes	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₁₃	#577	1	1	yes	characteristic
156	C₂×C₁₃	#578	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	Q₈×C₁₃	1	4	yes	complemented
161	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₁₃	1	2	yes	—
169	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	1	no	—
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	D₄×C₁₃	1	1	yes	characteristic
177	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	C₂³×C₁₃	1	2	yes	—
191	C₁₃	#3082	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	8	2	no	—
196	Q₈	C₂×Q₈×C₁₃	1	4	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	8	1	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	C₂×Q₈×C₁₃	1	2	yes	—
209	C₂×C₄	C₁₃×#41	1	2	yes	—
210	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₁₃×C₂²⋊Q₈	1	2	yes	—
217	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	1	no	—
226	C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
227	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
228	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂	C₁₃×#577	1	1	yes	characteristic
232	C₂	C₁₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂×D₄
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
3. C₁₃×#479	C₂	semidirect
4. C₁₃×#479	C₂	semidirect
5. C₁₃×#477	C₂	semidirect
6. C₁₃×#475	C₂	semidirect
44. C₂×Q₈×C₁₃	Q₈	semidirect
75. #3082	C₁₃	direct (N × H)
100. #479	C₂×C₁₃	semidirect
101. #479	C₂×C₁₃	semidirect
102. #477	C₂×C₁₃	semidirect
103. #475	C₂×C₁₃	semidirect
160. C₂×Q₈	Q₈×C₁₃	semidirect
191. C₁₃	#3082	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×#544	2
C₁₃×#519	2
C₁₃×#479	2
C₁₃×#477	2
C₁₃×#475	2
C₁₃×#428	2
C₁₃×#427	2
C₁₃×#424	2
C₁₃×#419	2
C₂×C₁₃×#103	2
#3082	13

Subgroups — C13×#3082 #13112140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3082 #13112140