Subgroups — C₁₃×#2994 #13112052

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#435	C₂	1	2	yes	maximal, complemented
6	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₁₃	C₂	1	2	yes	maximal, complemented
8	C₁₃×#433	C₂	1	2	yes	maximal, complemented
9	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#424	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#126	C₄	1	2	yes	—
16	C₁₃×#125	C₄	1	1	yes	characteristic
17	C₁₃×#124	C₄	1	1	yes	characteristic
18	C₁₃×#123	C₄	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
20	C₁₃×#121	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₁₃	C₂²	1	2	yes	—
23	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#117	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₁₃×#97	C₂²	1	2	yes	—
38	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₁₃×M₁₆	—	4	2	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
64	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
65	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
66	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	C₄²×C₁₃	C₂³	1	2	yes	—
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	#2994	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	D₄×C₁₃	—	2	4	no	—
78	D₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	#529	C₂×C₁₃	1	1	yes	characteristic
101	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#435	C₂×C₁₃	1	2	yes	complemented
105	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₈×D₄	C₂×C₁₃	1	2	yes	complemented
107	#433	C₂×C₁₃	1	2	yes	complemented
108	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#424	C₂×C₁₃	1	2	yes	complemented
110	C₂×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	#131	1	2	yes	—
117	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	#131	1	2	yes	—
126	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
129	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
130	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
131	#126	C₄×C₁₃	1	2	yes	—
132	#125	C₄×C₁₃	1	1	yes	characteristic
133	#124	C₄×C₁₃	1	1	yes	characteristic
134	#123	C₄×C₁₃	1	2	yes	—
135	C₂²⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
136	#121	C₄×C₁₃	1	1	yes	characteristic
137	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₄×D₄	C₂²×C₁₃	1	2	yes	—
139	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
140	#117	C₂²×C₁₃	1	2	yes	—
141	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
143	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
147	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
148	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
153	#97	C₂²×C₁₃	1	2	yes	—
154	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₁₃	1	2	yes	—
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	#520	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
162	C₂×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₂×D₄	C₂×C₄×C₁₃	1	2	yes	—
164	C₂×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₁₃	1	2	yes	—
176	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₃	1	2	yes	—
178	C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
179	C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
180	C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
182	C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
184	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
185	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
186	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
187	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
188	C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
189	C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
191	C₄²	C₂³×C₁₃	1	1	yes	characteristic
192	C₄²	C₂³×C₁₃	1	2	yes	—
193	C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₁₃	#2994	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	C₂²×C₄×C₁₃	1	2	yes	—
196	C₂³	C₂²×C₄×C₁₃	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
211	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
212	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	C₁₃×#131	1	2	yes	—
227	C₂²	C₂³×C₄×C₁₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	C₁₃×#131	1	2	yes	—
236	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
237	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	C₁₃×#520	1	1	yes	characteristic
241	C₂	C₂×C₁₃×#131	1	1	yes	characteristic
242	C₂	C₂×C₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₄
prime radical	C₁₃×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. C₈×Q₈×C₁₃	C₂	semidirect
4. C₁₃×#436	C₂	semidirect
5. C₁₃×#435	C₂	semidirect
6. C₈×D₄×C₁₃	C₂	semidirect
7. C₈×D₄×C₁₃	C₂	semidirect
8. C₁₃×#433	C₂	semidirect
9. C₁₃×#431	C₂	semidirect
10. C₁₃×#424	C₂	semidirect
11. C₂×C₁₃×C₄⋊C₈	C₂	semidirect
73. #2994	C₁₃	direct (N × H)
101. #446	C₂×C₁₃	semidirect
102. C₈×Q₈	C₂×C₁₃	semidirect
103. #436	C₂×C₁₃	semidirect
104. #435	C₂×C₁₃	semidirect
105. C₈×D₄	C₂×C₁₃	semidirect
106. C₈×D₄	C₂×C₁₃	semidirect
107. #433	C₂×C₁₃	semidirect
108. #431	C₂×C₁₃	semidirect
109. #424	C₂×C₁₃	semidirect
110. C₂×C₄⋊C₈	C₂×C₁₃	semidirect
194. C₁₃	#2994	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×#529	2
C₁₃×#446	2
C₈×Q₈×C₁₃	2
C₁₃×#436	2
C₁₃×#435	2
C₈×D₄×C₁₃	2
C₁₃×#433	2
C₁₃×#431	2
C₁₃×#424	2
C₂×C₁₃×C₄⋊C₈	2
#2994	13

Subgroups — C13×#2994 #13112052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2994 #13112052