Subgroups — C₁₃×#2677 #13111735

Order 1664.

544 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	C₂×C₁₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₁₃×#399	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#398	C₂	1	2	yes	maximal, complemented
5	C₁₃×#395	C₂	1	2	yes	maximal, complemented
6	C₁₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#388	C₂	1	2	yes	maximal
8	C₁₃×#387	C₂	1	2	yes	maximal, complemented
9	C₁₃×#385	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×C₄⋊Q₈	—	2	8	no	—
14	C₁₃×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
27	C₁₃×#95	C₂²	1	2	yes	—
28	C₁₃×#95	C₂²	1	1	yes	characteristic
29	C₁₃×#95	C₂²	1	2	yes	—
30	C₁₃×#95	C₂²	1	2	yes	—
31	C₁₃×#95	C₂²	1	2	yes	—
32	C₁₃×#95	C₂²	1	1	yes	characteristic
33	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₁₃	—	2	4	no	—
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂×Q₈×C₁₃	—	2	4	no	—
37	C₂×Q₈×C₁₃	—	4	2	no	—
38	C₂×Q₈×C₁₃	—	4	1	no	—
39	C₂²×C₄×C₁₃	—	2	4	no	—
40	C₂²×C₄×C₁₃	—	4	1	no	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₄⋊C₄	—	2	4	no	—
51	C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	1	no	—
55	C₁₃×C₄⋊C₄	D₄	1	8	yes	—
56	C₁₃×C₂²⋊C₄	—	2	4	no	—
57	C₁₃×C₂²⋊C₄	—	2	4	no	—
58	C₁₃×C₂²⋊C₄	—	4	1	no	—
59	C₁₃×C₂²⋊C₄	Q₈	1	4	yes	complemented
60	C₄²×C₁₃	—	2	4	no	—
61	#2677	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₁₃	—	2	2	no	—
63	C₂³×C₁₃	—	2	1	no	—
64	C₂³×C₁₃	—	2	1	no	—
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
68	Q₈×C₁₃	—	4	8	no	—
69	Q₈×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
93	C₂×C₄⋊Q₈	C₂×C₁₃	1	2	yes	complemented
94	C₂×C₂²⋊Q₈	C₂×C₁₃	1	2	yes	—
95	#399	C₂×C₁₃	1	1	yes	characteristic
96	#398	C₂×C₁₃	1	2	yes	complemented
97	#395	C₂×C₁₃	1	2	yes	complemented
98	#393	C₂×C₁₃	1	1	yes	characteristic
99	#388	C₂×C₁₃	1	2	yes	—
100	#387	C₂×C₁₃	1	2	yes	complemented
101	#385	C₂×C₁₃	1	1	yes	characteristic
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	2	4	no	—
104	C₂²×C₁₃	—	4	2	no	—
105	C₂²×C₁₃	—	4	1	no	—
106	C₂²×C₁₃	—	4	1	no	—
107	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
108	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	4	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	—	4	2	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	1	no	—
120	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
121	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
122	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	8	no	—
124	C₂²⋊Q₈	—	2	8	no	—
125	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
137	#95	C₂²×C₁₃	1	2	yes	—
138	#95	C₂²×C₁₃	1	1	yes	characteristic
139	#95	C₂²×C₁₃	1	2	yes	—
140	#95	C₂²×C₁₃	1	2	yes	—
141	#95	C₂²×C₁₃	1	2	yes	—
142	#95	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	4	2	no	—
144	C₂×C₁₃	#536	1	2	yes	—
145	C₂×C₁₃	#544	1	1	yes	characteristic
146	C₂×C₁₃	D₄²	1	1	yes	characteristic
147	C₂×C₁₃	D₄×Q₈	1	2	yes	—
148	C₂×C₁₃	#563	1	1	yes	characteristic
149	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
163	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄⋊C₄	D₄×C₁₃	1	8	yes	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	Q₈×C₁₃	1	4	yes	complemented
176	C₄²	—	2	4	no	—
177	C₁₃	#2677	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³	—	2	2	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
183	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
184	Q₈	—	4	8	no	—
185	Q₈	—	4	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
206	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
208	C₂×C₄	C₂×Q₈×C₁₃	1	4	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₂²×D₄×C₁₃	1	2	yes	—
215	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
216	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
218	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	1	no	—
227	C₂	—	4	2	no	—
228	C₂	C₁₃×#536	1	2	yes	—
229	C₂	C₁₃×#544	1	1	yes	characteristic
230	C₂	D₄²×C₁₃	1	1	yes	characteristic
231	C₂	D₄×Q₈×C₁₃	1	2	yes	—
232	C₂	C₁₃×#563	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
4. C₁₃×#398	C₂	semidirect
5. C₁₃×#395	C₂	semidirect
8. C₁₃×#387	C₂	semidirect
59. C₁₃×C₂²⋊C₄	Q₈	semidirect
61. #2677	C₁₃	direct (N × H)
93. C₂×C₄⋊Q₈	C₂×C₁₃	semidirect
96. #398	C₂×C₁₃	semidirect
97. #395	C₂×C₁₃	semidirect
100. #387	C₂×C₁₃	semidirect
175. C₂²⋊C₄	Q₈×C₁₃	semidirect
177. C₁₃	#2677	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×C₁₃×C₂²⋊Q₈	2
C₁₃×#399	2
C₁₃×#398	2
C₁₃×#395	2
C₁₃×#393	2
C₁₃×#388	2
C₁₃×#387	2
C₁₃×#385	2
#2677	13

Subgroups — C13×#2677 #13111735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2677 #13111735