Subgroups — C₁₃×#3277 #13112335

Order 1664.

476 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#137	C₂	1	2	yes	maximal, complemented
2	C₁₃×#548	C₂	1	2	yes	maximal, complemented
3	C₁₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#468	C₂	1	2	yes	maximal, complemented
5	C₁₃×#464	C₂	1	2	yes	maximal, complemented
6	C₁₃×#451	C₂	1	2	yes	maximal, complemented
7	C₁₃×#449	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	C₂²	1	2	yes	—
11	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₁₃×#137	—	2	8	no	—
13	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
20	C₁₃×#124	—	2	2	no	—
21	C₁₃×#124	C₂²	1	2	yes	—
22	C₁₃×C₂²⋊Q₈	—	2	2	no	—
23	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₁₃×#121	—	2	2	no	—
25	C₁₃×#121	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
27	C₄×D₄×C₁₃	—	2	2	no	—
28	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₁₃×#103	C₂²	1	2	yes	—
32	C₁₃×#103	C₂²	1	2	yes	—
33	C₁₃×#102	C₂²	1	2	yes	—
34	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
36	C₁₃×#41	—	2	4	no	—
37	C₁₃×#41	—	4	2	no	—
38	C₂×Q₈×C₁₃	—	2	2	no	—
39	C₂×Q₈×C₁₃	—	2	4	no	—
40	C₂×Q₈×C₁₃	—	4	2	no	—
41	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
42	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₁₃×Q₁₆	—	2	4	no	—
51	C₁₃×Q₁₆	—	4	1	no	—
52	C₁₃×Q₁₆	D₄	1	4	yes	complemented
53	C₁₃×SD₁₆	—	2	4	no	—
54	C₁₃×SD₁₆	—	2	4	no	—
55	C₁₃×D₈	—	4	1	no	—
56	C₁₃×M₁₆	—	2	4	no	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
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	#3277	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	C₂×D₄	1	2	yes	—
74	Q₈×C₁₃	—	2	2	no	—
75	Q₈×C₁₃	—	2	2	no	—
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	—	4	4	no	—
78	Q₈×C₁₃	C₂×D₄	1	4	yes	—
79	D₄×C₁₃	—	2	2	no	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	4	1	no	—
82	D₄×C₁₃	—	4	4	no	—
83	D₄×C₁₃	—	4	4	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	—	2	2	no	—
89	C₂×C₄×C₁₃	—	4	4	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
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	2	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×#137	C₂×C₁₃	1	2	yes	complemented
99	#548	C₂×C₁₃	1	2	yes	complemented
100	#495	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#468	C₂×C₁₃	1	2	yes	complemented
102	#464	C₂×C₁₃	1	2	yes	complemented
103	#451	C₂×C₁₃	1	2	yes	complemented
104	#449	C₂×C₁₃	1	2	yes	complemented
105	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	4	2	no	—
111	C₂²×C₁₃	—	4	2	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×#41	C₂²×C₁₃	1	2	yes	—
123	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
124	#137	—	2	8	no	—
125	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
126	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
129	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
130	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
132	#124	—	2	2	no	—
133	#124	C₂²×C₁₃	1	2	yes	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
136	#121	—	2	2	no	—
137	#121	C₂²×C₁₃	1	2	yes	—
138	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
141	#107	C₂²×C₁₃	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
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	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×C₁₃	—	4	2	no	—
149	C₂×C₁₃	—	8	2	no	—
150	C₂×C₁₃	D₄²	1	1	yes	characteristic
151	C₂×C₁₃	#575	1	1	yes	characteristic
152	C₂×C₁₃	#578	1	1	yes	characteristic
153	#41	—	2	4	no	—
154	#41	—	4	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
159	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
163	C₂×D₄	C₂³×C₁₃	1	2	yes	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	4	1	no	—
169	Q₁₆	D₄×C₁₃	1	4	yes	complemented
170	SD₁₆	—	2	4	no	—
171	SD₁₆	—	2	4	no	—
172	D₈	—	4	1	no	—
173	M₁₆	—	2	4	no	—
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	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
180	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	C₂³×C₁₃	1	1	yes	characteristic
188	C₁₃	#3277	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³	—	2	2	no	—
190	C₂³	C₂×D₄×C₁₃	1	2	yes	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	4	no	—
195	Q₈	C₂×D₄×C₁₃	1	4	yes	—
196	D₄	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	4	1	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	C₂×C₄	—	2	2	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	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
211	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×D₄×C₁₃	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
229	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	8	2	no	—
232	C₂	D₄²×C₁₃	1	1	yes	characteristic
233	C₂	C₁₃×#575	1	1	yes	characteristic
234	C₂	C₁₃×#578	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 ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#137	C₂	semidirect
2. C₁₃×#548	C₂	semidirect
3. C₁₃×#495	C₂	semidirect
4. C₁₃×#468	C₂	semidirect
5. C₁₃×#464	C₂	semidirect
6. C₁₃×#451	C₂	semidirect
7. C₁₃×#449	C₂	semidirect
8. C₄×C₁₃×Q₁₆	C₂	semidirect
9. C₁₃×#436	C₂	semidirect
52. C₁₃×Q₁₆	D₄	semidirect
71. #3277	C₁₃	direct (N × H)
98. C₂×#137	C₂×C₁₃	semidirect
99. #548	C₂×C₁₃	semidirect
100. #495	C₂×C₁₃	semidirect
101. #468	C₂×C₁₃	semidirect
102. #464	C₂×C₁₃	semidirect
103. #451	C₂×C₁₃	semidirect
104. #449	C₂×C₁₃	semidirect
105. C₄×Q₁₆	C₂×C₁₃	semidirect
106. #436	C₂×C₁₃	semidirect
169. Q₁₆	D₄×C₁₃	semidirect
188. C₁₃	#3277	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×#137	2
C₁₃×#548	2
C₁₃×#495	2
C₁₃×#468	2
C₁₃×#464	2
C₁₃×#451	2
C₁₃×#449	2
C₄×C₁₃×Q₁₆	2
C₁₃×#436	2
#3277	13

Subgroups — C13×#3277 #13112335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3277 #13112335