Subgroups — C₂×C₁₃×#2819 #15723019

Order 3328.

3342 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#124	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2819	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#394	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#393	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#392	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₁₃	C₂²	1	4	yes	—
10	C₂×C₁₃×#124	—	2	16	no	—
11	C₂×C₁₃×#124	C₂²	1	16	yes	complemented
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	8	yes	—
16	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#394	C₂²	1	32	yes	complemented
18	C₁₃×#393	C₂²	1	32	yes	complemented
19	C₁₃×#392	C₂²	1	32	yes	complemented
20	C₁₃×#382	C₂²	1	8	yes	complemented
21	C₂×C₁₃×#95	C₂²	1	4	yes	—
22	C₂×C₁₃×#95	C₂²	1	4	yes	—
23	C₂⁵×C₁₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₁₃	—	2	8	no	—
25	C₂²×Q₈×C₁₃	—	4	8	no	—
26	C₂²×Q₈×C₁₃	C₂³	1	8	yes	—
27	C₂²×D₄×C₁₃	—	2	8	no	—
28	C₂²×D₄×C₁₃	—	4	8	no	—
29	C₂²×D₄×C₁₃	—	4	16	no	—
30	C₂²×D₄×C₁₃	—	4	8	no	—
31	C₂²×D₄×C₁₃	C₂³	1	8	yes	—
32	C₂²×D₄×C₁₃	C₂³	1	16	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
37	C₁₃×#124	—	2	64	no	—
38	C₂×C₁₃×C₄⋊C₄	—	4	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
40	C₂×C₁₃×C₂²⋊C₄	—	2	16	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	16	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	4	16	no	—
43	C₂×C₁₃×C₂²⋊C₄	C₂³	1	16	yes	—
44	C₂×C₁₃×C₂²⋊C₄	C₂³	1	16	yes	—
45	C₂×C₁₃×C₂²⋊C₄	C₂³	1	32	yes	—
46	C₂×C₄²×C₁₃	D₄	1	8	yes	complemented
47	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
48	C₁₃×#95	C₂³	1	16	yes	—
49	C₁₃×#95	C₂³	1	16	yes	—
50	C₂×#2819	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₃	—	2	8	no	—
52	C₂⁴×C₁₃	—	2	16	no	—
53	C₂⁴×C₁₃	—	2	16	no	—
54	C₂⁴×C₁₃	—	2	8	no	—
55	C₂⁴×C₁₃	—	2	8	no	—
56	C₂⁴×C₁₃	C₂⁴	1	8	yes	—
57	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₁₃	—	2	16	no	—
59	C₂×Q₈×C₁₃	—	4	16	no	—
60	C₂×Q₈×C₁₃	—	4	32	no	—
61	C₂×D₄×C₁₃	—	2	16	no	—
62	C₂×D₄×C₁₃	—	4	16	no	—
63	C₂×D₄×C₁₃	—	4	32	no	—
64	C₂×D₄×C₁₃	—	4	16	no	—
65	C₂×D₄×C₁₃	—	4	32	no	—
66	C₂×D₄×C₁₃	—	4	64	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	8	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	8	no	—
71	C₂²×C₄×C₁₃	—	2	16	no	—
72	C₂²×C₄×C₁₃	—	2	16	no	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
80	C₁₃×C₄⋊C₄	—	4	8	no	—
81	C₁₃×C₂²⋊C₄	—	2	32	no	—
82	C₁₃×C₂²⋊C₄	—	2	32	no	—
83	C₁₃×C₂²⋊C₄	—	4	32	no	—
84	C₄²×C₁₃	C₂×D₄	1	16	yes	complemented
85	C₂²×#124	C₂×C₁₃	1	2	yes	complemented
86	#2819	C₂×C₁₃	1	16	yes	direct product factor
87	C₂×#394	C₂×C₁₃	1	4	yes	complemented
88	C₂×#393	C₂×C₁₃	1	4	yes	complemented
89	C₂×#392	C₂×C₁₃	1	4	yes	complemented
90	C₂×#382	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂³×C₁₃	—	2	8	no	—
92	C₂³×C₁₃	—	2	16	no	—
93	C₂³×C₁₃	—	2	16	no	—
94	C₂³×C₁₃	—	2	8	no	—
95	C₂³×C₁₃	—	2	8	no	—
96	C₂³×C₁₃	—	4	16	no	—
97	C₂³×C₁₃	—	4	16	no	—
98	C₂³×C₁₃	—	4	16	no	—
99	C₂³×C₁₃	—	4	32	no	—
100	C₂³×C₁₃	—	4	32	no	—
101	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
102	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
103	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
104	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
105	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
106	Q₈×C₁₃	—	4	32	no	—
107	D₄×C₁₃	—	4	32	no	—
108	D₄×C₁₃	—	4	64	no	—
109	C₂×C₄×C₁₃	—	2	16	no	—
110	C₂×C₄×C₁₃	—	2	8	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	8	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	8	no	—
115	C₂×C₄×C₁₃	—	2	16	no	—
116	C₂×C₄×C₁₃	—	2	16	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	4	16	no	—
119	C₂×C₄×C₁₃	—	4	16	no	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
122	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
123	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
124	C₂³×D₄	C₂²×C₁₃	1	4	yes	—
125	C₂×#124	—	2	16	no	—
126	C₂×#124	C₂²×C₁₃	1	16	yes	complemented
127	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
128	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
129	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
130	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	8	yes	—
131	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
132	#394	C₂²×C₁₃	1	32	yes	complemented
133	#393	C₂²×C₁₃	1	32	yes	complemented
134	#392	C₂²×C₁₃	1	32	yes	complemented
135	#382	C₂²×C₁₃	1	8	yes	complemented
136	C₂×#95	C₂²×C₁₃	1	4	yes	—
137	C₂×#95	C₂²×C₁₃	1	4	yes	—
138	C₂²×C₁₃	—	4	8	no	—
139	C₂²×C₁₃	—	4	8	no	—
140	C₂²×C₁₃	—	4	8	no	—
141	C₂²×C₁₃	—	4	16	no	—
142	C₂²×C₁₃	—	4	16	no	—
143	C₂²×C₁₃	—	8	32	no	—
144	C₂²×C₁₃	#535	1	8	yes	—
145	C₂²×C₁₃	#537	1	4	yes	—
146	C₂²×C₁₃	H(3,𝔽₄)	1	8	yes	—
147	C₂²×C₁₃	#562	1	8	yes	—
148	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	4	yes	—
150	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
152	C₄×C₁₃	—	2	8	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	4	8	no	—
155	C₄×C₁₃	—	4	8	no	—
156	C₂⁵	C₂³×C₁₃	1	4	yes	—
157	C₂²×Q₈	—	2	8	no	—
158	C₂²×Q₈	—	4	8	no	—
159	C₂²×Q₈	C₂³×C₁₃	1	8	yes	—
160	C₂²×D₄	—	2	8	no	—
161	C₂²×D₄	—	4	8	no	—
162	C₂²×D₄	—	4	16	no	—
163	C₂²×D₄	—	4	8	no	—
164	C₂²×D₄	C₂³×C₁₃	1	8	yes	—
165	C₂²×D₄	C₂³×C₁₃	1	16	yes	—
166	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
167	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
168	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
169	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
170	#124	—	2	64	no	—
171	C₂×C₄⋊C₄	—	4	4	no	—
172	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
173	C₂×C₂²⋊C₄	—	2	16	no	—
174	C₂×C₂²⋊C₄	—	2	16	no	—
175	C₂×C₂²⋊C₄	—	4	16	no	—
176	C₂×C₂²⋊C₄	C₂³×C₁₃	1	16	yes	—
177	C₂×C₂²⋊C₄	C₂³×C₁₃	1	16	yes	—
178	C₂×C₂²⋊C₄	C₂³×C₁₃	1	32	yes	—
179	C₂×C₄²	D₄×C₁₃	1	8	yes	complemented
180	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
181	#95	C₂³×C₁₃	1	16	yes	—
182	#95	C₂³×C₁₃	1	16	yes	—
183	C₂×C₁₃	—	8	8	no	—
184	C₂×C₁₃	#2819	1	8	yes	direct product factor
185	C₂×C₁₃	C₂×#535	1	2	yes	—
186	C₂×C₁₃	C₂×#537	1	1	yes	characteristic
187	C₂×C₁₃	C₂×H(3,𝔽₄)	1	2	yes	—
188	C₂×C₁₃	C₂×#562	1	2	yes	—
189	C₂⁴	—	2	8	no	—
190	C₂⁴	—	2	16	no	—
191	C₂⁴	—	2	16	no	—
192	C₂⁴	—	2	8	no	—
193	C₂⁴	—	2	8	no	—
194	C₂⁴	C₂⁴×C₁₃	1	8	yes	—
195	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
196	C₂×Q₈	—	2	16	no	—
197	C₂×Q₈	—	4	16	no	—
198	C₂×Q₈	—	4	32	no	—
199	C₂×D₄	—	2	16	no	—
200	C₂×D₄	—	4	16	no	—
201	C₂×D₄	—	4	32	no	—
202	C₂×D₄	—	4	16	no	—
203	C₂×D₄	—	4	32	no	—
204	C₂×D₄	—	4	64	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
214	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
216	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
217	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
218	C₄⋊C₄	—	4	8	no	—
219	C₂²⋊C₄	—	2	32	no	—
220	C₂²⋊C₄	—	2	32	no	—
221	C₂²⋊C₄	—	4	32	no	—
222	C₄²	C₂×D₄×C₁₃	1	16	yes	complemented
223	C₁₃	C₂×#2819	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	8	no	—
225	C₂³	—	2	16	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	—	4	16	no	—
231	C₂³	—	4	16	no	—
232	C₂³	—	4	32	no	—
233	C₂³	—	4	32	no	—
234	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
235	C₂³	C₁₃×H(5,𝔽₂)	1	8	yes	—
236	C₂³	C₁₃×H(5,𝔽₂)	1	2	yes	—
237	C₂³	C₁₃×2₋¹⁺⁴	1	2	yes	—
238	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
239	Q₈	—	4	32	no	—
240	D₄	—	4	32	no	—
241	D₄	—	4	64	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
254	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
255	C₂²	—	4	16	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	8	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	8	32	no	—
261	C₂²	C₁₃×#535	1	8	yes	—
262	C₂²	C₁₃×#537	1	4	yes	—
263	C₂²	C₁₃×H(3,𝔽₄)	1	8	yes	—
264	C₂²	C₁₃×#562	1	8	yes	—
265	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
266	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
267	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	4	yes	—
268	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
269	C₄	—	2	8	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	—	4	8	no	—
273	C₂	—	8	8	no	—
274	C₂	C₁₃×#2819	1	8	yes	direct product factor
275	C₂	C₂×C₁₃×#535	1	2	yes	—
276	C₂	C₂×C₁₃×#537	1	1	yes	characteristic
277	C₂	C₂×C₁₃×H(3,𝔽₄)	1	2	yes	—
278	C₂	C₂×C₁₃×#562	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×#124	C₂	semidirect
2. C₁₃×#2819	C₂	direct (N × H)
3. C₂×C₁₃×#394	C₂	semidirect
4. C₂×C₁₃×#393	C₂	semidirect
5. C₂×C₁₃×#392	C₂	semidirect
6. C₂×C₁₃×#382	C₂	semidirect
11. C₂×C₁₃×#124	C₂²	semidirect
17. C₁₃×#394	C₂²	semidirect
18. C₁₃×#393	C₂²	semidirect
19. C₁₃×#392	C₂²	semidirect
20. C₁₃×#382	C₂²	semidirect
46. C₂×C₄²×C₁₃	D₄	semidirect
50. C₂×#2819	C₁₃	direct (N × H)
84. C₄²×C₁₃	C₂×D₄	semidirect
85. C₂²×#124	C₂×C₁₃	semidirect
86. #2819	C₂×C₁₃	direct (N × H)
87. C₂×#394	C₂×C₁₃	semidirect
88. C₂×#393	C₂×C₁₃	semidirect
89. C₂×#392	C₂×C₁₃	semidirect
90. C₂×#382	C₂×C₁₃	semidirect
126. C₂×#124	C₂²×C₁₃	semidirect
132. #394	C₂²×C₁₃	semidirect
133. #393	C₂²×C₁₃	semidirect
134. #392	C₂²×C₁₃	semidirect
135. #382	C₂²×C₁₃	semidirect
179. C₂×C₄²	D₄×C₁₃	semidirect
184. C₂×C₁₃	#2819	direct (N × H)
222. C₄²	C₂×D₄×C₁₃	semidirect
223. C₁₃	C₂×#2819	direct (N × H)
274. C₂	C₁₃×#2819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#124	2
C₁₃×#2819	2
C₂×C₁₃×#394	2
C₂×C₁₃×#393	2
C₂×C₁₃×#392	2
C₂×C₁₃×#382	2
C₂×#2819	13

Subgroups — C2×C13×#2819 #15723019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2819 #15723019