Subgroups — C₁₃×#2814 #13111872

Order 1664.

478 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#398	C₂	1	2	yes	maximal, complemented
5	C₁₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#397	C₂	1	2	yes	maximal, complemented
7	C₁₃×#396	C₂	1	2	yes	maximal, complemented
8	C₁₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#394	C₂	1	2	yes	maximal
10	C₁₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#123	—	2	4	no	—
16	C₁₃×C₂²⋊Q₈	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₁₃×#95	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	1	yes	characteristic
33	C₁₃×#95	C₂²	1	2	yes	—
34	C₁₃×#95	C₂²	1	2	yes	—
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	—	4	2	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	D₄	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	—	4	2	no	—
66	C₁₃×C₂²⋊C₄	—	4	2	no	—
67	C₁₃×C₂²⋊C₄	—	4	2	no	—
68	#2814	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₃	—	2	1	no	—
70	C₂³×C₁₃	—	2	2	no	—
71	C₂³×C₁₃	—	2	2	no	—
72	C₂³×C₁₃	—	2	1	no	—
73	C₂³×C₁₃	—	2	1	no	—
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	C₂×D₄	1	2	yes	—
79	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
80	Q₈×C₁₃	—	4	4	no	—
81	D₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×#123	C₂×C₁₃	1	1	yes	characteristic
107	C₂×C₂²⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#398	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#398	C₂×C₁₃	1	2	yes	complemented
110	#397	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#397	C₂×C₁₃	1	2	yes	complemented
112	#396	C₂×C₁₃	1	2	yes	complemented
113	#395	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#394	C₂×C₁₃	1	2	yes	—
115	#393	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#380	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	4	2	no	—
121	C₂²×C₁₃	—	4	2	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	—	4	2	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
129	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
131	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	—	4	2	no	—
136	C₄×C₁₃	—	4	1	no	—
137	C₄×C₁₃	—	4	1	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	—	4	2	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
143	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
144	#123	—	2	4	no	—
145	C₂²⋊Q₈	—	2	4	no	—
146	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
160	#95	C₂²×C₁₃	1	1	yes	characteristic
161	#95	C₂²×C₁₃	1	1	yes	characteristic
162	#95	C₂²×C₁₃	1	2	yes	—
163	#95	C₂²×C₁₃	1	2	yes	—
164	#95	C₂²×C₁₃	1	2	yes	—
165	#95	C₂²×C₁₃	1	2	yes	—
166	C₂×C₁₃	—	4	2	no	—
167	C₂×C₁₃	—	8	1	no	—
168	C₂×C₁₃	#534	1	1	yes	characteristic
169	C₂×C₁₃	#536	1	1	yes	characteristic
170	C₂×C₁₃	#537	1	1	yes	characteristic
171	C₂×C₁₃	#562	1	2	yes	—
172	C₂×C₁₃	#563	1	2	yes	—
173	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
174	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	D₄×C₁₃	1	4	yes	—
181	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₁₃	#2814	1	1	yes	characteristic, direct product factor, Sylow
205	C₂³	—	2	1	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	1	no	—
214	C₂³	C₂×D₄×C₁₃	1	2	yes	—
215	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
216	Q₈	—	4	4	no	—
217	D₄	—	4	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
253	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
254	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
255	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
256	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	8	1	no	—
268	C₂	C₁₃×#534	1	1	yes	characteristic
269	C₂	C₁₃×#536	1	1	yes	characteristic
270	C₂	C₁₃×#537	1	1	yes	characteristic
271	C₂	C₁₃×#562	1	2	yes	—
272	C₂	C₁₃×#563	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
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
2. C₂×C₁₃×C₂²⋊Q₈	C₂	semidirect
3. C₁₃×#398	C₂	semidirect
4. C₁₃×#398	C₂	semidirect
5. C₁₃×#397	C₂	semidirect
6. C₁₃×#397	C₂	semidirect
7. C₁₃×#396	C₂	semidirect
8. C₁₃×#395	C₂	semidirect
10. C₁₃×#393	C₂	semidirect
11. C₁₃×#380	C₂	semidirect
68. #2814	C₁₃	direct (N × H)
107. C₂×C₂²⋊Q₈	C₂×C₁₃	semidirect
108. #398	C₂×C₁₃	semidirect
109. #398	C₂×C₁₃	semidirect
110. #397	C₂×C₁₃	semidirect
111. #397	C₂×C₁₃	semidirect
112. #396	C₂×C₁₃	semidirect
113. #395	C₂×C₁₃	semidirect
115. #393	C₂×C₁₃	semidirect
116. #380	C₂×C₁₃	semidirect
204. C₁₃	#2814	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×#123	2
C₂×C₁₃×C₂²⋊Q₈	2
C₁₃×#398	2
C₁₃×#397	2
C₁₃×#396	2
C₁₃×#395	2
C₁₃×#394	2
C₁₃×#393	2
C₁₃×#380	2
#2814	13

Subgroups — C13×#2814 #13111872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2814 #13111872