Subgroups — C₁₃×#23819 #15710158

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2100	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2073	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2041	C₂	1	2	yes	maximal
6	C₁₃×#2025	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×#102	C₂	1	4	yes	maximal, complemented
8	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₁₃	—	2	2	no	—
12	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×#102	C₂²	1	4	yes	—
14	C₂×C₁₃×#102	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#390	—	2	2	no	—
17	C₁₃×#390	C₂²	1	2	yes	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#387	—	2	2	no	—
20	C₁₃×#384	C₂²	1	1	yes	characteristic
21	C₁₃×#382	C₂²	1	1	yes	characteristic
22	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄³×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#340	C₂²	1	4	yes	—
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#336	C₂²	1	2	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	—	2	2	no	—
30	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
33	C₁₃×#127	—	2	4	no	—
34	C₁₃×#125	—	2	4	no	—
35	C₄×D₄×C₁₃	—	2	8	no	—
36	C₄×D₄×C₁₃	—	4	8	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#102	—	2	8	no	—
52	C₁₃×#102	—	2	8	no	—
53	C₄×C₈×C₁₃	—	2	4	no	—
54	C₁₃×#95	—	2	2	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	2	no	—
57	#23819	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	8	no	—
60	C₂×D₄×C₁₃	—	2	4	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	#41	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	4	4	no	—
70	C₂²×C₄×C₁₃	—	4	8	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	2	8	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	4	4	no	—
84	C₁₃×C₄⋊C₄	—	4	4	no	—
85	C₁₃×C₄⋊C₄	#41	1	4	yes	—
86	C₁₃×C₂²⋊C₄	—	4	8	no	—
87	C₁₃×C₂²⋊C₄	—	4	4	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	4	4	no	—
93	C₄²×C₁₃	#41	1	4	yes	—
94	C₄×#127	C₂×C₁₃	1	1	yes	characteristic
95	C₄×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#2100	C₂×C₁₃	1	2	yes	complemented
97	#2073	C₂×C₁₃	1	2	yes	complemented
98	#2041	C₂×C₁₃	1	2	yes	—
99	#2025	C₂×C₁₃	1	2	yes	complemented
100	C₄×#102	C₂×C₁₃	1	4	yes	complemented
101	#1744	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	—	2	4	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	—	4	4	no	—
106	C₂³×C₁₃	—	4	4	no	—
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	D₄×C₁₃	—	2	8	no	—
109	D₄×C₁₃	—	4	16	no	—
110	D₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	8	8	no	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#124	1	2	yes	—
128	C₂×C₄×C₁₃	#135	1	4	yes	—
129	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
133	C₈×C₁₃	—	4	4	no	—
134	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄×D₄	—	2	2	no	—
137	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
138	C₂×#102	C₂²×C₁₃	1	4	yes	—
139	C₂×#102	C₂²×C₁₃	1	4	yes	—
140	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
141	#390	—	2	2	no	—
142	#390	C₂²×C₁₃	1	2	yes	—
143	#389	C₂²×C₁₃	1	2	yes	—
144	#387	—	2	2	no	—
145	#384	C₂²×C₁₃	1	1	yes	characteristic
146	#382	C₂²×C₁₃	1	1	yes	characteristic
147	C₄×C₄⋊C₄	—	2	2	no	—
148	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
149	C₄³	C₂²×C₁₃	1	1	yes	characteristic
150	#340	C₂²×C₁₃	1	4	yes	—
151	#340	C₂²×C₁₃	1	4	yes	—
152	#336	C₂²×C₁₃	1	2	yes	—
153	#336	C₂²×C₁₃	1	2	yes	—
154	C₂²×C₁₃	—	4	8	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	8	4	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	2	no	—
159	C₂²×C₁₃	#525	1	1	yes	characteristic
160	C₂²×C₁₃	#525	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#135	1	2	yes	—
163	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
164	C₄×C₁₃	—	2	4	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	#490	1	4	yes	—
171	C₄×C₁₃	#529	1	4	yes	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
176	#127	—	2	4	no	—
177	#125	—	2	4	no	—
178	C₄×D₄	—	2	8	no	—
179	C₄×D₄	—	4	8	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
186	C₂×C₂²⋊C₄	—	2	4	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	D₄×C₁₃	1	2	yes	—
190	C₂×C₄²	D₄×C₁₃	1	2	yes	—
191	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	#102	—	2	8	no	—
195	#102	—	2	8	no	—
196	C₄×C₈	—	2	4	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₂×C₁₃	—	8	4	no	—
201	C₂×C₁₃	#2398	1	1	yes	characteristic
202	C₂×C₁₃	#3105	1	2	yes	—
203	C₂×C₁₃	#3106	1	2	yes	—
204	C₂×C₁₃	#3131	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#490	1	1	yes	characteristic
206	C₂⁴	—	2	2	no	—
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	C₁₃×#41	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	C₁₃×#41	1	4	yes	—
234	C₂²⋊C₄	—	4	8	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	4	no	—
241	C₄²	C₁₃×#41	1	4	yes	—
242	C₁₃	#23819	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
249	D₄	—	2	8	no	—
250	D₄	—	4	16	no	—
251	D₄	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	8	no	—
267	C₂×C₄	C₁₃×#124	1	2	yes	—
268	C₂×C₄	C₁₃×#124	1	2	yes	—
269	C₂×C₄	C₁₃×#135	1	4	yes	—
270	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
271	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
272	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
273	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
274	C₈	—	4	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	C₁₃×#525	1	1	yes	characteristic
281	C₂²	C₁₃×#525	1	1	yes	characteristic
282	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
283	C₂²	C₂×C₁₃×#135	1	2	yes	—
284	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	C₁₃×#490	1	4	yes	—
292	C₄	C₁₃×#529	1	4	yes	—
293	C₂	—	8	4	no	—
294	C₂	C₁₃×#2398	1	1	yes	characteristic
295	C₂	C₁₃×#3105	1	2	yes	—
296	C₂	C₁₃×#3106	1	2	yes	—
297	C₂	C₁₃×#3131	1	1	yes	characteristic
298	C₂	C₂×C₁₃×#490	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×#125	C₂	semidirect
3. C₁₃×#2100	C₂	semidirect
4. C₁₃×#2073	C₂	semidirect
6. C₁₃×#2025	C₂	semidirect
7. C₄×C₁₃×#102	C₂	semidirect
8. C₁₃×#1744	C₂	semidirect
57. #23819	C₁₃	direct (N × H)
95. C₄×#125	C₂×C₁₃	semidirect
96. #2100	C₂×C₁₃	semidirect
97. #2073	C₂×C₁₃	semidirect
99. #2025	C₂×C₁₃	semidirect
100. C₄×#102	C₂×C₁₃	semidirect
101. #1744	C₂×C₁₃	semidirect
242. C₁₃	#23819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#127	2
C₄×C₁₃×#125	2
C₁₃×#2100	2
C₁₃×#2073	2
C₁₃×#2041	2
C₁₃×#2025	2
C₄×C₁₃×#102	2
C₁₃×#1744	2
#23819	13

Subgroups — C13×#23819 #15710158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23819 #15710158