Subgroups — C₁₃×#21284 #15707623

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2319	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1972	C₂	1	2	yes	maximal
4	C₁₃×#1943	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1934	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1762	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#102	C₂²	1	4	yes	—
15	C₂×C₁₃×#102	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#390	C₂²	1	4	yes	—
21	C₁₃×#389	C₂²	1	4	yes	—
22	C₁₃×#388	—	2	4	no	—
23	C₁₃×#381	—	2	2	no	—
24	C₁₃×#380	—	2	2	no	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₁₃×#376	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	4	no	—
32	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₃	—	2	4	no	—
37	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₄²×C₁₃	—	2	2	no	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#107	C₂×C₄	1	4	yes	—
48	C₁₃×#106	—	2	4	no	—
49	C₁₃×#102	—	2	8	no	—
50	C₁₃×#102	C₂×C₄	1	8	yes	—
51	C₁₃×#97	—	2	4	no	—
52	C₁₃×#97	—	2	2	no	—
53	C₄×C₈×C₁₃	—	2	4	no	—
54	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	4	no	—
57	C₁₃×#95	D₄	1	2	yes	—
58	C₁₃×#95	D₄	1	2	yes	—
59	#21284	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	8	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	8	4	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×D₈	—	2	8	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	#41	1	4	yes	—
80	C₂×C₈×C₁₃	#41	1	4	yes	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	8	4	no	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₁₃×C₂²⋊C₄	—	2	8	no	—
87	C₁₃×C₂²⋊C₄	—	8	4	no	—
88	C₁₃×C₂²⋊C₄	—	8	4	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	8	2	no	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
95	#2319	C₂×C₁₃	1	2	yes	complemented
96	#1972	C₂×C₁₃	1	2	yes	—
97	#1943	C₂×C₁₃	1	2	yes	complemented
98	#1934	C₂×C₁₃	1	4	yes	complemented
99	C₄×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1762	C₂×C₁₃	1	2	yes	complemented
101	#1726	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	—	2	4	no	—
103	C₂³×C₁₃	—	4	4	no	—
104	C₂³×C₁₃	—	4	4	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	—	4	2	no	—
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	D₄×C₁₃	—	2	4	no	—
109	D₄×C₁₃	—	2	4	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	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	8	8	no	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	#117	1	4	yes	—
131	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
133	C₄×D₈	C₄×C₁₃	1	8	yes	complemented
134	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×#106	C₂²×C₁₃	1	2	yes	—
136	C₂×#102	C₂²×C₁₃	1	4	yes	—
137	C₂×#102	C₂²×C₁₃	1	2	yes	—
138	C₂×#97	C₂²×C₁₃	1	2	yes	—
139	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	#390	C₂²×C₁₃	1	4	yes	—
143	#389	C₂²×C₁₃	1	4	yes	—
144	#388	—	2	4	no	—
145	#381	—	2	2	no	—
146	#380	—	2	2	no	—
147	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
148	#376	C₂²×C₁₃	1	1	yes	characteristic
149	#340	C₂²×C₁₃	1	2	yes	—
150	#340	C₂²×C₁₃	1	4	yes	—
151	C₂²×C₁₃	—	4	4	no	—
152	C₂²×C₁₃	—	4	4	no	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	—	8	4	no	—
155	C₂²×C₁₃	—	8	4	no	—
156	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	#532	1	2	yes	—
158	C₂²×C₁₃	#576	1	2	yes	—
159	C₂²×C₁₃	#577	1	2	yes	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	—	4	4	no	—
164	C₄×C₁₃	—	8	4	no	—
165	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
167	C₄×C₁₃	#529	1	2	yes	—
168	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
169	C₂³×C₄	—	2	2	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
172	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
175	C₄×D₄	—	2	4	no	—
176	C₄×D₄	C₂×C₄×C₁₃	1	8	yes	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₂²⋊C₄	—	2	4	no	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	#107	C₂×C₄×C₁₃	1	4	yes	—
187	#106	—	2	4	no	—
188	#102	—	2	8	no	—
189	#102	C₂×C₄×C₁₃	1	8	yes	—
190	#97	—	2	4	no	—
191	#97	—	2	2	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
194	#95	—	2	2	no	—
195	#95	—	2	4	no	—
196	#95	D₄×C₁₃	1	2	yes	—
197	#95	D₄×C₁₃	1	2	yes	—
198	C₂×C₁₃	—	8	4	no	—
199	C₂×C₁₃	#2293	1	1	yes	characteristic
200	C₂×C₁₃	#2948	1	1	yes	characteristic
201	C₂×C₁₃	#2952	1	1	yes	characteristic
202	C₂×C₁₃	#3179	1	4	yes	—
203	C₂⁴	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	8	4	no	—
212	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
216	D₈	—	2	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
221	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂×C₈	C₁₃×#41	1	4	yes	—
223	C₂×C₈	C₁₃×#41	1	4	yes	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	8	4	no	—
228	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	8	4	no	—
231	C₂²⋊C₄	—	8	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	8	2	no	—
235	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
236	C₄²	C₁₃×#41	1	2	yes	—
237	C₁₃	#21284	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
244	D₄	—	2	4	no	—
245	D₄	—	2	4	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
262	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
263	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	4	no	—
266	C₈	C₁₃×#117	1	4	yes	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
273	C₂²	C₁₃×#532	1	2	yes	—
274	C₂²	C₁₃×#576	1	2	yes	—
275	C₂²	C₁₃×#577	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
282	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
283	C₄	C₁₃×#529	1	2	yes	—
284	C₂	—	8	4	no	—
285	C₂	C₁₃×#2293	1	1	yes	characteristic
286	C₂	C₁₃×#2948	1	1	yes	characteristic
287	C₂	C₁₃×#2952	1	1	yes	characteristic
288	C₂	C₁₃×#3179	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2319	C₂	semidirect
4. C₁₃×#1943	C₂	semidirect
5. C₁₃×#1934	C₂	semidirect
6. C₄×C₁₃×#106	C₂	semidirect
7. C₁₃×#1762	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
11. C₄×C₁₃×D₈	C₄	semidirect
59. #21284	C₁₃	direct (N × H)
95. #2319	C₂×C₁₃	semidirect
97. #1943	C₂×C₁₃	semidirect
98. #1934	C₂×C₁₃	semidirect
99. C₄×#106	C₂×C₁₃	semidirect
100. #1762	C₂×C₁₃	semidirect
101. #1726	C₂×C₁₃	semidirect
133. C₄×D₈	C₄×C₁₃	semidirect
237. C₁₃	#21284	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×C₄×C₁₃×D₈	2
C₁₃×#2319	2
C₁₃×#1972	2
C₁₃×#1943	2
C₁₃×#1934	2
C₄×C₁₃×#106	2
C₁₃×#1762	2
C₁₃×#1726	2
#21284	13

Subgroups — C13×#21284 #15707623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21284 #15707623