Subgroups — C₁₃×#13144 #15699483

Order 3328.

334 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×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1952	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
13	C₁₃×#428	—	2	1	no	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₁₃×#418	—	2	1	no	—
16	C₂×C₁₃×#100	—	2	1	no	—
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	C₁₃×#361	C₄	1	4	yes	complemented
19	C₁₃×#357	—	2	1	no	—
20	C₁₃×#357	—	2	1	no	—
21	C₁₃×#346	—	2	1	no	—
22	C₄×C₁₃×C₁₆	—	2	1	no	—
23	C₁₃×#343	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#117	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₁₃×M₃₂	—	2	1	no	—
38	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₂×C₁₃×C₁₆	—	2	1	no	—
42	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
43	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
44	C₁₃×#108	—	2	1	no	—
45	C₁₃×#108	—	2	1	no	—
46	C₁₃×#108	—	2	1	no	—
47	C₁₃×#108	C₂×C₄	1	2	yes	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	—	2	1	no	—
50	C₁₃×#107	—	4	1	no	—
51	C₁₃×#106	—	4	1	no	—
52	C₁₃×#102	—	4	1	no	—
53	C₁₃×#102	—	4	1	no	—
54	C₁₃×#100	—	4	2	no	—
55	C₁₃×#97	—	2	1	no	—
56	C₄×C₈×C₁₃	—	2	1	no	—
57	#13144	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂×D₄×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×D₈	—	2	2	no	—
68	C₁₃×D₈	—	2	2	no	—
69	C₁₃×D₈	—	4	1	no	—
70	C₁₃×D₈	—	4	1	no	—
71	C₁₃×M₁₆	—	2	1	no	—
72	C₁₃×M₁₆	—	2	2	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₁₃×M₁₆	—	4	1	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₁₃×M₁₆	SD₁₆	1	2	yes	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	—	4	1	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	4	1	no	—
87	C₁₃×C₄⋊C₄	—	4	1	no	—
88	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
90	C₁₃×C₂²⋊C₄	—	2	1	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₁₃×C₁₆	—	2	2	no	—
93	C₁₃×C₁₆	—	2	1	no	—
94	C₁₃×C₁₆	—	2	1	no	—
95	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
97	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#1978	C₂×C₁₃	1	1	yes	characteristic
99	#1952	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1387	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#1381	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	8	1	no	—
104	C₂³×C₁₃	—	8	1	no	—
105	C₂³×C₁₃	—	16	1	no	—
106	C₂³×C₁₃	—	16	1	no	—
107	C₂³×C₁₃	#135	1	1	yes	characteristic
108	D₄×C₁₃	—	4	2	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	8	1	no	—
111	D₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	8	1	no	—
115	C₂×C₄×C₁₃	—	16	1	no	—
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
122	C₈×C₁₃	—	2	2	no	—
123	C₈×C₁₃	—	4	2	no	—
124	C₈×C₁₃	—	8	1	no	—
125	C₈×C₁₃	#117	1	1	yes	characteristic
126	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₈×C₁₃	#123	1	1	yes	characteristic
128	C₈×C₁₃	#126	1	1	yes	characteristic
129	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
131	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
134	#428	—	2	1	no	—
135	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
136	#418	—	2	1	no	—
137	C₂×#100	—	2	1	no	—
138	#405	C₂²×C₁₃	1	1	yes	characteristic
139	#361	C₄×C₁₃	1	4	yes	complemented
140	#357	—	2	1	no	—
141	#357	—	2	1	no	—
142	#346	—	2	1	no	—
143	C₄×C₁₆	—	2	1	no	—
144	#343	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	—	2	1	no	—
147	C₂²×C₁₃	—	8	2	no	—
148	C₂²×C₁₃	—	16	2	no	—
149	C₂²×C₁₃	—	16	2	no	—
150	C₂²×C₁₃	—	16	1	no	—
151	C₂²×C₁₃	—	16	1	no	—
152	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
153	C₂²×C₁₃	#466	1	1	yes	characteristic
154	C₂²×C₁₃	#488	1	1	yes	characteristic
155	C₄×C₁₃	—	4	1	no	—
156	C₄×C₁₃	—	4	1	no	—
157	C₄×C₁₃	—	16	1	no	—
158	C₄×C₁₃	#388	1	1	yes	characteristic
159	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
160	C₄×C₁₃	#465	1	1	yes	characteristic
161	C₄×C₁₃	#486	1	1	yes	characteristic
162	C₂²×D₄	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	#117	D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	C₂×C₄×C₁₃	1	2	yes	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
182	#108	—	2	1	no	—
183	#108	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	C₂×C₄×C₁₃	1	2	yes	—
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#107	—	4	1	no	—
189	#106	—	4	1	no	—
190	#102	—	4	1	no	—
191	#102	—	4	1	no	—
192	#100	—	4	2	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	—	16	2	no	—
198	C₂×C₁₃	#1934	1	1	yes	characteristic
199	C₂×C₁₃	#2221	1	1	yes	characteristic
200	C₂×C₁₃	#2223	1	1	yes	characteristic
201	C₂⁴	—	4	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	D₈	—	2	2	no	—
211	D₈	—	2	2	no	—
212	D₈	—	4	1	no	—
213	D₈	—	4	1	no	—
214	M₁₆	—	2	1	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	C₁₃×SD₁₆	1	2	yes	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
226	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
227	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
228	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
232	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
233	C₂²⋊C₄	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	2	1	no	—
237	C₁₆	—	2	1	no	—
238	C₁₃	#13144	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	—	4	2	no	—
240	C₂³	—	8	1	no	—
241	C₂³	—	8	1	no	—
242	C₂³	—	16	1	no	—
243	C₂³	—	16	1	no	—
244	C₂³	C₁₃×#135	1	1	yes	characteristic
245	D₄	—	4	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	8	1	no	—
248	D₄	—	8	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	8	1	no	—
252	C₂×C₄	—	16	1	no	—
253	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
255	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	1	no	—
262	C₈	C₁₃×#117	1	1	yes	characteristic
263	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
264	C₈	C₁₃×#123	1	1	yes	characteristic
265	C₈	C₁₃×#126	1	1	yes	characteristic
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	16	2	no	—
270	C₂²	—	16	2	no	—
271	C₂²	—	16	1	no	—
272	C₂²	—	16	1	no	—
273	C₂²	C₄×C₁₃×SD₁₆	1	1	yes	characteristic
274	C₂²	C₁₃×#466	1	1	yes	characteristic
275	C₂²	C₁₃×#488	1	1	yes	characteristic
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	16	1	no	—
279	C₄	C₁₃×#388	1	1	yes	characteristic
280	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
281	C₄	C₁₃×#465	1	1	yes	characteristic
282	C₄	C₁₃×#486	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	16	2	no	—
286	C₂	C₁₃×#1934	1	1	yes	characteristic
287	C₂	C₁₃×#2221	1	1	yes	characteristic
288	C₂	C₁₃×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2108	C₂	semidirect
5. C₁₃×#1952	C₂	semidirect
6. C₁₃×#1387	C₂	semidirect
7. C₁₃×#1381	C₂	semidirect
18. C₁₃×#361	C₄	semidirect
57. #13144	C₁₃	direct (N × H)
97. #2108	C₂×C₁₃	semidirect
99. #1952	C₂×C₁₃	semidirect
100. #1387	C₂×C₁₃	semidirect
101. #1381	C₂×C₁₃	semidirect
139. #361	C₄×C₁₃	semidirect
238. C₁₃	#13144	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×#361	2
C₂×C₁₃×#357	2
C₁₃×#2108	2
C₁₃×#1978	2
C₁₃×#1952	2
C₁₃×#1387	2
C₁₃×#1381	2
#13144	13

Subgroups — C13×#13144 #15699483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13144 #15699483