Subgroups — C₁₃×#12818 #15699157

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 310 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×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	2	yes	maximal
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₄×C₁₃×D₈	—	2	1	no	—
10	C₄×C₁₃×D₈	—	2	1	no	—
11	C₄×C₁₃×D₈	C₄	1	2	yes	—
12	C₄×C₁₃×D₈	C₄	1	2	yes	—
13	C₂×C₁₃×#108	—	2	1	no	—
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₁₃×#423	—	2	1	no	—
17	C₂×C₁₃×#102	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊C₁₆	—	2	1	no	—
20	C₁₃×C₄⋊C₁₆	—	2	1	no	—
21	C₁₃×#361	—	2	4	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂³×C₄×C₁₃	—	4	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×D₈	D₄	1	2	yes	—
29	C₂×C₁₃×D₈	Q₈	1	2	yes	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₄×D₄×C₁₃	—	2	2	no	—
35	C₄×D₄×C₁₃	—	4	1	no	—
36	C₄×D₄×C₁₃	—	4	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₁₃×M₃₂	—	2	2	no	—
41	C₁₃×M₃₂	—	2	2	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₁₃×#108	—	2	2	no	—
44	C₁₃×#108	—	2	2	no	—
45	C₁₃×#108	—	4	2	no	—
46	C₁₃×#107	—	2	1	no	—
47	C₁₃×#107	C₂×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₈	—	4	1	no	—
49	C₁₃×C₄⋊C₈	—	4	1	no	—
50	C₁₃×#102	—	2	2	no	—
51	C₁₃×#102	—	2	2	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	#12818	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂×D₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	8	4	no	—
68	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₁₃×D₈	—	2	2	no	—
72	C₁₃×D₈	—	2	2	no	—
73	C₁₃×D₈	—	2	2	no	—
74	C₁₃×D₈	C₄⋊C₄	1	4	yes	—
75	C₁₃×M₁₆	—	4	2	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₁₃	D₈	1	1	yes	characteristic
95	C₄²×C₁₃	D₈	1	1	yes	characteristic
96	C₁₃×C₁₆	—	4	2	no	—
97	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
98	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂×#361	C₂×C₁₃	1	2	yes	—
100	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#1387	C₂×C₁₃	1	2	yes	complemented
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	—	8	2	no	—
106	C₂³×C₁₃	—	8	2	no	—
107	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	D₄×C₁₃	—	2	4	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	8	4	no	—
116	C₂×C₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	—	8	4	no	—
118	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	8	2	no	—
131	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₈×C₁₃	#117	1	1	yes	characteristic
133	C₈×C₁₃	#121	1	1	yes	characteristic
134	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄×D₄	—	2	1	no	—
137	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
138	C₄×D₈	—	2	1	no	—
139	C₄×D₈	—	2	1	no	—
140	C₄×D₈	C₄×C₁₃	1	2	yes	—
141	C₄×D₈	C₄×C₁₃	1	2	yes	—
142	C₂×#108	—	2	1	no	—
143	C₂×#108	C₂²×C₁₃	1	2	yes	—
144	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
145	#423	—	2	1	no	—
146	C₂×#102	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	1	no	—
149	C₄⋊C₁₆	—	2	1	no	—
150	#361	—	2	4	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	—	8	4	no	—
155	C₂²×C₁₃	—	8	4	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
159	C₂²×C₁₃	#461	1	1	yes	characteristic
160	C₂²×C₁₃	#476	1	1	yes	characteristic
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	#361	1	2	yes	complemented
166	C₄×C₁₃	#380	1	1	yes	characteristic
167	C₄×C₁₃	#417	1	1	yes	characteristic
168	C₄×C₁₃	#459	1	1	yes	characteristic
169	C₄×C₁₃	#474	1	1	yes	characteristic
170	C₂²×D₄	—	2	1	no	—
171	C₂³×C₄	—	4	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
176	C₂×D₈	D₄×C₁₃	1	2	yes	—
177	C₂×D₈	Q₈×C₁₃	1	2	yes	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	4	1	no	—
184	C₄×D₄	—	4	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	4	1	no	—
187	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
188	M₃₂	—	2	2	no	—
189	M₃₂	—	2	2	no	—
190	C₂×C₁₆	—	2	2	no	—
191	#108	—	2	2	no	—
192	#108	—	2	2	no	—
193	#108	—	4	2	no	—
194	#107	—	2	1	no	—
195	#107	C₂×C₄×C₁₃	1	2	yes	—
196	C₄⋊C₈	—	4	1	no	—
197	C₄⋊C₈	—	4	1	no	—
198	#102	—	2	2	no	—
199	#102	—	2	2	no	—
200	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
202	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
203	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₁₃	—	2	1	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	8	4	no	—
207	C₂×C₁₃	#1802	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
209	C₂×C₁₃	#2148	1	1	yes	characteristic
210	C₂⁴	—	4	1	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	8	4	no	—
221	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
222	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
224	D₈	—	2	2	no	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	D₈	C₁₃×C₄⋊C₄	1	4	yes	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
239	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
240	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
246	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₄²	C₁₃×D₈	1	1	yes	characteristic
248	C₄²	C₁₃×D₈	1	1	yes	characteristic
249	C₁₆	—	4	2	no	—
250	C₁₃	#12818	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
257	D₄	—	2	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
268	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
269	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
270	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
273	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
274	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
275	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	8	2	no	—
280	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
281	C₈	C₁₃×#117	1	1	yes	characteristic
282	C₈	C₁₃×#121	1	1	yes	characteristic
283	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	8	4	no	—
288	C₂²	—	8	4	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
292	C₂²	C₁₃×#461	1	1	yes	characteristic
293	C₂²	C₁₃×#476	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	2	no	—
298	C₄	C₁₃×#361	1	2	yes	complemented
299	C₄	C₁₃×#380	1	1	yes	characteristic
300	C₄	C₁₃×#417	1	1	yes	characteristic
301	C₄	C₁₃×#459	1	1	yes	characteristic
302	C₄	C₁₃×#474	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	4	no	—
306	C₂	C₁₃×#1802	1	1	yes	characteristic
307	C₂	C₂×C₁₃×#361	1	1	yes	characteristic
308	C₂	C₁₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
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 ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2150	C₂	semidirect
4. C₁₃×#1848	C₂	semidirect
5. C₁₃×#1387	C₂	semidirect
56. #12818	C₁₃	direct (N × H)
98. #2150	C₂×C₁₃	semidirect
100. #1848	C₂×C₁₃	semidirect
101. #1387	C₂×C₁₃	semidirect
165. C₄×C₁₃	#361	semidirect
250. C₁₃	#12818	direct (N × H)
298. C₄	C₁₃×#361	semidirect

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×#2150	2
C₂×C₁₃×#361	2
C₁₃×#1848	2
C₁₃×#1387	2
#12818	13

Subgroups — C13×#12818 #15699157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12818 #15699157