Subgroups — C₂×C₁₃×#2228 #15719961

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2228	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	—	2	1	no	—
11	C₂×C₄×D₄×C₁₃	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#499	—	2	2	no	—
15	C₁₃×#499	C₂²	1	4	yes	complemented
16	C₄×C₁₃×D₈	—	2	2	no	—
17	C₄×C₁₃×D₈	C₂²	1	4	yes	—
18	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#106	—	2	1	no	—
22	C₂×C₁₃×#102	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#367	C₂²	1	4	yes	complemented
25	C₁₃×#366	C₂²	1	4	yes	complemented
26	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	C₁₃×#357	C₂²	1	4	yes	—
28	C₁₃×#357	C₂²	1	4	yes	—
29	C₂²×D₄×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	—	4	1	no	—
31	C₂×C₁₃×D₈	—	2	1	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	Q₈	1	4	yes	—
34	C₂×C₁₃×D₈	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#125	—	2	4	no	—
38	C₁₃×#125	—	4	2	no	—
39	C₄×D₄×C₁₃	—	2	4	no	—
40	C₄×D₄×C₁₃	—	4	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₂×C₁₃×C₁₆	—	2	2	no	—
49	C₂×C₁₃×C₁₆	—	2	2	no	—
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
52	C₁₃×#107	—	2	1	no	—
53	C₁₃×#107	—	2	1	no	—
54	C₁₃×#107	—	2	1	no	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#107	C₂³	1	2	yes	—
57	C₁₃×#107	C₂³	1	2	yes	—
58	C₁₃×#106	—	2	4	no	—
59	C₁₃×#102	—	2	4	no	—
60	C₄×C₈×C₁₃	D₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	C₂×#2228	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	4	no	—
72	C₂²×C₄×C₁₃	—	8	4	no	—
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×D₈	—	2	4	no	—
77	C₁₃×D₈	—	2	2	no	—
78	C₁₃×D₈	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₃	#41	1	2	yes	—
84	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	4	1	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	4	no	—
93	C₁₃×C₂²⋊C₄	—	4	4	no	—
94	C₄²×C₁₃	D₈	1	2	yes	—
95	C₄²×C₁₃	C₂×D₄	1	2	yes	—
96	C₁₃×C₁₆	—	2	2	no	—
97	C₁₃×C₁₆	—	2	2	no	—
98	C₂×#499	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
100	#2228	C₂×C₁₃	1	8	yes	direct product factor
101	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
105	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
106	C₂³×C₁₃	—	4	2	no	—
107	C₂³×C₁₃	—	4	4	no	—
108	C₂³×C₁₃	—	8	2	no	—
109	C₂³×C₁₃	—	8	2	no	—
110	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
111	D₄×C₁₃	—	2	4	no	—
112	D₄×C₁₃	—	4	2	no	—
113	D₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	8	8	no	—
120	C₂×C₄×C₁₃	—	8	2	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#137	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
131	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#125	—	2	1	no	—
135	C₂×C₄×D₄	—	2	1	no	—
136	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
138	#499	—	2	2	no	—
139	#499	C₂²×C₁₃	1	4	yes	complemented
140	C₄×D₈	—	2	2	no	—
141	C₄×D₈	C₂²×C₁₃	1	4	yes	—
142	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×#106	—	2	1	no	—
146	C₂×#102	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	#367	C₂²×C₁₃	1	4	yes	complemented
149	#366	C₂²×C₁₃	1	4	yes	complemented
150	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
151	#357	C₂²×C₁₃	1	4	yes	—
152	#357	C₂²×C₁₃	1	4	yes	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	—	8	2	no	—
155	C₂²×C₁₃	—	8	2	no	—
156	C₂²×C₁₃	—	8	8	no	—
157	C₂²×C₁₃	#474	1	2	yes	—
158	C₂²×C₁₃	#508	1	2	yes	—
159	C₂²×C₁₃	#509	1	2	yes	—
160	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	#474	1	2	yes	—
167	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
169	C₂²×D₄	—	2	1	no	—
170	C₂³×C₄	—	4	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	Q₈×C₁₃	1	4	yes	—
174	C₂×D₈	C₂³×C₁₃	1	2	yes	—
175	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	#125	—	2	4	no	—
178	#125	—	4	2	no	—
179	C₄×D₄	—	2	4	no	—
180	C₄×D₄	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	4	1	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₂²⋊C₄	—	4	1	no	—
187	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
191	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
192	#107	—	2	1	no	—
193	#107	—	2	1	no	—
194	#107	—	2	1	no	—
195	#107	C₂³×C₁₃	1	2	yes	—
196	#107	C₂³×C₁₃	1	2	yes	—
197	#107	C₂³×C₁₃	1	2	yes	—
198	#106	—	2	4	no	—
199	#102	—	2	4	no	—
200	C₄×C₈	D₄×C₁₃	1	2	yes	—
201	C₄×C₈	C₂³×C₁₃	1	2	yes	—
202	C₂×C₁₃	—	8	4	no	—
203	C₂×C₁₃	#2228	1	4	yes	direct product factor
204	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
207	C₂⁴	—	4	1	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	8	4	no	—
217	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
218	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
220	D₈	—	2	4	no	—
221	D₈	—	2	2	no	—
222	D₈	C₂×Q₈×C₁₃	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
227	C₂×C₈	C₁₃×#41	1	2	yes	—
228	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	C₁₃×D₈	1	2	yes	—
239	C₄²	C₂×D₄×C₁₃	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₁₃	C₂×#2228	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³	—	4	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
261	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
262	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
263	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
264	C₂×C₄	C₁₃×#137	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
268	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
269	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
270	C₂²	—	4	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	8	no	—
274	C₂²	C₁₃×#474	1	2	yes	—
275	C₂²	C₁₃×#508	1	2	yes	—
276	C₂²	C₁₃×#509	1	2	yes	—
277	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	C₁₃×#474	1	2	yes	—
284	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
285	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
286	C₂	—	8	4	no	—
287	C₂	C₁₃×#2228	1	4	yes	direct product factor
288	C₂	C₂×C₁₃×#474	1	1	yes	characteristic
289	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
290	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
1. C₂×C₁₃×#499	C₂	semidirect
3. C₁₃×#2228	C₂	direct (N × H)
4. C₂×C₁₃×#367	C₂	semidirect
5. C₂×C₁₃×#366	C₂	semidirect
6. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
15. C₁₃×#499	C₂²	semidirect
24. C₁₃×#367	C₂²	semidirect
25. C₁₃×#366	C₂²	semidirect
26. C₁₃×C₄⋊C₁₆	C₂²	semidirect
62. C₂×#2228	C₁₃	direct (N × H)
98. C₂×#499	C₂×C₁₃	semidirect
100. #2228	C₂×C₁₃	direct (N × H)
101. C₂×#367	C₂×C₁₃	semidirect
102. C₂×#366	C₂×C₁₃	semidirect
103. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
139. #499	C₂²×C₁₃	semidirect
148. #367	C₂²×C₁₃	semidirect
149. #366	C₂²×C₁₃	semidirect
150. C₄⋊C₁₆	C₂²×C₁₃	semidirect
203. C₂×C₁₃	#2228	direct (N × H)
242. C₁₃	C₂×#2228	direct (N × H)
287. C₂	C₁₃×#2228	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×#499	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2228	2
C₂×C₁₃×#367	2
C₂×C₁₃×#366	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#357	2
C₂×#2228	13

Subgroups — C2×C13×#2228 #15719961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2228 #15719961