Subgroups — C₂×C₁₃×#2207 #15719915

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2207	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#127	—	2	1	no	—
11	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	—	2	8	no	—
13	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₁₃×#493	—	2	2	no	—
16	C₁₃×#493	C₂²	1	4	yes	—
17	C₄×C₁₃×Q₁₆	—	2	2	no	—
18	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	complemented
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#103	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₁₃×#357	C₂²	1	8	yes	complemented
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	2	no	—
26	C₂²×D₄×C₁₃	—	4	1	no	—
27	C₂²×D₄×C₁₃	—	4	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	D₄	1	4	yes	complemented
31	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
32	C₂×C₁₃×D₈	—	2	4	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	—	2	4	no	—
35	C₂×C₁₃×D₈	—	2	4	no	—
36	C₂×C₁₃×D₈	—	4	4	no	—
37	C₂×C₁₃×D₈	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×#127	—	2	4	no	—
41	C₁₃×#127	—	4	2	no	—
42	C₄×Q₈×C₁₃	—	2	4	no	—
43	C₄×Q₈×C₁₃	—	4	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄²×C₁₃	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₁₃×SD₃₂	—	2	16	no	—
49	C₂×C₁₃×C₁₆	—	2	4	no	—
50	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
51	C₁₃×#107	—	2	1	no	—
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×#103	—	2	4	no	—
54	C₄×C₈×C₁₃	D₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×#2207	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	—	4	1	no	—
61	C₂×Q₈×C₁₃	—	4	2	no	—
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	—	4	4	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	4	4	no	—
68	C₂×D₄×C₁₃	—	8	4	no	—
69	C₂×D₄×C₁₃	—	8	4	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₁₃×Q₁₆	—	2	4	no	—
76	C₁₃×Q₁₆	—	2	2	no	—
77	C₁₃×Q₁₆	C₂×D₄	1	4	yes	complemented
78	C₁₃×D₈	—	2	4	no	—
79	C₁₃×D₈	—	2	4	no	—
80	C₁₃×D₈	—	4	8	no	—
81	C₁₃×D₈	—	4	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₁₃×C₄⋊C₄	—	4	4	no	—
91	C₄²×C₁₃	—	4	4	no	—
92	C₄²×C₁₃	D₈	1	2	yes	—
93	C₄²×C₁₃	C₂×D₄	1	2	yes	—
94	C₁₃×C₁₆	—	2	4	no	—
95	C₂²×SD₃₂	C₂×C₁₃	1	2	yes	complemented
96	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
97	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#2207	C₂×C₁₃	1	8	yes	direct product factor
99	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂×#357	C₂×C₁₃	1	2	yes	complemented
101	C₂³×C₁₃	—	4	4	no	—
102	C₂³×C₁₃	—	8	4	no	—
103	C₂³×C₁₃	—	8	4	no	—
104	C₂³×C₁₃	—	8	4	no	—
105	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₁₃	—	2	4	no	—
107	Q₈×C₁₃	—	4	2	no	—
108	Q₈×C₁₃	—	4	4	no	—
109	D₄×C₁₃	—	4	4	no	—
110	D₄×C₁₃	—	4	4	no	—
111	D₄×C₁₃	—	8	8	no	—
112	D₄×C₁₃	—	8	4	no	—
113	D₄×C₁₃	—	8	8	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	8	2	no	—
119	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
120	C₂×C₄×C₁₃	#121	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₁₃	—	2	2	no	—
127	C₈×C₁₃	#121	1	2	yes	—
128	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
131	C₂²×D₈	—	2	1	no	—
132	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
133	C₂×#127	—	2	1	no	—
134	C₂×C₄×Q₈	—	2	1	no	—
135	C₂×SD₃₂	—	2	8	no	—
136	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	complemented
137	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
138	#493	—	2	2	no	—
139	#493	C₂²×C₁₃	1	4	yes	—
140	C₄×Q₁₆	—	2	2	no	—
141	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	complemented
142	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#103	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
146	#357	C₂²×C₁₃	1	8	yes	complemented
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	—	8	4	no	—
149	C₂²×C₁₃	—	8	4	no	—
150	C₂²×C₁₃	—	16	8	no	—
151	C₂²×C₁₃	#459	1	2	yes	—
152	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
153	C₂²×C₁₃	#509	1	2	yes	—
154	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	4	4	no	—
157	C₄×C₁₃	—	8	2	no	—
158	C₄×C₁₃	#459	1	2	yes	—
159	C₄×C₁₃	C₂×SD₃₂	1	4	yes	complemented
160	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
161	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	—	4	1	no	—
165	C₂²×D₄	—	4	1	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	D₄×C₁₃	1	4	yes	complemented
169	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	4	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	4	4	no	—
175	C₂×D₈	C₂³×C₁₃	1	4	yes	—
176	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	#127	—	2	4	no	—
179	#127	—	4	2	no	—
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	—	4	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	4	1	no	—
184	C₂×C₄²	—	4	1	no	—
185	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
186	SD₃₂	—	2	16	no	—
187	C₂×C₁₆	—	2	4	no	—
188	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
189	#107	—	2	1	no	—
190	#107	C₂³×C₁₃	1	2	yes	—
191	#103	—	2	4	no	—
192	C₄×C₈	D₄×C₁₃	1	2	yes	—
193	C₄×C₈	C₂³×C₁₃	1	2	yes	—
194	C₂×C₁₃	—	16	4	no	—
195	C₂×C₁₃	#2207	1	4	yes	direct product factor
196	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
197	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
198	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
199	C₂⁴	—	4	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	8	4	no	—
211	C₂×D₄	—	8	4	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
215	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	2	2	no	—
219	Q₁₆	C₂×D₄×C₁₃	1	4	yes	complemented
220	D₈	—	2	4	no	—
221	D₈	—	2	4	no	—
222	D₈	—	4	8	no	—
223	D₈	—	4	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
228	C₂×C₈	C₁₃×#41	1	2	yes	—
229	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₁₃×D₈	1	2	yes	—
235	C₄²	C₂×D₄×C₁₃	1	2	yes	—
236	C₁₆	—	2	4	no	—
237	C₁₃	C₂×#2207	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	4	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	4	2	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	D₄	—	8	8	no	—
249	D₄	—	8	4	no	—
250	D₄	—	8	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	C₁₃×SD₃₂	1	4	yes	complemented
257	C₂×C₄	C₁₃×#121	1	2	yes	—
258	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
259	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
260	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
261	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
262	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	C₁₃×#121	1	2	yes	—
265	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	16	8	no	—
271	C₂²	C₁₃×#459	1	2	yes	—
272	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
273	C₂²	C₁₃×#509	1	2	yes	—
274	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	2	no	—
278	C₄	C₁₃×#459	1	2	yes	—
279	C₄	C₂×C₁₃×SD₃₂	1	4	yes	complemented
280	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
281	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂	—	16	4	no	—
283	C₂	C₁₃×#2207	1	4	yes	direct product factor
284	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
285	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
286	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	C₂×C₁₃×#493	C₂
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. C₂²×C₁₃×SD₃₂	C₂	semidirect
3. C₂×C₄×C₁₃×Q₁₆	C₂	semidirect
4. C₁₃×#2207	C₂	direct (N × H)
5. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₁₃×#357	C₂	semidirect
13. C₂×C₁₃×SD₃₂	C₂²	semidirect
18. C₄×C₁₃×Q₁₆	C₂²	semidirect
22. C₁₃×C₄⋊C₁₆	C₂²	semidirect
23. C₁₃×#357	C₂²	semidirect
30. C₂×C₁₃×Q₁₆	D₄	semidirect
56. C₂×#2207	C₁₃	direct (N × H)
77. C₁₃×Q₁₆	C₂×D₄	semidirect
95. C₂²×SD₃₂	C₂×C₁₃	semidirect
97. C₂×C₄×Q₁₆	C₂×C₁₃	semidirect
98. #2207	C₂×C₁₃	direct (N × H)
99. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
100. C₂×#357	C₂×C₁₃	semidirect
119. C₂×C₄×C₁₃	SD₃₂	semidirect
136. C₂×SD₃₂	C₂²×C₁₃	semidirect
141. C₄×Q₁₆	C₂²×C₁₃	semidirect
145. C₄⋊C₁₆	C₂²×C₁₃	semidirect
146. #357	C₂²×C₁₃	semidirect
159. C₄×C₁₃	C₂×SD₃₂	semidirect
168. C₂×Q₁₆	D₄×C₁₃	semidirect
195. C₂×C₁₃	#2207	direct (N × H)
219. Q₁₆	C₂×D₄×C₁₃	semidirect
237. C₁₃	C₂×#2207	direct (N × H)
256. C₂×C₄	C₁₃×SD₃₂	semidirect
279. C₄	C₂×C₁₃×SD₃₂	semidirect
283. C₂	C₁₃×#2207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#493	2
C₂×C₄×C₁₃×Q₁₆	2
C₁₃×#2207	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#357	2
C₂×#2207	13

Subgroups — C2×C13×#2207 #15719915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2207 #15719915