Subgroups — C₂×C₁₃×#2206 #15719912

Order 3328.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2206	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×D₈	—	2	1	no	—
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#127	—	2	1	no	—
11	C₂×C₄×D₄×C₁₃	—	2	1	no	—
12	C₂×C₁₃×D₁₆	—	2	8	no	—
13	C₂×C₁₃×D₁₆	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	complemented
17	C₄×C₁₃×D₈	—	2	2	no	—
18	C₄×C₁₃×D₈	C₂²	1	4	yes	complemented
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#102	—	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×D₄×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	C₂³×C₄×C₁₃	—	4	1	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	4	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
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	—	4	4	no	—
36	C₂×C₁₃×D₈	D₄	1	4	yes	complemented
37	C₂×C₁₃×D₈	C₂³	1	4	yes	—
38	C₂×C₁₃×D₈	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×#127	—	2	4	no	—
42	C₁₃×#127	—	4	2	no	—
43	C₄×D₄×C₁₃	—	2	4	no	—
44	C₄×D₄×C₁₃	—	4	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₁₃×D₁₆	—	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×#102	—	2	4	no	—
54	C₄×C₈×C₁₃	D₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×#2206	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	4	1	no	—
58	C₂⁴×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	4	4	no	—
63	C₂×D₄×C₁₃	—	4	1	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	2	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	8	4	no	—
70	C₂×D₄×C₁₃	—	8	4	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₃	—	8	4	no	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₁₃×D₈	—	2	4	no	—
78	C₁₃×D₈	—	2	4	no	—
79	C₁₃×D₈	—	2	2	no	—
80	C₁₃×D₈	—	2	4	no	—
81	C₁₃×D₈	—	4	8	no	—
82	C₁₃×D₈	—	4	4	no	—
83	C₁₃×D₈	C₂×D₄	1	4	yes	complemented
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₂²⋊C₄	—	4	4	no	—
93	C₄²×C₁₃	D₈	1	2	yes	—
94	C₄²×C₁₃	C₂×D₄	1	2	yes	—
95	C₁₃×C₁₆	—	2	4	no	—
96	C₂²×D₁₆	C₂×C₁₃	1	2	yes	complemented
97	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#2206	C₂×C₁₃	1	8	yes	direct product factor
100	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂×#357	C₂×C₁₃	1	2	yes	complemented
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	4	no	—
104	C₂³×C₁₃	—	4	4	no	—
105	C₂³×C₁₃	—	8	2	no	—
106	C₂³×C₁₃	—	8	2	no	—
107	C₂³×C₁₃	—	8	4	no	—
108	C₂³×C₁₃	—	8	4	no	—
109	C₂³×C₁₃	—	8	4	no	—
110	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
111	D₄×C₁₃	—	2	4	no	—
112	D₄×C₁₃	—	4	4	no	—
113	D₄×C₁₃	—	4	2	no	—
114	D₄×C₁₃	—	4	4	no	—
115	D₄×C₁₃	—	4	4	no	—
116	D₄×C₁₃	—	8	8	no	—
117	D₄×C₁₃	—	8	4	no	—
118	D₄×C₁₃	—	8	8	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	8	8	no	—
123	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
124	C₂×C₄×C₁₃	#121	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	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	#121	1	2	yes	—
132	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂²×D₈	—	2	1	no	—
135	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
136	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#127	—	2	1	no	—
138	C₂×C₄×D₄	—	2	1	no	—
139	C₂×D₁₆	—	2	8	no	—
140	C₂×D₁₆	C₂²×C₁₃	1	8	yes	complemented
141	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
142	#493	—	2	2	no	—
143	#493	C₂²×C₁₃	1	4	yes	complemented
144	C₄×D₈	—	2	2	no	—
145	C₄×D₈	C₂²×C₁₃	1	4	yes	complemented
146	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#102	—	2	1	no	—
148	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
150	#357	C₂²×C₁₃	1	8	yes	complemented
151	C₂²×C₁₃	—	4	4	no	—
152	C₂²×C₁₃	—	8	2	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	8	no	—
155	C₂²×C₁₃	—	8	4	no	—
156	C₂²×C₁₃	—	8	4	no	—
157	C₂²×C₁₃	—	8	4	no	—
158	C₂²×C₁₃	—	16	8	no	—
159	C₂²×C₁₃	#459	1	2	yes	—
160	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
161	C₂²×C₁₃	#509	1	2	yes	—
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	#459	1	2	yes	—
166	C₄×C₁₃	C₂×D₁₆	1	4	yes	complemented
167	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
168	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂²×D₄	—	4	1	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	4	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	4	4	no	—
181	C₂×D₈	D₄×C₁₃	1	4	yes	complemented
182	C₂×D₈	C₂³×C₁₃	1	4	yes	—
183	C₂×D₈	C₂³×C₁₃	1	2	yes	—
184	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
186	#127	—	2	4	no	—
187	#127	—	4	2	no	—
188	C₄×D₄	—	2	4	no	—
189	C₄×D₄	—	4	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₂²⋊C₄	—	4	1	no	—
192	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
193	D₁₆	—	2	16	no	—
194	C₂×C₁₆	—	2	4	no	—
195	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
196	#107	—	2	1	no	—
197	#107	C₂³×C₁₃	1	2	yes	—
198	#102	—	2	4	no	—
199	C₄×C₈	D₄×C₁₃	1	2	yes	—
200	C₄×C₈	C₂³×C₁₃	1	2	yes	—
201	C₂×C₁₃	—	8	4	no	—
202	C₂×C₁₃	—	16	4	no	—
203	C₂×C₁₃	#2206	1	4	yes	direct product factor
204	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
205	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
207	C₂⁴	—	4	1	no	—
208	C₂⁴	—	4	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	8	4	no	—
220	C₂×D₄	—	8	4	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	8	4	no	—
224	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
225	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	D₈	—	2	4	no	—
228	D₈	—	2	4	no	—
229	D₈	—	2	2	no	—
230	D₈	—	2	4	no	—
231	D₈	—	4	8	no	—
232	D₈	—	4	4	no	—
233	D₈	C₂×D₄×C₁₃	1	4	yes	complemented
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
238	C₂×C₈	C₁₃×#41	1	2	yes	—
239	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	C₁₃×D₈	1	2	yes	—
244	C₄²	C₂×D₄×C₁₃	1	2	yes	—
245	C₁₆	—	2	4	no	—
246	C₁₃	C₂×#2206	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	—	8	4	no	—
253	C₂³	—	8	4	no	—
254	C₂³	—	8	4	no	—
255	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	4	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	4	no	—
261	D₄	—	8	8	no	—
262	D₄	—	8	4	no	—
263	D₄	—	8	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	C₁₃×D₁₆	1	4	yes	complemented
269	C₂×C₄	C₁₃×#121	1	2	yes	—
270	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
271	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
272	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
273	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	C₁₃×#121	1	2	yes	—
277	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
278	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
279	C₂²	—	4	4	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	8	no	—
283	C₂²	—	8	4	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	16	8	no	—
287	C₂²	C₁₃×#459	1	2	yes	—
288	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
289	C₂²	C₁₃×#509	1	2	yes	—
290	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	C₁₃×#459	1	2	yes	—
294	C₄	C₂×C₁₃×D₁₆	1	4	yes	complemented
295	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
296	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	—	16	4	no	—
299	C₂	C₁₃×#2206	1	4	yes	direct product factor
300	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
301	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
302	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
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×D₁₆	C₂	semidirect
2. C₂×C₁₃×#493	C₂	semidirect
3. C₂×C₄×C₁₃×D₈	C₂	semidirect
4. C₁₃×#2206	C₂	direct (N × H)
5. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₁₃×#357	C₂	semidirect
13. C₂×C₁₃×D₁₆	C₂²	semidirect
16. C₁₃×#493	C₂²	semidirect
18. C₄×C₁₃×D₈	C₂²	semidirect
22. C₁₃×C₄⋊C₁₆	C₂²	semidirect
23. C₁₃×#357	C₂²	semidirect
36. C₂×C₁₃×D₈	D₄	semidirect
56. C₂×#2206	C₁₃	direct (N × H)
83. C₁₃×D₈	C₂×D₄	semidirect
96. C₂²×D₁₆	C₂×C₁₃	semidirect
97. C₂×#493	C₂×C₁₃	semidirect
98. C₂×C₄×D₈	C₂×C₁₃	semidirect
99. #2206	C₂×C₁₃	direct (N × H)
100. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
101. C₂×#357	C₂×C₁₃	semidirect
123. C₂×C₄×C₁₃	D₁₆	semidirect
140. C₂×D₁₆	C₂²×C₁₃	semidirect
143. #493	C₂²×C₁₃	semidirect
145. C₄×D₈	C₂²×C₁₃	semidirect
149. C₄⋊C₁₆	C₂²×C₁₃	semidirect
150. #357	C₂²×C₁₃	semidirect
166. C₄×C₁₃	C₂×D₁₆	semidirect
181. C₂×D₈	D₄×C₁₃	semidirect
203. C₂×C₁₃	#2206	direct (N × H)
233. D₈	C₂×D₄×C₁₃	semidirect
246. C₁₃	C₂×#2206	direct (N × H)
268. C₂×C₄	C₁₃×D₁₆	semidirect
294. C₄	C₂×C₁₃×D₁₆	semidirect
299. C₂	C₁₃×#2206	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×D₁₆	2
C₂×C₁₃×#493	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2206	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#357	2
C₂×#2206	13

Subgroups — C2×C13×#2206 #15719912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2206 #15719912