Subgroups — C₂×C₁₃×#2221 #15719953

Order 3328.

678 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	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#472	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2221	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#361	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×D₁₆	—	2	1	no	—
11	C₂×C₁₃×D₁₆	—	2	1	no	—
12	C₂×C₁₃×D₁₆	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
14	C₁₃×#504	—	2	2	no	—
15	C₁₃×#504	C₂²	1	4	yes	complemented
16	C₂×C₁₃×M₃₂	—	2	1	no	—
17	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃×C₁₆	—	2	1	no	—
19	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
20	C₁₃×#472	—	2	4	no	—
21	C₁₃×#472	C₂²	1	8	yes	complemented
22	C₂×C₁₃×#108	C₂²	1	2	yes	—
23	C₂×C₁₃×#100	—	2	2	no	—
24	C₁₃×#368	C₂²	1	4	yes	complemented
25	C₁₃×#361	C₂²	1	8	yes	complemented
26	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₁₃×AGL(1,ℤ/8)	—	2	8	no	—
29	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
30	C₂×C₁₃×SD₁₆	—	4	2	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	—	4	2	no	—
37	C₂×C₁₃×D₈	C₂³	1	4	yes	—
38	C₁₃×#131	—	2	1	no	—
39	C₁₃×#131	—	2	1	no	—
40	C₁₃×#131	D₄	1	2	yes	—
41	C₁₃×#131	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂×C₁₃×M₁₆	—	2	2	no	—
44	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	—	2	1	no	—
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×D₁₆	—	2	4	no	—
48	C₁₃×D₁₆	—	2	4	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	4	no	—
51	C₁₃×M₃₂	C₂³	1	2	yes	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	—	2	4	no	—
54	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
55	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₁₃×#108	—	2	2	no	—
59	C₁₃×#108	C₂³	1	4	yes	—
60	C₁₃×#100	—	2	8	no	—
61	C₂×#2221	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	4	2	no	—
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	D₈	1	2	yes	—
66	C₁₃×#41	C₂×D₄	1	2	yes	—
67	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	—	2	1	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×SD₁₆	—	4	8	no	—
79	C₁₃×D₈	—	2	4	no	—
80	C₁₃×D₈	—	2	4	no	—
81	C₁₃×D₈	—	4	8	no	—
82	C₁₃×D₈	—	4	2	no	—
83	C₁₃×D₈	—	4	2	no	—
84	C₁₃×M₁₆	—	2	1	no	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₁₃×M₁₆	—	2	4	no	—
88	C₁₃×M₁₆	—	4	2	no	—
89	C₁₃×M₁₆	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	4	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₁₆	—	2	2	no	—
101	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
102	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
103	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₂×#504	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂×#472	C₂×C₁₃	1	2	yes	complemented
106	#2221	C₂×C₁₃	1	8	yes	direct product factor
107	C₂×#368	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#361	C₂×C₁₃	1	2	yes	complemented
109	C₂³×C₁₃	—	2	1	no	—
110	C₂³×C₁₃	—	4	4	no	—
111	C₂³×C₁₃	—	8	2	no	—
112	C₂³×C₁₃	—	8	2	no	—
113	C₂³×C₁₃	—	16	4	no	—
114	C₂³×C₁₃	#135	1	1	yes	characteristic
115	Q₈×C₁₃	—	2	1	no	—
116	Q₈×C₁₃	C₂×D₈	1	2	yes	—
117	D₄×C₁₃	—	2	2	no	—
118	D₄×C₁₃	—	2	2	no	—
119	D₄×C₁₃	—	2	1	no	—
120	D₄×C₁₃	—	4	4	no	—
121	D₄×C₁₃	—	4	4	no	—
122	D₄×C₁₃	—	8	2	no	—
123	D₄×C₁₃	—	8	2	no	—
124	D₄×C₁₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	4	4	no	—
136	C₈×C₁₃	#121	1	1	yes	characteristic
137	C₈×C₁₃	#121	1	1	yes	characteristic
138	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×AGL(1,ℤ/8)	—	2	2	no	—
141	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
142	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×D₁₆	—	2	1	no	—
144	C₂×D₁₆	—	2	1	no	—
145	C₂×D₁₆	—	2	2	no	—
146	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
147	#504	—	2	2	no	—
148	#504	C₂²×C₁₃	1	4	yes	complemented
149	C₂×M₃₂	—	2	1	no	—
150	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
151	C₂²×C₁₆	—	2	1	no	—
152	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
153	#472	—	2	4	no	—
154	#472	C₂²×C₁₃	1	8	yes	complemented
155	C₂×#108	C₂²×C₁₃	1	2	yes	—
156	C₂×#100	—	2	2	no	—
157	#368	C₂²×C₁₃	1	4	yes	complemented
158	#361	C₂²×C₁₃	1	8	yes	complemented
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	4	2	no	—
162	C₂²×C₁₃	—	8	4	no	—
163	C₂²×C₁₃	—	16	4	no	—
164	C₂²×C₁₃	—	16	4	no	—
165	C₂²×C₁₃	—	16	4	no	—
166	C₂²×C₁₃	#466	1	1	yes	characteristic
167	C₂²×C₁₃	#466	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	#466	1	1	yes	characteristic
171	C₄×C₁₃	#466	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
173	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
174	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
175	C₂²×D₄	—	2	2	no	—
176	AGL(1,ℤ/8)	—	2	8	no	—
177	AGL(1,ℤ/8)	—	4	4	no	—
178	C₂×SD₁₆	—	4	2	no	—
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	—	2	2	no	—
184	C₂×D₈	—	4	2	no	—
185	C₂×D₈	C₂³×C₁₃	1	4	yes	—
186	#131	—	2	1	no	—
187	#131	—	2	1	no	—
188	#131	D₄×C₁₃	1	2	yes	—
189	#131	C₂³×C₁₃	1	2	yes	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
195	D₁₆	—	2	4	no	—
196	D₁₆	—	2	4	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	4	no	—
199	M₃₂	C₂³×C₁₃	1	2	yes	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
205	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
206	#108	—	2	2	no	—
207	#108	C₂³×C₁₃	1	4	yes	—
208	#100	—	2	8	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	4	2	no	—
212	C₂×C₁₃	—	16	4	no	—
213	C₂×C₁₃	#2221	1	2	yes	direct product factor
214	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
215	C₂⁴	—	4	2	no	—
216	#41	—	2	1	no	—
217	#41	—	2	1	no	—
218	#41	C₁₃×D₈	1	2	yes	—
219	#41	C₂×D₄×C₁₃	1	2	yes	—
220	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	SD₁₆	—	4	8	no	—
232	D₈	—	2	4	no	—
233	D₈	—	2	4	no	—
234	D₈	—	4	8	no	—
235	D₈	—	4	2	no	—
236	D₈	—	4	2	no	—
237	M₁₆	—	2	1	no	—
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	C₂×D₄×C₁₃	1	2	yes	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
251	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₁₆	—	2	2	no	—
254	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
255	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
256	C₁₃	C₂×#2221	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	2	1	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	—	16	4	no	—
262	C₂³	C₁₃×#135	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	C₂×C₁₃×D₈	1	2	yes	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	2	1	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	C₂×C₁₃×D₈	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
277	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
278	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
280	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	4	no	—
284	C₈	C₁₃×#121	1	1	yes	characteristic
285	C₈	C₁₃×#121	1	1	yes	characteristic
286	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	16	4	no	—
294	C₂²	—	16	4	no	—
295	C₂²	C₁₃×#466	1	1	yes	characteristic
296	C₂²	C₁₃×#466	1	1	yes	characteristic
297	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	C₁₃×#466	1	1	yes	characteristic
300	C₄	C₁₃×#466	1	1	yes	characteristic
301	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
302	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	2	no	—
306	C₂	—	16	4	no	—
307	C₂	C₁₃×#2221	1	2	yes	direct product factor
308	C₂	C₂×C₁₃×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ 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×#504	C₂	semidirect
3. C₂×C₁₃×#472	C₂	semidirect
4. C₁₃×#2221	C₂	direct (N × H)
5. C₂×C₁₃×#368	C₂	semidirect
6. C₂×C₁₃×#361	C₂	semidirect
13. C₂×C₁₃×D₁₆	C₂²	semidirect
15. C₁₃×#504	C₂²	semidirect
21. C₁₃×#472	C₂²	semidirect
24. C₁₃×#368	C₂²	semidirect
25. C₁₃×#361	C₂²	semidirect
61. C₂×#2221	C₁₃	direct (N × H)
103. C₂²×D₁₆	C₂×C₁₃	semidirect
104. C₂×#504	C₂×C₁₃	semidirect
105. C₂×#472	C₂×C₁₃	semidirect
106. #2221	C₂×C₁₃	direct (N × H)
107. C₂×#368	C₂×C₁₃	semidirect
108. C₂×#361	C₂×C₁₃	semidirect
146. C₂×D₁₆	C₂²×C₁₃	semidirect
148. #504	C₂²×C₁₃	semidirect
154. #472	C₂²×C₁₃	semidirect
157. #368	C₂²×C₁₃	semidirect
158. #361	C₂²×C₁₃	semidirect
213. C₂×C₁₃	#2221	direct (N × H)
256. C₁₃	C₂×#2221	direct (N × H)
307. C₂	C₁₃×#2221	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×#504	2
C₂×C₁₃×#472	2
C₁₃×#2221	2
C₂×C₁₃×#368	2
C₂×C₁₃×#361	2
C₂×#2221	13

Subgroups — C2×C13×#2221 #15719953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2221 #15719953