Subgroups — C₂×C₁₃×#2188 #15719852

Order 3328.

1126 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#467	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2188	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	8	no	—
9	C₂²×C₁₃×Q₁₆	—	2	1	no	—
10	C₂²×C₁₃×Q₁₆	—	2	2	no	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
13	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₁₃×Q₃₂	—	2	8	no	—
17	C₂×C₁₃×Q₃₂	C₂²	1	8	yes	complemented
18	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
19	C₁₃×#467	—	2	2	no	—
20	C₁₃×#467	C₂²	1	4	yes	—
21	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#103	—	2	1	no	—
23	C₁₃×#358	C₂²	1	8	yes	complemented
24	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃	—	2	8	no	—
26	C₂²×Q₈×C₁₃	—	2	4	no	—
27	C₂²×Q₈×C₁₃	—	2	2	no	—
28	C₂²×Q₈×C₁₃	—	2	1	no	—
29	C₂²×Q₈×C₁₃	—	4	4	no	—
30	C₂²×Q₈×C₁₃	—	4	2	no	—
31	C₂²×Q₈×C₁₃	—	4	1	no	—
32	C₂³×C₄×C₁₃	—	4	1	no	—
33	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₁₃×Q₁₆	—	2	8	no	—
35	C₂×C₁₃×Q₁₆	—	2	16	no	—
36	C₂×C₁₃×Q₁₆	—	2	16	no	—
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	—	2	2	no	—
39	C₂×C₁₃×Q₁₆	—	2	4	no	—
40	C₂×C₁₃×Q₁₆	—	2	4	no	—
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	D₄	1	8	yes	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
45	C₂²×C₈×C₁₃	—	2	4	no	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	D₄	1	2	yes	—
48	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×C₂²⋊Q₈	—	2	4	no	—
51	C₁₃×C₂²⋊Q₈	—	4	2	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
55	C₁₃×Q₃₂	—	2	16	no	—
56	C₂×C₁₃×C₁₆	—	2	4	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#103	—	2	4	no	—
61	C₂×#2188	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
63	C₂×Q₈×C₁₃	—	2	16	no	—
64	C₂×Q₈×C₁₃	—	2	8	no	—
65	C₂×Q₈×C₁₃	—	2	4	no	—
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	4	8	no	—
68	C₂×Q₈×C₁₃	—	4	8	no	—
69	C₂×Q₈×C₁₃	—	4	16	no	—
70	C₂×Q₈×C₁₃	—	4	4	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	4	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	4	8	no	—
79	C₂²×C₄×C₁₃	—	4	4	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	D₈	1	2	yes	—
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₁₃×Q₁₆	—	2	16	no	—
88	C₁₃×Q₁₆	—	2	8	no	—
89	C₁₃×Q₁₆	—	2	4	no	—
90	C₁₃×Q₁₆	—	2	4	no	—
91	C₁₃×Q₁₆	—	4	2	no	—
92	C₁₃×Q₁₆	C₂×D₄	1	8	yes	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	4	no	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	4	no	—
103	C₁₃×C₂²⋊C₄	—	4	4	no	—
104	C₁₃×C₁₆	—	2	4	no	—
105	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
106	C₂²×Q₃₂	C₂×C₁₃	1	2	yes	complemented
107	C₂×#467	C₂×C₁₃	1	1	yes	characteristic
108	#2188	C₂×C₁₃	1	8	yes	direct product factor
109	C₂×#358	C₂×C₁₃	1	2	yes	complemented
110	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
111	C₂³×C₁₃	—	2	2	no	—
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	Q₃₂	1	4	yes	complemented
114	C₂³×C₁₃	C₂×D₈	1	2	yes	—
115	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	8	no	—
117	Q₈×C₁₃	—	4	16	no	—
118	Q₈×C₁₃	—	4	4	no	—
119	Q₈×C₁₃	—	4	8	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	4	16	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	8	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	#120	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	#120	1	2	yes	—
141	C₈×C₁₃	C₂²×D₄	1	2	yes	—
142	C₂³×Q₈	—	2	1	no	—
143	C₂²×Q₁₆	—	2	8	no	—
144	C₂²×Q₁₆	—	2	1	no	—
145	C₂²×Q₁₆	—	2	2	no	—
146	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
147	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
148	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
149	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₂²⋊Q₈	—	2	1	no	—
151	C₂×Q₃₂	—	2	8	no	—
152	C₂×Q₃₂	C₂²×C₁₃	1	8	yes	complemented
153	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
154	#467	—	2	2	no	—
155	#467	C₂²×C₁₃	1	4	yes	—
156	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#103	—	2	1	no	—
158	#358	C₂²×C₁₃	1	8	yes	complemented
159	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	8	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	#447	1	2	yes	—
164	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
165	C₂²×C₁₃	C₂×Q₃₂	1	4	yes	complemented
166	C₂²×C₁₃	#510	1	2	yes	—
167	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	4	8	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	#447	1	2	yes	—
173	C₄×C₁₃	C₂×#120	1	1	yes	characteristic
174	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂²×Q₈	—	2	8	no	—
176	C₂²×Q₈	—	2	4	no	—
177	C₂²×Q₈	—	2	2	no	—
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×Q₈	—	4	4	no	—
180	C₂²×Q₈	—	4	1	no	—
181	C₂²×Q₈	—	4	2	no	—
182	C₂³×C₄	—	4	1	no	—
183	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
184	C₂×Q₁₆	—	2	8	no	—
185	C₂×Q₁₆	—	2	16	no	—
186	C₂×Q₁₆	—	2	16	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	—	2	4	no	—
190	C₂×Q₁₆	—	2	4	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	D₄×C₁₃	1	8	yes	—
193	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
194	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₁₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₂²⋊Q₈	—	2	4	no	—
201	C₂²⋊Q₈	—	4	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	4	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	Q₃₂	—	2	16	no	—
206	C₂×C₁₆	—	2	4	no	—
207	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
208	#107	—	2	1	no	—
209	#107	C₂³×C₁₃	1	2	yes	—
210	#103	—	2	4	no	—
211	C₂×C₁₃	—	2	4	no	—
212	C₂×C₁₃	#2188	1	4	yes	direct product factor
213	C₂×C₁₃	C₂×#447	1	1	yes	characteristic
214	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
216	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
217	C₂×Q₈	—	2	16	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	8	no	—
222	C₂×Q₈	—	4	16	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	8	no	—
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
238	C₂²×C₄	C₁₃×D₈	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	Q₁₆	—	2	16	no	—
242	Q₁₆	—	2	8	no	—
243	Q₁₆	—	2	4	no	—
244	Q₁₆	—	2	4	no	—
245	Q₁₆	—	4	2	no	—
246	Q₁₆	C₂×D₄×C₁₃	1	8	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
253	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₁₆	—	2	4	no	—
259	C₁₃	C₂×#2188	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	C₁₃×Q₃₂	1	4	yes	complemented
263	C₂³	C₂×C₁₃×D₈	1	2	yes	—
264	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	Q₈	—	4	16	no	—
267	Q₈	—	4	8	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	8	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	16	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	C₁₃×#120	1	2	yes	—
284	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
285	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
286	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
287	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	C₁₃×#120	1	2	yes	—
290	C₈	C₂²×D₄×C₁₃	1	2	yes	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	C₁₃×#447	1	2	yes	—
295	C₂²	C₂×C₁₃×Q₃₂	1	2	yes	—
296	C₂²	C₂×C₁₃×Q₃₂	1	4	yes	complemented
297	C₂²	C₁₃×#510	1	2	yes	—
298	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	8	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	C₁₃×#447	1	2	yes	—
304	C₄	C₂×C₁₃×#120	1	1	yes	characteristic
305	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	C₁₃×#2188	1	4	yes	direct product factor
308	C₂	C₂×C₁₃×#447	1	1	yes	characteristic
309	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
310	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×Q₃₂	C₂	semidirect
4. C₁₃×#2188	C₂	direct (N × H)
5. C₂×C₁₃×#358	C₂	semidirect
17. C₂×C₁₃×Q₃₂	C₂²	semidirect
23. C₁₃×#358	C₂²	semidirect
61. C₂×#2188	C₁₃	direct (N × H)
106. C₂²×Q₃₂	C₂×C₁₃	semidirect
108. #2188	C₂×C₁₃	direct (N × H)
109. C₂×#358	C₂×C₁₃	semidirect
113. C₂³×C₁₃	Q₃₂	semidirect
152. C₂×Q₃₂	C₂²×C₁₃	semidirect
158. #358	C₂²×C₁₃	semidirect
165. C₂²×C₁₃	C₂×Q₃₂	semidirect
212. C₂×C₁₃	#2188	direct (N × H)
259. C₁₃	C₂×#2188	direct (N × H)
262. C₂³	C₁₃×Q₃₂	semidirect
296. C₂²	C₂×C₁₃×Q₃₂	semidirect
307. C₂	C₁₃×#2188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂²×C₁₃×Q₃₂	2
C₂×C₁₃×#467	2
C₁₃×#2188	2
C₂×C₁₃×#358	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2188	13

Subgroups — C2×C13×#2188 #15719852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2188 #15719852