Subgroups — C₂×#13113131 #15828537

Order 3328.

866 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13113131	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₄	1	2	yes	—
11	C₁₃×#431	C₄	1	4	yes	—
12	C₁₃×#431	C₂²	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊#1055	C₄	1	2	yes	—
18	#11718586	C₄	1	4	yes	—
19	#11718586	C₂²	1	4	yes	—
20	C₂×C₄⋊#1055	C₄	1	2	yes	—
21	C₄×#6099	C₄	1	4	yes	—
22	C₄×#6099	C₂²	1	4	yes	—
23	C₂×#88043	C₄	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718047	C₂²	1	4	yes	complemented
29	#11718046	C₂²	1	4	yes	complemented
30	#11718046	C₂²	1	4	yes	complemented
31	C₈×#1055	C₂²	1	4	yes	complemented
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×#6099	—	2	8	no	—
46	C₂×#6099	C₂×C₄	1	8	yes	—
47	C₂²×#1055	C₂×C₄	1	2	yes	—
48	C₂²×#1055	C₂×C₄	1	2	yes	—
49	C₂²×#1055	C₂×C₄	1	2	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
52	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂²⋊#1055	C₂×C₄	1	8	yes	—
58	C₄⋊#1055	C₂×C₄	1	8	yes	—
59	#88043	C₂×C₄	1	8	yes	—
60	C₄×#1055	C₂×C₄	1	2	yes	—
61	C₄×#1055	C₂×C₄	1	2	yes	—
62	C₄×#1055	C₂×C₄	1	2	yes	—
63	C₄×#1055	C₂×C₄	1	2	yes	—
64	C₄×#1055	C₂³	1	2	yes	—
65	C₄×#1055	C₂³	1	2	yes	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₄×#1055	C₂³	1	2	yes	—
68	C₂×#1453	—	13	1	no	maximal, Sylow
69	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
75	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
78	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₃	C₄²	1	2	yes	—
90	C₄²×C₁₃	C₄²	1	2	yes	—
91	C₄²×C₁₃	C₄²	1	2	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
99	#6099	—	2	16	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	4	no	—
103	C₂×#1055	M₁₆	1	8	yes	—
104	C₂×#1055	C₂²×C₄	1	4	yes	—
105	C₂×#1055	C₂²×C₄	1	4	yes	—
106	C₂×#1055	C₂²×C₄	1	4	yes	—
107	C₂×#1055	C₂²×C₄	1	4	yes	—
108	C₂×#431	—	13	1	no	—
109	C₂×#431	D₁₃	1	1	yes	characteristic
110	C₂×C₄×M₁₆	—	13	1	no	—
111	#1453	—	13	8	no	—
112	C₂×#322	—	13	1	no	—
113	C₂×#322	—	13	1	no	—
114	C₂×#322	—	13	1	no	—
115	C₂×C₈²	—	13	1	no	—
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
120	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₁₃	#131	1	4	yes	—
138	C₂×C₄×C₁₃	#131	1	4	yes	—
139	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₁₃	—	2	4	no	—
143	C₈×C₁₃	—	2	4	no	—
144	#1055	—	2	4	no	—
145	#1055	—	2	4	no	—
146	#1055	—	2	4	no	—
147	#1055	C₂×M₁₆	1	8	yes	—
148	C₂²×M₁₆	—	13	2	no	—
149	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
150	#431	—	13	4	no	—
151	#431	—	13	4	no	—
152	#431	Dic₁₃	1	4	yes	—
153	#431	C₂×D₁₃	1	4	yes	—
154	C₂×C₄⋊C₈	—	13	2	no	—
155	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
156	C₂×C₂²⋊C₈	—	13	2	no	—
157	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
158	C₄×M₁₆	—	13	4	no	—
159	C₄×M₁₆	—	13	4	no	—
160	C₂×#97	—	13	2	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	—	13	1	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
167	#322	—	13	4	no	—
168	#322	—	13	4	no	—
169	#322	—	13	4	no	—
170	C₈²	—	13	4	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	4	4	no	—
175	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
176	C₂²×C₁₃	#405	1	2	yes	—
177	C₂²×C₁₃	#405	1	2	yes	—
178	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
183	C₄×C₁₃	#405	1	2	yes	—
184	C₄×C₁₃	#405	1	2	yes	—
185	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
186	C₄×C₁₃	C₂×#131	1	2	yes	—
187	C₄×C₁₃	C₂×#131	1	2	yes	—
188	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
189	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
190	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
191	C₂×M₁₆	—	13	8	no	—
192	C₂×M₁₆	—	26	8	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₂²×C₈	—	13	2	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
198	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
199	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₁₃	1	2	yes	—
201	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₁₃	1	2	yes	—
203	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
205	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
206	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
207	C₄⋊C₈	—	13	8	no	—
208	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
209	C₂²⋊C₈	—	13	8	no	—
210	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
211	#97	—	13	8	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	—	13	2	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	—	13	2	no	—
216	C₄×C₈	—	13	2	no	—
217	C₄×C₈	—	13	2	no	—
218	C₄×C₈	—	13	2	no	—
219	C₄×C₈	—	13	2	no	—
220	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
221	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
222	C₄×C₈	C₂²×D₁₃	1	2	yes	—
223	C₄×C₈	C₂²×D₁₃	1	2	yes	—
224	C₂×C₁₃	—	4	2	no	—
225	C₂×C₁₃	#1453	1	4	yes	complemented
226	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
227	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
228	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
229	C₂⁴	C₄×Dic₁₃	1	1	yes	characteristic
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
235	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
238	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
245	M₁₆	—	26	16	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	13	8	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	—	13	4	no	—
251	C₂×C₈	—	13	4	no	—
252	C₂×C₈	—	13	4	no	—
253	C₂×C₈	—	26	4	no	—
254	C₂×C₈	—	26	4	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
257	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
258	C₄²	C₄×Dic₁₃	1	2	yes	—
259	C₄²	C₄×Dic₁₃	1	2	yes	—
260	C₄²	C₄×Dic₁₃	1	2	yes	—
261	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₁₃	1	2	yes	—
264	C₄²	C₂×C₄×D₁₃	1	2	yes	—
265	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
266	C₄²	C₂²×Dic₁₃	1	2	yes	—
267	C₄²	C₂³×D₁₃	1	1	yes	characteristic
268	C₁₃	C₂×#1453	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
273	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	#88155	1	4	yes	—
284	C₂×C₄	M₁₆×D₁₃	1	4	yes	—
285	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
291	C₂×C₄	#88202	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	13	8	no	—
298	C₈	—	26	4	no	—
299	C₈	—	26	4	no	—
300	C₈	—	26	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	#11718658	1	2	yes	—
306	C₂²	#11718762	1	2	yes	—
307	C₂²	M₁₆×Dic₁₃	1	2	yes	—
308	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	#11718658	1	2	yes	—
313	C₄	#11718762	1	2	yes	—
314	C₄	M₁₆×Dic₁₃	1	2	yes	—
315	C₄	C₂×#88155	1	2	yes	—
316	C₄	C₂×M₁₆×D₁₃	1	2	yes	—
317	C₄	C₂×#88202	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	#13113131	1	4	yes	direct product factor
320	C₂	C₂×M₁₆×Dic₁₃	1	1	yes	characteristic
321	C₂	C₂×#11718762	1	1	yes	characteristic
322	C₂	C₂×#11718658	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	C₂×#1453
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13113131	C₂	direct (N × H)
4. C₂×#11718046	C₂	semidirect
5. C₂×#11718046	C₂	semidirect
6. C₂×#11718047	C₂	semidirect
7. C₂×C₈×#1055	C₂	semidirect
28. #11718047	C₂²	semidirect
29. #11718046	C₂²	semidirect
30. #11718046	C₂²	semidirect
31. C₈×#1055	C₂²	semidirect
225. C₂×C₁₃	#1453	semidirect
268. C₁₃	C₂×#1453	semidirect
319. C₂	#13113131	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×#11718586	2
C₂×C₄×#6099	2
#13113131	2
C₂×#11718046	2
C₂×#11718047	2
C₂×C₈×#1055	2
C₂×C₁₃×#431	2
C₂×#1453	13

Subgroups — C2×#13113131 #15828537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113131 #15828537