Subgroups — C₂×#13113713 #15832481

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718491	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#6095	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718499	C₂	1	1	yes	maximal, characteristic
4	#13113713	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718076	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718075	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#6095	C₄	1	1	yes	characteristic
10	C₂²×#6094	—	2	1	no	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88061	C₄	1	1	yes	characteristic
17	C₂×#88060	—	2	1	no	—
18	C₂×#88057	C₄	1	1	yes	characteristic
19	C₂×#88056	C₄	1	1	yes	characteristic
20	C₂×#88054	—	2	1	no	—
21	C₂×C₈×Dic₁₃	—	2	1	no	—
22	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	#11718499	—	2	2	no	—
25	#11718499	C₂²	1	4	yes	—
26	#11718491	—	2	2	no	—
27	#11718491	C₂²	1	4	yes	complemented
28	C₄×#6095	C₄	1	4	yes	—
29	C₄×#6095	C₂²	1	4	yes	—
30	C₁₃×#335	C₂²	1	4	yes	complemented
31	#11718076	C₂²	1	4	yes	complemented
32	#11718075	C₂²	1	4	yes	—
33	#11718046	C₂²	1	4	yes	complemented
34	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₃	—	2	1	no	—
36	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂²×C₈×C₁₃	—	2	1	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6103	—	2	1	no	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#6101	—	2	1	no	—
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	D₄	1	1	yes	characteristic
49	C₂²×#1055	D₄	1	1	yes	characteristic
50	C₂×#6095	—	2	4	no	—
51	C₂×#6095	C₂×C₄	1	4	yes	—
52	C₂×#6094	—	2	4	no	—
53	C₂×#6094	—	4	4	no	—
54	C₄×D₅₂	—	2	1	no	—
55	C₄×D₅₂	—	2	1	no	—
56	C₄×D₅₂	C₂×C₄	1	2	yes	—
57	C₄×D₅₂	C₂³	1	2	yes	—
58	C₄×Dic₂₆	—	2	1	no	—
59	C₄×Dic₂₆	—	2	1	no	—
60	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₈	—	2	1	no	—
63	C₁₃×C₄⋊C₈	—	2	1	no	—
64	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88061	C₂×C₄	1	4	yes	—
69	#88060	—	2	4	no	—
70	#88057	C₂×C₄	1	4	yes	—
71	#88056	C₂×C₄	1	4	yes	—
72	#88054	—	2	4	no	—
73	C₈×Dic₁₃	—	2	4	no	—
74	C₄×#1055	—	2	1	no	—
75	C₄×#1055	D₄	1	2	yes	—
76	C₄×#1055	C₂³	1	2	yes	—
77	C₄×#1055	C₂³	1	2	yes	—
78	C₂×#1588	—	13	1	no	maximal, Sylow
79	C₂³×D₁₃	—	2	1	no	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
83	C₂²×Dic₁₃	—	2	1	no	—
84	C₂²×Dic₁₃	—	2	1	no	—
85	C₂²×Dic₁₃	—	2	1	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×D₅₂	—	2	2	no	—
88	C₂×D₅₂	—	2	1	no	—
89	C₂×D₅₂	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₁₃	—	2	2	no	—
91	C₂×C₄×D₁₃	—	2	2	no	—
92	C₂×C₄×D₁₃	—	4	2	no	—
93	C₂×C₄×D₁₃	—	4	2	no	—
94	C₂×Dic₂₆	—	2	1	no	—
95	C₂×Dic₂₆	—	2	2	no	—
96	C₂×Dic₂₆	M₁₆	1	4	yes	complemented
97	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	4	2	no	—
103	C₂×C₈×C₁₃	—	4	2	no	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6103	—	2	2	no	—
111	#6103	—	2	2	no	—
112	C₄⋊Dic₁₃	—	2	1	no	—
113	C₄⋊Dic₁₃	—	2	1	no	—
114	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
116	#6101	—	2	2	no	—
117	#6101	—	2	2	no	—
118	C₄×Dic₁₃	—	2	2	no	—
119	C₄×Dic₁₃	—	2	2	no	—
120	C₂×#1055	—	2	4	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	C₂×D₄	1	2	yes	—
125	C₂×#1055	C₂×D₄	1	2	yes	—
126	#6095	—	2	8	no	—
127	#6094	—	4	8	no	—
128	C₂×C₈×Q₈	—	13	1	no	—
129	C₂×C₄×SD₁₆	—	13	1	no	—
130	C₂×#436	—	13	1	no	—
131	#1588	—	13	8	no	—
132	C₂×#335	D₁₃	1	1	yes	characteristic, complemented
133	C₂×Q₈⋊C₈	—	13	1	no	—
134	C₂×D₄⋊C₈	—	13	1	no	—
135	C₂×#322	—	13	1	no	—
136	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₁₃	—	2	2	no	—
138	C₂²×D₁₃	—	4	2	no	—
139	C₂²×D₁₃	—	4	2	no	—
140	C₂²×D₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₁₃	#131	1	2	yes	—
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₃	#137	1	2	yes	—
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	4	no	—
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	—	4	2	no	—
157	C₂×Dic₁₃	—	4	2	no	—
158	D₅₂	—	2	2	no	—
159	D₅₂	—	2	2	no	—
160	D₅₂	—	4	2	no	—
161	C₄×D₁₃	—	4	2	no	—
162	C₄×D₁₃	—	4	2	no	—
163	C₄×D₁₃	—	4	4	no	—
164	Dic₂₆	—	2	2	no	—
165	Dic₂₆	—	2	4	no	—
166	Dic₂₆	C₂×M₁₆	1	4	yes	complemented
167	C₈×C₁₃	—	2	2	no	—
168	C₈×C₁₃	—	2	2	no	—
169	C₈×C₁₃	—	4	2	no	—
170	C₈×C₁₃	—	4	2	no	—
171	#1055	—	2	4	no	—
172	#1055	—	2	2	no	—
173	#1055	—	2	2	no	—
174	C₂²×SD₁₆	—	13	1	no	—
175	C₂²×M₁₆	—	26	1	no	—
176	C₂×C₄×Q₈	—	13	1	no	—
177	C₂×C₄×D₄	—	13	1	no	—
178	C₈×Q₈	—	13	4	no	—
179	C₈×Q₈	—	26	2	no	—
180	C₄×SD₁₆	—	13	4	no	—
181	C₄×SD₁₆	—	13	4	no	—
182	#436	—	13	4	no	—
183	#436	—	26	2	no	—
184	C₂×#106	—	13	1	no	—
185	C₂×C₄⋊C₈	—	26	1	no	—
186	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
188	C₂×#103	—	13	1	no	—
189	C₂×#102	—	13	1	no	—
190	C₂×C₂²⋊C₈	—	26	1	no	—
191	C₂×C₄×C₈	—	13	1	no	—
192	C₂×C₄×C₈	—	13	1	no	—
193	C₂×C₄×C₈	—	26	1	no	—
194	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
195	#335	C₂×D₁₃	1	4	yes	complemented
196	Q₈⋊C₈	—	13	4	no	—
197	D₄⋊C₈	—	13	4	no	—
198	#322	—	13	4	no	—
199	C₂²×C₁₃	#435	1	2	yes	—
200	C₂²×C₁₃	#440	1	2	yes	—
201	C₂²×C₁₃	#443	1	2	yes	—
202	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂×D₁₃	—	4	2	no	—
204	C₂×D₁₃	—	4	2	no	—
205	C₂×D₁₃	—	4	2	no	—
206	C₂×D₁₃	—	8	4	no	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	#435	1	2	yes	—
210	C₄×C₁₃	#440	1	2	yes	—
211	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
212	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
213	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
215	Dic₁₃	—	2	4	no	—
216	Dic₁₃	—	4	2	no	—
217	Dic₁₃	—	4	2	no	—
218	C₂²×Q₈	—	13	1	no	—
219	C₂²×D₄	—	13	1	no	—
220	C₂³×C₄	—	26	1	no	—
221	C₂×SD₁₆	—	13	4	no	—
222	C₂×SD₁₆	—	26	4	no	—
223	C₂×M₁₆	—	26	4	no	—
224	C₂×M₁₆	—	52	4	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	13	1	no	—
228	C₂²×C₈	—	13	1	no	—
229	C₂²×C₈	—	26	1	no	—
230	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
232	C₄×Q₈	—	13	2	no	—
233	C₄×Q₈	—	13	2	no	—
234	C₄×Q₈	—	26	1	no	—
235	C₄×Q₈	—	26	1	no	—
236	C₄×D₄	—	13	2	no	—
237	C₄×D₄	—	13	2	no	—
238	C₄×D₄	—	26	1	no	—
239	C₄×D₄	—	26	1	no	—
240	C₂×C₄⋊C₄	—	13	1	no	—
241	C₂×C₄⋊C₄	—	13	1	no	—
242	C₂×C₄⋊C₄	—	26	1	no	—
243	C₂×C₂²⋊C₄	—	26	1	no	—
244	C₂×C₄²	—	26	1	no	—
245	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
246	#106	—	13	4	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	26	4	no	—
250	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
251	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
252	#103	—	13	4	no	—
253	#102	—	13	4	no	—
254	C₂²⋊C₈	—	26	4	no	—
255	C₄×C₈	—	13	2	no	—
256	C₄×C₈	—	13	2	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	26	4	no	—
259	C₄×C₈	—	26	1	no	—
260	C₄×C₈	C₄×D₁₃	1	2	yes	—
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₂×C₁₃	#1588	1	4	yes	complemented
263	C₂×C₁₃	C₂×#435	1	1	yes	characteristic
264	C₂×C₁₃	C₂×#440	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
266	D₁₃	—	8	2	no	—
267	C₂⁴	—	26	1	no	—
268	C₂×Q₈	—	13	4	no	—
269	C₂×Q₈	—	13	2	no	—
270	C₂×Q₈	—	26	1	no	—
271	C₂×Q₈	—	26	2	no	—
272	C₂×D₄	—	13	2	no	—
273	C₂×D₄	—	26	1	no	—
274	C₂×D₄	—	26	2	no	—
275	C₂×D₄	—	26	2	no	—
276	C₂²×C₄	—	26	1	no	—
277	C₂²×C₄	—	26	1	no	—
278	C₂²×C₄	—	26	1	no	—
279	C₂²×C₄	—	26	2	no	—
280	C₂²×C₄	—	26	2	no	—
281	C₂²×C₄	—	52	2	no	—
282	C₂²×C₄	—	52	2	no	—
283	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
285	C₂²×C₄	#6131	1	1	yes	characteristic
286	SD₁₆	—	26	8	no	—
287	M₁₆	—	52	8	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	13	2	no	—
295	C₂×C₈	—	13	2	no	—
296	C₂×C₈	—	26	4	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
301	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
302	C₄⋊C₄	—	13	2	no	—
303	C₄⋊C₄	—	13	2	no	—
304	C₄⋊C₄	—	26	1	no	—
305	C₄⋊C₄	—	26	1	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	26	2	no	—
308	C₂²⋊C₄	—	26	2	no	—
309	C₂²⋊C₄	—	26	2	no	—
310	C₄²	—	26	2	no	—
311	C₄²	—	26	2	no	—
312	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
313	C₄²	D₄×D₁₃	1	1	yes	characteristic
314	C₄²	#6131	1	1	yes	characteristic
315	C₄²	C₂³×D₁₃	1	1	yes	characteristic
316	C₁₃	C₂×#1588	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	26	2	no	—
318	C₂³	—	52	2	no	—
319	C₂³	—	52	2	no	—
320	C₂³	—	52	2	no	—
321	C₂³	#88147	1	1	yes	characteristic
322	Q₈	—	13	4	no	—
323	Q₈	—	26	2	no	—
324	Q₈	—	26	4	no	—
325	D₄	—	26	2	no	—
326	D₄	—	26	2	no	—
327	D₄	—	52	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	26	4	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	52	2	no	—
334	C₂×C₄	—	52	2	no	—
335	C₂×C₄	—	52	2	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	—	52	4	no	—
338	C₂×C₄	#88147	1	1	yes	characteristic
339	C₂×C₄	#88147	1	1	yes	characteristic
340	C₂×C₄	#88147	1	1	yes	characteristic
341	C₂×C₄	#88155	1	2	yes	—
342	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
343	C₂×C₄	#88165	1	2	yes	—
344	C₂×C₄	#88172	1	2	yes	—
345	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
347	C₂×C₄	C₂×#6131	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	26	4	no	—
353	C₈	—	26	2	no	—
354	C₈	—	26	2	no	—
355	C₂²	—	52	2	no	—
356	C₂²	—	52	2	no	—
357	C₂²	—	52	2	no	—
358	C₂²	—	104	4	no	—
359	C₂²	#11718497	1	2	yes	—
360	C₂²	#11718550	1	2	yes	—
361	C₂²	#11718556	1	2	yes	—
362	C₂²	C₂×#88147	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	26	4	no	—
366	C₄	—	52	2	no	—
367	C₄	—	52	2	no	—
368	C₄	#11718497	1	2	yes	—
369	C₄	#11718550	1	2	yes	—
370	C₄	C₂×#88155	1	1	yes	characteristic
371	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
372	C₄	C₂×#88165	1	1	yes	characteristic
373	C₄	C₂×#88172	1	1	yes	characteristic
374	C₂	—	104	2	no	—
375	C₂	#13113713	1	4	yes	direct product factor
376	C₂	C₂×#11718497	1	1	yes	characteristic
377	C₂	C₂×#11718550	1	1	yes	characteristic
378	C₂	C₂×#11718556	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	C₂×#1588
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. C₂×#11718491	C₂	semidirect
4. #13113713	C₂	direct (N × H)
5. C₂×#11718076	C₂	semidirect
7. C₂×#11718046	C₂	semidirect
8. C₂×C₁₃×#335	C₂	semidirect
27. #11718491	C₂²	semidirect
30. C₁₃×#335	C₂²	semidirect
31. #11718076	C₂²	semidirect
33. #11718046	C₂²	semidirect
96. C₂×Dic₂₆	M₁₆	semidirect
132. C₂×#335	D₁₃	semidirect
166. Dic₂₆	C₂×M₁₆	semidirect
195. #335	C₂×D₁₃	semidirect
262. C₂×C₁₃	#1588	semidirect
316. C₁₃	C₂×#1588	semidirect
375. C₂	#13113713	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×#11718491	2
C₂×C₄×#6095	2
C₂×#11718499	2
#13113713	2
C₂×#11718076	2
C₂×#11718075	2
C₂×#11718046	2
C₂×C₁₃×#335	2
C₂×#1588	13

Subgroups — C2×#13113713 #15832481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113713 #15832481