Subgroups — C₂×#13114461 #15850635

Order 3328.

914 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718739	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718754	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718355	C₂	1	1	yes	maximal, characteristic
6	#13114461	C₂	1	8	yes	maximal, direct product factor
7	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂²×#6106	—	2	2	no	—
12	C₂²×#6106	—	2	2	no	—
13	C₂×#88128	C₂²	1	1	yes	characteristic
14	C₂×#88123	C₂²	1	1	yes	characteristic
15	C₁₃×#498	—	2	2	no	—
16	C₁₃×#498	C₂²	1	4	yes	complemented
17	C₂×C₁₃×#106	—	2	2	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88061	—	2	2	no	—
20	#11718754	—	2	2	no	—
21	#11718754	C₂²	1	4	yes	complemented
22	#11718746	—	2	2	no	—
23	#11718746	—	2	2	no	—
24	#11718746	C₂²	1	4	yes	—
25	#11718746	C₂²	1	4	yes	—
26	#11718739	—	2	2	no	—
27	#11718739	C₂²	1	4	yes	complemented
28	C₂×#88050	—	2	2	no	—
29	C₂×#88047	—	2	2	no	—
30	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
31	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
32	#11718355	—	2	2	no	—
33	#11718355	C₂²	1	4	yes	—
34	#11718046	C₂²	1	4	yes	complemented
35	C₂²×Q₈×C₁₃	—	2	1	no	—
36	C₂²×Q₈×C₁₃	—	2	1	no	—
37	C₂²×D₅₂	—	2	1	no	—
38	C₂²×D₅₂	—	2	1	no	—
39	C₂²×D₅₂	—	2	1	no	—
40	C₂²×Dic₂₆	—	2	1	no	—
41	C₂²×C₈×C₁₃	D₄	1	2	yes	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×#6106	—	2	8	no	—
56	C₂×#6106	—	2	8	no	—
57	C₂×#6106	—	4	8	no	—
58	C₂×#6106	—	4	8	no	—
59	C₂×C₄⋊Dic₁₃	—	2	1	no	—
60	C₂×C₄⋊Dic₁₃	—	2	1	no	—
61	C₂²×#1055	D₄	1	2	yes	—
62	C₂²×#1055	D₄	1	2	yes	—
63	#88128	—	2	1	no	—
64	#88128	—	2	1	no	—
65	#88128	—	2	1	no	—
66	#88128	C₂³	1	2	yes	—
67	#88123	—	2	1	no	—
68	#88123	—	2	1	no	—
69	#88123	—	2	1	no	—
70	#88123	C₂³	1	2	yes	—
71	C₁₃×#106	—	2	8	no	—
72	C₄×C₈×C₁₃	D₄	1	2	yes	—
73	C₄×C₈×C₁₃	C₂³	1	2	yes	—
74	#88061	—	2	8	no	—
75	#88050	—	2	8	no	—
76	#88047	—	2	8	no	—
77	C₄×#1055	D₄	1	2	yes	—
78	C₄×#1055	D₄	1	2	yes	—
79	C₄×#1055	C₂³	1	2	yes	—
80	C₄×#1055	C₂³	1	2	yes	—
81	C₂×#1714	—	13	1	no	maximal, Sylow
82	C₂³×D₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	C₂×Q₈×C₁₃	—	4	1	no	—
86	C₂×Q₈×C₁₃	—	4	1	no	—
87	C₂×Q₈×C₁₃	—	4	4	no	—
88	C₂×Q₈×C₁₃	—	4	4	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×Dic₁₃	—	4	1	no	—
95	C₂×D₅₂	—	2	2	no	—
96	C₂×D₅₂	—	2	2	no	—
97	C₂×D₅₂	—	2	2	no	—
98	C₂×D₅₂	—	4	4	no	—
99	C₂×D₅₂	—	4	4	no	—
100	C₂×D₅₂	—	4	4	no	—
101	C₂×D₅₂	—	4	1	no	—
102	C₂×D₅₂	—	4	1	no	—
103	C₂×D₅₂	—	4	1	no	—
104	C₂×Dic₂₆	—	2	2	no	—
105	C₂×Dic₂₆	—	4	4	no	—
106	C₂×Dic₂₆	—	4	1	no	—
107	C₂×C₈×C₁₃	—	2	4	no	—
108	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₄⋊C₄	—	4	1	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₄⋊C₄	—	4	1	no	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	#6106	—	4	16	no	—
122	#6106	—	4	16	no	—
123	C₄⋊Dic₁₃	—	2	2	no	—
124	C₄⋊Dic₁₃	—	2	2	no	—
125	C₄⋊Dic₁₃	—	4	1	no	—
126	C₄⋊Dic₁₃	—	4	1	no	—
127	C₂×#1055	—	2	4	no	—
128	C₂×#1055	SD₁₆	1	8	yes	complemented
129	C₂×#1055	C₂×D₄	1	4	yes	—
130	C₂×#1055	C₂×D₄	1	4	yes	—
131	C₂×#500	—	13	1	no	—
132	C₂×#498	D₁₃	1	1	yes	characteristic, complemented
133	C₂×#492	—	13	1	no	—
134	C₂×#492	—	13	1	no	—
135	C₂×#487	—	13	1	no	—
136	C₂×#486	—	13	1	no	—
137	#1714	—	13	8	no	—
138	C₂×#322	—	13	1	no	—
139	C₂³×C₁₃	#127	1	1	yes	characteristic
140	C₂²×D₁₃	—	4	2	no	—
141	C₂²×D₁₃	—	8	2	no	—
142	C₂²×D₁₃	—	8	2	no	—
143	C₂²×D₁₃	—	8	2	no	—
144	Q₈×C₁₃	—	4	4	no	—
145	Q₈×C₁₃	—	4	4	no	—
146	Q₈×C₁₃	—	8	2	no	—
147	Q₈×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	—	8	2	no	—
152	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
153	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
154	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
156	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
157	C₂×C₄×C₁₃	#137	1	4	yes	—
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×Dic₁₃	—	4	2	no	—
162	C₂×Dic₁₃	—	8	2	no	—
163	D₅₂	—	4	4	no	—
164	D₅₂	—	4	4	no	—
165	D₅₂	—	4	4	no	—
166	D₅₂	—	8	2	no	—
167	D₅₂	—	8	2	no	—
168	D₅₂	—	8	2	no	—
169	Dic₂₆	—	4	4	no	—
170	Dic₂₆	—	8	2	no	—
171	C₈×C₁₃	—	2	4	no	—
172	#1055	—	2	4	no	—
173	#1055	C₂×SD₁₆	1	8	yes	complemented
174	C₂²×SD₁₆	—	26	2	no	—
175	C₂²×SD₁₆	—	26	2	no	—
176	C₂×C₄⋊Q₈	—	13	1	no	—
177	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
179	C₂×#127	—	13	1	no	—
180	#500	—	13	4	no	—
181	#500	—	26	2	no	—
182	#498	—	2	2	no	—
183	#498	C₂×D₁₃	1	4	yes	complemented
184	#492	—	13	4	no	—
185	#492	—	13	4	no	—
186	#492	—	26	2	no	—
187	#492	—	26	2	no	—
188	#487	—	13	4	no	—
189	#487	—	26	2	no	—
190	#486	—	13	4	no	—
191	#486	—	26	2	no	—
192	C₂×#107	—	26	2	no	—
193	C₂×#106	—	2	2	no	—
194	C₂×#103	—	26	2	no	—
195	C₂×#102	—	26	2	no	—
196	C₂×C₄×C₈	—	13	1	no	—
197	C₂×C₄×C₈	—	13	1	no	—
198	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
199	#322	—	13	4	no	—
200	C₂²×C₁₃	#492	1	2	yes	—
201	C₂²×C₁₃	#496	1	2	yes	—
202	C₂²×C₁₃	#497	1	2	yes	—
203	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
204	C₂×D₁₃	—	8	2	no	—
205	C₂×D₁₃	—	8	2	no	—
206	C₂×D₁₃	—	8	2	no	—
207	C₂×D₁₃	—	16	4	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	—	8	2	no	—
210	C₄×C₁₃	#492	1	2	yes	—
211	C₄×C₁₃	#496	1	2	yes	—
212	C₄×C₁₃	#497	1	2	yes	—
213	C₄×C₁₃	C₂²×SD₁₆	1	2	yes	—
214	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
215	C₄×C₁₃	C₂×#137	1	2	yes	—
216	Dic₁₃	—	8	2	no	—
217	C₂²×Q₈	—	2	1	no	—
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×Q₈	—	26	1	no	—
220	C₂²×D₄	—	26	1	no	—
221	C₂²×D₄	—	26	1	no	—
222	C₂²×D₄	—	26	1	no	—
223	C₂×SD₁₆	—	26	8	no	—
224	C₂×SD₁₆	—	26	8	no	—
225	C₂×SD₁₆	—	52	8	no	—
226	C₂×SD₁₆	—	52	8	no	—
227	C₂²×C₈	—	13	2	no	—
228	C₂²×C₈	—	13	2	no	—
229	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	13	2	no	—
237	C₄⋊Q₈	—	26	1	no	—
238	C₄⋊Q₈	—	26	1	no	—
239	C₄⋊Q₈	—	26	1	no	—
240	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
241	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
242	#127	—	13	2	no	—
243	#127	—	26	1	no	—
244	#127	—	26	1	no	—
245	#127	—	26	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	26	1	no	—
251	C₂×C₄⋊C₄	—	26	1	no	—
252	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
253	#107	—	26	8	no	—
254	#106	—	2	8	no	—
255	#103	—	26	8	no	—
256	#102	—	26	8	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	13	2	no	—
260	C₄×C₈	—	13	2	no	—
261	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
262	C₄×C₈	C₂²×D₁₃	1	2	yes	—
263	C₂×C₁₃	#1714	1	4	yes	complemented
264	C₂×C₁₃	C₂×#492	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#496	1	1	yes	characteristic
266	C₂×C₁₃	C₂×#497	1	1	yes	characteristic
267	D₁₃	—	16	2	no	—
268	C₂⁴	—	52	1	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	4	4	no	—
274	C₂×Q₈	—	4	4	no	—
275	C₂×Q₈	—	26	2	no	—
276	C₂×Q₈	—	52	1	no	—
277	C₂×Q₈	—	52	4	no	—
278	C₂×D₄	—	26	2	no	—
279	C₂×D₄	—	26	2	no	—
280	C₂×D₄	—	26	2	no	—
281	C₂×D₄	—	52	4	no	—
282	C₂×D₄	—	52	4	no	—
283	C₂×D₄	—	52	1	no	—
284	C₂×D₄	—	52	1	no	—
285	C₂×D₄	—	52	1	no	—
286	C₂×D₄	—	52	4	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	52	1	no	—
290	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	SD₁₆	—	52	16	no	—
294	SD₁₆	—	52	16	no	—
295	C₂×C₈	—	2	4	no	—
296	C₂×C₈	—	13	8	no	—
297	C₂×C₈	—	13	4	no	—
298	C₂×C₈	—	13	4	no	—
299	C₂×C₈	—	26	4	no	—
300	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	52	1	no	—
312	C₄⋊C₄	—	52	1	no	—
313	C₄²	D₄×D₁₃	1	1	yes	characteristic
314	C₄²	D₄×D₁₃	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₄²	C₂³×D₁₃	1	1	yes	characteristic
317	C₁₃	C₂×#1714	1	1	yes	characteristic, complemented, Sylow
318	C₂³	—	52	2	no	—
319	C₂³	—	104	2	no	—
320	C₂³	—	104	2	no	—
321	C₂³	—	104	2	no	—
322	C₂³	#88194	1	1	yes	characteristic
323	Q₈	—	4	4	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	8	2	no	—
326	Q₈	—	8	2	no	—
327	Q₈	—	52	4	no	—
328	Q₈	—	104	2	no	—
329	D₄	—	52	4	no	—
330	D₄	—	52	4	no	—
331	D₄	—	52	4	no	—
332	D₄	—	104	2	no	—
333	D₄	—	104	2	no	—
334	D₄	—	104	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	—	104	2	no	—
341	C₂×C₄	SD₁₆×D₁₃	1	4	yes	—
342	C₂×C₄	#88168	1	4	yes	—
343	C₂×C₄	#88194	1	1	yes	characteristic
344	C₂×C₄	#88194	1	1	yes	characteristic
345	C₂×C₄	#88194	1	1	yes	characteristic
346	C₂×C₄	#88196	1	4	yes	—
347	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	13	8	no	—
352	C₈	—	26	4	no	—
353	C₂²	—	104	2	no	—
354	C₂²	—	104	2	no	—
355	C₂²	—	104	2	no	—
356	C₂²	—	208	4	no	—
357	C₂²	#11718747	1	2	yes	—
358	C₂²	#11718836	1	2	yes	—
359	C₂²	#11718837	1	2	yes	—
360	C₂²	C₂×#88194	1	1	yes	characteristic
361	C₄	—	8	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	104	2	no	—
364	C₄	#11718747	1	2	yes	—
365	C₄	#11718836	1	2	yes	—
366	C₄	#11718837	1	2	yes	—
367	C₄	C₂×SD₁₆×D₁₃	1	2	yes	—
368	C₄	C₂×#88168	1	2	yes	—
369	C₄	C₂×#88196	1	2	yes	—
370	C₂	—	208	2	no	—
371	C₂	#13114461	1	4	yes	direct product factor
372	C₂	C₂×#11718836	1	1	yes	characteristic
373	C₂	C₂×#11718837	1	1	yes	characteristic
374	C₂	C₂×#11718747	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	C₂×#1714
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
1. C₂×#11718739	C₂	semidirect
4. C₂×#11718754	C₂	semidirect
6. #13114461	C₂	direct (N × H)
7. C₂×#11718046	C₂	semidirect
8. C₂×C₁₃×#498	C₂	semidirect
16. C₁₃×#498	C₂²	semidirect
21. #11718754	C₂²	semidirect
27. #11718739	C₂²	semidirect
34. #11718046	C₂²	semidirect
128. C₂×#1055	SD₁₆	semidirect
132. C₂×#498	D₁₃	semidirect
173. #1055	C₂×SD₁₆	semidirect
183. #498	C₂×D₁₃	semidirect
263. C₂×C₁₃	#1714	semidirect
317. C₁₃	C₂×#1714	semidirect
371. C₂	#13114461	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×#11718739	2
C₂×#11718746	2
C₂×#11718754	2
C₂×#11718355	2
#13114461	2
C₂×#11718046	2
C₂×C₁₃×#498	2
C₂×#1714	13

Subgroups — C2×#13114461 #15850635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114461 #15850635