Subgroups — C₂×#13114499 #15850686

Order 3328.

850 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718740	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718754	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718749	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718749	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718355	C₂	1	1	yes	maximal, characteristic
6	#13114499	C₂	1	8	yes	maximal, direct product factor
7	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	—	2	2	no	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#88128	C₂²	1	1	yes	characteristic
13	C₂×#88123	C₂²	1	1	yes	characteristic
14	C₁₃×#494	—	2	2	no	—
15	C₁₃×#494	C₂²	1	4	yes	complemented
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88061	—	2	2	no	—
18	#11718754	—	2	2	no	—
19	#11718754	C₂²	1	4	yes	complemented
20	#11718749	—	2	2	no	—
21	#11718749	—	2	2	no	—
22	#11718749	C₂²	1	4	yes	—
23	#11718749	C₂²	1	4	yes	—
24	#11718740	—	2	2	no	—
25	#11718740	C₂²	1	4	yes	complemented
26	C₂×#88050	—	2	2	no	—
27	C₂×#88049	—	2	2	no	—
28	C₂×#88049	—	2	2	no	—
29	C₂×#88048	—	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	C₂×C₁₃×Q₁₆	—	2	8	no	—
42	C₂×C₁₃×Q₁₆	—	4	8	no	—
43	C₂²×C₈×C₁₃	D₄	1	2	yes	—
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	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊Dic₁₃	—	2	1	no	—
58	C₂×C₄⋊Dic₁₃	—	2	1	no	—
59	C₂²×#1055	D₄	1	2	yes	—
60	C₂²×#1055	D₄	1	2	yes	—
61	#88128	—	2	1	no	—
62	#88128	—	2	1	no	—
63	#88128	—	2	1	no	—
64	#88128	C₂³	1	2	yes	—
65	#88123	—	2	1	no	—
66	#88123	—	2	1	no	—
67	#88123	—	2	1	no	—
68	#88123	C₂³	1	2	yes	—
69	C₄×C₈×C₁₃	D₄	1	2	yes	—
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88061	—	2	8	no	—
72	#88050	—	2	8	no	—
73	#88049	—	2	8	no	—
74	#88049	—	2	8	no	—
75	#88048	—	2	8	no	—
76	C₄×#1055	D₄	1	2	yes	—
77	C₄×#1055	D₄	1	2	yes	—
78	C₄×#1055	C₂³	1	2	yes	—
79	C₄×#1055	C₂³	1	2	yes	—
80	C₂×#1720	—	13	1	no	maximal, Sylow
81	C₂³×D₁₃	—	4	1	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	4	1	no	—
85	C₂×Q₈×C₁₃	—	4	1	no	—
86	C₂×Q₈×C₁₃	—	4	4	no	—
87	C₂×Q₈×C₁₃	—	4	4	no	—
88	C₂²×C₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×Dic₁₃	—	4	1	no	—
94	C₂×D₅₂	—	2	2	no	—
95	C₂×D₅₂	—	2	2	no	—
96	C₂×D₅₂	—	2	2	no	—
97	C₂×D₅₂	—	4	4	no	—
98	C₂×D₅₂	—	4	4	no	—
99	C₂×D₅₂	—	4	4	no	—
100	C₂×D₅₂	—	4	1	no	—
101	C₂×D₅₂	—	4	1	no	—
102	C₂×D₅₂	—	4	1	no	—
103	C₂×Dic₂₆	—	2	2	no	—
104	C₂×Dic₂₆	—	4	4	no	—
105	C₂×Dic₂₆	—	4	1	no	—
106	C₁₃×Q₁₆	—	4	16	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	C₄⋊Dic₁₃	—	2	2	no	—
122	C₄⋊Dic₁₃	—	2	2	no	—
123	C₄⋊Dic₁₃	—	4	1	no	—
124	C₄⋊Dic₁₃	—	4	1	no	—
125	C₂×#1055	—	2	4	no	—
126	C₂×#1055	—	2	4	no	—
127	C₂×#1055	C₂×D₄	1	4	yes	—
128	C₂×#1055	C₂×D₄	1	4	yes	—
129	C₂×#498	—	13	1	no	—
130	C₂×#494	D₁₃	1	1	yes	characteristic, complemented
131	C₂×#487	—	13	1	no	—
132	C₂×#486	—	13	1	no	—
133	C₂×#486	—	13	1	no	—
134	C₂×#486	—	13	1	no	—
135	#1720	—	13	8	no	—
136	C₂×#322	—	13	1	no	—
137	C₂³×C₁₃	#127	1	1	yes	characteristic
138	C₂²×D₁₃	—	4	2	no	—
139	C₂²×D₁₃	—	8	2	no	—
140	C₂²×D₁₃	—	8	2	no	—
141	C₂²×D₁₃	—	8	2	no	—
142	Q₈×C₁₃	—	4	4	no	—
143	Q₈×C₁₃	—	4	4	no	—
144	Q₈×C₁₃	—	8	2	no	—
145	Q₈×C₁₃	—	8	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	8	2	no	—
149	C₂×C₄×C₁₃	—	8	2	no	—
150	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
153	C₂×C₄×C₁₃	#135	1	4	yes	—
154	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
155	C₂×C₄×C₁₃	#137	1	4	yes	—
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	8	2	no	—
161	D₅₂	—	4	4	no	—
162	D₅₂	—	4	4	no	—
163	D₅₂	—	4	4	no	—
164	D₅₂	—	8	2	no	—
165	D₅₂	—	8	2	no	—
166	D₅₂	—	8	2	no	—
167	Dic₂₆	—	4	4	no	—
168	Dic₂₆	—	8	2	no	—
169	C₈×C₁₃	—	2	4	no	—
170	#1055	—	2	4	no	—
171	#1055	—	2	4	no	—
172	C₂²×Q₁₆	—	2	2	no	—
173	C₂×C₄⋊Q₈	—	13	1	no	—
174	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
175	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#127	—	13	1	no	—
177	#498	—	13	4	no	—
178	#498	—	26	2	no	—
179	#494	—	2	2	no	—
180	#494	C₂×D₁₃	1	4	yes	complemented
181	#487	—	13	4	no	—
182	#487	—	26	2	no	—
183	#486	—	13	4	no	—
184	#486	—	13	4	no	—
185	#486	—	13	4	no	—
186	#486	—	26	2	no	—
187	#486	—	26	2	no	—
188	#486	—	26	2	no	—
189	C₂×#106	—	26	2	no	—
190	C₂×#103	—	26	2	no	—
191	C₂×#102	—	26	2	no	—
192	C₂×#102	—	26	2	no	—
193	C₂×#102	—	26	2	no	—
194	C₂×C₄×C₈	—	13	1	no	—
195	C₂×C₄×C₈	—	13	1	no	—
196	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
197	#322	—	13	4	no	—
198	C₂²×C₁₃	#495	1	2	yes	—
199	C₂²×C₁₃	#496	1	2	yes	—
200	C₂²×C₁₃	#497	1	2	yes	—
201	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
202	C₂×D₁₃	—	8	2	no	—
203	C₂×D₁₃	—	8	2	no	—
204	C₂×D₁₃	—	8	2	no	—
205	C₂×D₁₃	—	16	4	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	#495	1	2	yes	—
209	C₄×C₁₃	#496	1	2	yes	—
210	C₄×C₁₃	#497	1	2	yes	—
211	C₄×C₁₃	C₂×#135	1	2	yes	—
212	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
213	C₄×C₁₃	C₂×#137	1	2	yes	—
214	Dic₁₃	—	8	2	no	—
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×Q₈	—	2	1	no	—
217	C₂²×Q₈	—	26	1	no	—
218	C₂²×D₄	—	26	1	no	—
219	C₂²×D₄	—	26	1	no	—
220	C₂²×D₄	—	26	1	no	—
221	C₂×Q₁₆	—	2	8	no	—
222	C₂×Q₁₆	—	4	8	no	—
223	C₂²×C₈	—	13	2	no	—
224	C₂²×C₈	—	13	2	no	—
225	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	13	2	no	—
233	C₄⋊Q₈	—	26	1	no	—
234	C₄⋊Q₈	—	26	1	no	—
235	C₄⋊Q₈	—	26	1	no	—
236	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
237	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
238	#127	—	13	2	no	—
239	#127	—	26	1	no	—
240	#127	—	26	1	no	—
241	#127	—	26	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	26	1	no	—
247	C₂×C₄⋊C₄	—	26	1	no	—
248	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
249	#106	—	26	8	no	—
250	#103	—	26	8	no	—
251	#102	—	26	8	no	—
252	#102	—	26	8	no	—
253	#102	—	26	8	no	—
254	C₄×C₈	—	13	2	no	—
255	C₄×C₈	—	13	2	no	—
256	C₄×C₈	—	13	2	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
259	C₄×C₈	C₂²×D₁₃	1	2	yes	—
260	C₂×C₁₃	#1720	1	4	yes	complemented
261	C₂×C₁₃	C₂×#495	1	1	yes	characteristic
262	C₂×C₁₃	C₂×#496	1	1	yes	characteristic
263	C₂×C₁₃	C₂×#497	1	1	yes	characteristic
264	D₁₃	—	16	2	no	—
265	C₂⁴	—	52	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	—	4	4	no	—
272	C₂×Q₈	—	26	2	no	—
273	C₂×Q₈	—	52	1	no	—
274	C₂×Q₈	—	52	4	no	—
275	C₂×D₄	—	26	2	no	—
276	C₂×D₄	—	26	2	no	—
277	C₂×D₄	—	26	2	no	—
278	C₂×D₄	—	52	4	no	—
279	C₂×D₄	—	52	4	no	—
280	C₂×D₄	—	52	4	no	—
281	C₂×D₄	—	52	1	no	—
282	C₂×D₄	—	52	1	no	—
283	C₂×D₄	—	52	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	52	1	no	—
287	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
288	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
290	Q₁₆	—	4	16	no	—
291	C₂×C₈	—	2	4	no	—
292	C₂×C₈	—	13	4	no	—
293	C₂×C₈	—	13	4	no	—
294	C₂×C₈	—	26	4	no	—
295	C₂×C₈	—	26	4	no	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	26	2	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	52	1	no	—
308	C₄⋊C₄	—	52	1	no	—
309	C₄²	D₄×D₁₃	1	1	yes	characteristic
310	C₄²	D₄×D₁₃	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
312	C₄²	C₂³×D₁₃	1	1	yes	characteristic
313	C₁₃	C₂×#1720	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	52	2	no	—
315	C₂³	—	104	2	no	—
316	C₂³	—	104	2	no	—
317	C₂³	—	104	2	no	—
318	C₂³	#88194	1	1	yes	characteristic
319	Q₈	—	4	4	no	—
320	Q₈	—	4	4	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	52	4	no	—
324	Q₈	—	104	2	no	—
325	D₄	—	52	4	no	—
326	D₄	—	52	4	no	—
327	D₄	—	52	4	no	—
328	D₄	—	104	2	no	—
329	D₄	—	104	2	no	—
330	D₄	—	104	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	52	2	no	—
336	C₂×C₄	—	104	2	no	—
337	C₂×C₄	#88172	1	4	yes	—
338	C₂×C₄	#88173	1	4	yes	—
339	C₂×C₄	#88186	1	4	yes	—
340	C₂×C₄	#88194	1	1	yes	characteristic
341	C₂×C₄	#88194	1	1	yes	characteristic
342	C₂×C₄	#88194	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
344	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	26	4	no	—
348	C₈	—	26	4	no	—
349	C₂²	—	104	2	no	—
350	C₂²	—	104	2	no	—
351	C₂²	—	104	2	no	—
352	C₂²	—	208	4	no	—
353	C₂²	#11718735	1	2	yes	—
354	C₂²	#11718851	1	2	yes	—
355	C₂²	#11718852	1	2	yes	—
356	C₂²	C₂×#88194	1	1	yes	characteristic
357	C₄	—	8	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	104	2	no	—
360	C₄	#11718735	1	2	yes	—
361	C₄	#11718851	1	2	yes	—
362	C₄	#11718852	1	2	yes	—
363	C₄	C₂×#88172	1	2	yes	—
364	C₄	C₂×#88173	1	2	yes	—
365	C₄	C₂×#88186	1	2	yes	—
366	C₂	—	208	2	no	—
367	C₂	#13114499	1	4	yes	direct product factor
368	C₂	C₂×#11718852	1	1	yes	characteristic
369	C₂	C₂×#11718735	1	1	yes	characteristic
370	C₂	C₂×#11718851	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	C₂×#1720
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
1. C₂×#11718740	C₂	semidirect
2. C₂×#11718754	C₂	semidirect
6. #13114499	C₂	direct (N × H)
7. C₂×#11718046	C₂	semidirect
8. C₂×C₁₃×#494	C₂	semidirect
15. C₁₃×#494	C₂²	semidirect
19. #11718754	C₂²	semidirect
25. #11718740	C₂²	semidirect
34. #11718046	C₂²	semidirect
130. C₂×#494	D₁₃	semidirect
180. #494	C₂×D₁₃	semidirect
260. C₂×C₁₃	#1720	semidirect
313. C₁₃	C₂×#1720	semidirect
367. C₂	#13114499	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×#11718740	2
C₂×#11718754	2
C₂×#11718749	2
C₂×#11718355	2
#13114499	2
C₂×#11718046	2
C₂×C₁₃×#494	2
C₂×#1720	13

Subgroups — C2×#13114499 #15850686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114499 #15850686