Subgroups — C₂×#13127719 #16743367

Order 3328.

1830 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719427	C₂	1	2	yes	maximal
2	#13127719	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718535	C₂	1	2	yes	maximal
4	C₂×#11718536	C₂	1	4	yes	maximal
5	C₂×#11718752	C₂	1	2	yes	maximal
6	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
8	C₂×#11718534	C₂	1	2	yes	maximal
9	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×#88198	—	2	4	no	—
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	#11719427	—	2	8	no	—
14	#11719427	C₂²	1	16	yes	—
15	C₂×#88142	—	2	2	no	—
16	C₂×#88142	C₂²	1	4	yes	—
17	C₂×#88141	—	2	2	no	—
18	C₂×#88141	C₂²	1	4	yes	—
19	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
21	C₁₃×#500	C₂²	1	8	yes	—
22	C₂×C₁₃×#107	C₂²	1	2	yes	—
23	C₂×C₁₃×#107	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₂×#88058	C₂²	1	1	yes	characteristic
26	C₂×#88056	C₂²	1	2	yes	—
27	C₂×#88054	C₂²	1	2	yes	—
28	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
29	#11718752	C₂²	1	16	yes	—
30	C₂×#88050	C₂²	1	4	yes	—
31	C₂×#88047	C₂²	1	4	yes	—
32	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
33	#11718536	C₂²	1	32	yes	—
34	#11718535	C₂²	1	16	yes	—
35	#11718534	C₂²	1	16	yes	—
36	C₄×Dic₅₂	C₂²	1	8	yes	—
37	C₈×Dic₂₆	C₂²	1	8	yes	—
38	C₂²×Q₈×C₁₃	—	2	2	no	—
39	C₂²×Dic₂₆	—	2	4	no	—
40	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
41	C₂²×Dic₂₆	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	4	no	—
45	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#88198	—	2	32	no	—
50	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
52	C₂×#6101	—	2	2	no	—
53	C₂×#6101	—	2	4	no	—
54	C₂×#6101	—	2	4	no	—
55	C₂×#6101	D₄	1	2	yes	—
56	C₂×C₄×Dic₁₃	—	2	2	no	—
57	C₂×C₄×Dic₁₃	—	2	4	no	—
58	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₂×Dic₅₂	Q₈	1	8	yes	complemented
61	C₂×Dic₅₂	C₂³	1	4	yes	—
62	#88142	—	2	16	no	—
63	#88142	—	2	8	no	—
64	#88142	C₂³	1	16	yes	—
65	#88141	—	2	16	no	—
66	#88141	—	2	8	no	—
67	#88141	C₂³	1	16	yes	—
68	C₄×Dic₂₆	—	2	4	no	—
69	C₄×Dic₂₆	D₄	1	4	yes	—
70	C₄×Dic₂₆	C₂³	1	8	yes	—
71	C₄×Dic₂₆	C₂³	1	4	yes	—
72	C₁₃×#107	C₂³	1	8	yes	—
73	C₁₃×#107	C₂³	1	8	yes	—
74	C₄×C₈×C₁₃	C₂³	1	4	yes	—
75	#88058	C₂³	1	4	yes	—
76	#88056	C₂³	1	8	yes	—
77	#88054	C₂³	1	8	yes	—
78	C₈×Dic₁₃	C₂³	1	8	yes	—
79	#88050	C₂³	1	16	yes	—
80	#88047	C₂³	1	16	yes	—
81	C₄⋊#1055	C₂³	1	4	yes	—
82	C₂×Q₈×Q₁₆	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	8	no	—
84	C₂×Q₈×C₁₃	—	8	4	no	—
85	C₂²×C₄×C₁₃	—	2	4	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₁₃	—	2	2	no	—
90	C₂²×Dic₁₃	—	2	2	no	—
91	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
92	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
93	C₂×Dic₂₆	—	2	2	no	—
94	C₂×Dic₂₆	—	2	4	no	—
95	C₂×Dic₂₆	—	2	16	no	—
96	C₂×Dic₂₆	—	8	8	no	—
97	C₂×Dic₂₆	Q₁₆	1	8	yes	complemented
98	C₂×Dic₂₆	C₂×D₄	1	4	yes	—
99	C₂×Dic₂₆	C₂×Q₈	1	8	yes	—
100	C₂×Dic₂₆	C₂⁴	1	4	yes	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
104	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
105	C₁₃×C₄⋊C₄	—	2	8	no	—
106	C₁₃×C₄⋊C₄	—	2	4	no	—
107	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
108	C₄²×C₁₃	C₂×D₄	1	2	yes	—
109	C₄²×C₁₃	C₂⁴	1	2	yes	—
110	C₄⋊Dic₁₃	—	2	2	no	—
111	C₄⋊Dic₁₃	C₂×D₄	1	4	yes	—
112	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
113	#6101	—	2	8	no	—
114	#6101	—	2	16	no	—
115	#6101	—	2	16	no	—
116	#6101	C₂×D₄	1	8	yes	—
117	C₄×Dic₁₃	—	2	8	no	—
118	C₄×Dic₁₃	—	2	16	no	—
119	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
120	C₄×Dic₁₃	C₂⁴	1	4	yes	—
121	C₂×#1055	—	2	4	no	—
122	C₂×#1055	C₂⁴	1	4	yes	—
123	Dic₅₂	C₂×Q₈	1	16	yes	complemented
124	C₂×Q₈²	—	13	2	no	—
125	Q₈×Q₁₆	—	13	16	no	—
126	C₂×#500	—	13	2	no	—
127	C₂×#500	D₁₃	1	1	yes	characteristic
128	C₂×#477	—	13	4	no	—
129	C₂×#477	—	13	2	no	—
130	C₂×C₈×Q₈	—	13	1	no	—
131	C₂×C₄×Q₁₆	—	13	2	no	—
132	C₂×C₄×Q₁₆	—	13	1	no	—
133	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	Q₈×C₁₃	—	8	8	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	8	no	—
137	C₂×C₄×C₁₃	—	4	8	no	—
138	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
139	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
144	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
145	C₂×Dic₁₃	—	2	8	no	—
146	C₂×Dic₁₃	—	2	4	no	—
147	C₂×Dic₁₃	—	2	4	no	—
148	C₂×Dic₁₃	—	2	4	no	—
149	C₂×Dic₁₃	—	4	4	no	—
150	C₂×Dic₁₃	C₂×Q₁₆	1	8	yes	—
151	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
152	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
153	Dic₂₆	—	2	8	no	—
154	Dic₂₆	—	2	4	no	—
155	Dic₂₆	—	8	16	no	—
156	Dic₂₆	C₂×Q₁₆	1	16	yes	complemented
157	Dic₂₆	C₂²×Q₈	1	8	yes	—
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
160	#1055	—	2	4	no	—
161	C₂²×Q₁₆	—	13	1	no	—
162	Q₈²	—	13	16	no	—
163	Q₈²	—	26	8	no	—
164	C₂×C₄⋊Q₈	—	13	4	no	—
165	C₂×C₄⋊Q₈	—	26	2	no	—
166	C₂×C₄⋊Q₈	—	26	4	no	—
167	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
168	C₂×C₄×Q₈	—	13	1	no	—
169	C₂×C₄×Q₈	—	13	4	no	—
170	C₂×C₄×Q₈	—	13	2	no	—
171	C₂×C₄×Q₈	—	26	2	no	—
172	#500	—	13	16	no	—
173	#500	C₂×D₁₃	1	8	yes	—
174	#477	—	13	16	no	—
175	#477	—	13	32	no	—
176	C₈×Q₈	—	13	8	no	—
177	C₄×Q₁₆	—	13	16	no	—
178	C₄×Q₁₆	—	13	8	no	—
179	C₂×#107	—	13	1	no	—
180	C₂×#107	—	13	4	no	—
181	C₂×#107	C₂×D₁₃	1	2	yes	—
182	C₂×#107	C₂×D₁₃	1	2	yes	—
183	C₂×C₄⋊C₈	—	13	2	no	—
184	C₂×C₄⋊C₈	—	13	1	no	—
185	C₂×#103	—	13	2	no	—
186	C₂×#103	—	13	4	no	—
187	C₂×C₄×C₈	—	13	2	no	—
188	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
189	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
190	C₂²×C₁₃	C₂²×Q₁₆	1	2	yes	—
191	C₂²×C₁₃	#576	1	2	yes	—
192	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	4	8	no	—
195	C₄×C₁₃	D₄×Q₈	1	2	yes	—
196	C₄×C₁₃	C₂²×Q₁₆	1	4	yes	—
197	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
198	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
199	Dic₁₃	—	2	8	no	—
200	Dic₁₃	—	2	4	no	—
201	Dic₁₃	—	4	4	no	—
202	Dic₁₃	C₂²×Q₁₆	1	8	yes	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	13	1	no	—
205	C₂²×Q₈	—	13	2	no	—
206	C₂²×Q₈	—	26	4	no	—
207	C₂×Q₁₆	—	13	8	no	—
208	C₂×Q₁₆	—	13	4	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	4	no	—
213	C₄⋊Q₈	—	13	16	no	—
214	C₄⋊Q₈	—	26	8	no	—
215	C₄⋊Q₈	—	26	16	no	—
216	C₄⋊Q₈	—	26	32	no	—
217	C₄⋊Q₈	C₂²×D₁₃	1	8	yes	—
218	C₄×Q₈	—	13	4	no	—
219	C₄×Q₈	—	13	4	no	—
220	C₄×Q₈	—	13	16	no	—
221	C₄×Q₈	—	13	8	no	—
222	C₄×Q₈	—	26	8	no	—
223	C₄×Q₈	—	26	16	no	—
224	C₄×Q₈	—	26	4	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	13	2	no	—
227	C₂×C₄⋊C₄	—	13	1	no	—
228	C₂×C₄⋊C₄	—	13	2	no	—
229	C₂×C₄⋊C₄	—	26	4	no	—
230	C₂×C₄⋊C₄	—	26	4	no	—
231	C₂×C₄⋊C₄	—	26	2	no	—
232	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
233	C₂×C₄²	—	13	2	no	—
234	C₂×C₄²	—	26	4	no	—
235	C₂×C₄²	—	26	2	no	—
236	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
237	#107	—	13	4	no	—
238	#107	—	13	16	no	—
239	#107	C₂²×D₁₃	1	8	yes	—
240	#107	C₂²×D₁₃	1	8	yes	—
241	C₄⋊C₈	—	13	8	no	—
242	C₄⋊C₈	—	13	4	no	—
243	#103	—	13	8	no	—
244	#103	—	13	16	no	—
245	C₄×C₈	—	13	8	no	—
246	C₄×C₈	C₂²×D₁₃	1	4	yes	—
247	C₂×C₁₃	Q₈×Q₁₆	1	4	yes	complemented
248	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
249	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#576	1	1	yes	characteristic
251	C₂×Q₈	—	2	8	no	—
252	C₂×Q₈	—	8	4	no	—
253	C₂×Q₈	—	13	8	no	—
254	C₂×Q₈	—	13	4	no	—
255	C₂×Q₈	—	13	8	no	—
256	C₂×Q₈	—	13	4	no	—
257	C₂×Q₈	—	26	16	no	—
258	C₂×Q₈	—	26	2	no	—
259	C₂×Q₈	—	26	4	no	—
260	C₂×Q₈	—	104	8	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂²×C₄	—	13	2	no	—
264	C₂²×C₄	—	26	2	no	—
265	C₂²×C₄	—	26	2	no	—
266	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
269	Q₁₆	—	13	16	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	13	4	no	—
272	C₂×C₈	—	26	4	no	—
273	C₂×C₈	Q₈×D₁₃	1	4	yes	—
274	C₂×C₈	C₂³×D₁₃	1	2	yes	—
275	C₂×C₈	C₂³×D₁₃	1	2	yes	—
276	C₄⋊C₄	—	2	8	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	13	8	no	—
279	C₄⋊C₄	—	13	4	no	—
280	C₄⋊C₄	—	13	4	no	—
281	C₄⋊C₄	—	26	16	no	—
282	C₄⋊C₄	—	26	16	no	—
283	C₄⋊C₄	—	26	2	no	—
284	C₄⋊C₄	—	26	8	no	—
285	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
286	C₄²	—	13	4	no	—
287	C₄²	—	13	4	no	—
288	C₄²	—	26	16	no	—
289	C₄²	—	26	8	no	—
290	C₄²	D₄×D₁₃	1	2	yes	—
291	C₄²	C₂³×D₁₃	1	2	yes	—
292	C₁₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
293	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
294	Q₈	—	8	8	no	—
295	Q₈	—	13	16	no	—
296	Q₈	—	13	8	no	—
297	Q₈	—	26	8	no	—
298	Q₈	—	26	4	no	—
299	Q₈	—	104	16	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	13	8	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	26	8	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	52	4	no	—
311	C₂×C₄	Q₁₆×D₁₃	1	4	yes	—
312	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
315	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
316	C₂×C₄	#88254	1	2	yes	—
317	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	26	4	no	—
320	C₈	C₂×Q₈×D₁₃	1	4	yes	—
321	C₂²	#11719432	1	2	yes	—
322	C₂²	#11719462	1	2	yes	—
323	C₂²	C₂×Q₁₆×D₁₃	1	2	yes	—
324	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	13	8	no	—
328	C₄	—	26	4	no	—
329	C₄	—	26	8	no	—
330	C₄	—	52	4	no	—
331	C₄	#11719432	1	2	yes	—
332	C₄	C₂×Q₁₆×D₁₃	1	4	yes	—
333	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
334	C₄	C₂×#88254	1	1	yes	characteristic
335	C₂	#13127719	1	4	yes	direct product factor
336	C₂	C₂×#11719432	1	1	yes	characteristic
337	C₂	C₂²×Q₁₆×D₁₃	1	1	yes	characteristic
338	C₂	C₂×#11719462	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #13127719	C₂	direct (N × H)
60. C₂×Dic₅₂	Q₈	semidirect
97. C₂×Dic₂₆	Q₁₆	semidirect
123. Dic₅₂	C₂×Q₈	semidirect
156. Dic₂₆	C₂×Q₁₆	semidirect
247. C₂×C₁₃	Q₈×Q₁₆	semidirect
292. C₁₃	C₂×Q₈×Q₁₆	semidirect
335. C₂	#13127719	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×#11719427	2
#13127719	2
C₂×#11718535	2
C₂×#11718536	2
C₂×#11718752	2
C₂×C₈×Dic₂₆	2
C₂×C₄×Dic₅₂	2
C₂×#11718534	2
C₂×C₁₃×#500	2
C₂×Q₈×Q₁₆	13

Subgroups — C2×#13127719 #16743367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127719 #16743367