Subgroups — C₂×#13116670 #16016144

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718801	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718805	C₂	1	1	yes	maximal, characteristic
4	#13116670	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718100	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₂×#88204	C₄	1	1	yes	characteristic
11	C₂×#88155	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#88077	C₄	1	1	yes	characteristic
15	#11718805	C₄	1	4	yes	—
16	#11718805	C₂²	1	4	yes	—
17	#11718801	—	2	2	no	—
18	#11718801	C₂²	1	4	yes	complemented
19	C₂×#88065	C₄	1	1	yes	characteristic
20	C₂×#88062	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#88052	—	2	1	no	—
23	C₂×#88052	C₂²	1	1	yes	characteristic
24	C₂²×#6090	—	2	1	no	—
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718568	—	2	2	no	—
27	#11718568	C₂²	1	4	yes	complemented
28	C₂×#88038	—	2	1	no	—
29	C₂×C₁₆×D₁₃	—	2	1	no	—
30	C₁₃×#350	C₂²	1	4	yes	complemented
31	#11718103	C₂²	1	4	yes	complemented
32	#11718100	C₂²	1	4	yes	complemented
33	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂×#6127	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#131	—	2	1	no	—
37	C₁₃×#131	—	2	1	no	—
38	C₁₃×#131	C₂×C₄	1	2	yes	—
39	C₁₃×#131	C₂³	1	2	yes	—
40	C₂×C₁₃×M₁₆	—	2	1	no	—
41	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	—	2	1	no	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	#88204	C₈	1	4	yes	—
45	#88204	C₂×C₄	1	4	yes	—
46	C₂×#6107	C₈	1	1	yes	characteristic
47	C₂×#6106	C₈	1	1	yes	characteristic
48	C₂×#6105	C₈	1	1	yes	characteristic
49	C₂×#6104	C₈	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×#6099	C₂×C₄	1	1	yes	characteristic
52	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
53	#88155	—	2	1	no	—
54	#88155	—	2	1	no	—
55	#88155	C₂×C₄	1	2	yes	—
56	#88155	C₂³	1	2	yes	—
57	C₂×#6094	—	2	1	no	—
58	C₂×C₈×D₁₃	—	2	1	no	—
59	C₁₃×M₃₂	—	2	1	no	—
60	C₁₃×M₃₂	C₂³	1	2	yes	—
61	C₂×C₁₃×C₁₆	—	2	1	no	—
62	C₂×C₁₃×C₁₆	—	2	1	no	—
63	C₂×C₁₃×C₁₆	—	2	1	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
65	#88077	C₂×C₄	1	4	yes	—
66	#88065	C₂×C₄	1	4	yes	—
67	#88062	C₂×C₄	1	4	yes	—
68	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂³	1	2	yes	—
70	#88052	—	2	1	no	—
71	#88052	—	2	4	no	—
72	#88052	C₂³	1	2	yes	—
73	C₂×#6090	—	2	1	no	—
74	C₂×#6090	—	2	4	no	—
75	C₂×#6090	D₄	1	1	yes	characteristic
76	C₂×#6090	D₄	1	1	yes	characteristic
77	C₂×#6090	D₄	1	2	yes	—
78	C₂×#6090	C₂³	1	2	yes	—
79	#88038	—	2	4	no	—
80	C₁₆×D₁₃	—	2	4	no	—
81	C₂×#2170	—	13	1	no	maximal, Sylow
82	C₁₃×#41	—	2	1	no	—
83	C₁₃×#41	—	2	1	no	—
84	C₁₃×#41	C₂×C₈	1	2	yes	—
85	C₁₃×#41	C₂²×C₄	1	2	yes	—
86	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
87	C₂×D₄×C₁₃	—	2	1	no	—
88	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	—	2	1	no	—
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	—	2	1	no	—
92	C₂²×Dic₁₃	—	2	1	no	—
93	#6127	—	2	1	no	—
94	#6127	—	2	1	no	—
95	#6127	C₂×C₈	1	2	yes	—
96	#6127	C₂²×C₄	1	2	yes	—
97	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	—	2	1	no	—
99	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
100	C₁₃×M₁₆	—	2	1	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₃	—	2	1	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
113	#6107	C₂×C₈	1	4	yes	—
114	#6106	C₂×C₈	1	4	yes	—
115	#6105	C₂×C₈	1	4	yes	—
116	#6104	C₂×C₈	1	4	yes	—
117	C₄×Dic₁₃	—	2	1	no	—
118	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
119	#6099	—	2	1	no	—
120	#6099	C₂²×C₄	1	2	yes	—
121	C₂×#1055	—	2	1	no	—
122	C₂×#1055	C₂×C₈	1	1	yes	characteristic
123	C₂×#1055	C₂×C₈	1	1	yes	characteristic
124	C₂×#1055	C₂×C₈	1	2	yes	—
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	#6094	—	2	2	no	—
127	#6094	—	2	2	no	—
128	C₈×D₁₃	—	2	2	no	—
129	C₈×D₁₃	—	2	2	no	—
130	C₁₃×C₁₆	—	2	2	no	—
131	C₁₃×C₁₆	—	2	2	no	—
132	#6090	—	2	2	no	—
133	#6090	C₂×D₄	1	2	yes	—
134	#6090	C₂×D₄	1	2	yes	—
135	C₂×#504	—	13	1	no	—
136	C₂×#504	—	13	1	no	—
137	C₂×#443	—	13	1	no	—
138	#2170	—	13	8	no	—
139	C₂×#364	—	13	1	no	—
140	C₂×#350	—	13	1	no	—
141	C₂×#350	D₁₃	1	1	yes	characteristic, complemented
142	C₂×C₄×C₁₆	—	13	1	no	—
143	C₂³×C₁₃	—	2	1	no	—
144	C₂³×C₁₃	#131	1	1	yes	characteristic
145	C₂²×D₁₃	—	2	1	no	—
146	Q₈×C₁₃	—	2	1	no	—
147	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
148	D₄×C₁₃	—	2	2	no	—
149	D₄×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	2	1	no	—
151	D₄×C₁₃	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₁₃	—	2	1	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	2	2	no	—
162	C₂²⋊D₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	2	4	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	2	1	no	—
166	D₅₂	—	2	1	no	—
167	D₅₂	C₂²×C₈	1	2	yes	—
168	C₄×D₁₃	—	2	2	no	—
169	C₄×D₁₃	—	2	2	no	—
170	Dic₂₆	—	2	1	no	—
171	Dic₂₆	C₂²×C₈	1	2	yes	—
172	C₈×C₁₃	—	2	2	no	—
173	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
174	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
175	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
176	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
177	#1055	—	2	2	no	—
178	#1055	C₂²×C₈	1	2	yes	—
179	#1055	C₂²×C₈	1	2	yes	—
180	C₂×#135	—	13	1	no	—
181	C₂×#131	—	13	1	no	—
182	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
183	#504	—	13	4	no	—
184	#504	—	13	4	no	—
185	#504	—	26	2	no	—
186	#504	—	26	2	no	—
187	C₂×M₃₂	—	13	1	no	—
188	C₂×M₃₂	—	26	1	no	—
189	C₂×M₃₂	—	26	1	no	—
190	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
191	C₂²×C₁₆	—	13	1	no	—
192	C₂²×C₁₆	—	26	1	no	—
193	C₂²×C₁₆	—	26	1	no	—
194	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
195	#443	—	13	4	no	—
196	#443	—	13	4	no	—
197	C₂×#108	—	13	1	no	—
198	C₂×#104	—	13	1	no	—
199	C₂×#104	—	13	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	#364	—	13	4	no	—
202	#350	—	13	4	no	—
203	#350	C₂×D₁₃	1	4	yes	complemented
204	C₄×C₁₆	—	13	4	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	—	2	2	no	—
207	C₂²×C₁₃	—	4	2	no	—
208	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
209	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
210	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
211	C₂×D₁₃	—	2	2	no	—
212	C₂×D₁₃	—	4	2	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
215	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
216	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
218	Dic₁₃	—	2	4	no	—
219	Dic₁₃	—	2	2	no	—
220	C₂×#41	—	13	1	no	—
221	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
222	#135	—	13	4	no	—
223	#135	—	13	4	no	—
224	C₂×Q₁₆	—	13	1	no	—
225	C₂×SD₁₆	—	13	1	no	—
226	C₂×SD₁₆	—	13	1	no	—
227	C₂×D₈	—	13	1	no	—
228	#131	—	2	1	no	—
229	#131	—	2	1	no	—
230	#131	—	13	2	no	—
231	#131	—	13	2	no	—
232	#131	—	26	1	no	—
233	#131	—	26	1	no	—
234	#131	C₄×D₁₃	1	2	yes	—
235	#131	C₂²×D₁₃	1	2	yes	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	13	1	no	—
238	C₂×M₁₆	—	26	1	no	—
239	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	26	1	no	—
243	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×C₄²	—	13	1	no	—
245	M₃₂	—	2	1	no	—
246	M₃₂	—	13	2	no	—
247	M₃₂	—	26	4	no	—
248	M₃₂	—	26	4	no	—
249	M₃₂	—	26	1	no	—
250	M₃₂	C₂²×D₁₃	1	2	yes	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	13	2	no	—
255	C₂×C₁₆	—	13	1	no	—
256	C₂×C₁₆	—	13	1	no	—
257	C₂×C₁₆	—	13	2	no	—
258	C₂×C₁₆	—	26	4	no	—
259	C₂×C₁₆	—	26	4	no	—
260	C₂×C₁₆	—	26	1	no	—
261	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
262	#108	—	13	4	no	—
263	#104	—	13	4	no	—
264	#104	—	13	4	no	—
265	C₄×C₈	—	13	2	no	—
266	C₄×C₈	—	13	2	no	—
267	C₂×C₁₃	—	2	1	no	—
268	C₂×C₁₃	—	2	1	no	—
269	C₂×C₁₃	—	4	2	no	—
270	C₂×C₁₃	#2170	1	2	yes	complemented
271	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
272	D₁₃	—	4	2	no	—
273	#41	—	2	1	no	—
274	#41	—	2	1	no	—
275	#41	—	13	2	no	—
276	#41	—	13	2	no	—
277	#41	—	26	1	no	—
278	#41	—	26	1	no	—
279	#41	C₈×D₁₃	1	2	yes	—
280	#41	C₂×C₄×D₁₃	1	2	yes	—
281	C₂×Q₈	—	13	1	no	—
282	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	13	1	no	—
285	C₂×D₄	—	26	1	no	—
286	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	26	1	no	—
289	C₂²×C₄	—	26	1	no	—
290	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
291	Q₁₆	—	13	4	no	—
292	SD₁₆	—	13	4	no	—
293	SD₁₆	—	13	4	no	—
294	D₈	—	13	4	no	—
295	M₁₆	—	2	1	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	13	2	no	—
299	M₁₆	—	26	1	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	2	no	—
302	M₁₆	C₂×C₄×D₁₃	1	2	yes	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	2	no	—
309	C₂×C₈	—	13	2	no	—
310	C₂×C₈	—	26	1	no	—
311	C₂×C₈	—	26	2	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
315	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
316	C₂×C₈	#6129	1	1	yes	characteristic
317	C₂×C₈	#6129	1	1	yes	characteristic
318	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
319	C₄²	—	13	2	no	—
320	C₄²	—	26	1	no	—
321	C₁₆	—	2	2	no	—
322	C₁₆	—	2	2	no	—
323	C₁₆	—	13	2	no	—
324	C₁₆	—	13	2	no	—
325	C₁₆	—	26	2	no	—
326	C₁₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
327	C₂³	—	2	1	no	—
328	C₂³	—	26	1	no	—
329	C₂³	#88161	1	1	yes	characteristic
330	Q₈	—	2	1	no	—
331	Q₈	—	13	2	no	—
332	Q₈	—	26	1	no	—
333	Q₈	C₂×C₈×D₁₃	1	2	yes	—
334	D₄	—	2	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	2	1	no	—
337	D₄	—	13	2	no	—
338	D₄	—	26	2	no	—
339	D₄	—	26	2	no	—
340	D₄	—	26	1	no	—
341	D₄	C₂×C₈×D₁₃	1	2	yes	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	26	4	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	—	26	1	no	—
348	C₂×C₄	—	26	2	no	—
349	C₂×C₄	—	26	2	no	—
350	C₂×C₄	#88135	1	1	yes	characteristic
351	C₂×C₄	#88135	1	1	yes	characteristic
352	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
353	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
354	C₂×C₄	#88161	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	13	2	no	—
358	C₈	—	13	2	no	—
359	C₈	—	26	2	no	—
360	C₈	#88135	1	1	yes	characteristic
361	C₈	#88135	1	1	yes	characteristic
362	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
363	C₈	C₂×#6129	1	1	yes	characteristic
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	26	2	no	—
368	C₂²	—	52	2	no	—
369	C₂²	#11718386	1	1	yes	characteristic
370	C₂²	#11718386	1	1	yes	characteristic
371	C₂²	C₂×#88161	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	26	4	no	—
374	C₄	—	26	2	no	—
375	C₄	#11718386	1	1	yes	characteristic
376	C₄	#11718386	1	1	yes	characteristic
377	C₄	C₂×#88135	1	1	yes	characteristic
378	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	4	2	no	—
382	C₂	—	52	2	no	—
383	C₂	#13116670	1	2	yes	direct product factor
384	C₂	C₂×#11718386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#350	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×#88204	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#350	D₁₃
hypocenter	C₁₃	C₂×#2170
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂×#11718568	C₂	semidirect
2. C₂×#11718801	C₂	semidirect
4. #13116670	C₂	direct (N × H)
5. C₂×#11718103	C₂	semidirect
6. C₂×#11718100	C₂	semidirect
7. C₂×C₁₆×Dic₁₃	C₂	semidirect
8. C₂×C₁₃×#350	C₂	semidirect
18. #11718801	C₂²	semidirect
27. #11718568	C₂²	semidirect
30. C₁₃×#350	C₂²	semidirect
31. #11718103	C₂²	semidirect
32. #11718100	C₂²	semidirect
33. C₁₆×Dic₁₃	C₂²	semidirect
141. C₂×#350	D₁₃	semidirect
203. #350	C₂×D₁₃	semidirect
270. C₂×C₁₃	#2170	semidirect
326. C₁₃	C₂×#2170	semidirect
383. C₂	#13116670	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×#11718568	2
C₂×#11718801	2
C₂×#11718805	2
#13116670	2
C₂×#11718103	2
C₂×#11718100	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×#350	2
C₂×#2170	13

Subgroups — C2×#13116670 #16016144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116670 #16016144