Subgroups — C₂×#13117057 #16017811

Order 3328.

552 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718529	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718554	C₂	1	1	yes	maximal, characteristic
3	#13117057	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718079	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718109	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718106	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×M₁₆×D₁₃	—	2	1	no	—
10	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
11	C₂²×#6094	C₂²	1	1	yes	characteristic
12	C₂×#88151	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
16	C₂×#88062	—	2	1	no	—
17	C₂×#88059	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	1	yes	characteristic
19	C₂×#88050	—	2	1	no	—
20	#11718554	—	2	2	no	—
21	#11718554	C₂²	1	4	yes	—
22	#11718529	—	2	2	no	—
23	#11718529	C₂²	1	4	yes	—
24	C₁₃×#365	C₂²	1	4	yes	complemented
25	#11718109	C₂²	1	4	yes	complemented
26	#11718106	C₂²	1	4	yes	—
27	#11718088	C₂²	1	4	yes	complemented
28	#11718079	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
30	C₂²×Dic₂₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×#6103	—	4	1	no	—
35	C₂×C₄⋊Dic₁₃	—	4	1	no	—
36	C₂×#6099	—	2	1	no	—
37	C₂²×#1055	D₄	1	1	yes	characteristic
38	M₁₆×D₁₃	—	2	4	no	—
39	M₁₆×D₁₃	—	4	2	no	—
40	C₂×Dic₅₂	—	2	1	no	—
41	C₂×Dic₅₂	—	2	1	no	—
42	C₂×Dic₅₂	—	2	1	no	—
43	C₂×Dic₅₂	—	2	1	no	—
44	C₂×Dic₅₂	—	2	1	no	—
45	C₂×Dic₅₂	C₂³	1	2	yes	—
46	C₂×#6094	—	2	1	no	—
47	C₂×#6094	—	2	1	no	—
48	C₂×#6094	—	2	1	no	—
49	C₂×#6094	—	2	1	no	—
50	C₂×#6094	—	4	1	no	—
51	C₂×#6094	D₄	1	2	yes	—
52	C₂×#6094	C₂³	1	2	yes	—
53	C₂×C₈×D₁₃	—	4	1	no	—
54	#88151	—	2	4	no	—
55	#88151	—	4	2	no	—
56	C₁₃×M₃₂	—	2	1	no	—
57	C₁₃×M₃₂	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	—	2	1	no	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	C₂³	1	2	yes	—
61	C₁₃×#106	—	2	1	no	—
62	C₁₃×#106	C₂³	1	2	yes	—
63	#88062	—	2	4	no	—
64	#88059	—	2	1	no	—
65	#88059	C₂³	1	2	yes	—
66	#88052	—	2	1	no	—
67	#88052	C₂³	1	2	yes	—
68	C₂×#6090	—	2	1	no	—
69	#88050	—	2	4	no	—
70	C₂×#2239	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	4	1	no	—
75	C₂²×Dic₁₃	SD₁₆	1	1	yes	characteristic
76	C₂×C₄×D₁₃	—	2	1	no	—
77	C₂×C₄×D₁₃	—	2	1	no	—
78	C₂×C₄×D₁₃	—	2	1	no	—
79	C₂×C₄×D₁₃	—	2	1	no	—
80	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
81	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
82	C₂×Dic₂₆	—	2	2	no	—
83	C₂×Dic₂₆	—	4	1	no	—
84	C₂×Dic₂₆	—	4	1	no	—
85	C₂×Dic₂₆	—	4	1	no	—
86	C₂×Dic₂₆	—	4	1	no	—
87	C₂×Dic₂₆	—	4	1	no	—
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	#6103	—	4	4	no	—
100	C₄⋊Dic₁₃	—	4	4	no	—
101	#6099	—	2	2	no	—
102	#6099	—	4	1	no	—
103	C₂×#1055	—	2	1	no	—
104	C₂×#1055	—	2	1	no	—
105	C₂×#1055	—	4	1	no	—
106	C₂×#1055	C₂×D₄	1	2	yes	—
107	Dic₅₂	—	2	2	no	—
108	Dic₅₂	—	2	2	no	—
109	Dic₅₂	—	4	1	no	—
110	Dic₅₂	—	4	1	no	—
111	#6094	—	2	2	no	—
112	#6094	—	2	2	no	—
113	#6094	—	2	2	no	—
114	#6094	—	2	2	no	—
115	#6094	—	4	4	no	—
116	C₈×D₁₃	—	4	4	no	—
117	C₁₃×C₁₆	—	2	2	no	—
118	#6090	—	2	2	no	—
119	C₂×#470	—	13	1	no	—
120	C₂×#430	—	13	1	no	—
121	#2239	—	13	8	no	—
122	C₂×#365	—	13	1	no	—
123	C₂×#365	D₁₃	1	1	yes	characteristic, complemented
124	C₂×#362	—	13	1	no	—
125	C₂×#362	—	13	1	no	—
126	C₂×#349	—	13	1	no	—
127	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂²×D₁₃	—	2	1	no	—
129	C₂²×D₁₃	—	2	1	no	—
130	C₂²×D₁₃	—	4	2	no	—
131	C₂²×D₁₃	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	2	1	no	—
141	C₂×Dic₁₃	—	2	1	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	8	1	no	—
144	C₂×Dic₁₃	—	8	1	no	—
145	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
146	C₄×D₁₃	—	2	2	no	—
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	2	2	no	—
150	Dic₂₆	—	4	2	no	—
151	Dic₂₆	—	4	2	no	—
152	Dic₂₆	—	8	1	no	—
153	Dic₂₆	—	8	1	no	—
154	C₈×C₁₃	—	4	2	no	—
155	C₈×C₁₃	#123	1	1	yes	characteristic
156	C₈×C₁₃	#123	1	1	yes	characteristic
157	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	#1055	—	2	2	no	—
160	#1055	—	4	2	no	—
161	C₂²×Q₁₆	—	13	1	no	—
162	C₂²×M₁₆	—	13	1	no	—
163	C₂²×M₁₆	—	26	1	no	—
164	C₂×C₂²⋊Q₈	—	26	1	no	—
165	C₂×M₃₂	—	13	1	no	—
166	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
167	#470	—	13	4	no	—
168	#470	—	26	2	no	—
169	#430	—	13	4	no	—
170	#430	—	26	2	no	—
171	C₂×#108	—	13	1	no	—
172	C₂×#108	—	26	1	no	—
173	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
174	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
175	C₂×#103	—	26	1	no	—
176	#365	—	13	4	no	—
177	#365	C₂×D₁₃	1	4	yes	complemented
178	#362	—	13	4	no	—
179	#362	—	13	4	no	—
180	#349	—	13	4	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	#481	1	1	yes	characteristic
183	C₂²×C₁₃	#481	1	1	yes	characteristic
184	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
185	C₂×D₁₃	—	2	2	no	—
186	C₂×D₁₃	—	4	2	no	—
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	—	8	1	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	#481	1	1	yes	characteristic
191	C₄×C₁₃	#481	1	1	yes	characteristic
192	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
193	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	Dic₁₃	—	2	2	no	—
195	Dic₁₃	—	8	2	no	—
196	C₂²×Q₈	—	26	1	no	—
197	C₂³×C₄	—	13	1	no	—
198	C₂×Q₁₆	—	13	2	no	—
199	C₂×Q₁₆	—	26	1	no	—
200	C₂×Q₁₆	—	26	1	no	—
201	C₂×Q₁₆	—	26	1	no	—
202	C₂×Q₁₆	—	26	1	no	—
203	C₂×Q₁₆	—	26	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	13	2	no	—
206	C₂×M₁₆	—	13	2	no	—
207	C₂×M₁₆	—	26	1	no	—
208	C₂×M₁₆	—	26	1	no	—
209	C₂×M₁₆	—	26	1	no	—
210	C₂×M₁₆	—	26	1	no	—
211	C₂×M₁₆	—	26	4	no	—
212	C₂×M₁₆	—	26	1	no	—
213	C₂×M₁₆	—	52	1	no	—
214	C₂×M₁₆	—	52	2	no	—
215	C₂²×C₈	—	13	1	no	—
216	C₂²×C₈	—	52	1	no	—
217	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₂²⋊Q₈	—	26	4	no	—
219	C₂²⋊Q₈	—	52	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	52	1	no	—
222	C₂×C₂²⋊C₄	—	52	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	13	2	no	—
225	M₃₂	—	26	1	no	—
226	M₃₂	C₂²×D₁₃	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	26	1	no	—
229	#108	—	2	1	no	—
230	#108	—	13	2	no	—
231	#108	—	26	4	no	—
232	#108	—	26	1	no	—
233	#108	C₂²×D₁₃	1	2	yes	—
234	#106	—	2	1	no	—
235	#106	C₂²×D₁₃	1	2	yes	—
236	#103	—	26	4	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	2	1	no	—
239	C₂×C₁₃	#2239	1	2	yes	complemented
240	C₂×C₁₃	C₂×#481	1	1	yes	characteristic
241	D₁₃	—	4	2	no	—
242	C₂⁴	—	13	1	no	—
243	C₂×Q₈	—	26	2	no	—
244	C₂×Q₈	—	52	1	no	—
245	C₂×Q₈	—	52	1	no	—
246	C₂×Q₈	—	52	1	no	—
247	C₂×Q₈	—	52	1	no	—
248	C₂×Q₈	—	52	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	2	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	26	1	no	—
254	C₂²×C₄	—	26	1	no	—
255	C₂²×C₄	—	26	1	no	—
256	C₂²×C₄	—	26	1	no	—
257	C₂²×C₄	—	52	1	no	—
258	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
259	Q₁₆	—	26	2	no	—
260	Q₁₆	—	26	2	no	—
261	Q₁₆	—	52	1	no	—
262	Q₁₆	—	52	1	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	26	2	no	—
266	M₁₆	—	26	2	no	—
267	M₁₆	—	26	2	no	—
268	M₁₆	—	26	2	no	—
269	M₁₆	—	26	2	no	—
270	M₁₆	—	52	4	no	—
271	M₁₆	—	52	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	26	1	no	—
275	C₂×C₈	—	26	1	no	—
276	C₂×C₈	—	52	4	no	—
277	C₂×C₈	—	52	1	no	—
278	C₂×C₈	#6127	1	1	yes	characteristic
279	C₂×C₈	#6127	1	1	yes	characteristic
280	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
281	C₂×C₈	#6131	1	1	yes	characteristic
282	C₂×C₈	#6131	1	1	yes	characteristic
283	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	52	4	no	—
287	C₂²⋊C₄	—	52	4	no	—
288	C₁₆	—	2	2	no	—
289	C₁₆	—	26	2	no	—
290	C₁₃	C₂×#2239	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	13	2	no	—
292	C₂³	—	26	1	no	—
293	C₂³	—	26	1	no	—
294	C₂³	—	52	2	no	—
295	C₂³	SD₁₆×D₁₃	1	1	yes	characteristic
296	Q₈	—	52	2	no	—
297	Q₈	—	52	2	no	—
298	Q₈	—	104	1	no	—
299	Q₈	—	104	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	52	2	no	—
310	C₂×C₄	—	104	1	no	—
311	C₂×C₄	—	104	1	no	—
312	C₂×C₄	#88148	1	1	yes	characteristic
313	C₂×C₄	#88148	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
315	C₂×C₄	#88168	1	1	yes	characteristic
316	C₂×C₄	#88168	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	26	2	no	—
320	C₈	—	52	2	no	—
321	C₈	#88148	1	1	yes	characteristic
322	C₈	#88148	1	1	yes	characteristic
323	C₈	C₂×#6127	1	1	yes	characteristic
324	C₈	C₂×#6131	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	26	2	no	—
327	C₂²	—	52	2	no	—
328	C₂²	—	52	2	no	—
329	C₂²	—	104	1	no	—
330	C₂²	#11718524	1	1	yes	characteristic
331	C₂²	#11718524	1	1	yes	characteristic
332	C₂²	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	26	2	no	—
335	C₄	—	104	2	no	—
336	C₄	#11718524	1	1	yes	characteristic
337	C₄	#11718524	1	1	yes	characteristic
338	C₄	C₂×#88148	1	1	yes	characteristic
339	C₄	C₂×#88168	1	1	yes	characteristic
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	52	2	no	—
343	C₂	#13117057	1	2	yes	direct product factor
344	C₂	C₂×#11718524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718524
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#365	C₂
socle	C₂²×C₁₃	#481
prime radical	C₂³×D₁₃	SD₁₆
hypercenter	C₂×#365	D₁₃
hypocenter	C₁₃	C₂×#2239
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
3. #13117057	C₂	direct (N × H)
4. C₂×#11718079	C₂	semidirect
5. C₂×#11718109	C₂	semidirect
6. C₂×#11718088	C₂	semidirect
8. C₂×C₁₃×#365	C₂	semidirect
24. C₁₃×#365	C₂²	semidirect
25. #11718109	C₂²	semidirect
27. #11718088	C₂²	semidirect
28. #11718079	C₂²	semidirect
123. C₂×#365	D₁₃	semidirect
177. #365	C₂×D₁₃	semidirect
239. C₂×C₁₃	#2239	semidirect
290. C₁₃	C₂×#2239	semidirect
343. C₂	#13117057	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×#11718529	2
C₂×#11718554	2
#13117057	2
C₂×#11718079	2
C₂×#11718109	2
C₂×#11718088	2
C₂×#11718106	2
C₂×C₁₃×#365	2
C₂×#2239	13

Subgroups — C2×#13117057 #16017811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117057 #16017811