Subgroups — C₂×#13116732 #16016358

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718808	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718378	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116732	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718079	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718080	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718058	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	—	2	1	no	—
11	C₂×#88158	C₂²	1	1	yes	characteristic
12	C₂×#88155	—	2	1	no	—
13	C₁₃×#442	—	2	2	no	—
14	C₁₃×#442	C₂²	1	4	yes	complemented
15	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#102	—	2	1	no	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×#88067	C₄	1	2	yes	complemented
19	#11718808	C₄	1	4	yes	complemented
20	#11718808	C₂²	1	4	yes	—
21	C₂×#88066	C₄	1	2	yes	complemented
22	C₂×#88059	C₂²	1	1	yes	characteristic
23	C₂×#88052	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂×#88045	—	2	1	no	—
26	#11718378	—	2	2	no	—
27	#11718378	C₂²	1	4	yes	complemented
28	#11718155	C₂²	1	4	yes	complemented
29	#11718080	C₂²	1	4	yes	complemented
30	#11718079	C₂²	1	4	yes	complemented
31	#11718058	C₂²	1	4	yes	complemented
32	C₂²×D₄×C₁₃	—	2	1	no	—
33	C₂³×C₄×C₁₃	—	4	1	no	—
34	C₂×#6127	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₁₃	—	2	4	no	—
44	C₄×D₄×C₁₃	—	4	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₂×#6099	—	2	1	no	—
49	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic, complemented
50	#88158	—	2	1	no	—
51	#88158	—	2	1	no	—
52	#88158	C₂×C₄	1	2	yes	complemented
53	#88158	C₂³	1	2	yes	—
54	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂×#6095	—	2	1	no	—
56	#88155	—	2	4	no	—
57	#88155	—	4	2	no	—
58	C₂×#6094	—	4	1	no	—
59	C₂×C₈×D₁₃	—	4	1	no	—
60	C₁₃×#106	—	2	1	no	—
61	C₁₃×#106	C₂³	1	2	yes	—
62	C₁₃×#102	—	2	4	no	—
63	C₁₃×#97	D₄	1	2	yes	—
64	C₁₃×#97	C₂³	1	2	yes	—
65	#88067	C₂×C₄	1	8	yes	complemented
66	#88066	C₂×C₄	1	8	yes	complemented
67	#88059	—	2	1	no	—
68	#88059	C₂³	1	2	yes	—
69	#88052	—	2	1	no	—
70	#88052	C₂×C₄	1	2	yes	complemented
71	#88052	C₂³	1	2	yes	—
72	#88052	C₂³	1	2	yes	—
73	C₂×#6090	—	2	1	no	—
74	C₂×#6090	C₂×C₄	1	2	yes	complemented
75	#88045	—	2	4	no	—
76	C₂×AGL(1,ℤ/16)	—	13	1	no	maximal, Sylow
77	C₂⁴×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	4	4	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	8	4	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₂²⋊D₁₃	—	4	1	no	—
90	#6127	—	2	2	no	—
91	#6127	—	2	2	no	—
92	#6127	—	4	1	no	—
93	#6127	—	4	1	no	—
94	C₂×D₅₂	—	2	1	no	—
95	C₂×C₄×D₁₃	—	4	1	no	—
96	C₂×Dic₂₆	—	2	1	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	—	2	2	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₁₃×D₈	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
108	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	4	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	#6099	—	2	2	no	—
116	#6099	—	4	1	no	—
117	C₂×#1055	—	4	1	no	—
118	Dic₅₂	—	2	1	no	—
119	Dic₅₂	C₂²×C₄	1	2	yes	complemented
120	D₁₀₄	—	2	1	no	—
121	D₁₀₄	C₂²×C₄	1	2	yes	complemented
122	#6095	—	2	2	no	—
123	#6095	—	2	2	no	—
124	#6094	—	4	4	no	—
125	C₈×D₁₃	—	4	4	no	—
126	#6090	—	2	2	no	—
127	#6090	C₂²×C₄	1	4	yes	complemented
128	C₂×#509	—	13	1	no	—
129	C₂×#444	—	13	1	no	—
130	C₂×#442	D₁₃	1	1	yes	characteristic, complemented
131	AGL(1,ℤ/16)	—	13	8	no	—
132	C₂×#365	—	13	1	no	—
133	C₂×#361	—	13	1	no	—
134	C₂×#360	—	13	1	no	—
135	C₂×#347	—	13	1	no	—
136	C₂³×C₁₃	—	4	2	no	—
137	C₂³×C₁₃	—	4	2	no	—
138	C₂³×C₁₃	—	4	2	no	—
139	C₂³×C₁₃	—	8	2	no	—
140	C₂³×C₁₃	—	8	2	no	—
141	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂²×D₁₃	—	4	1	no	—
143	D₄×C₁₃	—	2	4	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	8	8	no	—
151	C₂×C₄×C₁₃	—	8	2	no	—
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂²⋊D₁₃	—	4	2	no	—
159	C₂²⋊D₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	4	1	no	—
161	D₅₂	—	2	2	no	—
162	D₅₂	—	4	1	no	—
163	C₄×D₁₃	—	4	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	4	1	no	—
167	C₈×C₁₃	—	2	2	no	—
168	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
169	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
171	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
172	#1055	—	4	2	no	—
173	C₂×#135	—	13	1	no	—
174	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
175	C₂×#131	—	26	1	no	—
176	C₂×C₄×D₄	—	2	1	no	—
177	#509	—	13	4	no	—
178	#509	—	13	4	no	—
179	C₂×SD₃₂	—	13	2	no	—
180	C₂×D₁₆	—	13	2	no	—
181	C₂×M₃₂	—	13	1	no	—
182	C₂×M₃₂	—	13	1	no	—
183	#444	—	13	4	no	—
184	#444	—	26	2	no	—
185	#442	—	2	2	no	—
186	#442	C₂×D₁₃	1	4	yes	complemented
187	C₂×#108	—	13	1	no	—
188	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
189	C₂×#104	—	26	1	no	—
190	C₂×#102	—	2	1	no	—
191	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
192	#365	—	13	4	no	—
193	#361	—	13	4	no	—
194	#360	—	13	4	no	—
195	#347	—	13	4	no	—
196	C₂²×C₁₃	—	2	2	no	—
197	C₂²×C₁₃	—	4	4	no	—
198	C₂²×C₁₃	—	8	2	no	—
199	C₂²×C₁₃	—	8	2	no	—
200	C₂²×C₁₃	—	8	4	no	—
201	C₂²×C₁₃	—	8	4	no	—
202	C₂²×C₁₃	#442	1	1	yes	characteristic
203	C₂²×C₁₃	#442	1	1	yes	characteristic
204	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×D₁₃	—	4	2	no	—
206	C₂×D₁₃	—	8	2	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	#442	1	1	yes	characteristic
210	C₄×C₁₃	#442	1	1	yes	characteristic
211	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
212	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	Dic₁₃	—	4	2	no	—
214	C₂×#41	—	26	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	C₂³×C₄	—	4	1	no	—
217	#135	—	13	2	no	—
218	#135	—	13	2	no	—
219	#135	—	26	1	no	—
220	#135	—	26	1	no	—
221	C₂×Q₁₆	—	13	1	no	—
222	C₂×SD₁₆	—	26	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	—	13	1	no	—
227	C₂×D₈	C₄×D₁₃	1	2	yes	—
228	C₂×D₈	C₄×D₁₃	1	2	yes	—
229	C₂×D₈	C₂²×D₁₃	1	2	yes	—
230	#131	—	26	4	no	—
231	#131	—	52	2	no	—
232	C₂×M₁₆	—	26	1	no	—
233	C₂×M₁₆	—	52	1	no	—
234	C₂²×C₈	—	52	1	no	—
235	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
236	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	4	2	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₂²⋊C₄	—	4	1	no	—
241	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
242	SD₃₂	—	13	8	no	—
243	D₁₆	—	13	8	no	—
244	M₃₂	—	13	2	no	—
245	M₃₂	—	13	2	no	—
246	M₃₂	—	13	2	no	—
247	M₃₂	—	26	1	no	—
248	C₂×C₁₆	—	13	2	no	—
249	C₂×C₁₆	—	26	1	no	—
250	#108	—	13	2	no	—
251	#108	—	26	1	no	—
252	#106	—	2	1	no	—
253	#106	C₂²×D₁₃	1	2	yes	—
254	#104	—	26	4	no	—
255	#102	—	2	4	no	—
256	#97	C₂²⋊D₁₃	1	2	yes	—
257	#97	C₂²×D₁₃	1	2	yes	—
258	C₂×C₁₃	—	2	1	no	—
259	C₂×C₁₃	—	2	1	no	—
260	C₂×C₁₃	—	8	4	no	—
261	C₂×C₁₃	AGL(1,ℤ/16)	1	2	yes	complemented
262	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
263	D₁₃	—	8	2	no	—
264	C₂⁴	—	4	1	no	—
265	#41	—	26	2	no	—
266	#41	—	26	2	no	—
267	#41	—	52	1	no	—
268	#41	—	52	1	no	—
269	C₂×Q₈	—	26	1	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	26	1	no	—
277	C₂×D₄	—	52	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	4	4	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	8	4	no	—
282	C₂²×C₄	—	52	1	no	—
283	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
284	Q₁₆	—	13	2	no	—
285	Q₁₆	—	26	1	no	—
286	SD₁₆	—	26	2	no	—
287	SD₁₆	—	26	2	no	—
288	D₈	—	2	2	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	13	2	no	—
292	D₈	—	26	1	no	—
293	D₈	C₂×C₄×D₁₃	1	4	yes	—
294	M₁₆	—	26	2	no	—
295	M₁₆	—	52	4	no	—
296	M₁₆	—	52	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	52	4	no	—
300	C₂×C₈	—	52	1	no	—
301	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₂×C₈	#6127	1	1	yes	characteristic
304	C₂×C₈	#6127	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
306	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	4	4	no	—
311	C₄²	—	2	1	no	—
312	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
313	C₁₆	—	13	4	no	—
314	C₁₆	—	26	2	no	—
315	C₁₃	C₂×AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	8	2	no	—
320	C₂³	—	8	2	no	—
321	C₂³	—	52	1	no	—
322	C₂³	#88186	1	1	yes	characteristic
323	Q₈	—	26	2	no	—
324	Q₈	—	52	1	no	—
325	D₄	—	2	4	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	26	2	no	—
330	D₄	—	52	2	no	—
331	D₄	—	52	2	no	—
332	D₄	—	52	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	8	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	—	52	2	no	—
341	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
342	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
343	C₂×C₄	#88186	1	1	yes	characteristic
344	C₂×C₄	#88203	1	1	yes	characteristic
345	C₂×C₄	#88203	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	52	2	no	—
349	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
350	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
351	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
352	C₈	C₂×#6127	1	1	yes	characteristic
353	C₂²	—	2	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	4	no	—
358	C₂²	—	8	4	no	—
359	C₂²	—	52	2	no	—
360	C₂²	—	104	2	no	—
361	C₂²	#11718675	1	1	yes	characteristic
362	C₂²	#11718675	1	1	yes	characteristic
363	C₂²	C₂×#88186	1	1	yes	characteristic
364	C₄	—	4	2	no	—
365	C₄	—	8	2	no	—
366	C₄	—	52	2	no	—
367	C₄	#11718675	1	1	yes	characteristic
368	C₄	#11718675	1	1	yes	characteristic
369	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
370	C₄	C₂×#88203	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	8	4	no	—
374	C₂	—	104	2	no	—
375	C₂	#13116732	1	2	yes	direct product factor
376	C₂	C₂×#11718675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718675
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#442	C₂
socle	C₂²×C₁₃	#442
prime radical	C₂×#11718808	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#442	D₁₃
hypocenter	C₁₃	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#442

Complements

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

N	H ≅ G/N	type
2. C₂×#11718378	C₂	semidirect
3. #13116732	C₂	direct (N × H)
4. C₂×#11718079	C₂	semidirect
5. C₂×#11718155	C₂	semidirect
6. C₂×#11718080	C₂	semidirect
7. C₂×#11718058	C₂	semidirect
8. C₂×C₁₃×#442	C₂	semidirect
14. C₁₃×#442	C₂²	semidirect
18. C₂×#88067	C₄	semidirect
19. #11718808	C₄	semidirect
21. C₂×#88066	C₄	semidirect
27. #11718378	C₂²	semidirect
28. #11718155	C₂²	semidirect
29. #11718080	C₂²	semidirect
30. #11718079	C₂²	semidirect
31. #11718058	C₂²	semidirect
49. C₂×Dic₅₂	C₂×C₄	semidirect
52. #88158	C₂×C₄	semidirect
54. C₂×D₁₀₄	C₂×C₄	semidirect
65. #88067	C₂×C₄	semidirect
66. #88066	C₂×C₄	semidirect
70. #88052	C₂×C₄	semidirect
74. C₂×#6090	C₂×C₄	semidirect
119. Dic₅₂	C₂²×C₄	semidirect
121. D₁₀₄	C₂²×C₄	semidirect
127. #6090	C₂²×C₄	semidirect
130. C₂×#442	D₁₃	semidirect
186. #442	C₂×D₁₃	semidirect
261. C₂×C₁₃	AGL(1,ℤ/16)	semidirect
315. C₁₃	C₂×AGL(1,ℤ/16)	semidirect
375. C₂	#13116732	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×#11718808	2
C₂×#11718378	2
#13116732	2
C₂×#11718079	2
C₂×#11718155	2
C₂×#11718080	2
C₂×#11718058	2
C₂×C₁₃×#442	2
C₂×AGL(1,ℤ/16)	13

Subgroups — C2×#13116732 #16016358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116732 #16016358