Subgroups — C₂×#13131224 #16807167

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×#11719096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11719162	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131224	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718233	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718270	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718304	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718485	C₂	1	1	yes	maximal, characteristic
9	C₂×#88204	C₄	1	1	yes	characteristic
10	C₂×#88202	C₂²	1	1	yes	characteristic
11	C₂×#88161	C₂²	1	1	yes	characteristic
12	#11719162	—	2	2	no	—
13	#11719162	C₂²	1	4	yes	complemented
14	C₂×#88106	—	2	1	no	—
15	C₂×#88106	C₂²	1	1	yes	characteristic
16	C₂×#88106	C₂²	1	1	yes	characteristic
17	C₂²×#6092	—	2	1	no	—
18	C₂²×#6092	C₂²	1	1	yes	characteristic
19	C₂²×#6092	C₂²	1	1	yes	characteristic
20	#11719096	—	2	2	no	—
21	#11719096	C₂²	1	4	yes	complemented
22	C₂×#88098	—	2	1	no	—
23	C₂×D₁₃⋊C₁₆	—	2	1	no	—
24	C₂×C₁₃×#104	C₄	1	1	yes	characteristic
25	C₂×#88062	C₄	1	1	yes	characteristic
26	C₂×#88045	C₄	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718485	C₄	1	4	yes	—
29	#11718485	C₂²	1	4	yes	—
30	#11718304	C₂²	1	4	yes	complemented
31	#11718270	C₂²	1	4	yes	complemented
32	#11718233	C₂²	1	4	yes	complemented
33	C₄×#6092	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	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88204	C₈	1	4	yes	—
39	#88204	C₂×C₄	1	4	yes	—
40	#88202	—	2	1	no	—
41	#88202	—	2	1	no	—
42	#88202	C₂×C₄	1	2	yes	—
43	#88202	C₂³	1	2	yes	—
44	C₂×#6107	C₈	1	1	yes	characteristic
45	C₂×#6106	C₈	1	1	yes	characteristic
46	C₂×#6105	C₈	1	1	yes	characteristic
47	C₂×#6104	C₈	1	1	yes	characteristic
48	C₂×#6099	—	2	1	no	—
49	C₂×#6099	C₂×C₄	1	1	yes	characteristic
50	C₂²×#1055	—	2	1	no	—
51	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
52	C₂²×#1055	C₂³	1	1	yes	characteristic
53	#88161	—	2	1	no	—
54	#88161	—	2	1	no	—
55	#88161	C₂×C₄	1	2	yes	—
56	#88161	C₂³	1	2	yes	—
57	C₂×#6094	—	2	1	no	—
58	C₂×C₈×D₁₃	—	2	1	no	—
59	#88106	—	2	1	no	—
60	#88106	—	2	1	no	—
61	#88106	—	2	4	no	—
62	#88106	C₂³	1	2	yes	—
63	#88106	C₂³	1	2	yes	—
64	C₂×#6092	—	2	1	no	—
65	C₂×#6092	—	2	1	no	—
66	C₂×#6092	—	2	1	no	—
67	C₂×#6092	—	2	1	no	—
68	C₂×#6092	—	2	4	no	—
69	C₂×#6092	D₄	1	1	yes	characteristic
70	C₂×#6092	D₄	1	1	yes	characteristic
71	C₂×#6092	D₄	1	2	yes	—
72	C₂×#6092	C₂³	1	2	yes	—
73	C₂×#6092	C₂³	1	2	yes	—
74	#88098	—	2	4	no	—
75	D₁₃⋊C₁₆	—	2	4	no	—
76	C₁₃×#104	C₂×C₄	1	4	yes	—
77	#88062	C₂×C₄	1	4	yes	—
78	#88045	C₂×C₄	1	4	yes	—
79	C₄×#1055	C₂×C₄	1	2	yes	—
80	C₄×#1055	C₂³	1	2	yes	—
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	—	2	1	no	—
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	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	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
105	#6107	C₂×C₈	1	4	yes	—
106	#6106	C₂×C₈	1	4	yes	—
107	#6105	C₂×C₈	1	4	yes	—
108	#6104	C₂×C₈	1	4	yes	—
109	#6099	—	2	1	no	—
110	#6099	—	2	2	no	—
111	#6099	—	2	2	no	—
112	#6099	C₂²×C₄	1	2	yes	—
113	C₂×#1055	—	2	1	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	C₂×C₈	1	1	yes	characteristic
117	C₂×#1055	C₂×C₈	1	1	yes	characteristic
118	C₂×#1055	C₂×C₈	1	2	yes	—
119	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
120	C₂×#1055	C₂²×C₄	1	2	yes	—
121	C₂×#1055	C₂×D₄	1	1	yes	characteristic
122	C₂×#1055	C₂×D₄	1	1	yes	characteristic
123	C₂×#1055	#41	1	1	yes	characteristic
124	C₂×#1055	#41	1	1	yes	characteristic
125	C₂×#1055	C₂⁴	1	1	yes	characteristic
126	#6094	—	2	2	no	—
127	#6094	—	2	2	no	—
128	C₈×D₁₃	—	2	2	no	—
129	C₈×D₁₃	—	2	2	no	—
130	#6092	—	2	2	no	—
131	#6092	—	2	2	no	—
132	#6092	—	2	2	no	—
133	#6092	C₂×D₄	1	2	yes	—
134	#6092	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	—	13	1	no	—
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	4	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	2	2	no	—
164	C₂²⋊D₁₃	—	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	#1055	—	2	2	no	—
174	#1055	C₄×D₄	1	1	yes	characteristic
175	#1055	C₄×D₄	1	1	yes	characteristic
176	#1055	C₂²×C₈	1	2	yes	—
177	#1055	C₂²×C₈	1	2	yes	—
178	#1055	C₂²×D₄	1	1	yes	characteristic
179	#1055	C₂×#41	1	1	yes	characteristic
180	C₂×#135	—	13	1	no	—
181	C₂×#131	—	13	1	no	—
182	C₂×#131	—	13	1	no	—
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	—	13	1	no	—
189	C₂×M₃₂	—	26	1	no	—
190	C₂×M₃₂	—	26	1	no	—
191	C₂²×C₁₆	—	13	1	no	—
192	C₂²×C₁₆	—	13	1	no	—
193	C₂²×C₁₆	—	26	1	no	—
194	C₂²×C₁₆	—	26	1	no	—
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	#261	1	1	yes	characteristic
200	C₂×C₄×C₈	—	13	1	no	—
201	#364	—	13	4	no	—
202	#350	—	13	4	no	—
203	#350	—	13	4	no	—
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	4	no	—
214	C₄×C₁₃	—	2	2	no	—
215	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
216	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
217	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
219	Dic₁₃	—	2	2	no	—
220	C₂×#41	—	13	1	no	—
221	C₂×#41	C₂×#261	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	—	13	2	no	—
229	#131	—	13	2	no	—
230	#131	—	13	2	no	—
231	#131	—	13	2	no	—
232	#131	—	26	1	no	—
233	#131	—	26	1	no	—
234	#131	—	26	1	no	—
235	#131	—	26	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	26	1	no	—
238	C₂×M₁₆	—	26	1	no	—
239	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
240	C₂²×C₈	—	13	1	no	—
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	26	1	no	—
243	C₂²×C₈	—	26	1	no	—
244	C₂×C₄²	C₂×#261	1	1	yes	characteristic
245	M₃₂	—	13	2	no	—
246	M₃₂	—	13	2	no	—
247	M₃₂	—	26	4	no	—
248	M₃₂	—	26	4	no	—
249	M₃₂	—	26	1	no	—
250	M₃₂	—	26	1	no	—
251	C₂×C₁₆	—	13	1	no	—
252	C₂×C₁₆	—	13	1	no	—
253	C₂×C₁₆	—	13	2	no	—
254	C₂×C₁₆	—	13	2	no	—
255	C₂×C₁₆	—	13	2	no	—
256	C₂×C₁₆	—	26	4	no	—
257	C₂×C₁₆	—	26	4	no	—
258	C₂×C₁₆	—	26	1	no	—
259	C₂×C₁₆	—	26	1	no	—
260	C₂×C₁₆	—	26	1	no	—
261	C₂×C₁₆	—	26	1	no	—
262	#108	—	13	4	no	—
263	#104	—	13	4	no	—
264	#104	C₂×#261	1	4	yes	—
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	D₁₃⋊C₈	1	2	yes	—
280	#41	C₂²×#261	1	2	yes	—
281	C₂×Q₈	—	13	1	no	—
282	C₂×Q₈	D₁₃⋊C₈	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	D₁₃⋊C₈	1	1	yes	characteristic
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	26	1	no	—
290	C₂²×C₄	C₂²×#261	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	—	13	2	no	—
297	M₁₆	—	26	2	no	—
298	M₁₆	—	26	2	no	—
299	M₁₆	—	26	2	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	1	no	—
302	M₁₆	C₂²×#261	1	2	yes	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	13	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	2	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	1	no	—
309	C₂×C₈	—	13	1	no	—
310	C₂×C₈	—	13	1	no	—
311	C₂×C₈	—	13	2	no	—
312	C₂×C₈	—	13	1	no	—
313	C₂×C₈	—	13	1	no	—
314	C₂×C₈	—	26	2	no	—
315	C₂×C₈	—	26	2	no	—
316	C₂×C₈	—	26	2	no	—
317	C₂×C₈	—	26	2	no	—
318	C₂×C₈	—	26	1	no	—
319	C₄²	—	2	1	no	—
320	C₄²	C₂²×#261	1	2	yes	—
321	C₁₆	—	13	2	no	—
322	C₁₆	—	13	2	no	—
323	C₁₆	—	26	2	no	—
324	C₁₆	—	26	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	#88238	1	1	yes	characteristic
330	Q₈	—	2	1	no	—
331	Q₈	—	13	2	no	—
332	Q₈	—	26	1	no	—
333	Q₈	C₂×D₁₃⋊C₈	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×D₁₃⋊C₈	1	2	yes	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	26	1	no	—
348	C₂×C₄	—	26	2	no	—
349	C₂×C₄	—	26	2	no	—
350	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
351	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
352	C₂×C₄	#88238	1	1	yes	characteristic
353	C₂×C₄	D₄×#261	1	1	yes	characteristic
354	C₂×C₄	D₄×#261	1	1	yes	characteristic
355	C₂×C₄	C₂³×#261	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	13	2	no	—
358	C₈	—	13	2	no	—
359	C₈	—	13	1	no	—
360	C₈	—	13	1	no	—
361	C₈	—	13	1	no	—
362	C₈	—	13	1	no	—
363	C₈	—	26	2	no	—
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	#11719063	1	1	yes	characteristic
370	C₂²	#11719063	1	1	yes	characteristic
371	C₂²	C₂×#88238	1	1	yes	characteristic
372	C₄	—	2	4	no	—
373	C₄	—	2	2	no	—
374	C₄	—	26	2	no	—
375	C₄	#11719063	1	1	yes	characteristic
376	C₄	#11719063	1	1	yes	characteristic
377	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
378	C₄	C₂×D₄×#261	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	#13131224	1	2	yes	direct product factor
384	C₂	C₂×#11719063	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#104	C₄
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×#88204	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#104	#261
hypocenter	C₁₃	C₂×#2170
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. C₂×#11719162	C₂	semidirect
3. #13131224	C₂	direct (N × H)
4. C₂×#11718233	C₂	semidirect
5. C₂×#11718270	C₂	semidirect
6. C₂×#11718304	C₂	semidirect
7. C₂×C₄×#6092	C₂	semidirect
13. #11719162	C₂²	semidirect
21. #11719096	C₂²	semidirect
30. #11718304	C₂²	semidirect
31. #11718270	C₂²	semidirect
32. #11718233	C₂²	semidirect
33. C₄×#6092	C₂²	semidirect
270. C₂×C₁₃	#2170	semidirect
326. C₁₃	C₂×#2170	semidirect
383. C₂	#13131224	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×#11719096	2
C₂×#11719162	2
#13131224	2
C₂×#11718233	2
C₂×#11718270	2
C₂×#11718304	2
C₂×C₄×#6092	2
C₂×#11718485	2
C₂×#2170	13

Subgroups — C2×#13131224 #16807167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131224 #16807167