Subgroups — C₂×#20243010 #26203991

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689521	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689496	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689496	C₂	1	1	yes	maximal, characteristic
5	C₂×#15689474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15689470	C₂	1	1	yes	maximal, characteristic
7	#20243010	C₂	1	8	yes	maximal, direct product factor
8	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625325	—	2	2	no	—
10	C₂²×#10625325	—	2	2	no	—
11	C₂×#13094024	C₂²	1	1	yes	characteristic
12	C₂×#13094024	C₂²	1	1	yes	characteristic
13	C₂×#13094024	C₂²	1	1	yes	characteristic
14	C₂×#13094021	C₂²	1	1	yes	characteristic
15	#15689528	—	2	2	no	—
16	#15689528	C₂²	1	4	yes	complemented
17	#15689521	—	2	2	no	—
18	#15689521	C₂²	1	4	yes	complemented
19	#15689496	—	2	2	no	—
20	#15689496	—	2	2	no	—
21	#15689496	C₂²	1	4	yes	—
22	#15689496	C₂²	1	4	yes	—
23	#15689474	—	2	2	no	—
24	#15689474	C₂²	1	4	yes	complemented
25	#15689470	—	2	2	no	—
26	#15689470	C₂²	1	4	yes	—
27	C₂×#13093958	—	2	2	no	—
28	C₂×#13093956	—	2	2	no	—
29	C₂×#13093949	—	2	2	no	—
30	C₂×#13093946	—	2	2	no	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#322	C₂²	1	4	yes	complemented
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×Dic₉₄	—	2	1	no	—
37	C₂²×Dic₉₄	—	2	1	no	—
38	C₂²×D₁₈₈	—	2	1	no	—
39	C₂²×D₁₈₈	—	2	1	no	—
40	C₂²×D₁₈₈	—	2	1	no	—
41	C₂×#10625325	—	2	8	no	—
42	C₂×#10625325	—	2	8	no	—
43	C₂×#10625325	—	4	8	no	—
44	C₂×#10625325	—	4	8	no	—
45	#13094024	—	2	1	no	—
46	#13094024	—	2	1	no	—
47	#13094024	—	2	1	no	—
48	#13094024	—	2	1	no	—
49	#13094024	—	2	1	no	—
50	#13094024	—	2	1	no	—
51	#13094024	—	2	1	no	—
52	#13094024	—	2	1	no	—
53	#13094024	—	2	1	no	—
54	#13094024	C₂³	1	2	yes	—
55	#13094024	C₂³	1	2	yes	—
56	#13094024	C₂³	1	2	yes	—
57	#13094021	—	2	1	no	—
58	#13094021	—	2	1	no	—
59	#13094021	—	2	1	no	—
60	#13094021	C₂³	1	2	yes	—
61	C₂×C₄⋊Dic₄₇	—	2	1	no	—
62	C₂×C₄⋊Dic₄₇	—	2	1	no	—
63	C₂×C₄⋊Dic₄₇	—	2	1	no	—
64	C₂×C₄⋊Dic₄₇	—	2	1	no	—
65	C₂×C₄⋊Dic₄₇	—	2	1	no	—
66	C₂×C₄⋊Dic₄₇	—	2	1	no	—
67	#13093958	—	2	8	no	—
68	#13093956	—	2	8	no	—
69	#13093949	—	2	8	no	—
70	#13093946	—	2	8	no	—
71	C₂²×C₈×C₄₇	D₄	1	2	yes	—
72	C₂²×C₈×C₄₇	D₄	1	2	yes	—
73	C₂²×C₈×C₄₇	D₄	1	2	yes	—
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₇	D₄	1	2	yes	—
76	C₄×C₈×C₄₇	D₄	1	2	yes	—
77	C₄×C₈×C₄₇	D₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	2	yes	—
79	C₄×C₈×C₄₇	C₂³	1	2	yes	—
80	C₄×C₈×C₄₇	C₂³	1	2	yes	—
81	C₂³×D₄₇	—	4	1	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	4	1	no	—
86	C₂²×Dic₄₇	—	4	1	no	—
87	C₂²×Dic₄₇	—	4	1	no	—
88	C₂×D₁₈₈	—	2	2	no	—
89	C₂×D₁₈₈	—	2	2	no	—
90	C₂×D₁₈₈	—	2	2	no	—
91	C₂×D₁₈₈	—	4	4	no	—
92	C₂×D₁₈₈	—	4	4	no	—
93	C₂×D₁₈₈	—	4	4	no	—
94	C₂×D₁₈₈	—	4	1	no	—
95	C₂×D₁₈₈	—	4	1	no	—
96	C₂×D₁₈₈	—	4	1	no	—
97	C₂×Dic₉₄	—	2	2	no	—
98	C₂×Dic₉₄	—	2	2	no	—
99	C₂×Dic₉₄	—	2	2	no	—
100	C₂×Dic₉₄	—	4	4	no	—
101	C₂×Dic₉₄	—	4	4	no	—
102	C₂×Dic₉₄	—	4	4	no	—
103	C₂×Dic₉₄	—	4	1	no	—
104	C₂×Dic₉₄	—	4	1	no	—
105	C₂×Dic₉₄	—	4	1	no	—
106	C₂×C₈×C₄₇	—	2	4	no	—
107	C₂×C₈×C₄₇	—	2	4	no	—
108	C₂×C₈×C₄₇	SD₁₆	1	8	yes	complemented
109	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
111	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	C₄⋊Dic₄₇	—	4	1	no	—
123	C₄⋊Dic₄₇	—	4	1	no	—
124	C₄⋊Dic₄₇	—	4	1	no	—
125	C₄⋊Dic₄₇	—	4	1	no	—
126	C₄⋊Dic₄₇	—	4	1	no	—
127	C₄⋊Dic₄₇	—	4	1	no	—
128	#10625325	—	4	16	no	—
129	#10625325	—	4	16	no	—
130	C₂³×C₄₇	#127	1	1	yes	characteristic
131	C₂²×D₄₇	—	4	2	no	—
132	C₂²×D₄₇	—	8	2	no	—
133	C₂²×D₄₇	—	8	2	no	—
134	C₂²×D₄₇	—	8	2	no	—
135	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
139	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
140	C₂×C₄×C₄₇	#137	1	4	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	8	2	no	—
148	C₂×Dic₄₇	—	8	2	no	—
149	C₂×Dic₄₇	—	8	2	no	—
150	D₁₈₈	—	4	4	no	—
151	D₁₈₈	—	4	4	no	—
152	D₁₈₈	—	4	4	no	—
153	D₁₈₈	—	8	2	no	—
154	D₁₈₈	—	8	2	no	—
155	D₁₈₈	—	8	2	no	—
156	Dic₉₄	—	4	4	no	—
157	Dic₉₄	—	4	4	no	—
158	Dic₉₄	—	4	4	no	—
159	Dic₉₄	—	8	2	no	—
160	Dic₉₄	—	8	2	no	—
161	Dic₉₄	—	8	2	no	—
162	C₈×C₄₇	—	2	4	no	—
163	C₈×C₄₇	—	2	4	no	—
164	C₈×C₄₇	C₂×SD₁₆	1	8	yes	complemented
165	C₂×#1714	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	#492	1	2	yes	—
167	C₂²×C₄₇	#496	1	2	yes	—
168	C₂²×C₄₇	#497	1	2	yes	—
169	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	8	2	no	—
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	16	4	no	—
174	C₄×C₄₇	#492	1	2	yes	—
175	C₄×C₄₇	#496	1	2	yes	—
176	C₄×C₄₇	#497	1	2	yes	—
177	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
178	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₄×C₄₇	C₂×#137	1	2	yes	—
180	Dic₄₇	—	8	2	no	—
181	Dic₄₇	—	8	2	no	—
182	Dic₄₇	—	8	2	no	—
183	C₂×#500	—	47	1	no	—
184	C₂×#498	—	47	1	no	—
185	C₂×#492	—	47	1	no	—
186	C₂×#492	—	47	1	no	—
187	C₂×#487	—	47	1	no	—
188	C₂×#486	—	47	1	no	—
189	#1714	—	47	8	no	—
190	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
191	C₂×C₄₇	#1714	1	4	yes	complemented
192	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
193	C₂×C₄₇	C₂×#496	1	1	yes	characteristic
194	C₂×C₄₇	C₂×#497	1	1	yes	characteristic
195	D₄₇	—	16	2	no	—
196	C₂²×SD₁₆	—	94	2	no	—
197	C₂²×SD₁₆	—	94	2	no	—
198	C₂×C₄⋊Q₈	—	47	1	no	—
199	C₂×C₄⋊Q₈	—	47	1	no	—
200	C₂×C₄⋊Q₈	—	47	1	no	—
201	C₂×#127	—	47	1	no	—
202	#500	—	47	4	no	—
203	#500	—	94	2	no	—
204	#498	—	47	4	no	—
205	#498	—	94	2	no	—
206	#492	—	47	4	no	—
207	#492	—	47	4	no	—
208	#492	—	94	2	no	—
209	#492	—	94	2	no	—
210	#487	—	47	4	no	—
211	#487	—	94	2	no	—
212	#486	—	47	4	no	—
213	#486	—	94	2	no	—
214	C₂×#107	—	94	2	no	—
215	C₂×#106	—	94	2	no	—
216	C₂×#103	—	94	2	no	—
217	C₂×#102	—	94	2	no	—
218	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
219	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
221	#322	C₂×D₄₇	1	4	yes	complemented
222	C₄₇	C₂×#1714	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	—	94	1	no	—
224	C₂²×Q₈	—	94	1	no	—
225	C₂²×Q₈	—	94	1	no	—
226	C₂²×D₄	—	94	1	no	—
227	C₂²×D₄	—	94	1	no	—
228	C₂²×D₄	—	94	1	no	—
229	C₂×SD₁₆	—	94	8	no	—
230	C₂×SD₁₆	—	94	8	no	—
231	C₂×SD₁₆	—	188	8	no	—
232	C₂×SD₁₆	—	188	8	no	—
233	C₂²×C₈	D₁₈₈	1	2	yes	—
234	C₂²×C₈	D₁₈₈	1	2	yes	—
235	C₂²×C₈	D₁₈₈	1	2	yes	—
236	C₄⋊Q₈	—	47	2	no	—
237	C₄⋊Q₈	—	47	2	no	—
238	C₄⋊Q₈	—	47	2	no	—
239	C₄⋊Q₈	—	94	1	no	—
240	C₄⋊Q₈	—	94	1	no	—
241	C₄⋊Q₈	—	94	1	no	—
242	C₄⋊Q₈	—	94	1	no	—
243	C₄⋊Q₈	—	94	1	no	—
244	C₄⋊Q₈	—	94	1	no	—
245	C₄⋊Q₈	—	94	1	no	—
246	C₄⋊Q₈	—	94	1	no	—
247	C₄⋊Q₈	—	94	1	no	—
248	#127	—	47	2	no	—
249	#127	—	94	1	no	—
250	#127	—	94	1	no	—
251	#127	—	94	1	no	—
252	C₂×C₄⋊C₄	—	94	1	no	—
253	C₂×C₄⋊C₄	—	94	1	no	—
254	C₂×C₄⋊C₄	—	94	1	no	—
255	C₂×C₄⋊C₄	—	94	1	no	—
256	C₂×C₄⋊C₄	—	94	1	no	—
257	C₂×C₄⋊C₄	—	94	1	no	—
258	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
259	#107	—	94	8	no	—
260	#106	—	94	8	no	—
261	#103	—	94	8	no	—
262	#102	—	94	8	no	—
263	C₄×C₈	D₁₈₈	1	2	yes	—
264	C₄×C₈	D₁₈₈	1	2	yes	—
265	C₄×C₈	D₁₈₈	1	2	yes	—
266	C₄×C₈	C₂²×D₄₇	1	2	yes	—
267	C₄×C₈	C₂²×D₄₇	1	2	yes	—
268	C₄×C₈	C₂²×D₄₇	1	2	yes	—
269	C₂⁴	—	188	1	no	—
270	C₂×Q₈	—	94	2	no	—
271	C₂×Q₈	—	94	2	no	—
272	C₂×Q₈	—	94	2	no	—
273	C₂×Q₈	—	188	4	no	—
274	C₂×Q₈	—	188	4	no	—
275	C₂×Q₈	—	188	4	no	—
276	C₂×Q₈	—	188	1	no	—
277	C₂×Q₈	—	188	1	no	—
278	C₂×Q₈	—	188	1	no	—
279	C₂×D₄	—	94	2	no	—
280	C₂×D₄	—	94	2	no	—
281	C₂×D₄	—	94	2	no	—
282	C₂×D₄	—	188	1	no	—
283	C₂×D₄	—	188	1	no	—
284	C₂×D₄	—	188	1	no	—
285	C₂×D₄	—	188	4	no	—
286	C₂×D₄	—	188	4	no	—
287	C₂×D₄	—	188	4	no	—
288	C₂²×C₄	—	188	1	no	—
289	C₂²×C₄	—	188	1	no	—
290	C₂²×C₄	—	188	1	no	—
291	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
294	SD₁₆	—	188	16	no	—
295	SD₁₆	—	188	16	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	#10625325	1	8	yes	complemented
299	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
300	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
301	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	188	1	no	—
309	C₄⋊C₄	—	188	1	no	—
310	C₄⋊C₄	—	188	1	no	—
311	C₄⋊C₄	—	188	1	no	—
312	C₄⋊C₄	—	188	1	no	—
313	C₄⋊C₄	—	188	1	no	—
314	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
315	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
316	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
317	C₄²	C₂³×D₄₇	1	1	yes	characteristic
318	C₂³	—	376	2	no	—
319	C₂³	—	376	2	no	—
320	C₂³	—	376	2	no	—
321	C₂³	—	188	2	no	—
322	C₂³	#13094021	1	1	yes	characteristic
323	Q₈	—	376	2	no	—
324	Q₈	—	376	2	no	—
325	Q₈	—	376	2	no	—
326	Q₈	—	188	4	no	—
327	Q₈	—	188	4	no	—
328	Q₈	—	188	4	no	—
329	D₄	—	376	2	no	—
330	D₄	—	376	2	no	—
331	D₄	—	376	2	no	—
332	D₄	—	188	4	no	—
333	D₄	—	188	4	no	—
334	D₄	—	188	4	no	—
335	C₂×C₄	—	376	2	no	—
336	C₂×C₄	—	376	2	no	—
337	C₂×C₄	—	376	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	#13094021	1	1	yes	characteristic
342	C₂×C₄	#13094021	1	1	yes	characteristic
343	C₂×C₄	#13094021	1	1	yes	characteristic
344	C₂×C₄	C₂×#10625325	1	4	yes	—
345	C₂×C₄	#13094052	1	4	yes	—
346	C₂×C₄	#13094057	1	4	yes	—
347	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	2	4	no	—
352	C₈	C₂×#10625325	1	8	yes	complemented
353	C₂²	—	376	2	no	—
354	C₂²	—	376	2	no	—
355	C₂²	—	376	2	no	—
356	C₂²	—	752	4	no	—
357	C₂²	#15689496	1	2	yes	—
358	C₂²	#15689512	1	2	yes	—
359	C₂²	#15689517	1	2	yes	—
360	C₂²	C₂×#13094021	1	1	yes	characteristic
361	C₄	—	376	2	no	—
362	C₄	—	376	2	no	—
363	C₄	—	376	2	no	—
364	C₄	#15689496	1	2	yes	—
365	C₄	#15689512	1	2	yes	—
366	C₄	#15689517	1	2	yes	—
367	C₄	C₂²×#10625325	1	2	yes	—
368	C₄	C₂×#13094052	1	2	yes	—
369	C₄	C₂×#13094057	1	2	yes	—
370	C₂	—	752	2	no	—
371	C₂	#20243010	1	4	yes	direct product factor
372	C₂	C₂×#15689496	1	1	yes	characteristic
373	C₂	C₂×#15689512	1	1	yes	characteristic
374	C₂	C₂×#15689517	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	C₂×#1714
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
2. C₂×#15689521	C₂	semidirect
5. C₂×#15689474	C₂	semidirect
7. #20243010	C₂	direct (N × H)
8. C₂×C₄₇×#322	C₂	semidirect
16. #15689528	C₂²	semidirect
18. #15689521	C₂²	semidirect
24. #15689474	C₂²	semidirect
34. C₄₇×#322	C₂²	semidirect
108. C₂×C₈×C₄₇	SD₁₆	semidirect
164. C₈×C₄₇	C₂×SD₁₆	semidirect
190. C₂×#322	D₄₇	semidirect
191. C₂×C₄₇	#1714	semidirect
221. #322	C₂×D₄₇	semidirect
222. C₄₇	C₂×#1714	semidirect
298. C₂×C₈	#10625325	semidirect
352. C₈	C₂×#10625325	semidirect
371. C₂	#20243010	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×#15689528	2
C₂×#15689521	2
C₂×#15689496	2
C₂×#15689474	2
C₂×#15689470	2
#20243010	2
C₂×C₄₇×#322	2
C₂×#1714	47

Subgroups — C2×#20243010 #26203991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243010 #26203991