Subgroups — C₂×#20243008 #26203989

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×#15689526	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689495	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689474	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15689469	C₂	1	1	yes	maximal, characteristic
6	#20243008	C₂	1	8	yes	maximal, direct product factor
7	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#492	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625335	—	2	2	no	—
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094024	C₂²	1	1	yes	characteristic
12	#15689526	—	2	2	no	—
13	#15689526	C₂²	1	4	yes	complemented
14	#15689519	—	2	2	no	—
15	#15689519	C₂²	1	4	yes	complemented
16	#15689495	—	2	2	no	—
17	#15689495	C₂²	1	4	yes	—
18	#15689474	—	2	2	no	—
19	#15689474	C₂²	1	4	yes	complemented
20	#15689469	—	2	2	no	—
21	#15689469	C₂²	1	4	yes	—
22	C₂×#13093957	—	2	2	no	—
23	C₂×#13093955	—	2	2	no	—
24	C₂×#13093949	—	2	2	no	—
25	C₂×#13093945	—	2	2	no	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	#15688845	C₂²	1	4	yes	complemented
29	C₂²×SD₁₆×C₄₇	—	2	2	no	—
30	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#492	—	2	2	no	—
33	C₄₇×#492	C₂²	1	4	yes	—
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×Dic₉₄	—	2	1	no	—
37	C₂×#10625335	—	2	8	no	—
38	C₂×#10625335	—	4	8	no	—
39	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
40	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
41	#13094024	—	2	1	no	—
42	#13094024	—	2	1	no	—
43	#13094024	—	2	1	no	—
44	#13094024	—	2	1	no	—
45	#13094024	—	2	1	no	—
46	#13094024	—	2	1	no	—
47	#13094024	C₂³	1	2	yes	—
48	#13094024	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	#13093957	—	2	8	no	—
54	#13093955	—	2	8	no	—
55	#13093949	—	2	8	no	—
56	#13093945	—	2	8	no	—
57	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
58	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
59	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
60	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
61	C₂²×Q₈×C₄₇	—	2	1	no	—
62	C₂²×D₄×C₄₇	—	2	1	no	—
63	C₂²×D₄×C₄₇	—	2	1	no	—
64	C₂²×D₄×C₄₇	—	2	1	no	—
65	C₂×SD₁₆×C₄₇	—	2	8	no	—
66	C₂×SD₁₆×C₄₇	—	4	8	no	—
67	C₂²×C₈×C₄₇	D₄	1	2	yes	—
68	C₄⋊Q₈×C₄₇	—	2	1	no	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	—	2	1	no	—
71	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
72	#127×C₄₇	—	2	1	no	—
73	#127×C₄₇	—	2	1	no	—
74	#127×C₄₇	—	2	1	no	—
75	#127×C₄₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	D₄	1	2	yes	—
80	C₄×C₈×C₄₇	C₂³	1	2	yes	—
81	C₂⁴×C₄₇	—	4	1	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	—	4	1	no	—
84	C₂×Q₈×C₄₇	—	4	4	no	—
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	2	2	no	—
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	4	1	no	—
89	C₂×D₄×C₄₇	—	4	1	no	—
90	C₂×D₄×C₄₇	—	4	1	no	—
91	C₂×D₄×C₄₇	—	4	4	no	—
92	C₂×D₄×C₄₇	—	4	4	no	—
93	C₂×D₄×C₄₇	—	4	4	no	—
94	C₂²×C₄×C₄₇	—	4	1	no	—
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×Dic₄₇	—	4	1	no	—
99	C₂²×Dic₄₇	—	4	1	no	—
100	C₂×Dic₉₄	—	2	2	no	—
101	C₂×Dic₉₄	—	2	2	no	—
102	C₂×Dic₉₄	—	4	4	no	—
103	C₂×Dic₉₄	—	4	4	no	—
104	C₂×Dic₉₄	—	4	1	no	—
105	C₂×Dic₉₄	—	4	1	no	—
106	SD₁₆×C₄₇	—	4	16	no	—
107	C₂×C₈×C₄₇	—	2	4	no	—
108	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	4	1	no	—
112	C₄⋊C₄×C₄₇	—	4	1	no	—
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
117	#10625335	—	4	16	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	C₂×C₄₇⋊C₈	—	2	4	no	—
127	C₂×C₄₇⋊C₈	SD₁₆	1	8	yes	complemented
128	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
129	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
130	C₂³×C₄₇	—	4	2	no	—
131	C₂³×C₄₇	—	8	2	no	—
132	C₂³×C₄₇	—	8	2	no	—
133	C₂³×C₄₇	—	8	2	no	—
134	C₂³×C₄₇	#127	1	1	yes	characteristic
135	Q₈×C₄₇	—	4	4	no	—
136	Q₈×C₄₇	—	8	2	no	—
137	D₄×C₄₇	—	4	4	no	—
138	D₄×C₄₇	—	4	4	no	—
139	D₄×C₄₇	—	4	4	no	—
140	D₄×C₄₇	—	8	2	no	—
141	D₄×C₄₇	—	8	2	no	—
142	D₄×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	8	2	no	—
145	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
149	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
150	C₂×C₄×C₄₇	#137	1	4	yes	—
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	—	8	2	no	—
157	C₂×Dic₄₇	—	8	2	no	—
158	Dic₉₄	—	4	4	no	—
159	Dic₉₄	—	4	4	no	—
160	Dic₉₄	—	8	2	no	—
161	Dic₉₄	—	8	2	no	—
162	C₈×C₄₇	—	2	4	no	—
163	C₄₇⋊C₈	—	2	4	no	—
164	C₄₇⋊C₈	C₂×SD₁₆	1	8	yes	complemented
165	C₂×#1714	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	8	2	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	—	8	2	no	—
169	C₂²×C₄₇	—	16	4	no	—
170	C₂²×C₄₇	#492	1	2	yes	—
171	C₂²×C₄₇	#496	1	2	yes	—
172	C₂²×C₄₇	#497	1	2	yes	—
173	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
174	C₄×C₄₇	—	8	2	no	—
175	C₄×C₄₇	#492	1	2	yes	—
176	C₄×C₄₇	#496	1	2	yes	—
177	C₄×C₄₇	#497	1	2	yes	—
178	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
179	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄×C₄₇	C₂×#137	1	2	yes	—
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	D₄₇	1	1	yes	characteristic
187	C₂×#487	—	47	1	no	—
188	C₂×#486	—	47	1	no	—
189	#1714	—	47	8	no	—
190	C₂×#322	—	47	1	no	—
191	C₂×C₄₇	—	16	2	no	—
192	C₂×C₄₇	#1714	1	4	yes	complemented
193	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
194	C₂×C₄₇	C₂×#496	1	1	yes	characteristic
195	C₂×C₄₇	C₂×#497	1	1	yes	characteristic
196	C₂²×SD₁₆	—	2	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	C₂×D₄₇	1	1	yes	characteristic
201	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
202	#500	—	47	4	no	—
203	#500	—	94	2	no	—
204	#498	—	47	4	no	—
205	#498	—	94	2	no	—
206	#492	—	2	2	no	—
207	#492	—	47	4	no	—
208	#492	—	94	2	no	—
209	#492	C₂×D₄₇	1	4	yes	—
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	—	47	1	no	—
219	C₂×C₄×C₈	—	47	1	no	—
220	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
221	#322	—	47	4	no	—
222	C₄₇	C₂×#1714	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	—	94	1	no	—
225	C₂²×Q₈	—	94	1	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂²×D₄	—	2	1	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂×SD₁₆	—	2	8	no	—
230	C₂×SD₁₆	—	4	8	no	—
231	C₂×SD₁₆	—	94	8	no	—
232	C₂×SD₁₆	—	188	8	no	—
233	C₂²×C₈	—	47	2	no	—
234	C₂²×C₈	—	47	2	no	—
235	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	47	2	no	—
240	C₄⋊Q₈	—	47	2	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	C₂²×D₄₇	1	2	yes	—
248	#127	—	2	1	no	—
249	#127	—	2	1	no	—
250	#127	—	2	1	no	—
251	#127	C₂²×D₄₇	1	2	yes	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	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	—	47	2	no	—
264	C₄×C₈	—	47	2	no	—
265	C₄×C₈	—	47	2	no	—
266	C₄×C₈	—	47	2	no	—
267	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
268	C₄×C₈	C₂²×D₄₇	1	2	yes	—
269	C₂⁴	—	4	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	94	2	no	—
274	C₂×Q₈	—	94	2	no	—
275	C₂×Q₈	—	188	4	no	—
276	C₂×Q₈	—	188	4	no	—
277	C₂×Q₈	—	188	1	no	—
278	C₂×Q₈	—	188	1	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	4	no	—
286	C₂×D₄	—	4	4	no	—
287	C₂×D₄	—	4	4	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	188	1	no	—
290	C₂²×C₄	—	188	1	no	—
291	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	SD₁₆	—	4	16	no	—
295	SD₁₆	—	188	16	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	47	8	no	—
298	C₂×C₈	—	47	4	no	—
299	C₂×C₈	—	47	4	no	—
300	C₂×C₈	—	94	4	no	—
301	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	94	2	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	D₄×D₄₇	1	1	yes	characteristic
315	C₄²	D₄×D₄₇	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
317	C₄²	C₂³×D₄₇	1	1	yes	characteristic
318	C₂³	—	4	2	no	—
319	C₂³	—	8	2	no	—
320	C₂³	—	8	2	no	—
321	C₂³	—	8	2	no	—
322	C₂³	#13094020	1	1	yes	characteristic
323	Q₈	—	4	4	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	376	2	no	—
326	Q₈	—	376	2	no	—
327	Q₈	—	188	4	no	—
328	Q₈	—	188	4	no	—
329	D₄	—	4	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	8	2	no	—
333	D₄	—	8	2	no	—
334	D₄	—	8	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	376	2	no	—
338	C₂×C₄	—	376	2	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	#13094020	1	1	yes	characteristic
342	C₂×C₄	#13094020	1	1	yes	characteristic
343	C₂×C₄	#13094020	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
345	C₂×C₄	#13094050	1	4	yes	—
346	C₂×C₄	#13094053	1	4	yes	—
347	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	47	8	no	—
352	C₈	—	94	4	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	16	4	no	—
357	C₂²	#15689494	1	2	yes	—
358	C₂²	#15689510	1	2	yes	—
359	C₂²	#15689514	1	2	yes	—
360	C₂²	C₂×#13094020	1	1	yes	characteristic
361	C₄	—	8	2	no	—
362	C₄	—	376	2	no	—
363	C₄	—	376	2	no	—
364	C₄	#15689494	1	2	yes	—
365	C₄	#15689510	1	2	yes	—
366	C₄	#15689514	1	2	yes	—
367	C₄	C₂×SD₁₆×D₄₇	1	2	yes	—
368	C₄	C₂×#13094050	1	2	yes	—
369	C₄	C₂×#13094053	1	2	yes	—
370	C₂	—	16	2	no	—
371	C₂	#20243008	1	4	yes	direct product factor
372	C₂	C₂×#15689494	1	1	yes	characteristic
373	C₂	C₂×#15689510	1	1	yes	characteristic
374	C₂	C₂×#15689514	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#492	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#127×C₄₇	C₂²
hypercenter	C₂×#492	D₄₇
hypocenter	C₄₇	C₂×#1714
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#492

Complements

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

N	H ≅ G/N	type
1. C₂×#15689526	C₂	semidirect
2. C₂×#15689519	C₂	semidirect
4. C₂×#15689474	C₂	semidirect
6. #20243008	C₂	direct (N × H)
7. C₂×#15688845	C₂	semidirect
13. #15689526	C₂²	semidirect
15. #15689519	C₂²	semidirect
19. #15689474	C₂²	semidirect
28. #15688845	C₂²	semidirect
127. C₂×C₄₇⋊C₈	SD₁₆	semidirect
164. C₄₇⋊C₈	C₂×SD₁₆	semidirect
192. C₂×C₄₇	#1714	semidirect
222. C₄₇	C₂×#1714	semidirect
371. C₂	#20243008	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×#15689526	2
C₂×#15689519	2
C₂×#15689495	2
C₂×#15689474	2
C₂×#15689469	2
#20243008	2
C₂×#15688845	2
C₂×C₄₇×#492	2
C₂×#1714	47

Subgroups — C2×#20243008 #26203989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243008 #26203989