Subgroups — C₂×#20243045 #26204039

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689471	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689471	C₂	1	1	yes	maximal, characteristic
5	C₂×#15689470	C₂	1	1	yes	maximal, characteristic
6	#20243045	C₂	1	8	yes	maximal, direct product factor
7	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625337	—	2	2	no	—
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094021	C₂²	1	1	yes	characteristic
12	#15689519	—	2	2	no	—
13	#15689519	C₂²	1	4	yes	complemented
14	#15689502	—	2	2	no	—
15	#15689502	C₂²	1	4	yes	complemented
16	#15689471	—	2	2	no	—
17	#15689471	—	2	2	no	—
18	#15689471	C₂²	1	4	yes	—
19	#15689471	C₂²	1	4	yes	—
20	#15689470	—	2	2	no	—
21	#15689470	C₂²	1	4	yes	—
22	C₂×#13093955	—	2	2	no	—
23	C₂×#13093946	—	2	2	no	—
24	C₂×#13093944	—	2	2	no	—
25	C₂×#13093944	—	2	2	no	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	#15688844	C₂²	1	4	yes	complemented
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#487	—	2	2	no	—
32	C₄₇×#487	C₂²	1	4	yes	complemented
33	C₂×#103×C₄₇	—	2	2	no	—
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×D₁₈₈	—	2	1	no	—
37	C₂²×D₁₈₈	—	2	1	no	—
38	C₂²×D₁₈₈	—	2	1	no	—
39	C₂×#10625337	—	2	8	no	—
40	C₂×#10625337	—	4	8	no	—
41	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
42	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
43	#13094024	—	2	1	no	—
44	#13094024	—	2	1	no	—
45	#13094024	—	2	1	no	—
46	#13094024	C₂³	1	2	yes	—
47	#13094021	—	2	1	no	—
48	#13094021	—	2	1	no	—
49	#13094021	—	2	1	no	—
50	#13094021	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	#13093955	—	2	8	no	—
54	#13093946	—	2	8	no	—
55	#13093944	—	2	8	no	—
56	#13093944	—	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×Q₈×C₄₇	—	2	1	no	—
63	C₂²×C₈×C₄₇	D₄	1	2	yes	—
64	C₄⋊Q₈×C₄₇	—	2	1	no	—
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	C₄⋊Q₈×C₄₇	—	2	1	no	—
67	C₄⋊Q₈×C₄₇	—	2	1	no	—
68	C₄⋊Q₈×C₄₇	—	2	1	no	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
71	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	#103×C₄₇	—	2	8	no	—
78	C₄×C₈×C₄₇	D₄	1	2	yes	—
79	C₄×C₈×C₄₇	C₂³	1	2	yes	—
80	C₂³×D₄₇	—	4	1	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	—	4	1	no	—
84	C₂×Q₈×C₄₇	—	4	1	no	—
85	C₂×Q₈×C₄₇	—	4	4	no	—
86	C₂×Q₈×C₄₇	—	4	4	no	—
87	C₂²×C₄×C₄₇	—	4	1	no	—
88	C₂²×C₄×C₄₇	—	4	1	no	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₇	—	4	1	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	2	2	no	—
95	C₂×D₁₈₈	—	2	2	no	—
96	C₂×D₁₈₈	—	4	4	no	—
97	C₂×D₁₈₈	—	4	4	no	—
98	C₂×D₁₈₈	—	4	4	no	—
99	C₂×D₁₈₈	—	4	1	no	—
100	C₂×D₁₈₈	—	4	1	no	—
101	C₂×D₁₈₈	—	4	1	no	—
102	C₂×Dic₉₄	—	2	2	no	—
103	C₂×Dic₉₄	—	4	4	no	—
104	C₂×Dic₉₄	—	4	1	no	—
105	C₂×C₈×C₄₇	—	2	4	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
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	C₄⋊C₄×C₄₇	—	4	1	no	—
114	C₄⋊C₄×C₄₇	—	4	1	no	—
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
119	#10625337	—	4	16	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	C₂×C₄₇⋊C₈	—	2	4	no	—
125	C₂×C₄₇⋊C₈	—	2	4	no	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
127	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
128	C₂³×C₄₇	#127	1	1	yes	characteristic
129	C₂²×D₄₇	—	4	2	no	—
130	C₂²×D₄₇	—	8	2	no	—
131	C₂²×D₄₇	—	8	2	no	—
132	C₂²×D₄₇	—	8	2	no	—
133	Q₈×C₄₇	—	4	4	no	—
134	Q₈×C₄₇	—	4	4	no	—
135	Q₈×C₄₇	—	8	2	no	—
136	Q₈×C₄₇	—	8	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	8	2	no	—
140	C₂×C₄×C₄₇	—	8	2	no	—
141	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
143	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#135	1	4	yes	—
145	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
146	C₂×C₄×C₄₇	#137	1	4	yes	—
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	8	2	no	—
152	D₁₈₈	—	4	4	no	—
153	D₁₈₈	—	4	4	no	—
154	D₁₈₈	—	4	4	no	—
155	D₁₈₈	—	8	2	no	—
156	D₁₈₈	—	8	2	no	—
157	D₁₈₈	—	8	2	no	—
158	Dic₉₄	—	4	4	no	—
159	Dic₉₄	—	8	2	no	—
160	C₈×C₄₇	—	2	4	no	—
161	C₄₇⋊C₈	—	2	4	no	—
162	C₄₇⋊C₈	—	2	4	no	—
163	C₂×#1720	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	#495	1	2	yes	—
165	C₂²×C₄₇	#496	1	2	yes	—
166	C₂²×C₄₇	#497	1	2	yes	—
167	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	8	2	no	—
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	16	4	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#495	1	2	yes	—
175	C₄×C₄₇	#496	1	2	yes	—
176	C₄×C₄₇	#497	1	2	yes	—
177	C₄×C₄₇	C₂×#135	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	C₂×#498	—	47	1	no	—
182	C₂×#494	—	47	1	no	—
183	C₂×#487	D₄₇	1	1	yes	characteristic, complemented
184	C₂×#486	—	47	1	no	—
185	C₂×#486	—	47	1	no	—
186	C₂×#486	—	47	1	no	—
187	#1720	—	47	8	no	—
188	C₂×#322	—	47	1	no	—
189	C₂×C₄₇	#1720	1	4	yes	complemented
190	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
191	C₂×C₄₇	C₂×#496	1	1	yes	characteristic
192	C₂×C₄₇	C₂×#497	1	1	yes	characteristic
193	D₄₇	—	16	2	no	—
194	C₂²×Q₁₆	—	94	2	no	—
195	C₂×C₄⋊Q₈	—	47	1	no	—
196	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
197	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#127	—	47	1	no	—
199	#498	—	47	4	no	—
200	#498	—	94	2	no	—
201	#494	—	47	4	no	—
202	#494	—	94	2	no	—
203	#487	—	2	2	no	—
204	#487	C₂×D₄₇	1	4	yes	complemented
205	#486	—	47	4	no	—
206	#486	—	47	4	no	—
207	#486	—	47	4	no	—
208	#486	—	94	2	no	—
209	#486	—	94	2	no	—
210	#486	—	94	2	no	—
211	C₂×#106	—	94	2	no	—
212	C₂×#103	—	2	2	no	—
213	C₂×#102	—	94	2	no	—
214	C₂×#102	—	94	2	no	—
215	C₂×#102	—	94	2	no	—
216	C₂×C₄×C₈	—	47	1	no	—
217	C₂×C₄×C₈	—	47	1	no	—
218	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
219	#322	—	47	4	no	—
220	C₄₇	C₂×#1720	1	1	yes	characteristic, complemented, Sylow
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×Q₈	—	2	1	no	—
223	C₂²×Q₈	—	94	1	no	—
224	C₂²×D₄	—	94	1	no	—
225	C₂²×D₄	—	94	1	no	—
226	C₂²×D₄	—	94	1	no	—
227	C₂×Q₁₆	—	94	8	no	—
228	C₂×Q₁₆	—	188	8	no	—
229	C₂²×C₈	—	47	2	no	—
230	C₂²×C₈	—	47	2	no	—
231	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	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	C₂²×D₄₇	1	2	yes	—
243	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
244	#127	—	47	2	no	—
245	#127	—	94	1	no	—
246	#127	—	94	1	no	—
247	#127	—	94	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	94	1	no	—
253	C₂×C₄⋊C₄	—	94	1	no	—
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#106	—	94	8	no	—
256	#103	—	2	8	no	—
257	#102	—	94	8	no	—
258	#102	—	94	8	no	—
259	#102	—	94	8	no	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	—	47	2	no	—
262	C₄×C₈	—	47	2	no	—
263	C₄×C₈	—	47	2	no	—
264	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
265	C₄×C₈	C₂²×D₄₇	1	2	yes	—
266	C₂⁴	—	188	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	4	4	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	94	2	no	—
274	C₂×Q₈	—	188	4	no	—
275	C₂×Q₈	—	188	1	no	—
276	C₂×D₄	—	94	2	no	—
277	C₂×D₄	—	94	2	no	—
278	C₂×D₄	—	94	2	no	—
279	C₂×D₄	—	188	1	no	—
280	C₂×D₄	—	188	1	no	—
281	C₂×D₄	—	188	1	no	—
282	C₂×D₄	—	188	4	no	—
283	C₂×D₄	—	188	4	no	—
284	C₂×D₄	—	188	4	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	188	1	no	—
288	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
291	Q₁₆	—	188	16	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	47	4	no	—
294	C₂×C₈	—	47	4	no	—
295	C₂×C₈	—	94	4	no	—
296	C₂×C₈	—	94	4	no	—
297	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	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	—	188	1	no	—
309	C₄⋊C₄	—	188	1	no	—
310	C₄²	D₄×D₄₇	1	1	yes	characteristic
311	C₄²	D₄×D₄₇	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₄²	C₂³×D₄₇	1	1	yes	characteristic
314	C₂³	—	376	2	no	—
315	C₂³	—	376	2	no	—
316	C₂³	—	376	2	no	—
317	C₂³	—	188	2	no	—
318	C₂³	#13094020	1	1	yes	characteristic
319	Q₈	—	4	4	no	—
320	Q₈	—	4	4	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	376	2	no	—
324	Q₈	—	188	4	no	—
325	D₄	—	376	2	no	—
326	D₄	—	376	2	no	—
327	D₄	—	376	2	no	—
328	D₄	—	188	4	no	—
329	D₄	—	188	4	no	—
330	D₄	—	188	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	376	2	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	#13094020	1	1	yes	characteristic
338	C₂×C₄	#13094020	1	1	yes	characteristic
339	C₂×C₄	#13094020	1	1	yes	characteristic
340	C₂×C₄	#13094044	1	4	yes	—
341	C₂×C₄	#13094051	1	4	yes	—
342	C₂×C₄	#13094056	1	4	yes	—
343	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
344	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	94	4	no	—
348	C₈	—	94	4	no	—
349	C₂²	—	376	2	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	376	2	no	—
352	C₂²	—	752	4	no	—
353	C₂²	#15689505	1	2	yes	—
354	C₂²	#15689509	1	2	yes	—
355	C₂²	#15689516	1	2	yes	—
356	C₂²	C₂×#13094020	1	1	yes	characteristic
357	C₄	—	8	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	376	2	no	—
360	C₄	#15689505	1	2	yes	—
361	C₄	#15689509	1	2	yes	—
362	C₄	#15689516	1	2	yes	—
363	C₄	C₂×#13094044	1	2	yes	—
364	C₄	C₂×#13094051	1	2	yes	—
365	C₄	C₂×#13094056	1	2	yes	—
366	C₂	—	752	2	no	—
367	C₂	#20243045	1	4	yes	direct product factor
368	C₂	C₂×#15689505	1	1	yes	characteristic
369	C₂	C₂×#15689509	1	1	yes	characteristic
370	C₂	C₂×#15689516	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#487	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	C₂×#487	D₄₇
hypocenter	C₄₇	C₂×#1720
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
1. C₂×#15689519	C₂	semidirect
2. C₂×#15689502	C₂	semidirect
6. #20243045	C₂	direct (N × H)
7. C₂×#15688844	C₂	semidirect
8. C₂×C₄₇×#487	C₂	semidirect
13. #15689519	C₂²	semidirect
15. #15689502	C₂²	semidirect
28. #15688844	C₂²	semidirect
32. C₄₇×#487	C₂²	semidirect
183. C₂×#487	D₄₇	semidirect
189. C₂×C₄₇	#1720	semidirect
204. #487	C₂×D₄₇	semidirect
220. C₄₇	C₂×#1720	semidirect
367. C₂	#20243045	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×#15689519	2
C₂×#15689502	2
C₂×#15689471	2
C₂×#15689470	2
#20243045	2
C₂×#15688844	2
C₂×C₄₇×#487	2
C₂×#1720	47

Subgroups — C2×#20243045 #26204039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243045 #26204039