Subgroups — C₂×#20243001 #26203955

Order 12032.

802 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689473	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	#20243001	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×#13094021	C₂²	1	1	yes	characteristic
11	#15689475	—	2	2	no	—
12	#15689475	C₂²	1	4	yes	complemented
13	#15689473	—	2	2	no	—
14	#15689473	C₂²	1	4	yes	complemented
15	#15689471	—	2	2	no	—
16	#15689471	—	2	2	no	—
17	#15689471	C₂²	1	4	yes	—
18	#15689471	C₂²	1	4	yes	—
19	#15689470	—	2	2	no	—
20	#15689470	C₂²	1	4	yes	—
21	C₂×#13093948	—	2	2	no	—
22	C₂×#13093947	—	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	C₈×C₄₇⋊C₈	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	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	C₂³	1	2	yes	—
45	#13094021	—	2	1	no	—
46	#13094021	—	2	1	no	—
47	#13094021	—	2	1	no	—
48	#13094021	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	#13093948	—	2	8	no	—
52	#13093947	—	2	8	no	—
53	#13093946	—	2	8	no	—
54	#13093944	—	2	8	no	—
55	#13093944	—	2	8	no	—
56	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
57	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
58	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
59	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
60	C₂²×Q₈×C₄₇	—	2	1	no	—
61	C₂²×Q₈×C₄₇	—	2	1	no	—
62	C₂²×C₈×C₄₇	D₄	1	2	yes	—
63	C₄⋊Q₈×C₄₇	—	2	1	no	—
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	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
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×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#103×C₄₇	—	2	8	no	—
77	C₄×C₈×C₄₇	D₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	2	yes	—
79	C₂³×D₄₇	—	4	1	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	4	1	no	—
83	C₂×Q₈×C₄₇	—	4	1	no	—
84	C₂×Q₈×C₄₇	—	4	4	no	—
85	C₂×Q₈×C₄₇	—	4	4	no	—
86	C₂²×C₄×C₄₇	—	4	1	no	—
87	C₂²×C₄×C₄₇	—	4	1	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	4	1	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	2	2	no	—
95	C₂×D₁₈₈	—	4	4	no	—
96	C₂×D₁₈₈	—	4	4	no	—
97	C₂×D₁₈₈	—	4	4	no	—
98	C₂×D₁₈₈	—	4	1	no	—
99	C₂×D₁₈₈	—	4	1	no	—
100	C₂×D₁₈₈	—	4	1	no	—
101	C₂×Dic₉₄	—	2	2	no	—
102	C₂×Dic₉₄	—	4	4	no	—
103	C₂×Dic₉₄	—	4	1	no	—
104	C₂×C₈×C₄₇	—	2	4	no	—
105	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
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	—	4	1	no	—
111	C₄⋊C₄×C₄₇	—	4	1	no	—
112	C₄⋊C₄×C₄₇	—	4	1	no	—
113	C₄⋊C₄×C₄₇	—	4	1	no	—
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	4	1	no	—
121	C₄⋊Dic₄₇	—	4	1	no	—
122	C₂×C₄₇⋊C₈	—	2	4	no	—
123	C₂×C₄₇⋊C₈	—	2	4	no	—
124	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
126	C₂³×C₄₇	#127	1	1	yes	characteristic
127	C₂²×D₄₇	—	4	2	no	—
128	C₂²×D₄₇	—	8	2	no	—
129	C₂²×D₄₇	—	8	2	no	—
130	C₂²×D₄₇	—	8	2	no	—
131	Q₈×C₄₇	—	4	4	no	—
132	Q₈×C₄₇	—	4	4	no	—
133	Q₈×C₄₇	—	8	2	no	—
134	Q₈×C₄₇	—	8	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	8	2	no	—
138	C₂×C₄×C₄₇	—	8	2	no	—
139	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#135	1	4	yes	—
143	C₂×C₄×C₄₇	#135	1	4	yes	—
144	C₂×C₄×C₄₇	#135	1	4	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₄₇	—	4	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	—	8	2	no	—
158	C₈×C₄₇	—	2	4	no	—
159	C₄₇⋊C₈	—	2	4	no	—
160	C₄₇⋊C₈	—	2	4	no	—
161	C₂×#1711	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	#495	1	2	yes	—
163	C₂²×C₄₇	#495	1	2	yes	—
164	C₂²×C₄₇	#495	1	2	yes	—
165	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
166	C₂×D₄₇	—	8	2	no	—
167	C₂×D₄₇	—	8	2	no	—
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	16	4	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	#495	1	2	yes	—
173	C₄×C₄₇	#495	1	2	yes	—
174	C₄×C₄₇	#495	1	2	yes	—
175	C₄×C₄₇	C₂×#135	1	2	yes	—
176	C₄×C₄₇	C₂×#135	1	2	yes	—
177	C₄×C₄₇	C₂×#135	1	2	yes	—
178	Dic₄₇	—	8	2	no	—
179	C₂×#487	—	47	1	no	—
180	C₂×#487	—	47	1	no	—
181	C₂×#487	D₄₇	1	1	yes	characteristic, complemented
182	C₂×#486	—	47	1	no	—
183	C₂×#486	—	47	1	no	—
184	C₂×#486	—	47	1	no	—
185	#1711	—	47	8	no	—
186	C₂×C₈²	—	47	1	no	—
187	C₂×C₄₇	#1711	1	4	yes	complemented
188	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
189	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
190	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
191	D₄₇	—	16	2	no	—
192	C₂×C₄⋊Q₈	—	47	1	no	—
193	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
195	C₂×#127	—	47	1	no	—
196	#487	—	2	2	no	—
197	#487	—	47	4	no	—
198	#487	—	47	4	no	—
199	#487	—	94	2	no	—
200	#487	—	94	2	no	—
201	#487	C₂×D₄₇	1	4	yes	complemented
202	#486	—	47	4	no	—
203	#486	—	47	4	no	—
204	#486	—	47	4	no	—
205	#486	—	94	2	no	—
206	#486	—	94	2	no	—
207	#486	—	94	2	no	—
208	C₂×#103	—	2	2	no	—
209	C₂×#103	—	94	2	no	—
210	C₂×#103	—	94	2	no	—
211	C₂×#102	—	94	2	no	—
212	C₂×#102	—	94	2	no	—
213	C₂×#102	—	94	2	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	C₂×C₄×C₈	—	47	1	no	—
216	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
217	C₈²	—	47	4	no	—
218	C₄₇	C₂×#1711	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	94	1	no	—
222	C₂²×D₄	—	94	1	no	—
223	C₂²×D₄	—	94	1	no	—
224	C₂²×D₄	—	94	1	no	—
225	C₂²×C₈	—	47	2	no	—
226	C₂²×C₈	—	47	2	no	—
227	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	47	2	no	—
235	C₄⋊Q₈	—	94	1	no	—
236	C₄⋊Q₈	—	94	1	no	—
237	C₄⋊Q₈	—	94	1	no	—
238	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
239	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
240	#127	—	47	2	no	—
241	#127	—	94	1	no	—
242	#127	—	94	1	no	—
243	#127	—	94	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄⋊C₄	—	94	1	no	—
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	#103	—	2	8	no	—
252	#103	—	94	8	no	—
253	#103	—	94	8	no	—
254	#102	—	94	8	no	—
255	#102	—	94	8	no	—
256	#102	—	94	8	no	—
257	C₄×C₈	—	47	2	no	—
258	C₄×C₈	—	47	2	no	—
259	C₄×C₈	—	47	2	no	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
262	C₄×C₈	C₂²×D₄₇	1	2	yes	—
263	C₂⁴	—	188	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	4	4	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	94	2	no	—
271	C₂×Q₈	—	188	4	no	—
272	C₂×Q₈	—	188	1	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂×D₄	—	94	2	no	—
275	C₂×D₄	—	94	2	no	—
276	C₂×D₄	—	188	1	no	—
277	C₂×D₄	—	188	1	no	—
278	C₂×D₄	—	188	1	no	—
279	C₂×D₄	—	188	4	no	—
280	C₂×D₄	—	188	4	no	—
281	C₂×D₄	—	188	4	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	188	1	no	—
285	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	47	4	no	—
290	C₂×C₈	—	47	4	no	—
291	C₂×C₈	—	94	4	no	—
292	C₂×C₈	—	94	4	no	—
293	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	188	1	no	—
305	C₄⋊C₄	—	188	1	no	—
306	C₄²	D₄×D₄₇	1	1	yes	characteristic
307	C₄²	D₄×D₄₇	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₄²	C₂³×D₄₇	1	1	yes	characteristic
310	C₂³	—	376	2	no	—
311	C₂³	—	376	2	no	—
312	C₂³	—	376	2	no	—
313	C₂³	—	188	2	no	—
314	C₂³	#13094020	1	1	yes	characteristic
315	Q₈	—	4	4	no	—
316	Q₈	—	4	4	no	—
317	Q₈	—	8	2	no	—
318	Q₈	—	8	2	no	—
319	Q₈	—	376	2	no	—
320	Q₈	—	188	4	no	—
321	D₄	—	376	2	no	—
322	D₄	—	376	2	no	—
323	D₄	—	376	2	no	—
324	D₄	—	188	4	no	—
325	D₄	—	188	4	no	—
326	D₄	—	188	4	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	376	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	#13094020	1	1	yes	characteristic
334	C₂×C₄	#13094020	1	1	yes	characteristic
335	C₂×C₄	#13094020	1	1	yes	characteristic
336	C₂×C₄	#13094044	1	4	yes	—
337	C₂×C₄	#13094045	1	4	yes	—
338	C₂×C₄	#13094047	1	4	yes	—
339	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	94	4	no	—
344	C₈	—	94	4	no	—
345	C₂²	—	376	2	no	—
346	C₂²	—	376	2	no	—
347	C₂²	—	376	2	no	—
348	C₂²	—	752	4	no	—
349	C₂²	#15689504	1	2	yes	—
350	C₂²	#15689505	1	2	yes	—
351	C₂²	#15689508	1	2	yes	—
352	C₂²	C₂×#13094020	1	1	yes	characteristic
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	376	2	no	—
356	C₄	#15689504	1	2	yes	—
357	C₄	#15689505	1	2	yes	—
358	C₄	#15689508	1	2	yes	—
359	C₄	C₂×#13094044	1	2	yes	—
360	C₄	C₂×#13094045	1	2	yes	—
361	C₄	C₂×#13094047	1	2	yes	—
362	C₂	—	752	2	no	—
363	C₂	#20243001	1	4	yes	direct product factor
364	C₂	C₂×#15689504	1	1	yes	characteristic
365	C₂	C₂×#15689505	1	1	yes	characteristic
366	C₂	C₂×#15689508	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×#1711
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×#15689475	C₂	semidirect
2. C₂×#15689473	C₂	semidirect
6. #20243001	C₂	direct (N × H)
7. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
8. C₂×C₄₇×#487	C₂	semidirect
12. #15689475	C₂²	semidirect
14. #15689473	C₂²	semidirect
28. C₈×C₄₇⋊C₈	C₂²	semidirect
32. C₄₇×#487	C₂²	semidirect
181. C₂×#487	D₄₇	semidirect
187. C₂×C₄₇	#1711	semidirect
201. #487	C₂×D₄₇	semidirect
218. C₄₇	C₂×#1711	semidirect
363. C₂	#20243001	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×#15689475	2
C₂×#15689473	2
C₂×#15689471	2
C₂×#15689470	2
#20243001	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄₇×#487	2
C₂×#1711	47

Subgroups — C2×#20243001 #26203955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243001 #26203955