Subgroups — C₂×#20253899 #27085880

Order 12032.

1942 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689678	C₂	1	2	yes	maximal
2	#20253899	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689524	C₂	1	4	yes	maximal, complemented
5	C₂×#15689519	C₂	1	2	yes	maximal, complemented
6	C₂×#15689160	C₂	1	4	yes	maximal
7	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
10	#15689678	—	2	8	no	—
11	#15689678	C₂²	1	16	yes	—
12	C₂×#13094024	C₂²	1	2	yes	—
13	C₂×#13094015	C₂²	1	4	yes	—
14	C₂×#13094004	—	2	2	no	—
15	C₂×#13094000	—	2	2	no	—
16	C₂×Q₈×Dic₄₇	—	2	2	no	—
17	C₂×#13093989	—	2	2	no	—
18	C₂×#13093982	C₂²	1	4	yes	—
19	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
20	#15689528	C₂²	1	8	yes	complemented
21	#15689524	C₂²	1	32	yes	complemented
22	#15689519	C₂²	1	16	yes	complemented
23	#15689160	C₂²	1	32	yes	—
24	C₂×#13093958	C₂²	1	4	yes	—
25	C₂×#13093955	C₂²	1	8	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
28	C₄×C₈×D₄₇	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#500	C₂²	1	8	yes	complemented
31	C₂×#107×C₄₇	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×Dic₉₄	—	2	2	no	—
34	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂×C₈×D₄₇	Q₈	1	16	yes	complemented
39	C₂×C₈×D₄₇	C₂³	1	8	yes	—
40	#13094024	—	2	4	no	—
41	#13094024	C₂³	1	8	yes	—
42	#13094015	—	2	8	no	—
43	#13094015	C₂³	1	16	yes	—
44	#13094004	—	2	16	no	—
45	#13094000	—	2	16	no	—
46	Q₈×Dic₄₇	—	2	16	no	—
47	#13093989	—	2	16	no	—
48	#13093982	—	2	8	no	—
49	#13093982	C₂³	1	16	yes	—
50	C₂×C₄⋊Dic₄₇	—	2	2	no	—
51	C₂×C₄⋊Dic₄₇	—	2	4	no	—
52	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
53	C₂×Dic₄₇⋊C₄	—	2	4	no	—
54	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
55	C₂×C₄×Dic₄₇	—	2	4	no	—
56	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
57	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
58	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
59	C₄²×D₄₇	D₄	1	4	yes	—
60	C₄²×D₄₇	C₂³	1	4	yes	—
61	#13093958	C₂³	1	16	yes	—
62	#13093955	C₂³	1	32	yes	—
63	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
64	C₈×Dic₄₇	C₂³	1	8	yes	—
65	C₂²×Q₈×C₄₇	—	2	2	no	—
66	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₄₇	—	2	4	no	—
68	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#107×C₄₇	C₂³	1	16	yes	—
73	C₄×C₈×C₄₇	C₂³	1	4	yes	—
74	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₄₇	—	2	8	no	—
76	C₂×Q₈×C₄₇	—	4	8	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	4	no	—
82	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×D₄₇	—	2	4	no	—
86	C₂×C₄×D₄₇	—	2	4	no	—
87	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
88	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
89	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₄₇	C₂×Q₈	1	8	yes	—
91	C₂×C₄×D₄₇	C₂⁴	1	2	yes	—
92	C₂×Dic₉₄	—	2	8	no	—
93	C₂×Dic₉₄	—	4	8	no	—
94	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
95	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₄₇	—	2	8	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
99	C₄²×C₄₇	C₂×D₄	1	2	yes	—
100	C₄²×C₄₇	C₂⁴	1	2	yes	—
101	(C₂×D₄₇)⋊C₄	—	2	16	no	—
102	C₄⋊Dic₄₇	—	2	8	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	—	2	16	no	—
105	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
106	Dic₄₇⋊C₄	—	2	16	no	—
107	C₄×Dic₄₇	—	2	16	no	—
108	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
109	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
110	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
111	C₄×Dic₄₇	C₂⁴	1	2	yes	—
112	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
113	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
114	C₈×D₄₇	C₂×Q₈	1	32	yes	complemented
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₄₇	—	2	2	no	—
117	C₂²×D₄₇	—	2	2	no	—
118	C₂²×D₄₇	C₄⋊Q₈	1	4	yes	—
119	C₂²×D₄₇	C₂²×D₄	1	2	yes	—
120	Q₈×C₄₇	—	4	16	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	4	8	no	—
123	C₂×C₄×C₄₇	#135	1	4	yes	—
124	C₂×C₄×C₄₇	#135	1	4	yes	—
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
129	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	8	no	—
133	C₂×Dic₄₇	—	4	8	no	—
134	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
135	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
136	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
137	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
138	C₄×D₄₇	—	2	8	no	—
139	C₄×D₄₇	—	2	8	no	—
140	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
141	C₄×D₄₇	C₂²×Q₈	1	8	yes	—
142	Dic₉₄	—	4	16	no	—
143	C₈×C₄₇	C₂²×Q₈	1	8	yes	—
144	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
145	C₂×#3160	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
147	C₂²×C₄₇	C₂×#135	1	2	yes	—
148	C₂²×C₄₇	C₂×#135	1	2	yes	—
149	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
150	C₂×D₄₇	—	2	8	no	—
151	C₂×D₄₇	—	2	2	no	—
152	C₂×D₄₇	—	2	2	no	—
153	C₂×D₄₇	C₂×C₄⋊Q₈	1	4	yes	—
154	C₄×C₄₇	—	4	8	no	—
155	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
156	C₄×C₄₇	C₂×#135	1	4	yes	—
157	C₄×C₄₇	C₂×#135	1	4	yes	—
158	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
159	Dic₄₇	—	2	2	no	—
160	Dic₄₇	—	2	2	no	—
161	Dic₄₇	—	4	8	no	—
162	Dic₄₇	C₂×C₄⋊Q₈	1	4	yes	—
163	C₂×#533	—	47	2	no	—
164	#3160	—	47	16	no	—
165	C₂×#500	—	47	1	no	—
166	C₂×#500	D₄₇	1	1	yes	characteristic, complemented
167	C₂×#499	—	47	4	no	—
168	C₂×#498	—	47	2	no	—
169	C₂×#427	—	47	4	no	—
170	C₂²×C₄×C₈	—	47	1	no	—
171	C₂×C₄₇	#3160	1	4	yes	complemented
172	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
173	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
174	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
175	D₄₇	—	2	4	no	—
176	C₂³×C₈	—	47	2	no	—
177	#533	—	47	16	no	—
178	#533	—	94	8	no	—
179	C₂×C₄⋊Q₈	—	47	2	no	—
180	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
181	C₂×#125	—	47	4	no	—
182	C₂×C₂²⋊Q₈	—	94	2	no	—
183	C₂×C₂²⋊Q₈	—	94	2	no	—
184	C₂×C₄×Q₈	—	94	2	no	—
185	C₂×C₄×Q₈	—	94	2	no	—
186	C₂×#117	—	47	4	no	—
187	C₂²×C₄²	—	47	1	no	—
188	#500	—	47	8	no	—
189	#500	C₂×D₄₇	1	8	yes	complemented
190	#499	—	47	32	no	—
191	#498	—	47	16	no	—
192	#427	—	47	32	no	—
193	C₂×#107	—	47	4	no	—
194	C₂×#107	C₂×D₄₇	1	4	yes	—
195	C₂×#106	—	47	8	no	—
196	C₂×C₄×C₈	—	47	2	no	—
197	C₂×C₄×C₈	—	47	1	no	—
198	C₂×C₄×C₈	—	47	8	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
200	C₄₇	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	94	2	no	—
203	C₂³×C₄	—	47	1	no	—
204	C₂³×C₄	—	47	1	no	—
205	C₂³×C₄	—	47	1	no	—
206	C₂²×C₈	—	47	2	no	—
207	C₂²×C₈	—	47	16	no	—
208	C₂²×C₈	—	47	8	no	—
209	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	47	8	no	—
212	C₄⋊Q₈	—	94	4	no	—
213	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	—
214	#125	—	47	16	no	—
215	#125	—	94	8	no	—
216	C₂²⋊Q₈	—	94	16	no	—
217	C₂²⋊Q₈	—	94	16	no	—
218	C₄×Q₈	—	94	16	no	—
219	C₄×Q₈	—	94	16	no	—
220	#117	—	47	16	no	—
221	#117	—	94	8	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	47	4	no	—
224	C₂×C₄⋊C₄	—	94	4	no	—
225	C₂×C₄⋊C₄	—	94	4	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
228	C₂×C₂²⋊C₄	—	94	4	no	—
229	C₂×C₄²	—	47	1	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	—	47	1	no	—
232	C₂×C₄²	—	47	4	no	—
233	C₂×C₄²	—	47	4	no	—
234	C₂×C₄²	—	94	4	no	—
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	#107	—	47	16	no	—
237	#107	C₂²×D₄₇	1	16	yes	—
238	#106	—	47	32	no	—
239	C₄×C₈	—	47	8	no	—
240	C₄×C₈	—	47	4	no	—
241	C₄×C₈	C₂²×D₄₇	1	4	yes	—
242	C₂⁴	—	47	1	no	—
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	4	8	no	—
245	C₂×Q₈	—	94	8	no	—
246	C₂×Q₈	—	188	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	47	1	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	—	47	8	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	1	no	—
253	C₂²×C₄	—	47	1	no	—
254	C₂²×C₄	—	47	4	no	—
255	C₂²×C₄	—	47	4	no	—
256	C₂²×C₄	—	94	4	no	—
257	C₂²×C₄	—	94	4	no	—
258	C₂²×C₄	—	94	4	no	—
259	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	C₂×C₈	—	47	8	no	—
263	C₂×C₈	—	47	4	no	—
264	C₂×C₈	—	47	32	no	—
265	C₂×C₈	Q₈×D₄₇	1	8	yes	—
266	C₂×C₈	C₂³×D₄₇	1	4	yes	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	47	8	no	—
270	C₄⋊C₄	—	94	16	no	—
271	C₄⋊C₄	—	94	16	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	94	8	no	—
274	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
275	C₂²⋊C₄	—	94	16	no	—
276	C₄²	—	47	4	no	—
277	C₄²	—	47	4	no	—
278	C₄²	—	47	2	no	—
279	C₄²	—	47	2	no	—
280	C₄²	—	94	16	no	—
281	C₄²	D₄×D₄₇	1	2	yes	—
282	C₄²	C₂³×D₄₇	1	2	yes	—
283	C₂³	—	47	4	no	—
284	C₂³	—	47	2	no	—
285	C₂³	—	94	2	no	—
286	C₂³	—	94	2	no	—
287	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
288	Q₈	—	4	16	no	—
289	Q₈	—	188	16	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	47	4	no	—
293	C₂×C₄	—	47	2	no	—
294	C₂×C₄	—	47	8	no	—
295	C₂×C₄	—	47	8	no	—
296	C₂×C₄	—	47	2	no	—
297	C₂×C₄	—	47	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	94	8	no	—
301	C₂×C₄	—	94	8	no	—
302	C₂×C₄	—	94	8	no	—
303	C₂×C₄	—	188	8	no	—
304	C₂×C₄	#13094043	1	4	yes	—
305	C₂×C₄	#13094047	1	4	yes	—
306	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
308	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
310	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
311	C₈	—	47	8	no	—
312	C₈	C₂×Q₈×D₄₇	1	8	yes	—
313	C₂²	—	47	4	no	—
314	C₂²	—	94	2	no	—
315	C₂²	—	94	2	no	—
316	C₂²	—	94	8	no	—
317	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
318	C₂²	C₂×#13094043	1	2	yes	—
319	C₂²	C₂×#13094047	1	2	yes	—
320	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
321	C₄	—	4	8	no	—
322	C₄	—	47	4	no	—
323	C₄	—	94	2	no	—
324	C₄	—	94	2	no	—
325	C₄	—	188	8	no	—
326	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
327	C₄	C₂×#13094043	1	4	yes	—
328	C₄	C₂×#13094047	1	4	yes	—
329	C₄	C₂²×Q₈×D₄₇	1	2	yes	—
330	C₂	—	94	4	no	—
331	C₂	#20253899	1	4	yes	direct product factor
332	C₂	C₂×C₄⋊Q₈×D₄₇	1	1	yes	characteristic
333	C₂	C₂²×#13094043	1	1	yes	characteristic
334	C₂	C₂²×#13094047	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	C₂×#3160
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #20253899	C₂	direct (N × H)
3. C₂×#15689528	C₂	semidirect
4. C₂×#15689524	C₂	semidirect
5. C₂×#15689519	C₂	semidirect
8. C₂×C₄₇×#500	C₂	semidirect
20. #15689528	C₂²	semidirect
21. #15689524	C₂²	semidirect
22. #15689519	C₂²	semidirect
30. C₄₇×#500	C₂²	semidirect
38. C₂×C₈×D₄₇	Q₈	semidirect
114. C₈×D₄₇	C₂×Q₈	semidirect
166. C₂×#500	D₄₇	semidirect
171. C₂×C₄₇	#3160	semidirect
189. #500	C₂×D₄₇	semidirect
200. C₄₇	C₂×#3160	semidirect
331. C₂	#20253899	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×#15689678	2
#20253899	2
C₂×#15689528	2
C₂×#15689524	2
C₂×#15689519	2
C₂×#15689160	2
C₂×C₄×C₈×D₄₇	2
C₂×C₄₇×#500	2
C₂×#3160	47

Subgroups — C2×#20253899 #27085880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253899 #27085880