Subgroups — C₂×#20253901 #27085882

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×#15689679	C₂	1	2	yes	maximal
2	#20253901	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689527	C₂	1	2	yes	maximal, complemented
4	C₂×#15689525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15689522	C₂	1	2	yes	maximal, complemented
6	C₂×#15689520	C₂	1	2	yes	maximal, complemented
7	C₂×#15689160	C₂	1	2	yes	maximal
8	C₂×#15689158	C₂	1	2	yes	maximal
9	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
13	#15689679	—	2	8	no	—
14	#15689679	C₂²	1	16	yes	—
15	C₂×#13094023	C₂²	1	4	yes	—
16	C₂×#13094016	C₂²	1	2	yes	—
17	C₂×#13094002	—	2	4	no	—
18	C₂×#13093989	—	2	4	no	—
19	C₂×#13093982	C₂²	1	4	yes	—
20	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
21	#15689527	C₂²	1	16	yes	complemented
22	#15689525	C₂²	1	8	yes	complemented
23	#15689522	C₂²	1	16	yes	complemented
24	#15689520	C₂²	1	16	yes	complemented
25	#15689160	C₂²	1	16	yes	—
26	#15689158	C₂²	1	16	yes	—
27	C₂×#13093958	C₂²	1	2	yes	—
28	C₂×#13093957	C₂²	1	4	yes	—
29	C₂×#13093956	C₂²	1	2	yes	—
30	C₂×#13093955	C₂²	1	4	yes	—
31	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₇	C₂²	1	8	yes	—
35	C₂×#125×C₄₇	C₂²	1	2	yes	—
36	C₄₇×#499	C₂²	1	8	yes	complemented
37	C₂×#107×C₄₇	C₂²	1	2	yes	—
38	C₂×#106×C₄₇	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×Dic₉₄	—	2	4	no	—
41	C₂²×C₄×D₄₇	D₄	1	2	yes	—
42	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₄₇	Q₈	1	8	yes	complemented
46	C₂×C₈×D₄₇	Q₈	1	8	yes	complemented
47	C₂×C₈×D₄₇	C₂³	1	4	yes	—
48	C₂×C₈×D₄₇	C₂³	1	4	yes	—
49	#13094023	—	2	8	no	—
50	#13094023	C₂³	1	16	yes	—
51	#13094016	—	2	4	no	—
52	#13094016	C₂³	1	8	yes	—
53	#13094002	—	2	32	no	—
54	#13093989	—	2	32	no	—
55	#13093982	—	2	8	no	—
56	#13093982	C₂³	1	16	yes	—
57	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
58	C₂×Dic₄₇⋊C₄	—	2	4	no	—
59	C₂×Dic₄₇⋊C₄	—	2	4	no	—
60	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
61	C₂×C₄×Dic₄₇	—	2	4	no	—
62	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
63	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
64	C₄²×D₄₇	D₄	1	4	yes	—
65	C₄²×D₄₇	C₂³	1	4	yes	—
66	#13093958	C₂³	1	8	yes	—
67	#13093957	C₂³	1	16	yes	—
68	#13093956	C₂³	1	8	yes	—
69	#13093955	C₂³	1	16	yes	—
70	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
71	C₈×Dic₄₇	C₂³	1	4	yes	—
72	C₈×Dic₄₇	C₂³	1	4	yes	—
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	#125×C₄₇	—	2	4	no	—
76	#125×C₄₇	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	#107×C₄₇	C₂³	1	8	yes	—
81	#106×C₄₇	C₂³	1	8	yes	—
82	C₄×C₈×C₄₇	C₂³	1	4	yes	—
83	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	—	2	4	no	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	4	no	—
88	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₄₇	—	2	8	no	—
91	C₂×C₄×D₄₇	C₂×D₄	1	8	yes	—
92	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
94	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
95	C₂×C₄×D₄₇	C₂⁴	1	2	yes	—
96	C₂×Dic₉₄	—	2	16	no	—
97	C₂×Dic₉₄	—	4	16	no	—
98	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
99	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
101	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
102	C₄⋊C₄×C₄₇	—	2	16	no	—
103	C₄⋊C₄×C₄₇	—	2	4	no	—
104	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
105	C₄²×C₄₇	C₂×D₄	1	2	yes	—
106	C₄²×C₄₇	C₂⁴	1	2	yes	—
107	(C₂×D₄₇)⋊C₄	—	2	16	no	—
108	C₄⋊Dic₄₇	—	2	4	no	—
109	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
110	Dic₄₇⋊C₄	—	2	16	no	—
111	Dic₄₇⋊C₄	—	2	16	no	—
112	C₄×Dic₄₇	—	2	16	no	—
113	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
114	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
115	C₄×Dic₄₇	C₂⁴	1	2	yes	—
116	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
117	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
118	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
120	C₈×D₄₇	C₂×Q₈	1	16	yes	complemented
121	C₈×D₄₇	C₂×Q₈	1	16	yes	complemented
122	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×D₄₇	—	2	4	no	—
124	C₂²×D₄₇	C₄⋊Q₈	1	4	yes	—
125	C₂²×D₄₇	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	8	no	—
128	C₂×C₄×C₄₇	—	4	8	no	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
134	C₂×Dic₄₇	—	2	8	no	—
135	C₂×Dic₄₇	—	4	8	no	—
136	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
137	C₂×Dic₄₇	#135	1	8	yes	—
138	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
139	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
140	C₄×D₄₇	—	2	16	no	—
141	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
142	C₄×D₄₇	C₂²×Q₈	1	4	yes	—
143	C₄×D₄₇	C₂²×Q₈	1	4	yes	—
144	Dic₉₄	—	4	32	no	—
145	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
146	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
147	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	—
148	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	—
149	C₂×#3160	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
151	C₂²×C₄₇	C₂×#135	1	4	yes	—
152	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
153	C₂×D₄₇	—	2	8	no	—
154	C₂×D₄₇	—	2	4	no	—
155	C₂×D₄₇	C₂×C₄⋊Q₈	1	4	yes	—
156	C₄×C₄₇	—	2	4	no	—
157	C₄×C₄₇	—	4	8	no	—
158	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
159	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
160	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
161	Dic₄₇	—	4	8	no	—
162	Dic₄₇	C₂×C₄⋊Q₈	1	4	yes	—
163	Dic₄₇	C₂×#135	1	8	yes	—
164	C₂×#533	—	47	2	no	—
165	#3160	—	47	16	no	—
166	C₂×#500	—	47	2	no	—
167	C₂×#499	—	47	1	no	—
168	C₂×#499	—	47	2	no	—
169	C₂×#499	D₄₇	1	1	yes	characteristic, complemented
170	C₂×#498	—	47	2	no	—
171	C₂×#427	—	47	2	no	—
172	C₂×#427	—	47	2	no	—
173	C₂²×C₄×C₈	—	47	1	no	—
174	C₂×C₄₇	#3160	1	4	yes	complemented
175	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₄₇	C₂²×#135	1	2	yes	—
177	D₄₇	—	2	4	no	—
178	C₂³×C₈	—	47	1	no	—
179	C₂³×C₈	—	47	1	no	—
180	#533	—	47	16	no	—
181	#533	—	94	8	no	—
182	C₂×C₄⋊Q₈	—	47	4	no	—
183	C₂×#125	—	47	2	no	—
184	C₂×#125	C₂×D₄₇	1	2	yes	—
185	C₂×C₂²⋊Q₈	—	94	4	no	—
186	C₂×C₄×Q₈	—	94	4	no	—
187	C₂×#117	—	47	4	no	—
188	C₂²×C₄²	—	47	1	no	—
189	#500	—	47	16	no	—
190	#499	—	47	16	no	—
191	#499	—	47	8	no	—
192	#499	C₂×D₄₇	1	8	yes	complemented
193	#498	—	47	16	no	—
194	#427	—	47	16	no	—
195	#427	—	47	16	no	—
196	C₂×#107	—	47	2	no	—
197	C₂×#107	—	47	4	no	—
198	C₂×#107	C₂×D₄₇	1	2	yes	—
199	C₂×#106	—	47	2	no	—
200	C₂×#106	—	47	4	no	—
201	C₂×#106	C₂×D₄₇	1	2	yes	—
202	C₂×C₄×C₈	—	47	1	no	—
203	C₂×C₄×C₈	—	47	1	no	—
204	C₂×C₄×C₈	—	47	1	no	—
205	C₂×C₄×C₈	—	47	8	no	—
206	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
207	C₄₇	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	94	4	no	—
209	C₂³×C₄	—	47	1	no	—
210	C₂³×C₄	—	47	2	no	—
211	C₂²×C₈	—	47	1	no	—
212	C₂²×C₈	—	47	1	no	—
213	C₂²×C₈	—	47	8	no	—
214	C₂²×C₈	—	47	8	no	—
215	C₂²×C₈	—	47	4	no	—
216	C₂²×C₈	—	47	4	no	—
217	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	47	16	no	—
220	C₄⋊Q₈	—	94	8	no	—
221	#125	—	2	4	no	—
222	#125	—	47	8	no	—
223	#125	—	94	4	no	—
224	#125	C₂²×D₄₇	1	8	yes	—
225	C₂²⋊Q₈	—	94	32	no	—
226	C₄×Q₈	—	94	32	no	—
227	#117	—	47	16	no	—
228	#117	—	94	8	no	—
229	C₂×C₄⋊C₄	—	2	4	no	—
230	C₂×C₄⋊C₄	—	47	4	no	—
231	C₂×C₄⋊C₄	—	94	4	no	—
232	C₂×C₄⋊C₄	—	94	4	no	—
233	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
234	C₂×C₂²⋊C₄	—	94	4	no	—
235	C₂×C₄²	—	47	2	no	—
236	C₂×C₄²	—	47	1	no	—
237	C₂×C₄²	—	47	4	no	—
238	C₂×C₄²	—	47	4	no	—
239	C₂×C₄²	—	94	4	no	—
240	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
241	#107	—	47	8	no	—
242	#107	—	47	16	no	—
243	#107	C₂²×D₄₇	1	8	yes	—
244	#106	—	47	8	no	—
245	#106	—	47	16	no	—
246	#106	C₂²×D₄₇	1	8	yes	—
247	C₄×C₈	—	47	4	no	—
248	C₄×C₈	—	47	4	no	—
249	C₄×C₈	—	47	4	no	—
250	C₄×C₈	C₂²×D₄₇	1	4	yes	—
251	C₂⁴	—	47	1	no	—
252	C₂×Q₈	—	94	16	no	—
253	C₂×Q₈	—	188	16	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	47	1	no	—
256	C₂²×C₄	—	47	2	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	4	no	—
259	C₂²×C₄	—	47	4	no	—
260	C₂²×C₄	—	47	2	no	—
261	C₂²×C₄	—	47	8	no	—
262	C₂²×C₄	—	94	8	no	—
263	C₂²×C₄	—	94	4	no	—
264	C₂²×C₄	D₄×D₄₇	1	2	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	C₂×C₈	—	47	4	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	47	2	no	—
270	C₂×C₈	—	47	16	no	—
271	C₂×C₈	—	47	16	no	—
272	C₂×C₈	Q₈×D₄₇	1	4	yes	—
273	C₂×C₈	Q₈×D₄₇	1	4	yes	—
274	C₂×C₈	C₂³×D₄₇	1	2	yes	—
275	C₂×C₈	C₂³×D₄₇	1	2	yes	—
276	C₄⋊C₄	—	2	16	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	47	8	no	—
279	C₄⋊C₄	—	94	16	no	—
280	C₄⋊C₄	—	94	16	no	—
281	C₄⋊C₄	—	94	4	no	—
282	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
283	C₂²⋊C₄	—	94	16	no	—
284	C₄²	—	47	8	no	—
285	C₄²	—	47	2	no	—
286	C₄²	—	47	2	no	—
287	C₄²	—	94	16	no	—
288	C₄²	D₄×D₄₇	1	2	yes	—
289	C₄²	C₂³×D₄₇	1	2	yes	—
290	C₂³	—	47	4	no	—
291	C₂³	—	47	2	no	—
292	C₂³	—	94	4	no	—
293	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
294	Q₈	—	188	32	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	47	8	no	—
299	C₂×C₄	—	47	4	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	47	4	no	—
302	C₂×C₄	—	47	8	no	—
303	C₂×C₄	—	47	4	no	—
304	C₂×C₄	—	47	4	no	—
305	C₂×C₄	—	94	16	no	—
306	C₂×C₄	—	94	8	no	—
307	C₂×C₄	—	188	8	no	—
308	C₂×C₄	C₂×D₄×D₄₇	1	4	yes	—
309	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
311	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
312	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
313	C₈	—	47	4	no	—
314	C₈	—	47	4	no	—
315	C₈	C₂×Q₈×D₄₇	1	4	yes	—
316	C₈	C₂×Q₈×D₄₇	1	4	yes	—
317	C₂²	—	47	4	no	—
318	C₂²	—	94	4	no	—
319	C₂²	—	94	8	no	—
320	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
321	C₂²	#135×D₄₇	1	4	yes	—
322	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	4	8	no	—
325	C₄	—	47	8	no	—
326	C₄	—	47	4	no	—
327	C₄	—	188	8	no	—
328	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
329	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
330	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
331	C₂	—	94	4	no	—
332	C₂	#20253901	1	4	yes	direct product factor
333	C₂	C₂×C₄⋊Q₈×D₄₇	1	1	yes	characteristic
334	C₂	C₂×#135×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#499	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#499	D₄₇
hypocenter	C₄₇	C₂×#3160
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #20253901	C₂	direct (N × H)
3. C₂×#15689527	C₂	semidirect
4. C₂×#15689525	C₂	semidirect
5. C₂×#15689522	C₂	semidirect
6. C₂×#15689520	C₂	semidirect
10. C₂×C₄₇×#499	C₂	semidirect
21. #15689527	C₂²	semidirect
22. #15689525	C₂²	semidirect
23. #15689522	C₂²	semidirect
24. #15689520	C₂²	semidirect
36. C₄₇×#499	C₂²	semidirect
45. C₂×C₈×D₄₇	Q₈	semidirect
46. C₂×C₈×D₄₇	Q₈	semidirect
120. C₈×D₄₇	C₂×Q₈	semidirect
121. C₈×D₄₇	C₂×Q₈	semidirect
169. C₂×#499	D₄₇	semidirect
174. C₂×C₄₇	#3160	semidirect
192. #499	C₂×D₄₇	semidirect
207. C₄₇	C₂×#3160	semidirect
332. C₂	#20253901	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×#15689679	2
#20253901	2
C₂×#15689527	2
C₂×#15689525	2
C₂×#15689522	2
C₂×#15689520	2
C₂×#15689160	2
C₂×#15689158	2
C₂×C₄×C₈×D₄₇	2
C₂×C₄₇×#499	2
C₂×#3160	47

Subgroups — C2×#20253901 #27085882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253901 #27085882