Subgroups — C₂×#20256277 #27096733

Order 12032.

1766 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×#15689852	C₂	1	2	yes	maximal
2	#20256277	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689524	C₂	1	2	yes	maximal
5	C₂×#15689432	C₂	1	4	yes	maximal
6	C₂×#15689408	C₂	1	2	yes	maximal
7	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×Dic₄₇	C₂	1	2	yes	maximal
9	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
10	#15689852	—	2	8	no	—
11	#15689852	C₂²	1	16	yes	—
12	C₂×#13094024	—	2	2	no	—
13	C₂×#13094024	C₂²	1	2	yes	—
14	C₂×#13094015	—	2	4	no	—
15	C₂×#13094015	C₂²	1	4	yes	—
16	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
17	C₂×C₄×Dic₉₄	—	2	2	no	—
18	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
19	#15689528	C₂²	1	8	yes	—
20	#15689524	C₂²	1	16	yes	—
21	#15689432	C₂²	1	32	yes	—
22	#15689408	C₂²	1	16	yes	—
23	C₈×Dic₉₄	C₂²	1	8	yes	—
24	Q₁₆×Dic₄₇	C₂²	1	16	yes	—
25	C₂×#13093958	C₂²	1	2	yes	—
26	C₂×#13093958	C₂²	1	2	yes	—
27	C₂×#13093955	C₂²	1	4	yes	—
28	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
30	C₂×#13093948	C₂²	1	4	yes	—
31	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
34	C₄×Q₁₆×C₄₇	C₂²	1	8	yes	—
35	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#103×C₄₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×Dic₉₄	—	2	2	no	—
39	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	#13094024	—	2	16	no	—
42	#13094024	—	2	4	no	—
43	#13094024	C₂³	1	8	yes	—
44	#13094015	—	2	32	no	—
45	#13094015	—	2	8	no	—
46	#13094015	C₂³	1	16	yes	—
47	Q₈×Dic₄₇	—	2	8	no	—
48	Q₈×Dic₄₇	C₂³	1	16	yes	—
49	C₄×Dic₉₄	—	2	16	no	—
50	C₄×Dic₉₄	D₄	1	4	yes	—
51	C₄×Dic₉₄	C₂³	1	4	yes	—
52	C₂×C₄⋊Dic₄₇	—	2	2	no	—
53	C₂×C₄⋊Dic₄₇	—	2	4	no	—
54	C₂×C₄⋊Dic₄₇	—	2	4	no	—
55	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
57	C₂×Dic₄₇⋊C₄	—	2	4	no	—
58	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
59	C₂×C₄×Dic₄₇	—	2	4	no	—
60	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
61	#13093958	C₂³	1	8	yes	—
62	#13093958	C₂³	1	8	yes	—
63	#13093955	C₂³	1	16	yes	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
65	Dic₄₇⋊C₈	C₂³	1	8	yes	—
66	#13093948	C₂³	1	16	yes	—
67	C₈×Dic₄₇	C₂³	1	8	yes	—
68	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₄₇	Q₈	1	8	yes	complemented
70	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×Q₈×C₄₇	—	2	4	no	—
74	C₄×Q₈×C₄₇	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
77	C₂×C₄²×C₄₇	—	2	2	no	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	#107×C₄₇	C₂³	1	4	yes	—
80	#103×C₄₇	C₂³	1	8	yes	—
81	C₄×C₈×C₄₇	C₂³	1	4	yes	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	—	2	4	no	—
84	C₂×Q₈×C₄₇	C₂×Q₈	1	8	yes	—
85	C₂×Q₈×C₄₇	C₂⁴	1	4	yes	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	2	2	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	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	2	4	no	—
92	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
93	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
94	C₂×Dic₉₄	—	2	4	no	—
95	C₂×Dic₉₄	—	2	8	no	—
96	C₂×Dic₉₄	—	4	8	no	—
97	C₂×Dic₉₄	C₂×D₄	1	4	yes	—
98	Q₁₆×C₄₇	C₂×Q₈	1	16	yes	complemented
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
101	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
102	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
103	C₄⋊C₄×C₄₇	—	2	8	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	C₂⁴	1	4	yes	—
106	C₄²×C₄₇	—	2	8	no	—
107	C₄²×C₄₇	C₂×D₄	1	2	yes	—
108	C₄²×C₄₇	C₂⁴	1	2	yes	—
109	C₄⋊Dic₄₇	—	2	8	no	—
110	C₄⋊Dic₄₇	—	2	4	no	—
111	C₄⋊Dic₄₇	—	2	16	no	—
112	C₄⋊Dic₄₇	—	2	16	no	—
113	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
114	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
115	Dic₄₇⋊C₄	—	2	16	no	—
116	Dic₄₇⋊C₄	C₂×D₄	1	8	yes	—
117	C₄×Dic₄₇	—	2	16	no	—
118	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
119	C₄×Dic₄₇	C₂⁴	1	4	yes	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
122	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₄₇	—	2	8	no	—
124	Q₈×C₄₇	—	2	4	no	—
125	Q₈×C₄₇	C₂²×Q₈	1	8	yes	—
126	C₂×C₄×C₄₇	—	2	8	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	2	4	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	#135	1	4	yes	—
132	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
137	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	8	no	—
141	C₂×Dic₄₇	—	4	8	no	—
142	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
143	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
144	Dic₉₄	—	2	8	no	—
145	Dic₉₄	—	4	16	no	—
146	C₈×C₄₇	—	2	2	no	—
147	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
148	C₄₇⋊C₈	—	2	4	no	—
149	C₂×#3383	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
151	C₂²×C₄₇	C₂×#135	1	2	yes	—
152	C₂²×C₄₇	#578	1	2	yes	—
153	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
154	C₄×C₄₇	—	2	8	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	4	4	no	—
157	C₄×C₄₇	D₄×Q₈	1	2	yes	—
158	C₄×C₄₇	C₂×#135	1	4	yes	—
159	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
160	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₄₇	—	2	4	no	—
162	Dic₄₇	—	2	4	no	—
163	Dic₄₇	—	4	8	no	—
164	C₂×#557	—	47	2	no	—
165	#3383	—	47	16	no	—
166	C₂×#500	—	47	1	no	—
167	C₂×#499	—	47	2	no	—
168	C₂×#479	—	47	4	no	—
169	C₂×#475	—	47	2	no	—
170	C₂×C₈×Q₈	—	47	1	no	—
171	C₂×C₄×Q₁₆	—	47	2	no	—
172	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
173	C₂×C₄₇	#3383	1	4	yes	complemented
174	C₂×C₄₇	C₂×D₄×Q₈	1	1	yes	characteristic
175	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
177	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
178	#557	—	47	16	no	—
179	#557	—	94	8	no	—
180	C₂×C₄⋊Q₈	—	47	2	no	—
181	C₂×C₄⋊Q₈	—	94	2	no	—
182	C₂×#125	—	47	4	no	—
183	C₂×#125	—	94	4	no	—
184	C₂×C₄×Q₈	—	47	1	no	—
185	C₂×C₄×Q₈	—	47	4	no	—
186	C₂×C₄×Q₈	—	94	2	no	—
187	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
188	#500	—	47	8	no	—
189	#499	—	47	16	no	—
190	#479	—	47	32	no	—
191	#475	—	47	16	no	—
192	C₈×Q₈	—	47	8	no	—
193	C₄×Q₁₆	—	47	16	no	—
194	C₄×Q₁₆	C₂×D₄₇	1	8	yes	—
195	C₂×#107	—	47	2	no	—
196	C₂×#107	—	47	2	no	—
197	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#106	—	47	4	no	—
199	C₂×C₄⋊C₈	—	47	2	no	—
200	C₂×C₄⋊C₈	—	47	1	no	—
201	C₂×#103	—	47	4	no	—
202	C₂×#103	C₂×D₄₇	1	2	yes	—
203	C₂×C₄×C₈	—	47	2	no	—
204	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
205	C₄₇	C₂×#3383	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₈	—	47	1	no	—
207	C₂²×Q₈	—	94	2	no	—
208	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
209	C₂×Q₁₆	Dic₉₄	1	8	yes	complemented
210	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₄⋊Q₈	—	47	8	no	—
215	C₄⋊Q₈	—	94	16	no	—
216	C₄⋊Q₈	—	94	4	no	—
217	#125	—	47	16	no	—
218	#125	—	94	8	no	—
219	#125	—	94	32	no	—
220	C₄×Q₈	—	2	4	no	—
221	C₄×Q₈	—	47	4	no	—
222	C₄×Q₈	—	47	4	no	—
223	C₄×Q₈	—	47	16	no	—
224	C₄×Q₈	—	94	16	no	—
225	C₄×Q₈	—	94	8	no	—
226	C₄×Q₈	C₂²×D₄₇	1	8	yes	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	47	1	no	—
229	C₂×C₄⋊C₄	—	47	2	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	—	94	4	no	—
234	C₂×C₄⋊C₄	—	94	2	no	—
235	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	—	47	2	no	—
238	C₂×C₄²	—	94	4	no	—
239	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
240	#107	—	47	8	no	—
241	#107	—	47	8	no	—
242	#107	C₂²×D₄₇	1	4	yes	—
243	#106	—	47	16	no	—
244	C₄⋊C₈	—	47	8	no	—
245	C₄⋊C₈	—	47	4	no	—
246	#103	—	47	16	no	—
247	#103	C₂²×D₄₇	1	8	yes	—
248	C₄×C₈	—	47	8	no	—
249	C₄×C₈	C₂²×D₄₇	1	4	yes	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	47	4	no	—
253	C₂×Q₈	—	94	4	no	—
254	C₂×Q₈	—	94	8	no	—
255	C₂×Q₈	—	188	8	no	—
256	C₂×Q₈	C₂×Dic₉₄	1	8	yes	—
257	C₂×Q₈	C₂³×D₄₇	1	4	yes	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	47	2	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	94	4	no	—
263	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
264	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	Q₁₆	C₂×Dic₉₄	1	16	yes	complemented
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	47	4	no	—
269	C₂×C₈	—	94	4	no	—
270	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
271	C₂×C₈	C₂³×D₄₇	1	2	yes	—
272	C₂×C₈	C₂³×D₄₇	1	2	yes	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	47	8	no	—
276	C₄⋊C₄	—	47	4	no	—
277	C₄⋊C₄	—	47	8	no	—
278	C₄⋊C₄	—	94	16	no	—
279	C₄⋊C₄	—	94	16	no	—
280	C₄⋊C₄	—	94	16	no	—
281	C₄⋊C₄	—	94	4	no	—
282	C₄⋊C₄	—	94	8	no	—
283	C₄⋊C₄	C₂³×D₄₇	1	4	yes	—
284	C₄²	—	2	8	no	—
285	C₄²	—	47	4	no	—
286	C₄²	—	47	4	no	—
287	C₄²	—	94	16	no	—
288	C₄²	D₄×D₄₇	1	2	yes	—
289	C₄²	C₂³×D₄₇	1	2	yes	—
290	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	Q₈	—	2	4	no	—
293	Q₈	—	94	8	no	—
294	Q₈	—	188	16	no	—
295	Q₈	C₂²×Dic₉₄	1	8	yes	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	47	4	no	—
302	C₂×C₄	—	47	4	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	—	94	4	no	—
305	C₂×C₄	—	94	8	no	—
306	C₂×C₄	—	188	8	no	—
307	C₂×C₄	#13094047	1	4	yes	—
308	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
309	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
311	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
312	C₂×C₄	#13094071	1	2	yes	—
313	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	94	4	no	—
316	C₈	C₂²×Dic₉₄	1	4	yes	—
317	C₂²	D₄×Dic₉₄	1	2	yes	—
318	C₂²	C₂×#13094047	1	2	yes	—
319	C₂²	#15689937	1	2	yes	—
320	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
321	C₄	—	2	8	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	4	no	—
324	C₄	—	94	4	no	—
325	C₄	—	94	4	no	—
326	C₄	—	188	8	no	—
327	C₄	D₄×Dic₉₄	1	2	yes	—
328	C₄	C₂×#13094047	1	4	yes	—
329	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
330	C₄	C₂×#13094071	1	1	yes	characteristic
331	C₂	#20256277	1	4	yes	direct product factor
332	C₂	C₂×D₄×Dic₉₄	1	1	yes	characteristic
333	C₂	C₂²×#13094047	1	1	yes	characteristic
334	C₂	C₂×#15689937	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×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#3383
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20256277	C₂	direct (N × H)
69. C₂×Q₁₆×C₄₇	Q₈	semidirect
98. Q₁₆×C₄₇	C₂×Q₈	semidirect
173. C₂×C₄₇	#3383	semidirect
205. C₄₇	C₂×#3383	semidirect
209. C₂×Q₁₆	Dic₉₄	semidirect
266. Q₁₆	C₂×Dic₉₄	semidirect
331. C₂	#20256277	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×#15689852	2
#20256277	2
C₂×#15689528	2
C₂×#15689524	2
C₂×#15689432	2
C₂×#15689408	2
C₂×C₈×Dic₉₄	2
C₂×Q₁₆×Dic₄₇	2
C₂×C₄×Q₁₆×C₄₇	2
C₂×#3383	47

Subgroups — C2×#20256277 #27096733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256277 #27096733