Subgroups — C₂×#20253972 #27086020

Order 12032.

2006 conjugacy classes of subgroups.

The subgroup lattice has 342 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	#20253972	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689473	C₂	1	2	yes	maximal, complemented
5	C₂×#15689361	C₂	1	4	yes	maximal
6	C₂×Q₁₆×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#494	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	—	2	2	no	—
14	C₂×#13094004	—	2	2	no	—
15	C₂×#13094000	C₂²	1	4	yes	—
16	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
17	C₂×#13093989	—	2	2	no	—
18	C₂×#13093982	—	2	2	no	—
19	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
20	#15689528	C₂²	1	8	yes	complemented
21	#15689473	C₂²	1	16	yes	complemented
22	#15689361	C₂²	1	32	yes	—
23	Q₁₆×Dic₄₇	C₂²	1	32	yes	complemented
24	C₂×#13093958	C₂²	1	4	yes	—
25	C₂×#13093948	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	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
30	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
31	C₄₇×#494	C₂²	1	8	yes	complemented
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	—	2	8	no	—
39	C₂×C₈×D₄₇	C₂³	1	8	yes	—
40	#13094024	—	2	4	no	—
41	#13094024	C₂³	1	8	yes	—
42	#13094015	—	2	16	no	—
43	#13094004	—	2	16	no	—
44	#13094000	—	2	8	no	—
45	#13094000	C₂³	1	16	yes	—
46	Q₈×Dic₄₇	—	2	8	no	—
47	Q₈×Dic₄₇	C₂³	1	16	yes	—
48	#13093989	—	2	16	no	—
49	#13093982	—	2	16	no	—
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	#13093948	C₂³	1	32	yes	—
63	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
64	C₈×Dic₄₇	C₂³	1	8	yes	—
65	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
66	C₂×Q₁₆×C₄₇	—	2	16	no	—
67	C₂×Q₁₆×C₄₇	C₂³	1	16	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	—	2	4	no	—
70	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	4	yes	—
75	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₄₇	—	2	4	no	—
77	C₂×Q₈×C₄₇	—	2	16	no	—
78	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₄₇	—	2	4	no	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	4	no	—
84	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×D₄₇	—	2	4	no	—
88	C₂×C₄×D₄₇	—	2	4	no	—
89	C₂×C₄×D₄₇	Q₁₆	1	8	yes	complemented
90	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
92	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
93	C₂×C₄×D₄₇	C₂⁴	1	2	yes	—
94	C₂×Dic₉₄	—	2	8	no	—
95	C₂×Dic₉₄	—	4	8	no	—
96	Q₁₆×C₄₇	—	2	32	no	—
97	C₂×C₈×C₄₇	#41	1	8	yes	—
98	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
99	C₄⋊C₄×C₄₇	—	2	8	no	—
100	C₄⋊C₄×C₄₇	—	2	8	no	—
101	C₄²×C₄₇	C₂×D₄	1	2	yes	—
102	C₄²×C₄₇	C₂⁴	1	2	yes	—
103	(C₂×D₄₇)⋊C₄	—	2	16	no	—
104	C₄⋊Dic₄₇	—	2	8	no	—
105	C₄⋊Dic₄₇	—	2	4	no	—
106	C₄⋊Dic₄₇	—	2	16	no	—
107	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
108	Dic₄₇⋊C₄	—	2	16	no	—
109	C₄×Dic₄₇	—	2	16	no	—
110	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
111	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
112	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
113	C₄×Dic₄₇	C₂⁴	1	2	yes	—
114	C₂×C₄₇⋊C₈	—	2	4	no	—
115	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
116	C₈×D₄₇	—	2	16	no	—
117	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²×D₄₇	—	2	2	no	—
119	C₂²×D₄₇	—	2	2	no	—
120	C₂²×D₄₇	C₂×Q₁₆	1	4	yes	—
121	C₂²×D₄₇	C₂²×D₄	1	2	yes	—
122	Q₈×C₄₇	—	2	16	no	—
123	Q₈×C₄₇	—	4	8	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
127	C₂×C₄×C₄₇	#135	1	4	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
132	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	8	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₂×Dic₄₇	C₂×Q₁₆	1	4	yes	—
138	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
139	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
140	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
141	C₄×D₄₇	—	2	4	no	—
142	C₄×D₄₇	—	2	8	no	—
143	C₄×D₄₇	—	2	4	no	—
144	C₄×D₄₇	C₂×Q₁₆	1	16	yes	complemented
145	Dic₉₄	—	4	16	no	—
146	C₈×C₄₇	C₂×#41	1	8	yes	—
147	C₄₇⋊C₈	—	2	4	no	—
148	C₂×#3170	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	#532	1	2	yes	—
150	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
151	C₂²×C₄₇	C₂×#135	1	2	yes	—
152	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
153	C₂×D₄₇	—	2	8	no	—
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	—	2	2	no	—
156	C₂×D₄₇	C₂²×Q₁₆	1	4	yes	—
157	C₄×C₄₇	—	4	8	no	—
158	C₄×C₄₇	#532	1	2	yes	—
159	C₄×C₄₇	C₂²×Q₁₆	1	4	yes	—
160	C₄×C₄₇	C₂×#135	1	4	yes	—
161	C₄×C₄₇	C₂²×#41	1	2	yes	—
162	Dic₄₇	—	2	2	no	—
163	Dic₄₇	—	2	2	no	—
164	Dic₄₇	—	4	8	no	—
165	Dic₄₇	C₂²×Q₁₆	1	4	yes	—
166	C₂×#533	—	47	2	no	—
167	#3170	—	47	16	no	—
168	C₂×#500	—	47	1	no	—
169	C₂×#494	D₄₇	1	1	yes	characteristic, complemented
170	C₂×#487	—	47	2	no	—
171	C₂×#467	—	47	4	no	—
172	C₂×C₄×Q₁₆	—	47	4	no	—
173	C₂²×C₄×C₈	—	47	1	no	—
174	C₂×C₄₇	#3170	1	4	yes	complemented
175	C₂×C₄₇	C₂×#532	1	1	yes	characteristic
176	C₂×C₄₇	C₂³×Q₁₆	1	1	yes	characteristic
177	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
178	D₄₇	—	2	4	no	—
179	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
180	C₂³×C₈	—	47	2	no	—
181	#533	—	47	16	no	—
182	#533	—	94	8	no	—
183	C₂×C₄⋊Q₈	—	47	2	no	—
184	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
185	C₂×#125	—	94	2	no	—
186	C₂×C₂²⋊Q₈	—	47	4	no	—
187	C₂×C₂²⋊Q₈	—	94	2	no	—
188	C₂×C₄×Q₈	—	47	4	no	—
189	C₂×C₄×Q₈	—	94	2	no	—
190	C₂×#117	—	94	2	no	—
191	C₂²×C₄²	—	47	1	no	—
192	#500	—	47	8	no	—
193	#494	C₂×D₄₇	1	8	yes	complemented
194	#487	—	47	16	no	—
195	#467	—	47	32	no	—
196	C₄×Q₁₆	—	47	32	no	—
197	C₂×#107	—	47	4	no	—
198	C₂×#103	—	47	8	no	—
199	C₂×C₄×C₈	—	47	2	no	—
200	C₂×C₄×C₈	—	47	1	no	—
201	C₂×C₄×C₈	—	47	8	no	—
202	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
203	C₄₇	C₂×#3170	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	94	2	no	—
205	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
206	C₂³×C₄	—	47	1	no	—
207	C₂³×C₄	—	47	1	no	—
208	C₂³×C₄	—	47	1	no	—
209	C₂×Q₁₆	—	2	16	no	—
210	C₂×Q₁₆	C₂²×D₄₇	1	16	yes	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	47	8	no	—
213	C₂²×C₈	—	94	8	no	—
214	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
215	C₄⋊Q₈	—	2	4	no	—
216	C₄⋊Q₈	—	47	8	no	—
217	C₄⋊Q₈	—	94	4	no	—
218	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	—
219	#125	—	94	16	no	—
220	C₂²⋊Q₈	—	47	16	no	—
221	C₂²⋊Q₈	—	94	16	no	—
222	C₂²⋊Q₈	—	94	8	no	—
223	C₄×Q₈	—	47	16	no	—
224	C₄×Q₈	—	94	16	no	—
225	C₄×Q₈	—	94	8	no	—
226	#117	—	94	16	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	47	4	no	—
230	C₂×C₄⋊C₄	—	94	4	no	—
231	C₂×C₄⋊C₄	—	94	4	no	—
232	C₂×C₄⋊C₄	—	94	2	no	—
233	C₂×C₂²⋊C₄	—	94	4	no	—
234	C₂×C₄²	—	47	1	no	—
235	C₂×C₄²	—	47	1	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	16	no	—
242	#103	—	47	32	no	—
243	C₄×C₈	—	47	8	no	—
244	C₄×C₈	—	47	4	no	—
245	C₄×C₈	C₂²×D₄₇	1	4	yes	—
246	C₂⁴	—	47	1	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	16	no	—
249	C₂×Q₈	—	94	8	no	—
250	C₂×Q₈	—	188	8	no	—
251	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	47	1	no	—
254	C₂²×C₄	—	47	8	no	—
255	C₂²×C₄	—	47	4	no	—
256	C₂²×C₄	—	47	1	no	—
257	C₂²×C₄	—	47	1	no	—
258	C₂²×C₄	—	47	4	no	—
259	C₂²×C₄	—	47	2	no	—
260	C₂²×C₄	—	47	2	no	—
261	C₂²×C₄	—	94	4	no	—
262	C₂²×C₄	—	94	4	no	—
263	C₂²×C₄	—	94	4	no	—
264	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
267	Q₁₆	—	2	32	no	—
268	C₂×C₈	—	47	4	no	—
269	C₂×C₈	—	94	4	no	—
270	C₂×C₈	—	94	16	no	—
271	C₂×C₈	#10625352	1	8	yes	—
272	C₂×C₈	C₂³×D₄₇	1	4	yes	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	47	8	no	—
276	C₄⋊C₄	—	94	16	no	—
277	C₄⋊C₄	—	94	16	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	94	8	no	—
280	C₂²⋊C₄	—	94	16	no	—
281	C₄²	—	47	4	no	—
282	C₄²	—	47	2	no	—
283	C₄²	—	47	4	no	—
284	C₄²	—	47	2	no	—
285	C₄²	—	94	16	no	—
286	C₄²	D₄×D₄₇	1	2	yes	—
287	C₄²	C₂³×D₄₇	1	2	yes	—
288	C₂³	—	47	4	no	—
289	C₂³	—	47	2	no	—
290	C₂³	—	94	2	no	—
291	C₂³	—	94	2	no	—
292	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
293	Q₈	—	2	16	no	—
294	Q₈	—	4	8	no	—
295	Q₈	—	188	16	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	47	4	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	47	16	no	—
301	C₂×C₄	—	47	2	no	—
302	C₂×C₄	—	47	2	no	—
303	C₂×C₄	—	94	2	no	—
304	C₂×C₄	—	94	2	no	—
305	C₂×C₄	—	94	8	no	—
306	C₂×C₄	—	94	4	no	—
307	C₂×C₄	—	94	4	no	—
308	C₂×C₄	—	94	8	no	—
309	C₂×C₄	—	188	8	no	—
310	C₂×C₄	Q₁₆×D₄₇	1	4	yes	—
311	C₂×C₄	#13094047	1	4	yes	—
312	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
314	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂×#10625352	1	4	yes	—
316	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
317	C₈	—	94	4	no	—
318	C₈	C₂×#10625352	1	8	yes	—
319	C₂²	—	47	4	no	—
320	C₂²	—	94	8	no	—
321	C₂²	—	94	2	no	—
322	C₂²	—	94	2	no	—
323	C₂²	#15689671	1	2	yes	—
324	C₂²	C₂×Q₁₆×D₄₇	1	2	yes	—
325	C₂²	C₂×#13094047	1	2	yes	—
326	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
327	C₄	—	4	8	no	—
328	C₄	—	47	4	no	—
329	C₄	—	94	2	no	—
330	C₄	—	94	2	no	—
331	C₄	—	188	8	no	—
332	C₄	#15689671	1	2	yes	—
333	C₄	C₂×Q₁₆×D₄₇	1	4	yes	—
334	C₄	C₂×#13094047	1	4	yes	—
335	C₄	C₂²×#10625352	1	2	yes	—
336	C₂	—	94	4	no	—
337	C₂	#20253972	1	4	yes	direct product factor
338	C₂	C₂×#15689671	1	1	yes	characteristic
339	C₂	C₂²×Q₁₆×D₄₇	1	1	yes	characteristic
340	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×#494	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	C₂×#3170
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #20253972	C₂	direct (N × H)
3. C₂×#15689528	C₂	semidirect
4. C₂×#15689473	C₂	semidirect
6. C₂×Q₁₆×Dic₄₇	C₂	semidirect
8. C₂×C₄₇×#494	C₂	semidirect
20. #15689528	C₂²	semidirect
21. #15689473	C₂²	semidirect
23. Q₁₆×Dic₄₇	C₂²	semidirect
31. C₄₇×#494	C₂²	semidirect
89. C₂×C₄×D₄₇	Q₁₆	semidirect
144. C₄×D₄₇	C₂×Q₁₆	semidirect
169. C₂×#494	D₄₇	semidirect
174. C₂×C₄₇	#3170	semidirect
193. #494	C₂×D₄₇	semidirect
203. C₄₇	C₂×#3170	semidirect
337. C₂	#20253972	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
#20253972	2
C₂×#15689528	2
C₂×#15689473	2
C₂×#15689361	2
C₂×Q₁₆×Dic₄₇	2
C₂×C₄×C₈×D₄₇	2
C₂×C₄₇×#494	2
C₂×#3170	47

Subgroups — C2×#20253972 #27086020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253972 #27086020