Subgroups — C₂×#20255820 #27094783

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689846	C₂	1	2	yes	maximal
2	#20255820	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689463	C₂	1	4	yes	maximal, complemented
5	C₂×#15689408	C₂	1	2	yes	maximal, complemented
6	C₂×#15689361	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#15689160	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
11	#15689846	—	2	8	no	—
12	#15689846	C₂²	1	16	yes	—
13	C₂×#13094024	C₂²	1	2	yes	—
14	C₂×#13094019	—	2	4	no	—
15	C₂×#13094012	—	2	2	no	—
16	C₂×#13094000	C₂²	1	4	yes	—
17	C₂×C₄×Dic₉₄	—	2	2	no	—
18	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
19	C₂×#13093982	C₂²	1	4	yes	—
20	#15689528	C₂²	1	8	yes	complemented
21	#15689463	C₂²	1	32	yes	complemented
22	#15689408	C₂²	1	16	yes	complemented
23	#15689361	C₂²	1	16	yes	complemented
24	C₈×D₁₈₈	C₂²	1	8	yes	—
25	#15689160	C₂²	1	16	yes	complemented
26	C₂×#13093958	C₂²	1	2	yes	—
27	C₂×#13093958	C₂²	1	2	yes	—
28	C₂×#13093955	C₂²	1	4	yes	—
29	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂×#13093948	C₂²	1	4	yes	—
31	C₂×(C₂×D₄₇)⋊C₈	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	complemented
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×D₁₈₈	D₄	1	1	yes	characteristic
40	C₂²×C₄×D₄₇	C₂³	1	2	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₂×C₈×D₄₇	—	2	8	no	—
43	C₂×C₈×D₄₇	C₂³	1	8	yes	—
44	#13094024	—	2	4	no	—
45	#13094024	C₂³	1	8	yes	—
46	#13094019	—	2	32	no	—
47	#13094012	—	2	16	no	—
48	#13094000	—	2	8	no	—
49	#13094000	C₂³	1	16	yes	—
50	C₄×Dic₉₄	—	2	16	no	—
51	C₄×D₁₈₈	D₄	1	4	yes	—
52	C₄×D₁₈₈	C₂³	1	4	yes	—
53	#13093982	—	2	8	no	—
54	#13093982	C₂³	1	16	yes	—
55	C₂×C₄⋊Dic₄₇	—	2	2	no	—
56	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
57	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
58	C₂×Dic₄₇⋊C₄	—	2	4	no	—
59	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
60	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
61	C₂×(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
62	C₂×C₄×Dic₄₇	—	2	4	no	—
63	#13093958	C₂³	1	8	yes	—
64	#13093958	C₂³	1	8	yes	—
65	#13093955	C₂³	1	16	yes	—
66	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
67	#13093948	C₂³	1	16	yes	—
68	(C₂×D₄₇)⋊C₈	C₂³	1	8	yes	—
69	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₄₇	—	2	4	no	—
71	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₄₇	—	2	4	no	—
75	C₄×Q₈×C₄₇	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₄₇	—	2	2	no	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	#107×C₄₇	C₂³	1	4	yes	—
81	#103×C₄₇	C₂³	1	8	yes	—
82	C₄×C₈×C₄₇	C₂³	1	4	yes	—
83	C₂³×D₄₇	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂×Q₈×C₄₇	—	2	4	no	—
86	C₂×Q₈×C₄₇	—	2	4	no	—
87	C₂×Q₈×C₄₇	C₂⁴	1	4	yes	—
88	C₂²×C₄×C₄₇	—	2	2	no	—
89	C₂²×C₄×C₄₇	—	2	2	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	4	no	—
94	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
95	C₂×D₁₈₈	—	2	4	no	—
96	C₂×D₁₈₈	C₂×D₄	1	4	yes	—
97	C₂×C₄×D₄₇	—	2	4	no	—
98	C₂×C₄×D₄₇	—	2	4	no	—
99	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
100	C₂×C₄×D₄₇	C₂⁴	1	4	yes	—
101	C₂×Dic₉₄	—	2	8	no	—
102	C₂×Dic₉₄	—	4	8	no	—
103	Q₁₆×C₄₇	—	2	8	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	#41	1	4	yes	—
106	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
107	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
108	C₄⋊C₄×C₄₇	—	2	8	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	C₂⁴	1	4	yes	—
111	C₄²×C₄₇	—	2	8	no	—
112	C₄²×C₄₇	C₂×D₄	1	2	yes	—
113	C₄²×C₄₇	C₂⁴	1	2	yes	—
114	(C₂×D₄₇)⋊C₄	—	2	16	no	—
115	(C₂×D₄₇)⋊C₄	—	2	16	no	—
116	(C₂×D₄₇)⋊C₄	C₂×D₄	1	8	yes	—
117	C₄⋊Dic₄₇	—	2	8	no	—
118	C₄⋊Dic₄₇	—	2	4	no	—
119	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
120	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
121	Dic₄₇⋊C₄	—	2	16	no	—
122	C₄×Dic₄₇	—	2	16	no	—
123	C₂×C₄₇⋊C₈	—	2	4	no	—
124	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
125	C₈×D₄₇	—	2	16	no	—
126	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂²×D₄₇	—	2	4	no	—
128	C₂²×D₄₇	—	2	4	no	—
129	C₂²×D₄₇	—	2	4	no	—
130	C₂²×D₄₇	C₂²×D₄	1	4	yes	—
131	Q₈×C₄₇	—	2	4	no	—
132	Q₈×C₄₇	—	2	4	no	—
133	Q₈×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	4	no	—
136	C₂×C₄×C₄₇	—	2	4	no	—
137	C₂×C₄×C₄₇	—	4	4	no	—
138	C₂×C₄×C₄₇	—	4	4	no	—
139	C₂×C₄×C₄₇	#135	1	4	yes	—
140	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
144	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
145	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
146	C₂×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	2	8	no	—
148	C₂×Dic₄₇	—	4	8	no	—
149	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
150	D₁₈₈	—	2	8	no	—
151	C₄×D₄₇	—	2	8	no	—
152	C₄×D₄₇	—	2	4	no	—
153	C₄×D₄₇	—	2	4	no	—
154	Dic₉₄	—	4	16	no	—
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	C₂×#41	1	4	yes	—
157	C₄₇⋊C₈	—	2	4	no	—
158	C₂×#3342	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	#547	1	2	yes	—
160	C₂²×C₄₇	C₂×#135	1	2	yes	—
161	C₂²×C₄₇	#578	1	2	yes	—
162	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
163	C₂×D₄₇	—	2	4	no	—
164	C₂×D₄₇	—	2	4	no	—
165	C₂×D₄₇	—	2	4	no	—
166	C₂×D₄₇	—	4	8	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	4	4	no	—
169	C₄×C₄₇	—	4	4	no	—
170	C₄×C₄₇	#547	1	2	yes	—
171	C₄×C₄₇	C₂×#135	1	4	yes	—
172	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
173	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
174	Dic₄₇	—	2	4	no	—
175	Dic₄₇	—	4	8	no	—
176	C₂×#556	—	47	2	no	—
177	#3342	—	47	16	no	—
178	C₂×#500	—	47	1	no	—
179	C₂×#485	—	47	4	no	—
180	C₂×#475	—	47	2	no	—
181	C₂×#467	—	47	2	no	—
182	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
183	C₂×C₈×D₄	—	47	1	no	—
184	C₂×#427	—	47	2	no	—
185	C₂×C₄₇	#3342	1	4	yes	complemented
186	C₂×C₄₇	C₂×#547	1	1	yes	characteristic
187	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
189	D₄₇	—	4	4	no	—
190	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
191	C₂³×C₈	—	47	2	no	—
192	#556	—	47	16	no	—
193	#556	—	94	8	no	—
194	C₂×C₄⋊Q₈	—	47	2	no	—
195	C₂×#126	—	94	4	no	—
196	C₂×#124	—	94	2	no	—
197	C₂×C₂²⋊Q₈	—	47	4	no	—
198	C₂×C₄×Q₈	—	94	2	no	—
199	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
200	C₂×C₄×D₄	—	47	1	no	—
201	C₂×#117	—	47	4	no	—
202	#500	—	47	8	no	—
203	#485	—	47	32	no	—
204	#475	—	47	16	no	—
205	#467	—	47	16	no	—
206	C₄×Q₁₆	C₂×D₄₇	1	8	yes	complemented
207	C₈×D₄	—	47	8	no	—
208	#427	—	47	16	no	—
209	C₂×#107	—	47	2	no	—
210	C₂×#107	—	47	2	no	—
211	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
212	C₂×#106	—	47	4	no	—
213	C₂×C₄⋊C₈	—	47	1	no	—
214	C₂×#103	—	47	4	no	—
215	C₂×#103	C₂×D₄₇	1	2	yes	—
216	C₂×C₂²⋊C₈	—	47	2	no	—
217	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
218	C₄₇	C₂×#3342	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	94	2	no	—
220	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
221	C₂²×D₄	—	47	1	no	—
222	C₂³×C₄	—	47	2	no	—
223	C₂×Q₁₆	—	2	4	no	—
224	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
225	C₂²×C₈	—	47	2	no	—
226	C₂²×C₈	—	47	8	no	—
227	C₂²×C₈	—	94	8	no	—
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	C₄⋊Q₈	—	47	8	no	—
231	C₄⋊Q₈	—	94	4	no	—
232	#126	—	94	32	no	—
233	#124	—	94	16	no	—
234	C₂²⋊Q₈	—	47	16	no	—
235	C₂²⋊Q₈	—	94	8	no	—
236	C₄×Q₈	—	2	4	no	—
237	C₄×Q₈	—	94	16	no	—
238	C₄×Q₈	C₂²×D₄₇	1	8	yes	—
239	C₄×D₄	—	47	4	no	—
240	C₄×D₄	—	47	4	no	—
241	#117	—	47	16	no	—
242	#117	—	94	8	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	47	1	no	—
245	C₂×C₄⋊C₄	—	47	4	no	—
246	C₂×C₄⋊C₄	—	94	4	no	—
247	C₂×C₄⋊C₄	—	94	2	no	—
248	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
249	C₂×C₂²⋊C₄	—	47	2	no	—
250	C₂×C₂²⋊C₄	—	94	4	no	—
251	C₂×C₂²⋊C₄	—	94	4	no	—
252	C₂×C₄²	—	2	2	no	—
253	C₂×C₄²	—	94	4	no	—
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#107	—	47	8	no	—
256	#107	—	47	8	no	—
257	#107	C₂²×D₄₇	1	4	yes	—
258	#106	—	47	16	no	—
259	C₄⋊C₈	—	47	4	no	—
260	#103	—	47	16	no	—
261	#103	C₂²×D₄₇	1	8	yes	—
262	C₂²⋊C₈	—	47	8	no	—
263	C₄×C₈	C₂²×D₄₇	1	4	yes	—
264	C₂⁴	—	47	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	4	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	94	8	no	—
269	C₂×Q₈	—	188	8	no	—
270	C₂×Q₈	C₂³×D₄₇	1	4	yes	—
271	C₂×D₄	—	47	4	no	—
272	C₂×D₄	—	94	4	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	47	2	no	—
276	C₂²×C₄	—	47	4	no	—
277	C₂²×C₄	—	47	4	no	—
278	C₂²×C₄	—	94	4	no	—
279	C₂²×C₄	—	94	4	no	—
280	C₂²×C₄	—	94	4	no	—
281	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
283	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
284	Q₁₆	—	2	8	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	47	4	no	—
287	C₂×C₈	—	94	4	no	—
288	C₂×C₈	—	94	16	no	—
289	C₂×C₈	#10625350	1	4	yes	—
290	C₂×C₈	C₂³×D₄₇	1	2	yes	—
291	C₂×C₈	C₂³×D₄₇	1	2	yes	—
292	C₄⋊C₄	—	2	8	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	47	4	no	—
295	C₄⋊C₄	—	47	8	no	—
296	C₄⋊C₄	—	94	16	no	—
297	C₄⋊C₄	—	94	4	no	—
298	C₄⋊C₄	—	94	8	no	—
299	C₄⋊C₄	C₂³×D₄₇	1	4	yes	—
300	C₂²⋊C₄	—	47	8	no	—
301	C₂²⋊C₄	—	94	16	no	—
302	C₂²⋊C₄	—	94	16	no	—
303	C₄²	—	2	8	no	—
304	C₄²	—	94	16	no	—
305	C₄²	D₄×D₄₇	1	2	yes	—
306	C₄²	C₂³×D₄₇	1	2	yes	—
307	C₂³	—	47	4	no	—
308	C₂³	—	94	4	no	—
309	C₂³	—	94	4	no	—
310	C₂³	—	94	4	no	—
311	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
312	Q₈	—	2	4	no	—
313	Q₈	—	2	4	no	—
314	Q₈	—	4	4	no	—
315	Q₈	—	188	16	no	—
316	D₄	—	94	8	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	47	4	no	—
323	C₂×C₄	—	94	4	no	—
324	C₂×C₄	—	94	8	no	—
325	C₂×C₄	—	94	4	no	—
326	C₂×C₄	—	94	4	no	—
327	C₂×C₄	—	94	8	no	—
328	C₂×C₄	—	188	8	no	—
329	C₂×C₄	#13094047	1	4	yes	—
330	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
331	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
333	C₂×C₄	C₂×#10625350	1	2	yes	—
334	C₂×C₄	#13094071	1	2	yes	—
335	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	94	4	no	—
338	C₈	C₂×#10625350	1	4	yes	—
339	C₂²	—	94	4	no	—
340	C₂²	—	94	4	no	—
341	C₂²	—	94	4	no	—
342	C₂²	—	188	8	no	—
343	C₂²	#15689778	1	2	yes	—
344	C₂²	C₂×#13094047	1	2	yes	—
345	C₂²	#15689937	1	2	yes	—
346	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	4	4	no	—
349	C₄	—	4	4	no	—
350	C₄	—	94	4	no	—
351	C₄	—	188	8	no	—
352	C₄	#15689778	1	2	yes	—
353	C₄	C₂×#13094047	1	4	yes	—
354	C₄	C₂²×#10625350	1	1	yes	characteristic
355	C₄	C₂×#13094071	1	1	yes	characteristic
356	C₂	—	188	4	no	—
357	C₂	#20255820	1	4	yes	direct product factor
358	C₂	C₂×#15689778	1	1	yes	characteristic
359	C₂	C₂²×#13094047	1	1	yes	characteristic
360	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	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#3342
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. #20255820	C₂	direct (N × H)
3. C₂×#15689528	C₂	semidirect
4. C₂×#15689463	C₂	semidirect
5. C₂×#15689408	C₂	semidirect
6. C₂×#15689361	C₂	semidirect
8. C₂×#15689160	C₂	semidirect
9. C₂×C₄×Q₁₆×C₄₇	C₂	semidirect
20. #15689528	C₂²	semidirect
21. #15689463	C₂²	semidirect
22. #15689408	C₂²	semidirect
23. #15689361	C₂²	semidirect
25. #15689160	C₂²	semidirect
34. C₄×Q₁₆×C₄₇	C₂²	semidirect
182. C₂×C₄×Q₁₆	D₄₇	semidirect
185. C₂×C₄₇	#3342	semidirect
206. C₄×Q₁₆	C₂×D₄₇	semidirect
218. C₄₇	C₂×#3342	semidirect
357. C₂	#20255820	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×#15689846	2
#20255820	2
C₂×#15689528	2
C₂×#15689463	2
C₂×#15689408	2
C₂×#15689361	2
C₂×C₈×D₁₈₈	2
C₂×#15689160	2
C₂×C₄×Q₁₆×C₄₇	2
C₂×#3342	47

Subgroups — C2×#20255820 #27094783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20255820 #27094783