Subgroups — C₂×#20244859 #26245237

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093957	C₂	1	2	yes	maximal
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20244859	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	2	yes	maximal
5	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₇×#389	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
9	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×#13093957	—	2	8	no	—
12	C₂×#13093957	C₂²	1	8	yes	—
13	#15688993	—	2	2	no	—
14	#15688993	C₂²	1	4	yes	—
15	#15688885	C₂²	1	8	yes	—
16	#15688878	C₂²	1	4	yes	—
17	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#389	—	2	2	no	—
22	C₄₇×#389	C₂²	1	4	yes	—
23	C₂×#95×C₄₇	—	2	1	no	—
24	C₂³×Dic₄₇	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	4	4	no	—
35	C₂×C₄⋊Dic₄₇	—	4	4	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	#13093957	—	2	16	no	—
38	C₂³×C₄×C₄₇	—	2	1	no	—
39	C₂³×C₄×C₄₇	—	2	1	no	—
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
49	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₇	—	2	4	no	—
53	C₂×C₄²×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	#95×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	D₈	1	4	yes	—
70	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×C₄×C₄₇	#41	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	—	4	4	no	—
82	C₂²×Dic₄₇	—	4	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₄⋊C₄×C₄₇	—	4	8	no	—
88	C₄⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	8	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	—	4	8	no	—
96	C₂×C₄₇⋊C₈	—	2	16	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
100	C₂³×C₄₇	#121	1	2	yes	—
101	C₂³×C₄₇	#123	1	2	yes	—
102	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₄₇	C₂×D₈	1	2	yes	—
104	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
105	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂³×C₄₇	#137	1	2	yes	—
107	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	8	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	8	4	no	—
122	C₂×C₄×C₄₇	#121	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
124	C₂×C₄×C₄₇	#123	1	2	yes	—
125	C₂×C₄×C₄₇	#125	1	8	yes	—
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
128	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	8	8	no	—
137	C₄₇⋊C₈	—	2	8	no	—
138	C₂×#2083	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	#398	1	4	yes	—
140	C₂²×C₄₇	#459	1	4	yes	—
141	C₂²×C₄₇	#462	1	4	yes	—
142	C₂²×C₄₇	#480	1	4	yes	—
143	C₂²×C₄₇	#484	1	4	yes	—
144	C₂²×C₄₇	#500	1	4	yes	—
145	C₂²×C₄₇	#501	1	4	yes	—
146	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
150	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	4	no	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	#398	1	4	yes	—
158	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
159	C₄×C₄₇	C₂×#125	1	2	yes	—
160	Dic₄₇	—	8	4	no	—
161	C₂²×#107	—	47	2	no	—
162	C₂×#389	D₄₇	1	1	yes	characteristic
163	C₂×#384	—	47	1	no	—
164	#2083	—	47	8	no	—
165	C₂×#340	—	47	2	no	—
166	C₂×#336	—	47	1	no	—
167	C₂×C₄₇	#2083	1	8	yes	complemented
168	C₂×C₄₇	C₂×#398	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#484	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#501	1	1	yes	characteristic
175	C₂³×C₈	—	47	2	no	—
176	C₂²×C₄⋊C₄	—	2	1	no	—
177	C₂²×C₄⋊C₄	—	47	2	no	—
178	C₂²×C₄⋊C₄	—	94	1	no	—
179	C₂²×C₄⋊C₄	—	94	1	no	—
180	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#107	—	47	8	no	—
184	C₂×#107	—	94	8	no	—
185	#389	—	2	2	no	—
186	#389	C₂×D₄₇	1	4	yes	—
187	#384	—	47	4	no	—
188	#384	—	94	2	no	—
189	C₂×#95	—	2	1	no	—
190	#340	—	47	8	no	—
191	#336	—	47	4	no	—
192	C₄₇	C₂×#2083	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	—	2	1	no	—
195	C₂³×C₄	—	94	2	no	—
196	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	—	47	8	no	—
200	C₂²×C₈	—	47	4	no	—
201	C₂²×C₈	—	94	4	no	—
202	C₂²×C₈	—	94	4	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	4	4	no	—
209	C₂×C₄⋊C₄	—	47	4	no	—
210	C₂×C₄⋊C₄	—	94	4	no	—
211	C₂×C₄⋊C₄	—	94	4	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	4	no	—
214	C₂×C₄⋊C₄	—	94	4	no	—
215	C₂×C₄⋊C₄	—	188	4	no	—
216	C₂×C₄⋊C₄	—	188	4	no	—
217	C₂×C₄⋊C₄	D₁₈₈	1	4	yes	—
218	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
219	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
220	C₂×C₄²	—	2	4	no	—
221	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
223	#107	—	94	16	no	—
224	#95	—	2	4	no	—
225	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	—	188	4	no	—
240	C₂²×C₄	—	188	4	no	—
241	C₂²×C₄	—	188	4	no	—
242	C₂²×C₄	#10625334	1	4	yes	—
243	C₂²×C₄	#10625337	1	4	yes	—
244	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
245	C₂²×C₄	#10625350	1	2	yes	—
246	C₂²×C₄	#10625352	1	4	yes	—
247	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	C₂×C₈	—	47	8	no	—
253	C₂×C₈	—	94	16	no	—
254	C₂×C₈	—	94	4	no	—
255	C₂×C₈	—	94	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	8	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	94	4	no	—
263	C₄⋊C₄	—	188	8	no	—
264	C₄⋊C₄	—	188	4	no	—
265	C₄⋊C₄	—	188	8	no	—
266	C₄⋊C₄	C₂×D₁₈₈	1	4	yes	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₂³	#13093998	1	2	yes	—
270	C₂³	#13094005	1	2	yes	—
271	C₂³	#13094022	1	2	yes	—
272	C₂³	C₂×#10625334	1	2	yes	—
273	C₂³	C₂×#10625337	1	2	yes	—
274	C₂³	#13094049	1	2	yes	—
275	C₂³	#13094054	1	2	yes	—
276	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	376	8	no	—
292	C₂×C₄	—	188	4	no	—
293	C₂×C₄	—	188	4	no	—
294	C₂×C₄	—	188	4	no	—
295	C₂×C₄	#13093998	1	2	yes	—
296	C₂×C₄	#13094004	1	8	yes	—
297	C₂×C₄	#13094005	1	2	yes	—
298	C₂×C₄	#13094014	1	8	yes	—
299	C₂×C₄	#13094022	1	2	yes	—
300	C₂×C₄	C₂×#10625334	1	4	yes	—
301	C₂×C₄	C₂×#10625337	1	4	yes	—
302	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
303	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
304	C₂×C₄	C₂×#10625352	1	2	yes	—
305	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
306	C₈	—	94	8	no	—
307	C₂²	#15689053	1	4	yes	—
308	C₂²	#15689318	1	4	yes	—
309	C₂²	#15689333	1	4	yes	—
310	C₂²	#15689434	1	4	yes	—
311	C₂²	#15689456	1	4	yes	—
312	C₂²	#15689526	1	4	yes	—
313	C₂²	#15689533	1	4	yes	—
314	C₂²	C₂×#13093998	1	1	yes	characteristic
315	C₂²	C₂×#13094005	1	1	yes	characteristic
316	C₂²	C₂×#13094022	1	1	yes	characteristic
317	C₂²	C₂²×#10625334	1	1	yes	characteristic
318	C₂²	C₂²×#10625337	1	1	yes	characteristic
319	C₂²	C₂×#13094049	1	1	yes	characteristic
320	C₂²	C₂×#13094054	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	2	no	—
325	C₄	—	376	4	no	—
326	C₄	#15689053	1	4	yes	—
327	C₄	C₂×#13094004	1	2	yes	—
328	C₄	C₂×#13094014	1	2	yes	—
329	C₂	#20244859	1	8	yes	direct product factor
330	C₂	C₂×#15689053	1	1	yes	characteristic
331	C₂	C₂×#15689318	1	1	yes	characteristic
332	C₂	C₂×#15689333	1	1	yes	characteristic
333	C₂	C₂×#15689434	1	1	yes	characteristic
334	C₂	C₂×#15689456	1	1	yes	characteristic
335	C₂	C₂×#15689526	1	1	yes	characteristic
336	C₂	C₂×#15689533	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#389	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#389	D₄₇
hypocenter	C₄₇	C₂×#2083
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#389

Complements

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

N	H ≅ G/N	type
3. #20244859	C₂	direct (N × H)
167. C₂×C₄₇	#2083	semidirect
192. C₄₇	C₂×#2083	semidirect
329. C₂	#20244859	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13093957	2
C₂×#15688993	2
#20244859	2
C₂×#15688885	2
C₂×#15688878	2
C₂×C₄₇×#389	2
C₂×#2083	47

Subgroups — C2×#20244859 #26245237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244859 #26245237