Subgroups — C₂×#20244796 #26245019

Order 12032.

1150 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093948	C₂	1	2	yes	maximal
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20244796	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×#391	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×#13093948	—	2	8	no	—
12	C₂×#13093948	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	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#391	—	2	2	no	—
21	C₄₇×#391	C₂²	1	4	yes	—
22	C₂×#95×C₄₇	—	2	1	no	—
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	—	2	4	no	—
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	#13093948	—	2	16	no	—
38	C₂²×Q₈×C₄₇	—	2	1	no	—
39	C₂²×Q₈×C₄₇	—	2	2	no	—
40	C₂²×Q₈×C₄₇	—	2	4	no	—
41	C₂²×Q₈×C₄₇	—	2	4	no	—
42	C₂²×Q₈×C₄₇	D₄	1	4	yes	—
43	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₄₇	—	2	1	no	—
45	C₂³×C₄×C₄₇	—	2	1	no	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
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	#95×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₄₇	—	2	4	no	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	—	2	16	no	—
61	C₂×Q₈×C₄₇	—	2	4	no	—
62	C₂×Q₈×C₄₇	—	2	4	no	—
63	C₂×Q₈×C₄₇	—	4	2	no	—
64	C₂×Q₈×C₄₇	—	4	2	no	—
65	C₂×Q₈×C₄₇	—	4	8	no	—
66	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	—	4	2	no	—
78	C₂²×C₄×C₄₇	—	4	2	no	—
79	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
80	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	#41	1	4	yes	—
87	C₂²×C₄×C₄₇	#41	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₇	—	2	4	no	—
90	C₂²×Dic₄₇	—	4	4	no	—
91	C₂²×Dic₄₇	—	4	4	no	—
92	C₂²×Dic₄₇	—	4	4	no	—
93	C₄⋊C₄×C₄₇	—	2	4	no	—
94	C₄⋊C₄×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	—	2	4	no	—
96	C₄²×C₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	4	8	no	—
100	C₄⋊Dic₄₇	—	4	4	no	—
101	C₄⋊Dic₄₇	—	4	8	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	4	8	no	—
106	C₂³×C₄₇	#121	1	2	yes	—
107	C₂³×C₄₇	#123	1	2	yes	—
108	C₂³×C₄₇	#124	1	2	yes	—
109	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
110	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
111	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
112	C₂³×C₄₇	#137	1	2	yes	—
113	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₄₇	—	2	8	no	—
115	Q₈×C₄₇	—	4	4	no	—
116	Q₈×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	8	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	8	4	no	—
131	C₂×C₄×C₄₇	#121	1	2	yes	—
132	C₂×C₄×C₄₇	#123	1	2	yes	—
133	C₂×C₄×C₄₇	#123	1	8	yes	—
134	C₂×C₄×C₄₇	#124	1	2	yes	—
135	C₂×C₄×C₄₇	#126	1	8	yes	—
136	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
137	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₄₇	—	4	4	no	—
143	C₂×Dic₄₇	—	4	4	no	—
144	C₂×Dic₄₇	—	4	4	no	—
145	C₂×Dic₄₇	—	8	8	no	—
146	C₄₇⋊C₈	—	4	4	no	—
147	C₂×#2072	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	#397	1	4	yes	—
149	C₂²×C₄₇	#460	1	4	yes	—
150	C₂²×C₄₇	#462	1	4	yes	—
151	C₂²×C₄₇	#483	1	4	yes	—
152	C₂²×C₄₇	#484	1	4	yes	—
153	C₂²×C₄₇	#487	1	4	yes	—
154	C₂²×C₄₇	#489	1	4	yes	—
155	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
158	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
159	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	—	4	4	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	#397	1	4	yes	—
167	C₄×C₄₇	C₂×#123	1	2	yes	—
168	C₄×C₄₇	C₂×#126	1	2	yes	—
169	Dic₄₇	—	8	4	no	—
170	C₂²×#103	—	47	2	no	—
171	C₂×#391	D₄₇	1	1	yes	characteristic
172	C₂×#384	—	47	1	no	—
173	#2072	—	47	8	no	—
174	C₂×#340	—	47	2	no	—
175	C₂×#336	—	47	1	no	—
176	C₂×C₄₇	#2072	1	8	yes	complemented
177	C₂×C₄₇	C₂×#397	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#460	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#483	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#484	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#487	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#489	1	1	yes	characteristic
184	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
185	C₂³×C₈	—	47	2	no	—
186	C₂²×C₄⋊C₄	—	47	2	no	—
187	C₂²×C₄⋊C₄	—	94	1	no	—
188	C₂²×C₄⋊C₄	—	94	1	no	—
189	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
190	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
191	C₂×#103	—	47	8	no	—
192	C₂×#103	—	94	8	no	—
193	#391	—	2	2	no	—
194	#391	C₂×D₄₇	1	4	yes	—
195	#384	—	47	4	no	—
196	#384	—	94	2	no	—
197	C₂×#95	—	2	1	no	—
198	C₂×#95	—	2	1	no	—
199	#340	—	47	8	no	—
200	#336	—	47	4	no	—
201	C₄₇	C₂×#2072	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×Q₈	—	2	4	no	—
205	C₂²×Q₈	—	2	4	no	—
206	C₂²×Q₈	D₁₈₈	1	4	yes	—
207	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
208	C₂³×C₄	—	2	1	no	—
209	C₂³×C₄	—	2	1	no	—
210	C₂³×C₄	—	94	2	no	—
211	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
214	C₂²×C₈	—	47	4	no	—
215	C₂²×C₈	—	94	4	no	—
216	C₂²×C₈	—	94	4	no	—
217	C₂²×C₈	—	94	4	no	—
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	47	4	no	—
221	C₂×C₄⋊C₄	—	94	4	no	—
222	C₂×C₄⋊C₄	—	94	4	no	—
223	C₂×C₄⋊C₄	—	94	2	no	—
224	C₂×C₄⋊C₄	—	94	4	no	—
225	C₂×C₄⋊C₄	—	94	4	no	—
226	C₂×C₄⋊C₄	—	188	4	no	—
227	C₂×C₄⋊C₄	—	188	4	no	—
228	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
229	C₂×C₄²	—	2	4	no	—
230	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
231	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
232	#103	—	94	16	no	—
233	#95	—	2	4	no	—
234	#95	—	2	4	no	—
235	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	16	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	4	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	8	no	—
244	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	94	4	no	—
258	C₂²×C₄	—	188	4	no	—
259	C₂²×C₄	—	188	4	no	—
260	C₂²×C₄	—	188	4	no	—
261	C₂²×C₄	#10625336	1	4	yes	—
262	C₂²×C₄	#10625337	1	4	yes	—
263	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
264	C₂²×C₄	#10625350	1	2	yes	—
265	C₂²×C₄	#10625352	1	4	yes	—
266	C₂²×C₄	#10625352	1	4	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
268	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
269	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
271	C₂×C₈	—	94	4	no	—
272	C₂×C₈	—	94	4	no	—
273	C₂×C₈	—	94	4	no	—
274	C₂×C₈	—	188	8	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	94	4	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	188	8	no	—
280	C₄⋊C₄	—	188	4	no	—
281	C₄⋊C₄	—	188	8	no	—
282	C₄²	—	2	4	no	—
283	C₄²	—	2	4	no	—
284	C₂³	#13093998	1	2	yes	—
285	C₂³	#13094005	1	2	yes	—
286	C₂³	#13094011	1	2	yes	—
287	C₂³	C₂×#10625336	1	2	yes	—
288	C₂³	C₂×#10625337	1	2	yes	—
289	C₂³	#13094049	1	2	yes	—
290	C₂³	#13094054	1	2	yes	—
291	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
292	Q₈	—	2	8	no	—
293	Q₈	—	4	8	no	—
294	Q₈	—	4	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	376	8	no	—
310	C₂×C₄	—	188	4	no	—
311	C₂×C₄	—	188	4	no	—
312	C₂×C₄	—	188	4	no	—
313	C₂×C₄	#13093998	1	2	yes	—
314	C₂×C₄	#13094005	1	2	yes	—
315	C₂×C₄	#13094008	1	8	yes	—
316	C₂×C₄	#13094011	1	2	yes	—
317	C₂×C₄	#13094017	1	8	yes	—
318	C₂×C₄	C₂×#10625336	1	4	yes	—
319	C₂×C₄	C₂×#10625337	1	4	yes	—
320	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
321	C₂×C₄	C₂×#10625352	1	2	yes	—
322	C₂×C₄	C₂×#10625352	1	2	yes	—
323	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
324	C₈	—	188	4	no	—
325	C₂²	#15689048	1	4	yes	—
326	C₂²	#15689323	1	4	yes	—
327	C₂²	#15689333	1	4	yes	—
328	C₂²	#15689451	1	4	yes	—
329	C₂²	#15689456	1	4	yes	—
330	C₂²	#15689473	1	4	yes	—
331	C₂²	#15689482	1	4	yes	—
332	C₂²	C₂×#13093998	1	1	yes	characteristic
333	C₂²	C₂×#13094005	1	1	yes	characteristic
334	C₂²	C₂×#13094011	1	1	yes	characteristic
335	C₂²	C₂²×#10625336	1	1	yes	characteristic
336	C₂²	C₂²×#10625337	1	1	yes	characteristic
337	C₂²	C₂×#13094049	1	1	yes	characteristic
338	C₂²	C₂×#13094054	1	1	yes	characteristic
339	C₄	—	2	4	no	—
340	C₄	—	2	4	no	—
341	C₄	—	4	4	no	—
342	C₄	—	8	2	no	—
343	C₄	—	376	4	no	—
344	C₄	#15689048	1	4	yes	—
345	C₄	C₂×#13094008	1	2	yes	—
346	C₄	C₂×#13094017	1	2	yes	—
347	C₂	#20244796	1	8	yes	direct product factor
348	C₂	C₂×#15689048	1	1	yes	characteristic
349	C₂	C₂×#15689323	1	1	yes	characteristic
350	C₂	C₂×#15689333	1	1	yes	characteristic
351	C₂	C₂×#15689451	1	1	yes	characteristic
352	C₂	C₂×#15689456	1	1	yes	characteristic
353	C₂	C₂×#15689473	1	1	yes	characteristic
354	C₂	C₂×#15689482	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×#391	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#391	D₄₇
hypocenter	C₄₇	C₂×#2072
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#391

Complements

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

N	H ≅ G/N	type
3. #20244796	C₂	direct (N × H)
176. C₂×C₄₇	#2072	semidirect
201. C₄₇	C₂×#2072	semidirect
347. C₂	#20244796	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×#13093948	2
C₂×#15688993	2
#20244796	2
C₂×#15688885	2
C₂×#15688878	2
C₂×C₄₇×#391	2
C₂×#2072	47

Subgroups — C2×#20244796 #26245019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244796 #26245019