Subgroups — C₄×#15689496 #26224699

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689496	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625325	C₂	1	4	yes	maximal, complemented
3	C₄×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#13094021	C₂	1	1	yes	maximal, characteristic
5	#20244242	C₂	1	4	yes	maximal, complemented
6	#20244194	C₂	1	1	yes	maximal, characteristic
7	#20244190	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625325	C₂²	1	4	yes	—
11	C₂×#13094024	C₂²	1	1	yes	characteristic
12	C₂×#13094021	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
14	C₂×C₄×D₁₈₈	—	2	2	no	—
15	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
16	#15689496	C₄	1	8	yes	direct product factor
17	C₄×#10625325	—	2	16	no	—
18	C₂×#13093956	C₂²	1	4	yes	—
19	C₂×#13093949	C₂²	1	4	yes	—
20	C₂×#13093946	C₂²	1	4	yes	—
21	#15689029	C₂²	1	2	yes	—
22	#15689025	—	2	2	no	—
23	#15689025	C₂²	1	2	yes	—
24	#15689021	—	2	2	no	—
25	#15688993	C₂²	1	1	yes	characteristic
26	#15688993	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
32	C₄³×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×Dic₉₄	C₂³	1	2	yes	—
34	C₂²×D₁₈₈	—	2	2	no	—
35	C₂²×D₁₈₈	C₂³	1	2	yes	—
36	C₂²×C₄×D₄₇	—	2	2	no	—
37	C₂×#10625325	—	2	8	no	—
38	C₂×#10625325	—	2	8	no	—
39	C₂×#10625325	C₂×C₄	1	16	yes	complemented
40	#13094024	—	2	2	no	—
41	#13094024	C₂×C₄	1	4	yes	complemented
42	#13094021	—	2	2	no	—
43	#13094021	C₂×C₄	1	4	yes	—
44	C₄×Dic₉₄	—	2	8	no	—
45	C₄×D₁₈₈	—	2	8	no	—
46	C₄×D₁₈₈	—	4	8	no	—
47	C₂×C₄⋊Dic₄₇	—	2	2	no	—
48	C₂×C₄⋊Dic₄₇	—	2	2	no	—
49	C₂×C₄⋊Dic₄₇	—	2	2	no	—
50	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
52	C₂×Dic₄₇⋊C₄	—	2	2	no	—
53	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
54	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
55	C₂×C₄×Dic₄₇	—	2	2	no	—
56	#13093956	—	2	8	no	—
57	#13093949	—	2	8	no	—
58	#13093946	—	2	8	no	—
59	C₄₇⋊#95	—	2	4	no	—
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₇	D₄	1	2	yes	—
63	C₂×C₄²×C₄₇	D₄	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	—	2	4	no	—
68	C₄×C₈×C₄₇	—	2	2	no	—
69	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
70	C₄×C₈×C₄₇	D₄	1	8	yes	complemented
71	C₂³×D₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂×D₁₈₈	—	2	4	no	—
81	C₂×D₁₈₈	—	2	4	no	—
82	C₂×D₁₈₈	—	2	8	no	—
83	C₂×D₁₈₈	—	2	2	no	—
84	C₂×D₁₈₈	—	4	4	no	—
85	C₂×D₁₈₈	—	4	4	no	—
86	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄×D₄₇	—	4	4	no	—
88	C₂×C₄×D₄₇	—	4	8	no	—
89	C₂×Dic₉₄	—	2	4	no	—
90	C₂×Dic₉₄	—	2	4	no	—
91	C₂×Dic₉₄	—	2	2	no	—
92	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₄₇	#41	1	4	yes	—
99	C₄²×C₄₇	—	2	4	no	—
100	C₄²×C₄₇	—	2	2	no	—
101	C₄²×C₄₇	SD₁₆	1	8	yes	complemented
102	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₇	C₂×D₄	1	4	yes	—
104	C₄²×C₄₇	#41	1	2	yes	—
105	(C₂×D₄₇)⋊C₄	—	4	4	no	—
106	(C₂×D₄₇)⋊C₄	—	4	8	no	—
107	C₄⋊Dic₄₇	—	2	8	no	—
108	C₄⋊Dic₄₇	—	2	4	no	—
109	C₄⋊Dic₄₇	—	2	4	no	—
110	C₄⋊Dic₄₇	—	4	4	no	—
111	C₄⋊Dic₄₇	—	4	4	no	—
112	Dic₄₇⋊C₄	—	4	4	no	—
113	C₄×Dic₄₇	—	4	4	no	—
114	#10625325	—	2	32	no	—
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₄₇	—	2	4	no	—
117	C₂²×D₄₇	—	4	2	no	—
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	—	4	4	no	—
120	C₂²×D₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₄₇	#127	1	2	yes	—
126	C₂×C₄×C₄₇	#127	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
128	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
130	C₂×C₄×C₄₇	#135	1	4	yes	—
131	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	4	no	—
141	C₂×Dic₄₇	—	4	4	no	—
142	D₁₈₈	—	2	8	no	—
143	D₁₈₈	—	4	4	no	—
144	D₁₈₈	—	4	16	no	—
145	C₄×D₄₇	—	8	8	no	—
146	Dic₉₄	—	2	8	no	—
147	Dic₉₄	—	4	4	no	—
148	C₈×C₄₇	—	2	4	no	—
149	C₈×C₄₇	C₄×D₄	1	8	yes	complemented
150	C₄×#492	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
153	C₂²×C₄₇	#532	1	1	yes	characteristic
154	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
155	C₂²×C₄₇	C₂×#135	1	2	yes	—
156	C₂×D₄₇	—	4	4	no	—
157	C₂×D₄₇	—	4	8	no	—
158	C₂×D₄₇	—	8	2	no	—
159	C₂×D₄₇	—	8	2	no	—
160	C₂×D₄₇	—	8	4	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	C₄×SD₁₆	1	8	yes	complemented
164	C₄×C₄₇	#492	1	4	yes	complemented
165	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
166	C₄×C₄₇	#532	1	2	yes	—
167	Dic₄₇	—	4	4	no	—
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	—	8	2	no	—
170	C₂×#492	—	47	1	no	—
171	C₂×C₄×SD₁₆	—	47	4	no	—
172	C₄×C₄⋊Q₈	—	47	1	no	—
173	C₄×#127	—	47	1	no	—
174	#1968	—	47	4	no	—
175	#1957	—	47	1	no	—
176	#1956	—	47	1	no	—
177	#1844	—	47	1	no	—
178	C₄²×C₈	D₄₇	1	1	yes	characteristic, complemented
179	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
180	C₂×C₄₇	C₂×C₄×SD₁₆	1	2	yes	—
181	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
182	C₂×C₄₇	#3147	1	1	yes	characteristic
183	C₂×C₄₇	#3167	1	2	yes	—
184	D₄₇	—	8	4	no	—
185	C₂²×SD₁₆	—	47	4	no	—
186	C₂×C₄⋊Q₈	—	47	1	no	—
187	C₂×#127	—	47	1	no	—
188	C₂×C₄×Q₈	—	47	2	no	—
189	C₂×C₄×D₄	—	47	2	no	—
190	C₂×C₄×D₄	—	94	2	no	—
191	#492	—	47	8	no	—
192	C₄×SD₁₆	—	94	16	no	—
193	C₂×#106	—	47	4	no	—
194	C₂×#103	—	47	4	no	—
195	C₂×#102	—	47	4	no	—
196	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
200	#391	—	47	2	no	—
201	#390	—	47	2	no	—
202	#390	—	94	2	no	—
203	#389	—	94	2	no	—
204	#384	—	47	1	no	—
205	#384	—	47	1	no	—
206	C₄×C₄⋊C₄	—	94	2	no	—
207	C₄³	C₂×D₄₇	1	1	yes	characteristic
208	C₄₇	C₄×#492	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	47	2	no	—
210	C₂²×D₄	—	47	2	no	—
211	C₂²×D₄	—	94	2	no	—
212	C₂³×C₄	—	94	2	no	—
213	C₂×SD₁₆	—	47	16	no	—
214	C₂×SD₁₆	—	94	8	no	—
215	C₂×SD₁₆	—	94	8	no	—
216	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
218	C₄⋊Q₈	—	47	4	no	—
219	C₄⋊Q₈	—	94	2	no	—
220	#127	—	47	4	no	—
221	#127	—	94	2	no	—
222	C₄×Q₈	—	94	8	no	—
223	C₄×D₄	—	94	8	no	—
224	C₄×D₄	—	188	8	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	47	2	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₄⋊C₄	—	94	2	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₂²⋊C₄	—	94	2	no	—
232	C₂×C₂²⋊C₄	—	94	4	no	—
233	C₂×C₄²	—	94	2	no	—
234	C₂×C₄²	D₁₈₈	1	2	yes	—
235	C₂×C₄²	D₁₈₈	1	2	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	#106	—	94	8	no	—
240	#103	—	94	8	no	—
241	#102	—	94	8	no	—
242	C₄×C₈	—	2	4	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
245	C₄×C₈	D₁₈₈	1	8	yes	complemented
246	#95	—	94	4	no	—
247	C₂⁴	—	94	2	no	—
248	C₂×Q₈	—	47	4	no	—
249	C₂×Q₈	—	94	2	no	—
250	C₂×Q₈	—	94	4	no	—
251	C₂×Q₈	—	94	4	no	—
252	C₂×D₄	—	47	4	no	—
253	C₂×D₄	—	94	8	no	—
254	C₂×D₄	—	94	4	no	—
255	C₂×D₄	—	94	4	no	—
256	C₂×D₄	—	94	2	no	—
257	C₂×D₄	—	188	4	no	—
258	C₂×D₄	—	188	4	no	—
259	C₂²×C₄	—	94	2	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	—	188	4	no	—
263	C₂²×C₄	—	188	8	no	—
264	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
266	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
267	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	SD₁₆	—	94	32	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
273	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
274	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
275	C₂×C₈	#10625350	1	4	yes	—
276	C₄⋊C₄	—	94	8	no	—
277	C₄⋊C₄	—	94	4	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	188	4	no	—
280	C₄⋊C₄	—	188	4	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₂²⋊C₄	—	188	4	no	—
283	C₂²⋊C₄	—	188	8	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	188	4	no	—
287	C₄²	#10625325	1	8	yes	complemented
288	C₄²	C₂×C₄×D₄₇	1	2	yes	—
289	C₄²	C₂×D₁₈₈	1	4	yes	—
290	C₄²	#10625350	1	2	yes	—
291	C₂³	—	94	4	no	—
292	C₂³	—	188	4	no	—
293	C₂³	—	188	4	no	—
294	C₂³	—	188	2	no	—
295	C₂³	—	188	2	no	—
296	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
297	Q₈	—	94	8	no	—
298	Q₈	—	188	4	no	—
299	D₄	—	94	8	no	—
300	D₄	—	188	4	no	—
301	D₄	—	188	16	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	94	4	no	—
306	C₂×C₄	—	376	8	no	—
307	C₂×C₄	—	188	4	no	—
308	C₂×C₄	—	188	4	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
314	C₂×C₄	#13094021	1	2	yes	—
315	C₂×C₄	#13094021	1	2	yes	—
316	C₂×C₄	C₂×#10625325	1	4	yes	—
317	C₂×C₄	C₂×#10625325	1	4	yes	—
318	C₂×C₄	C₂×#10625325	1	4	yes	—
319	C₂×C₄	#13094046	1	4	yes	—
320	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
322	C₂×C₄	C₂×#10625350	1	2	yes	—
323	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	C₄×D₁₈₈	1	8	yes	complemented
326	C₂²	—	376	2	no	—
327	C₂²	—	376	2	no	—
328	C₂²	—	376	4	no	—
329	C₂²	—	188	4	no	—
330	C₂²	—	188	8	no	—
331	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
332	C₂²	C₂×#13094021	1	1	yes	characteristic
333	C₂²	#15689673	1	1	yes	characteristic
334	C₂²	C₂²×#10625325	1	2	yes	—
335	C₂²	C₂×#13094046	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	376	2	no	—
339	C₄	—	376	2	no	—
340	C₄	—	188	4	no	—
341	C₄	C₄×#10625325	1	8	yes	complemented
342	C₄	#15689496	1	4	yes	direct product factor
343	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
344	C₄	#15689673	1	2	yes	—
345	C₂	—	376	4	no	—
346	C₂	C₄×#13094021	1	1	yes	characteristic
347	C₂	C₂×C₄×#10625325	1	2	yes	—
348	C₂	C₂×#15689496	1	1	yes	characteristic
349	C₂	#20253816	1	1	yes	characteristic
350	C₂	#20253947	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#13094021	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₄×#492
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#10625325	C₂	semidirect
3. C₄×#13094024	C₂	semidirect
5. #20244242	C₂	semidirect
7. #20244190	C₂	semidirect
8. #20243545	C₂	semidirect
9. C₄²×C₈×C₄₇	C₂	semidirect
16. #15689496	C₄	direct (N × H)
39. C₂×#10625325	C₂×C₄	semidirect
41. #13094024	C₂×C₄	semidirect
69. C₄×C₈×C₄₇	C₂×C₄	semidirect
70. C₄×C₈×C₄₇	D₄	semidirect
101. C₄²×C₄₇	SD₁₆	semidirect
149. C₈×C₄₇	C₄×D₄	semidirect
163. C₄×C₄₇	C₄×SD₁₆	semidirect
164. C₄×C₄₇	#492	semidirect
178. C₄²×C₈	D₄₇	semidirect
208. C₄₇	C₄×#492	semidirect
244. C₄×C₈	C₄×D₄₇	semidirect
245. C₄×C₈	D₁₈₈	semidirect
287. C₄²	#10625325	semidirect
325. C₈	C₄×D₁₈₈	semidirect
341. C₄	C₄×#10625325	semidirect
342. C₄	#15689496	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×#15689496	2
C₂×C₄×#10625325	2
C₄×#13094024	2
C₄×#13094021	2
#20244242	2
#20244194	2
#20244190	2
#20243545	2
C₄²×C₈×C₄₇	2
C₄×#492	47

Subgroups — C4×#15689496 #26224699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689496 #26224699