Subgroups — C₄×#15689489 #26224353

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689489	C₂	1	1	yes	maximal, characteristic
2	C₄×#13094021	C₂	1	1	yes	maximal, characteristic
3	#20244358	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244211	C₂	1	1	yes	maximal, characteristic
5	#20244084	C₂	1	4	yes	maximal, complemented
6	#20243521	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13093944	C₂	1	4	yes	maximal, complemented
8	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094021	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₈₈	—	2	2	no	—
12	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
13	#15689489	C₄	1	8	yes	direct product factor
14	C₂×#13093944	C₂²	1	4	yes	—
15	C₂×#13093944	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093935	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	#15689025	—	2	2	no	—
21	#15689025	C₂²	1	2	yes	—
22	#15688993	C₂²	1	1	yes	characteristic
23	#15688885	C₂²	1	4	yes	—
24	#15688885	C₂²	1	4	yes	—
25	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#389	C₂²	1	2	yes	—
27	C₄₇×#387	—	2	2	no	—
28	C₄₇×#382	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×D₁₈₈	—	2	2	no	—
33	C₂²×D₁₈₈	C₂³	1	2	yes	—
34	C₂²×C₄×D₄₇	—	2	2	no	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13094021	—	2	2	no	—
38	#13094021	C₂×C₄	1	4	yes	—
39	C₄×D₁₈₈	—	2	8	no	—
40	C₄×D₁₈₈	—	4	8	no	—
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
44	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	#13093944	—	2	8	no	—
46	#13093944	C₂×C₄	1	16	yes	complemented
47	#13093935	—	2	2	no	—
48	#13093935	—	2	4	no	—
49	#13093935	C₂×C₄	1	4	yes	complemented
50	C₄×C₄₇⋊C₈	—	2	4	no	—
51	#125×C₄₇	—	2	2	no	—
52	#125×C₄₇	C₂×C₄	1	4	yes	complemented
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	D₄	1	2	yes	—
59	C₂×C₄²×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	#95×C₄₇	—	2	2	no	—
64	#95×C₄₇	—	2	2	no	—
65	#95×C₄₇	—	2	2	no	—
66	C₂³×D₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂×D₁₈₈	—	2	4	no	—
76	C₂×D₁₈₈	—	2	4	no	—
77	C₂×D₁₈₈	—	2	8	no	—
78	C₂×D₁₈₈	—	2	2	no	—
79	C₂×D₁₈₈	—	4	4	no	—
80	C₂×D₁₈₈	—	4	4	no	—
81	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
82	C₂×C₄×D₄₇	—	4	4	no	—
83	C₂×C₄×D₄₇	—	4	8	no	—
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	4	4	no	—
88	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	4	2	no	—
92	C₄²×C₄₇	—	4	4	no	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	#41	1	2	yes	—
95	C₄²×C₄₇	#41	1	4	yes	—
96	(C₂×D₄₇)⋊C₄	—	4	8	no	—
97	(C₂×D₄₇)⋊C₄	—	4	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	4	4	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	#41	1	4	yes	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂²×D₄₇	—	2	4	no	—
108	C₂²×D₄₇	—	4	2	no	—
109	C₂²×D₄₇	—	4	2	no	—
110	C₂²×D₄₇	—	4	4	no	—
111	C₂²×D₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₄₇	#124	1	2	yes	—
126	C₂×C₄×C₄₇	#124	1	2	yes	—
127	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	D₁₈₈	—	2	8	no	—
135	D₁₈₈	—	4	16	no	—
136	D₁₈₈	—	4	4	no	—
137	C₄×D₄₇	—	8	8	no	—
138	C₄₇⋊C₈	—	2	4	no	—
139	C₄₇⋊C₈	—	2	4	no	—
140	C₄×#490	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
143	C₂²×C₄₇	#532	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
145	C₂²×C₄₇	#575	1	2	yes	—
146	C₂×D₄₇	—	4	8	no	—
147	C₂×D₄₇	—	4	4	no	—
148	C₂×D₄₇	—	8	2	no	—
149	C₂×D₄₇	—	8	2	no	—
150	C₂×D₄₇	—	8	4	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	—	4	2	no	—
154	C₄×C₄₇	—	4	4	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	#490	1	4	yes	complemented
157	C₄×C₄₇	C₄×#41	1	2	yes	—
158	C₄×C₄₇	#529	1	2	yes	—
159	Dic₄₇	—	8	2	no	—
160	C₂×#490	—	47	1	no	—
161	C₄×#127	—	47	1	no	—
162	C₄×#125	D₄₇	1	1	yes	characteristic, complemented
163	#1992	—	47	1	no	—
164	#1961	—	47	1	no	—
165	#1936	—	47	4	no	—
166	#1836	—	47	1	no	—
167	C₄×#102	—	47	4	no	—
168	C₄×#97	—	47	1	no	—
169	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
170	C₂×C₄₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
171	C₂×C₄₇	C₂×#490	1	1	yes	characteristic
172	C₂×C₄₇	#3135	1	1	yes	characteristic
173	C₂×C₄₇	#3171	1	2	yes	—
174	D₄₇	—	8	4	no	—
175	C₂×#127	—	47	1	no	—
176	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
177	C₂×C₄×D₄	—	47	2	no	—
178	C₂×C₄×D₄	—	94	2	no	—
179	#490	—	47	8	no	—
180	C₂×#102	—	47	4	no	—
181	C₂×#102	—	47	4	no	—
182	C₂×#97	—	47	1	no	—
183	C₂×#97	—	47	2	no	—
184	C₂×#97	—	47	1	no	—
185	C₂×C₄×C₈	—	47	2	no	—
186	#390	—	47	2	no	—
187	#390	—	94	2	no	—
188	#389	C₂×D₄₇	1	2	yes	—
189	#387	—	2	2	no	—
190	#384	—	47	1	no	—
191	#382	C₂×D₄₇	1	1	yes	characteristic
192	C₄×C₄⋊C₄	—	2	2	no	—
193	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
194	C₄³	C₂×D₄₇	1	1	yes	characteristic
195	#340	—	47	4	no	—
196	#340	—	47	4	no	—
197	C₄₇	C₄×#490	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	47	2	no	—
199	C₂²×D₄	—	94	2	no	—
200	C₂³×C₄	—	94	2	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	—	47	2	no	—
203	#127	—	47	4	no	—
204	#127	—	94	2	no	—
205	#125	—	2	2	no	—
206	#125	C₄×D₄₇	1	4	yes	complemented
207	C₄×D₄	—	94	8	no	—
208	C₄×D₄	—	188	8	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	47	2	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
214	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
215	C₂×C₂²⋊C₄	—	94	2	no	—
216	C₂×C₂²⋊C₄	—	94	4	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
219	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#102	—	47	16	no	—
224	#102	—	94	8	no	—
225	#97	—	47	4	no	—
226	#97	—	94	4	no	—
227	#97	—	94	2	no	—
228	C₄×C₈	—	94	4	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	C₂⁴	—	94	2	no	—
233	C₂×D₄	—	47	4	no	—
234	C₂×D₄	—	94	8	no	—
235	C₂×D₄	—	94	4	no	—
236	C₂×D₄	—	94	2	no	—
237	C₂×D₄	—	94	4	no	—
238	C₂×D₄	—	188	4	no	—
239	C₂×D₄	—	188	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	94	2	no	—
243	C₂²×C₄	—	188	4	no	—
244	C₂²×C₄	—	188	8	no	—
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	—	94	2	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	94	4	no	—
261	C₄⋊C₄	—	188	4	no	—
262	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
263	C₂²⋊C₄	—	188	8	no	—
264	C₂²⋊C₄	—	188	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₂×C₄×D₄₇	1	2	yes	—
270	C₄²	#10625350	1	2	yes	—
271	C₄²	#10625354	1	4	yes	—
272	C₂³	—	94	4	no	—
273	C₂³	—	188	2	no	—
274	C₂³	—	188	4	no	—
275	C₂³	—	188	4	no	—
276	C₂³	—	188	2	no	—
277	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
278	D₄	—	94	8	no	—
279	D₄	—	188	16	no	—
280	D₄	—	188	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	376	8	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
297	C₂×C₄	#13094013	1	2	yes	—
298	C₂×C₄	#13094013	1	2	yes	—
299	C₂×C₄	#13094049	1	4	yes	—
300	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	#41×D₄₇	1	2	yes	—
302	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625354	1	2	yes	—
304	C₈	—	94	4	no	—
305	C₈	—	94	4	no	—
306	C₂²	—	376	2	no	—
307	C₂²	—	376	4	no	—
308	C₂²	—	376	2	no	—
309	C₂²	—	188	8	no	—
310	C₂²	—	188	4	no	—
311	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₂²	C₂×#13094013	1	1	yes	characteristic
313	C₂²	#15689669	1	1	yes	characteristic
314	C₂²	C₂×#13094049	1	2	yes	—
315	C₂²	#15689921	1	2	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	376	2	no	—
322	C₄	#15689489	1	4	yes	direct product factor
323	C₄	C₄×#10625354	1	2	yes	—
324	C₄	#15689660	1	2	yes	—
325	C₂	—	376	4	no	—
326	C₂	C₄×#13094013	1	1	yes	characteristic
327	C₂	C₄×#13094049	1	2	yes	—
328	C₂	C₂×#15689489	1	1	yes	characteristic
329	C₂	#20253736	1	1	yes	characteristic
330	C₂	#20253984	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#13094021	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#125	D₄₇
hypocenter	C₄₇	C₄×#490
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
3. #20244358	C₂	semidirect
5. #20244084	C₂	semidirect
6. #20243521	C₂	semidirect
7. C₄×#13093944	C₂	semidirect
8. C₄×#13093935	C₂	semidirect
9. C₄×#125×C₄₇	C₂	semidirect
13. #15689489	C₄	direct (N × H)
46. #13093944	C₂×C₄	semidirect
49. #13093935	C₂×C₄	semidirect
52. #125×C₄₇	C₂×C₄	semidirect
156. C₄×C₄₇	#490	semidirect
162. C₄×#125	D₄₇	semidirect
197. C₄₇	C₄×#490	semidirect
206. #125	C₄×D₄₇	semidirect
322. C₄	#15689489	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×#15689489	2
C₄×#13094021	2
#20244358	2
#20244211	2
#20244084	2
#20243521	2
C₄×#13093944	2
C₄×#13093935	2
C₄×#125×C₄₇	2
C₄×#490	47

Subgroups — C4×#15689489 #26224353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689489 #26224353