Subgroups — #26224802

Order 12032.

1238 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689236	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₃₇₆	C₂	1	2	yes	maximal, complemented
4	#20246113	C₂	1	2	yes	maximal, complemented
5	#20244264	C₂	1	2	yes	maximal, complemented
6	#20244254	C₂	1	2	yes	maximal, complemented
7	#20244212	C₂	1	2	yes	maximal, complemented
8	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	2	yes	—
11	C₂²×D₃₇₆	C₂²	1	2	yes	—
12	C₂×#13094021	C₂²	1	2	yes	—
13	C₂×C₄×D₁₈₈	C₂²	1	4	yes	—
14	#15689500	C₄	1	8	yes	complemented
15	#15689236	—	2	8	no	—
16	C₄×D₃₇₆	—	2	8	no	—
17	C₂×#13093958	C₂²	1	2	yes	—
18	C₂×#13093956	C₂²	1	2	yes	—
19	C₂×#13093946	C₂²	1	4	yes	—
20	C₂×#13093946	C₂²	1	4	yes	—
21	#15689025	C₂²	1	4	yes	—
22	#15689001	—	2	8	no	—
23	#15688989	C₂²	1	2	yes	—
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#376	C₂²	1	1	yes	characteristic
31	C₂²×D₁₈₈	—	2	4	no	—
32	C₂²×D₁₈₈	C₂³	1	4	yes	—
33	C₂²×C₄×D₄₇	—	2	4	no	—
34	C₂×D₃₇₆	—	2	4	no	—
35	C₂×D₃₇₆	—	2	4	no	—
36	C₂×D₃₇₆	—	2	4	no	—
37	C₂×D₃₇₆	—	2	4	no	—
38	C₂×D₃₇₆	C₂×C₄	1	8	yes	complemented
39	C₂×D₃₇₆	C₂×C₄	1	8	yes	complemented
40	#13094021	—	2	4	no	—
41	#13094021	C₂×C₄	1	8	yes	complemented
42	C₄×D₁₈₈	—	2	8	no	—
43	C₄×D₁₈₈	—	2	8	no	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
45	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
46	C₂×(C₂×D₄₇)⋊C₄	—	2	8	no	—
47	#13093958	—	2	4	no	—
48	#13093956	—	2	4	no	—
49	#13093946	—	2	8	no	—
50	#13093946	—	2	8	no	—
51	C₄₇⋊#95	—	2	4	no	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	#97×C₄₇	—	2	2	no	—
60	#97×C₄₇	—	2	2	no	—
61	#97×C₄₇	D₄	1	4	yes	complemented
62	C₄×C₈×C₄₇	—	2	2	no	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
64	C₄×C₈×C₄₇	D₄	1	4	yes	complemented
65	#95×C₄₇	D₄	1	4	yes	—
66	C₂³×D₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂×D₁₈₈	—	2	4	no	—
73	C₂×D₁₈₈	—	2	4	no	—
74	C₂×D₁₈₈	—	2	4	no	—
75	C₂×D₁₈₈	—	2	4	no	—
76	C₂×D₁₈₈	—	2	4	no	—
77	C₂×D₁₈₈	—	2	16	no	—
78	C₂×D₁₈₈	—	8	8	no	—
79	C₂×D₁₈₈	C₂²×C₄	1	8	yes	—
80	C₂×C₄×D₄₇	—	4	8	no	—
81	C₂×C₄×D₄₇	—	8	8	no	—
82	C₂×C₈×C₄₇	—	2	1	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	1	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₇	#41	1	2	yes	—
93	C₂×C₈×C₄₇	#41	1	2	yes	—
94	C₄²×C₄₇	—	2	1	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂×D₄	1	2	yes	—
98	C₄²×C₄₇	C₂×D₄	1	2	yes	—
99	C₄²×C₄₇	#41	1	2	yes	—
100	(C₂×D₄₇)⋊C₄	—	4	8	no	—
101	(C₂×D₄₇)⋊C₄	—	8	8	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	D₃₇₆	—	2	16	no	—
105	D₃₇₆	—	2	16	no	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂²×D₄₇	—	2	8	no	—
108	C₂²×D₄₇	—	4	4	no	—
109	C₂²×D₄₇	—	4	4	no	—
110	C₂²×D₄₇	—	4	8	no	—
111	C₂²×D₄₇	—	4	8	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₄₇	#127	1	4	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	8	no	—
126	D₁₈₈	—	2	8	no	—
127	D₁₈₈	—	2	8	no	—
128	D₁₈₈	—	4	4	no	—
129	D₁₈₈	—	4	4	no	—
130	D₁₈₈	—	8	16	no	—
131	C₄×D₄₇	—	8	16	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	C₄×D₄	1	4	yes	—
135	C₈×C₄₇	C₄×D₄	1	4	yes	—
136	#21697	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
139	C₂²×C₄₇	#532	1	1	yes	characteristic
140	C₂²×C₄₇	#576	1	4	yes	—
141	C₂×D₄₇	—	4	8	no	—
142	C₂×D₄₇	—	4	16	no	—
143	C₂×D₄₇	—	8	4	no	—
144	C₂×D₄₇	—	8	4	no	—
145	C₂×D₄₇	—	8	8	no	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₄₇	#532	1	1	yes	characteristic
152	C₄×C₄₇	#532	1	1	yes	characteristic
153	Dic₄₇	—	8	4	no	—
154	C₂×#493	—	47	1	no	—
155	C₂×#442	—	47	2	no	—
156	C₂×C₄×D₈	—	47	2	no	—
157	#2328	—	47	2	no	—
158	#1972	—	47	2	no	—
159	#1970	—	47	2	no	—
160	#1961	—	47	2	no	—
161	#1846	—	47	1	no	—
162	#1726	D₄₇	1	1	yes	characteristic, complemented
163	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
164	C₂×C₄₇	#2948	1	2	yes	—
165	C₂×C₄₇	#3155	1	2	yes	—
166	C₂×C₄₇	#3176	1	2	yes	—
167	D₄₇	—	8	8	no	—
168	C₂²×D₈	—	47	2	no	—
169	C₂²×D₈	—	47	2	no	—
170	C₂×#127	—	47	2	no	—
171	C₂×C₄×D₄	—	47	4	no	—
172	#493	—	47	8	no	—
173	#442	—	94	8	no	—
174	C₄×D₈	—	94	8	no	—
175	C₂×#107	—	47	2	no	—
176	C₂×#106	—	47	2	no	—
177	C₂×#102	—	47	4	no	—
178	C₂×#102	—	47	4	no	—
179	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
180	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
181	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#390	—	47	4	no	—
186	#386	—	94	8	no	—
187	#382	—	47	2	no	—
188	#376	C₂×D₄₇	1	1	yes	characteristic
189	C₄₇	#21697	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₄	—	47	4	no	—
191	C₂²×D₄	—	94	4	no	—
192	C₂³×C₄	—	94	4	no	—
193	C₂×D₈	—	47	8	no	—
194	C₂×D₈	—	47	8	no	—
195	C₂×D₈	—	94	4	no	—
196	C₂×D₈	—	94	4	no	—
197	C₂×D₈	—	94	4	no	—
198	C₂×D₈	—	94	4	no	—
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	#127	—	47	8	no	—
204	#127	—	94	4	no	—
205	C₄×D₄	—	94	8	no	—
206	C₄×D₄	—	94	8	no	—
207	C₂×C₄⋊C₄	—	47	4	no	—
208	C₂×C₂²⋊C₄	—	94	8	no	—
209	C₂×C₂²⋊C₄	—	94	4	no	—
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	#107	—	94	4	no	—
214	#106	—	94	4	no	—
215	#102	—	94	8	no	—
216	#102	—	94	8	no	—
217	#97	—	2	2	no	—
218	#97	—	2	2	no	—
219	#97	D₁₈₈	1	4	yes	complemented
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
222	C₄×C₈	D₁₈₈	1	4	yes	complemented
223	#95	—	94	4	no	—
224	#95	D₁₈₈	1	4	yes	—
225	C₂⁴	—	94	4	no	—
226	C₂×D₄	—	47	8	no	—
227	C₂×D₄	—	94	16	no	—
228	C₂×D₄	—	94	4	no	—
229	C₂×D₄	—	94	4	no	—
230	C₂×D₄	—	94	4	no	—
231	C₂×D₄	—	94	4	no	—
232	C₂×D₄	—	94	4	no	—
233	C₂×D₄	—	376	8	no	—
234	C₂²×C₄	—	94	4	no	—
235	C₂²×C₄	—	376	8	no	—
236	C₂²×C₄	—	188	8	no	—
237	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
238	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
239	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
240	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
241	D₈	—	94	16	no	—
242	D₈	—	94	16	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
249	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
250	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
251	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
252	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
253	C₂×C₈	#10625350	1	2	yes	—
254	C₂×C₈	#10625350	1	2	yes	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	—	94	4	no	—
257	C₂²⋊C₄	—	376	8	no	—
258	C₂²⋊C₄	—	188	8	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	C₂×C₄×D₄₇	1	2	yes	—
262	C₄²	C₂×D₁₈₈	1	2	yes	—
263	C₄²	C₂×D₁₈₈	1	2	yes	—
264	C₄²	#10625350	1	2	yes	—
265	C₂³	—	94	8	no	—
266	C₂³	—	188	4	no	—
267	C₂³	—	188	8	no	—
268	C₂³	—	188	8	no	—
269	C₂³	—	188	4	no	—
270	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
271	D₄	—	94	8	no	—
272	D₄	—	94	8	no	—
273	D₄	—	376	16	no	—
274	D₄	—	188	4	no	—
275	D₄	—	188	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	376	16	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	—	188	8	no	—
283	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
284	C₂×C₄	#13094021	1	4	yes	—
285	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	C₄×D₁₈₈	1	4	yes	—
294	C₈	C₄×D₁₈₈	1	4	yes	—
295	C₂²	—	376	8	no	—
296	C₂²	—	376	4	no	—
297	C₂²	—	376	4	no	—
298	C₂²	—	188	16	no	—
299	C₂²	—	188	8	no	—
300	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
301	C₂²	C₂×#13094021	1	1	yes	characteristic
302	C₂²	#15689673	1	1	yes	characteristic
303	C₂²	#15689928	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	376	4	no	—
308	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
309	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
310	C₄	#15689673	1	1	yes	characteristic
311	C₄	#15689673	1	1	yes	characteristic
312	C₂	—	376	8	no	—
313	C₂	C₄×#13094021	1	1	yes	characteristic
314	C₂	#20252241	1	2	yes	—
315	C₂	#20253874	1	2	yes	—
316	C₂	#20254038	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#15689500	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21697
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#15689236	C₂	semidirect
3. C₂×C₄×D₃₇₆	C₂	semidirect
4. #20246113	C₂	semidirect
5. #20244264	C₂	semidirect
6. #20244254	C₂	semidirect
7. #20244212	C₂	semidirect
8. #20243552	C₂	semidirect
9. C₄₇×#1726	C₂	semidirect
14. #15689500	C₄	semidirect
38. C₂×D₃₇₆	C₂×C₄	semidirect
39. C₂×D₃₇₆	C₂×C₄	semidirect
41. #13094021	C₂×C₄	semidirect
61. #97×C₄₇	D₄	semidirect
63. C₄×C₈×C₄₇	C₂×C₄	semidirect
64. C₄×C₈×C₄₇	D₄	semidirect
162. #1726	D₄₇	semidirect
189. C₄₇	#21697	semidirect
219. #97	D₁₈₈	semidirect
221. C₄×C₈	C₄×D₄₇	semidirect
222. C₄×C₈	D₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689500	2
C₂×#15689236	2
C₂×C₄×D₃₇₆	2
#20246113	2
#20244264	2
#20244254	2
#20244212	2
#20243552	2
C₄₇×#1726	2
#21697	47