Subgroups — #26369650

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093968	C₂	1	1	yes	maximal, characteristic
2	#20252255	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252240	C₂	1	1	yes	maximal, characteristic
4	#20245249	C₂	1	4	yes	maximal, complemented
5	#20245243	C₂	1	4	yes	maximal, complemented
6	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245109	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2140	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	#15689600	—	2	1	no	—
12	#15689598	—	2	1	no	—
13	C₂×#13093968	C₄	1	8	yes	complemented
14	C₂×#13093968	C₂²	1	2	yes	—
15	C₂×#13093968	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#15689235	—	2	2	no	—
18	#15689229	—	2	2	no	—
19	#15689217	C₂²	1	4	yes	complemented
20	D₈×Dic₄₇	C₂²	1	4	yes	—
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689131	—	2	1	no	—
23	#15689128	—	2	1	no	—
24	#15689078	C₂²	1	1	yes	characteristic
25	#15688950	C₂²	1	2	yes	complemented
26	#15688948	C₂²	1	2	yes	complemented
27	#15688928	C₂²	1	4	yes	complemented
28	#15688898	C₂²	1	2	yes	complemented
29	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
30	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#427	C₂²	1	1	yes	characteristic
33	C₄₇×#357	C₂²	1	4	yes	—
34	C₂²×Dic₉₄	—	2	1	no	—
35	C₂×Dic₁₈₈	—	2	1	no	—
36	C₂×Dic₁₈₈	—	2	2	no	—
37	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	complemented
38	C₂×Dic₁₈₈	C₂³	1	2	yes	—
39	C₂×#10625329	D₄	1	1	yes	characteristic
40	C₄×Dic₉₄	—	4	2	no	—
41	#13093989	—	2	4	no	—
42	D₄×Dic₄₇	—	2	4	no	—
43	D₄×Dic₄₇	—	4	2	no	—
44	#13093983	—	2	1	no	—
45	#13093983	D₄	1	1	yes	characteristic
46	#13093980	—	4	2	no	—
47	C₂×C₂²⋊Dic₄₇	—	4	2	no	—
48	#13093968	C₂×C₄	1	8	yes	complemented
49	#13093968	C₂×C₄	1	8	yes	complemented
50	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
51	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	#13093958	C₂³	1	2	yes	—
54	#13093956	—	2	1	no	—
55	#13093947	—	2	4	no	—
56	#13093945	—	2	4	no	—
57	#13093934	D₄	1	2	yes	—
58	C₈×Dic₄₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₄₇	—	2	1	no	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	2	no	—
62	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₂×D₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	#117×C₄₇	—	2	1	no	—
66	M₃₂×C₄₇	—	2	2	no	—
67	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
68	#107×C₄₇	C₂³	1	2	yes	—
69	#106×C₄₇	—	2	1	no	—
70	C₂⁴×C₄₇	—	4	1	no	—
71	C₂×D₄×C₄₇	—	2	4	no	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	4	1	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	4	1	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂×Dic₉₄	—	2	4	no	—
79	C₂×Dic₉₄	—	2	2	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	2	no	—
82	D₈×C₄₇	—	2	4	no	—
83	D₈×C₄₇	—	2	2	no	—
84	D₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	#41	1	2	yes	—
88	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₂²⋊C₄×C₄₇	—	4	1	no	—
91	C₄²×C₄₇	—	4	1	no	—
92	C₂²⋊Dic₄₇	—	4	1	no	—
93	C₂²⋊Dic₄₇	—	4	4	no	—
94	C₂²⋊Dic₄₇	D₈	1	2	yes	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	D₈	1	2	yes	—
97	Dic₄₇⋊C₄	—	4	4	no	—
98	C₄×Dic₄₇	—	4	1	no	—
99	C₄×Dic₄₇	—	4	4	no	—
100	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
101	#10625329	—	2	2	no	—
102	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
103	Dic₁₈₈	—	2	4	no	—
104	Dic₁₈₈	—	2	2	no	—
105	Dic₁₈₈	C₂²×C₄	1	4	yes	—
106	C₁₆×C₄₇	—	2	2	no	—
107	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₇	—	4	2	no	—
110	C₂³×C₄₇	—	4	4	no	—
111	C₂³×C₄₇	—	16	2	no	—
112	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
113	D₄×C₄₇	—	2	4	no	—
114	D₄×C₄₇	—	4	2	no	—
115	D₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#135	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	16	4	no	—
125	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
126	Dic₉₄	—	2	4	no	—
127	Dic₉₄	—	4	2	no	—
128	Dic₉₄	—	4	4	no	—
129	C₈×C₄₇	C₄×D₄	1	2	yes	—
130	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₄₇⋊C₈	—	2	2	no	—
133	#33457	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	—	8	4	no	—
137	C₂²×C₄₇	—	16	4	no	—
138	C₂²×C₄₇	C₄×D₈	1	2	yes	—
139	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₄₇	—	8	2	no	—
141	C₄×C₄₇	C₄×D₈	1	2	yes	—
142	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	4	4	no	—
146	Dic₄₇	—	8	2	no	—
147	C₂²×SD₃₂	—	47	1	no	—
148	#2950	—	47	1	no	—
149	#2948	—	47	1	no	—
150	#2175	—	47	4	no	—
151	C₄×SD₃₂	—	47	4	no	—
152	#2158	—	47	1	no	—
153	#2141	—	47	1	no	—
154	#2140	D₄₇	1	1	yes	characteristic
155	#2108	—	47	1	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	—	8	4	no	—
158	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
159	C₂×C₄₇	#3416	1	2	yes	—
160	C₂²×Q₁₆	—	47	1	no	—
161	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
162	#519	—	94	1	no	—
163	#518	—	94	1	no	—
164	C₂×SD₃₂	—	47	8	no	—
165	C₂×SD₃₂	—	47	2	no	—
166	C₂×SD₃₂	—	47	2	no	—
167	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
168	C₂²×C₁₆	—	47	1	no	—
169	#442	—	94	2	no	—
170	#441	—	94	2	no	—
171	C₄×Q₁₆	—	47	4	no	—
172	C₄×D₈	—	47	4	no	—
173	#427	—	47	1	no	—
174	#427	C₂×D₄₇	1	1	yes	characteristic
175	#419	—	94	1	no	—
176	#418	—	94	1	no	—
177	#405	—	47	1	no	—
178	#367	—	47	2	no	—
179	#366	—	47	2	no	—
180	#358	—	47	4	no	—
181	#357	C₂×D₄₇	1	4	yes	—
182	#346	—	47	2	no	—
183	C₄×C₁₆	—	47	2	no	—
184	C₄₇	#33457	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	94	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	47	4	no	—
188	C₂×Q₁₆	—	47	2	no	—
189	C₂×Q₁₆	—	94	1	no	—
190	C₂×Q₁₆	—	94	2	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	C₄×D₄₇	1	4	yes	—
194	C₂×D₈	C₂²×D₄₇	1	2	yes	—
195	C₂×M₁₆	—	47	1	no	—
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄×Q₈	—	94	4	no	—
198	C₄×Q₈	—	188	2	no	—
199	C₄×D₄	—	94	4	no	—
200	C₄×D₄	—	188	2	no	—
201	#117	—	2	1	no	—
202	#117	—	47	1	no	—
203	#117	—	94	1	no	—
204	#117	—	188	2	no	—
205	C₂×C₂²⋊C₄	—	188	2	no	—
206	SD₃₂	—	47	8	no	—
207	SD₃₂	—	47	8	no	—
208	M₃₂	—	2	2	no	—
209	C₂×C₁₆	—	47	4	no	—
210	C₂×C₁₆	—	47	1	no	—
211	C₂×C₁₆	—	47	1	no	—
212	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
213	#107	—	47	2	no	—
214	#107	C₂²×D₄₇	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	94	1	no	—
217	#103	—	94	4	no	—
218	#102	—	94	4	no	—
219	#97	—	47	2	no	—
220	C₄×C₈	—	47	2	no	—
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	94	4	no	—
223	C₂×Q₈	—	94	2	no	—
224	C₂×Q₈	—	188	1	no	—
225	C₂×Q₈	—	188	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	188	4	no	—
231	C₂²×C₄	—	188	1	no	—
232	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
233	Q₁₆	—	47	4	no	—
234	Q₁₆	—	94	4	no	—
235	Q₁₆	—	94	2	no	—
236	D₈	—	2	4	no	—
237	D₈	—	2	2	no	—
238	D₈	C₂×C₄×D₄₇	1	4	yes	—
239	M₁₆	—	94	2	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
242	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
243	C₂×C₈	#10625352	1	2	yes	—
244	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	47	2	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₄⋊C₄	—	188	4	no	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	47	2	no	—
251	C₂²⋊C₄	—	188	4	no	—
252	C₂²⋊C₄	—	188	1	no	—
253	C₄²	—	4	1	no	—
254	C₄²	—	47	2	no	—
255	C₄²	—	188	4	no	—
256	C₄²	—	188	1	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	47	2	no	—
259	C₁₆	—	47	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	D₈×D₄₇	1	1	yes	characteristic
264	Q₈	—	94	4	no	—
265	Q₈	—	188	2	no	—
266	Q₈	—	188	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	47	4	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	—	752	4	no	—
276	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
277	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
278	C₂×C₄	#13094045	1	2	yes	—
279	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
280	C₈	—	94	2	no	—
281	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
282	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
283	C₈	C₂×#10625352	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	#15689203	1	2	yes	—
289	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
290	C₄	—	8	2	no	—
291	C₄	—	94	4	no	—
292	C₄	—	376	2	no	—
293	C₄	—	188	4	no	—
294	C₄	#15689203	1	2	yes	—
295	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
296	C₄	C₂×#13094045	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×#15689203	1	1	yes	characteristic
300	C₂	#20256534	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2140	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2140	D₄₇
hypocenter	C₄₇	#33457
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2140

Complements

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

N	H ≅ G/N	type
2. #20252255	C₂	semidirect
4. #20245249	C₂	semidirect
5. #20245243	C₂	semidirect
6. #20245170	C₂	semidirect
7. #20245109	C₂	semidirect
8. #20244987	C₂	semidirect
13. C₂×#13093968	C₄	semidirect
19. #15689217	C₂²	semidirect
25. #15688950	C₂²	semidirect
26. #15688948	C₂²	semidirect
27. #15688928	C₂²	semidirect
28. #15688898	C₂²	semidirect
29. C₁₆×Dic₄₇	C₂²	semidirect
37. C₂×Dic₁₈₈	C₂×C₄	semidirect
48. #13093968	C₂×C₄	semidirect
49. #13093968	C₂×C₄	semidirect
50. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
184. C₄₇	#33457	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093968	2
#20252255	2
#20252240	2
#20245249	2
#20245243	2
#20245170	2
#20245109	2
#20244987	2
C₄₇×#2140	2
#33457	47