Subgroups — #26131506

Order 12032.

442 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252159	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688931	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688929	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688898	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	1	yes	characteristic
9	#15689591	—	2	1	no	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#15689217	C₄	1	2	yes	—
12	#15689210	—	2	2	no	—
13	#15689205	C₄	1	2	yes	—
14	C₂×#13093960	C₂²	1	1	yes	characteristic
15	#15689117	—	2	1	no	—
16	M₁₆×Dic₄₇	—	2	1	no	—
17	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	#15688931	C₄	1	4	yes	complemented
19	#15688929	C₄	1	4	yes	complemented
20	#15688898	—	2	2	no	—
21	#15688887	—	2	1	no	—
22	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#427	C₂²	1	1	yes	characteristic
25	C₂×#10625350	—	2	1	no	—
26	#13094046	—	2	1	no	—
27	#13094046	—	2	1	no	—
28	#13094046	C₂×C₄	1	2	yes	—
29	#13094046	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
31	C₂×#10625325	—	2	1	no	—
32	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
33	C₂×#10625329	—	2	1	no	—
34	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	#13093989	—	2	2	no	—
36	#13093986	—	2	2	no	—
37	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
38	#13093981	—	4	1	no	—
39	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
40	C₄²×D₄₇	—	4	1	no	—
41	C₂×C₄₇⋊C₁₆	—	2	1	no	—
42	C₂×C₄₇⋊C₁₆	—	2	1	no	—
43	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
44	#13093960	—	2	1	no	—
45	#13093960	C₂×C₄	1	2	yes	complemented
46	#13093947	—	2	2	no	—
47	#13093944	—	2	2	no	—
48	#13093934	—	4	1	no	—
49	C₈×Dic₄₇	—	4	1	no	—
50	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
51	C₈×Dic₄₇	D₄	1	2	yes	—
52	C₂×M₁₆×C₄₇	—	2	1	no	—
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	#117×C₄₇	—	2	1	no	—
55	C₂×C₁₆×C₄₇	—	2	1	no	—
56	C₂×C₁₆×C₄₇	—	2	1	no	—
57	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
58	#108×C₄₇	—	2	1	no	—
59	#108×C₄₇	C₂×C₄	1	2	yes	complemented
60	#107×C₄₇	C₂×C₄	1	2	yes	—
61	#106×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₇	—	4	1	no	—
64	C₂²×Dic₄₇	SD₁₆	1	2	yes	—
65	#10625350	—	2	2	no	—
66	#10625350	—	2	2	no	—
67	#10625350	—	4	1	no	—
68	#10625350	—	4	1	no	—
69	C₂×D₁₈₈	—	2	1	no	—
70	C₂×C₄×D₄₇	—	4	1	no	—
71	C₂×C₄×D₄₇	—	4	2	no	—
72	C₂×Dic₉₄	—	2	1	no	—
73	M₁₆×C₄₇	—	2	2	no	—
74	M₁₆×C₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	—	4	1	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	—	2	2	no	—
83	C₂²⋊C₄×C₄₇	—	4	1	no	—
84	C₄²×C₄₇	—	4	1	no	—
85	(C₂×D₄₇)⋊C₄	—	4	2	no	—
86	Dic₄₇⋊C₄	—	4	2	no	—
87	C₄×Dic₄₇	—	4	2	no	—
88	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₄₇	D₈	1	1	yes	characteristic
90	C₄×Dic₄₇	D₈	1	1	yes	characteristic
91	#10625329	—	2	2	no	—
92	#10625329	—	4	1	no	—
93	C₂×C₄₇⋊C₈	—	2	1	no	—
94	C₂×C₄₇⋊C₈	—	2	1	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
97	Dic₁₈₈	—	2	1	no	—
98	Dic₁₈₈	C₄²	1	2	yes	—
99	D₃₇₆	—	2	1	no	—
100	D₃₇₆	C₄²	1	2	yes	—
101	#10625325	—	2	2	no	—
102	#10625325	—	2	2	no	—
103	C₁₆×C₄₇	—	2	2	no	—
104	C₄₇⋊C₁₆	—	2	2	no	—
105	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂²×D₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
114	C₂²⋊D₄₇	—	4	2	no	—
115	C₂²⋊D₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	C₂×Dic₄₇	—	4	1	no	—
118	C₂×Dic₄₇	#102	1	2	yes	—
119	C₂×Dic₄₇	#102	1	2	yes	—
120	D₁₈₈	—	2	2	no	—
121	D₁₈₈	—	4	1	no	—
122	C₄×D₄₇	—	4	2	no	—
123	C₄×D₄₇	—	4	2	no	—
124	C₄×D₄₇	—	8	4	no	—
125	Dic₉₄	—	2	2	no	—
126	Dic₉₄	—	4	1	no	—
127	C₈×C₄₇	—	4	2	no	—
128	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₄₇	#117	1	1	yes	characteristic
130	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
131	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
132	C₄₇⋊C₈	—	2	2	no	—
133	C₄₇⋊C₈	—	4	2	no	—
134	#12640	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
137	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
138	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
139	C₂×D₄₇	—	4	2	no	—
140	C₂×D₄₇	—	8	2	no	—
141	C₄×C₄₇	—	8	2	no	—
142	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₄₇	#416	1	1	yes	characteristic
144	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
145	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
146	Dic₄₇	—	2	2	no	—
147	Dic₄₇	—	2	2	no	—
148	Dic₄₇	—	4	2	no	—
149	C₄×#135	—	47	1	no	—
150	C₂×#359	—	47	1	no	—
151	C₂×#359	—	47	1	no	—
152	C₂×#346	—	47	1	no	—
153	C₄×#108	—	47	1	no	—
154	#1382	—	47	1	no	—
155	#1382	D₄₇	1	1	yes	characteristic, complemented
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
159	C₂×C₄₇	#2143	1	2	yes	—
160	D₄₇	—	8	2	no	—
161	C₂×#135	—	47	1	no	—
162	C₄×#41	—	94	1	no	—
163	C₂²×C₁₆	—	47	1	no	—
164	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
165	C₄×Q₁₆	—	47	2	no	—
166	C₄×SD₁₆	—	94	2	no	—
167	C₄×D₈	—	47	2	no	—
168	C₂×#108	—	47	1	no	—
169	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
170	#427	C₂×D₄₇	1	1	yes	characteristic
171	#416	—	94	1	no	—
172	C₄×M₁₆	—	94	1	no	—
173	C₂×C₄×C₈	—	47	1	no	—
174	#359	—	47	4	no	—
175	#359	—	47	4	no	—
176	#346	—	94	2	no	—
177	#341	—	94	1	no	—
178	C₄₇	#12640	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	94	1	no	—
180	#135	—	47	2	no	—
181	#135	—	47	2	no	—
182	#135	—	94	1	no	—
183	#135	—	94	1	no	—
184	C₂×Q₁₆	—	47	1	no	—
185	C₂×SD₁₆	—	94	1	no	—
186	C₂×D₈	—	47	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	94	1	no	—
189	C₂²×C₈	—	47	1	no	—
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	C₄×Q₈	—	94	2	no	—
192	C₄×D₄	—	94	2	no	—
193	C₄×D₄	—	188	2	no	—
194	#117	—	2	1	no	—
195	#117	—	188	1	no	—
196	C₂×C₄²	—	47	1	no	—
197	C₂×C₄²	—	188	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	47	2	no	—
201	C₂×C₁₆	—	94	1	no	—
202	C₂×C₁₆	—	94	1	no	—
203	C₂×C₁₆	C₄×D₄₇	1	2	yes	complemented
204	#108	—	2	1	no	—
205	#108	—	47	2	no	—
206	#108	—	94	1	no	—
207	#108	C₄×D₄₇	1	2	yes	complemented
208	#107	C₄×D₄₇	1	2	yes	—
209	#106	—	2	1	no	—
210	#103	—	94	2	no	—
211	#102	—	94	2	no	—
212	#97	—	188	1	no	—
213	C₄×C₈	—	47	2	no	—
214	C₄×C₈	—	47	2	no	—
215	C₄×C₈	—	188	1	no	—
216	#41	—	94	2	no	—
217	#41	—	94	2	no	—
218	#41	—	188	1	no	—
219	#41	—	188	1	no	—
220	C₂×Q₈	—	94	1	no	—
221	C₂×D₄	—	94	1	no	—
222	C₂×D₄	—	188	1	no	—
223	C₂²×C₄	—	47	2	no	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	—	188	1	no	—
226	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
227	Q₁₆	—	47	2	no	—
228	Q₁₆	—	94	1	no	—
229	SD₁₆	—	94	2	no	—
230	SD₁₆	—	94	2	no	—
231	D₈	—	47	2	no	—
232	D₈	—	94	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	94	2	no	—
236	M₁₆	—	188	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	47	2	no	—
239	C₂×C₈	—	94	1	no	—
240	C₂×C₈	—	94	1	no	—
241	C₂×C₈	—	188	1	no	—
242	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
246	C₂×C₈	#10625352	1	1	yes	characteristic
247	C₂×C₈	#10625354	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	188	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	188	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	47	2	no	—
254	C₄²	—	47	1	no	—
255	C₄²	—	47	1	no	—
256	C₄²	—	188	2	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	94	2	no	—
259	C₂³	—	188	1	no	—
260	C₂³	SD₁₆×D₄₇	1	1	yes	characteristic
261	Q₈	—	94	2	no	—
262	Q₈	—	188	1	no	—
263	D₄	—	94	2	no	—
264	D₄	—	188	2	no	—
265	D₄	—	188	2	no	—
266	D₄	—	188	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	47	2	no	—
269	C₂×C₄	—	47	2	no	—
270	C₂×C₄	—	376	4	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	—	188	1	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	#13093986	1	1	yes	characteristic
278	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
279	C₂×C₄	#13094045	1	1	yes	characteristic
280	C₂×C₄	#13094047	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	94	2	no	—
283	C₈	—	188	2	no	—
284	C₈	C₄²×D₄₇	1	1	yes	characteristic
285	C₈	#13093982	1	1	yes	characteristic
286	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
287	C₈	#13093986	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	376	2	no	—
290	C₂²	—	188	2	no	—
291	C₂²	#102×D₄₇	1	1	yes	characteristic
292	C₂²	#15689208	1	1	yes	characteristic
293	C₂²	#15689210	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	94	2	no	—
296	C₄	—	94	2	no	—
297	C₄	—	188	2	no	—
298	C₄	#15688970	1	1	yes	characteristic
299	C₄	#15689118	1	1	yes	characteristic
300	C₄	#15689203	1	1	yes	characteristic
301	C₄	#15689205	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	376	2	no	—
305	C₂	#20243202	1	1	yes	characteristic
306	C₂	#20245114	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1382	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1382	D₄₇
hypocenter	C₄₇	#12640
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₂×#15688898	C₂	semidirect
5. #108×Dic₄₇	C₂	semidirect
6. #20241483	C₂	semidirect
7. C₄₇×#1382	C₂	semidirect
18. #15688931	C₄	semidirect
19. #15688929	C₄	semidirect
43. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
45. #13093960	C₂×C₄	semidirect
57. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
59. #108×C₄₇	C₂×C₄	semidirect
155. #1382	D₄₇	semidirect
178. C₄₇	#12640	semidirect
203. C₂×C₁₆	C₄×D₄₇	semidirect
207. #108	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252159	2
C₂×#15688931	2
C₂×#15688929	2
C₂×#15688898	2
#108×Dic₄₇	2
#20241483	2
C₄₇×#1382	2
#12640	47