Subgroups — #26220576

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246031	C₂	1	2	yes	maximal, complemented
2	#20244265	C₂	1	2	yes	maximal
3	#20244127	C₂	1	2	yes	maximal, complemented
4	#20244073	C₂	1	4	yes	maximal, complemented
5	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243213	C₂	1	2	yes	maximal, complemented
7	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093956	C₂²	1	2	yes	—
10	C₂×#13093945	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	1	yes	characteristic
12	C₂×#13093934	C₂²	1	2	yes	—
13	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
14	#15689026	C₂²	1	4	yes	—
15	#15689022	C₂²	1	4	yes	—
16	#15689016	—	2	4	no	—
17	#15688986	—	2	2	no	—
18	#15688983	—	2	2	no	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	#15688969	C₂²	1	1	yes	characteristic
21	#15688886	C₂²	1	2	yes	—
22	#15688885	C₂²	1	4	yes	—
23	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
25	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	#13093956	—	2	4	no	—
37	#13093945	—	2	8	no	—
38	#13093935	—	2	2	no	—
39	#13093934	—	2	4	no	—
40	C₈×Dic₄₇	—	2	4	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₄₇⋊#95	—	2	4	no	—
43	C₄₇⋊#95	D₄	1	2	yes	—
44	C₄₇⋊#95	D₄	1	2	yes	—
45	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₇	—	2	2	no	—
47	C₂×D₈×C₄₇	—	2	4	no	—
48	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₄×D₄×C₄₇	—	2	4	no	—
52	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#107×C₄₇	C₂×C₄	1	4	yes	—
57	#102×C₄₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
59	C₂⁴×C₄₇	—	2	2	no	—
60	C₂×D₄×C₄₇	—	2	4	no	—
61	C₂×D₄×C₄₇	—	2	4	no	—
62	C₂×D₄×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	8	4	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
72	D₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	#41	1	4	yes	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²⋊C₄×C₄₇	—	2	8	no	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	#41	1	2	yes	—
82	C₂²⋊Dic₄₇	—	8	4	no	—
83	C₂²⋊Dic₄₇	—	8	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	8	4	no	—
87	C₄×Dic₄₇	—	2	2	no	—
88	C₄×Dic₄₇	—	2	2	no	—
89	C₄×Dic₄₇	—	8	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	#41	1	4	yes	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	4	4	no	—
95	C₂³×C₄₇	—	4	4	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	D₄×C₄₇	—	2	4	no	—
100	D₄×C₄₇	—	2	4	no	—
101	D₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	8	8	no	—
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	#117	1	4	yes	—
121	C₄₇⋊C₈	—	2	4	no	—
122	#21284	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	4	4	no	—
124	C₂²×C₄₇	—	4	4	no	—
125	C₂²×C₄₇	—	4	4	no	—
126	C₂²×C₄₇	—	8	4	no	—
127	C₂²×C₄₇	—	8	4	no	—
128	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₇	#532	1	2	yes	—
130	C₂²×C₄₇	#576	1	2	yes	—
131	C₂²×C₄₇	#577	1	2	yes	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
136	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
137	C₄×C₄₇	#529	1	2	yes	—
138	Dic₄₇	—	4	4	no	—
139	Dic₄₇	—	8	4	no	—
140	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
141	#2319	—	47	2	no	—
142	#1972	—	47	2	no	—
143	#1943	—	47	2	no	—
144	#1934	—	47	4	no	—
145	C₄×#106	—	47	1	no	—
146	#1762	—	47	2	no	—
147	#1726	—	47	1	no	—
148	C₂×C₄₇	—	8	4	no	—
149	C₂×C₄₇	#2293	1	1	yes	characteristic
150	C₂×C₄₇	#2948	1	1	yes	characteristic
151	C₂×C₄₇	#2952	1	1	yes	characteristic
152	C₂×C₄₇	#3179	1	4	yes	—
153	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
155	C₄×D₈	Dic₄₇	1	8	yes	complemented
156	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#106	—	47	2	no	—
158	C₂×#102	—	47	4	no	—
159	C₂×#102	C₂×D₄₇	1	2	yes	—
160	C₂×#97	—	47	2	no	—
161	C₂×#97	—	47	1	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
164	#390	—	47	4	no	—
165	#389	—	47	4	no	—
166	#388	—	94	4	no	—
167	#381	—	94	2	no	—
168	#380	—	94	2	no	—
169	C₄×C₄⋊C₄	—	47	2	no	—
170	#376	—	47	1	no	—
171	#340	—	47	2	no	—
172	#340	—	47	4	no	—
173	C₄₇	#21284	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₄×D₄	—	2	4	no	—
182	C₄×D₄	C₂×Dic₄₇	1	8	yes	—
183	C₂×C₄⋊C₄	—	47	4	no	—
184	C₂×C₄⋊C₄	—	94	4	no	—
185	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	94	4	no	—
188	C₂×C₂²⋊C₄	—	94	4	no	—
189	C₂×C₄²	—	47	2	no	—
190	C₂×C₄²	—	94	2	no	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#107	C₂×Dic₄₇	1	4	yes	—
193	#106	—	94	4	no	—
194	#102	—	94	8	no	—
195	#102	C₂×Dic₄₇	1	8	yes	—
196	#97	—	94	4	no	—
197	#97	—	94	2	no	—
198	C₄×C₈	—	94	4	no	—
199	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
200	#95	—	47	2	no	—
201	#95	—	47	2	no	—
202	#95	—	94	4	no	—
203	#95	—	94	2	no	—
204	C₂⁴	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	8	4	no	—
212	C₂²×C₄	—	47	4	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	D₈	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	47	4	no	—
220	C₂×C₈	—	94	2	no	—
221	C₂×C₈	—	94	2	no	—
222	C₂×C₈	#10625350	1	4	yes	—
223	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
224	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	94	4	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₄⋊C₄	—	376	4	no	—
229	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	376	4	no	—
232	C₂²⋊C₄	—	376	4	no	—
233	C₄²	—	94	2	no	—
234	C₄²	—	94	2	no	—
235	C₄²	—	376	2	no	—
236	C₄²	#10625352	1	2	yes	—
237	C₄²	C₂²×Dic₄₇	1	2	yes	—
238	C₂³	—	2	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
244	D₄	—	2	4	no	—
245	D₄	—	2	4	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	—	94	4	no	—
253	C₂×C₄	—	94	4	no	—
254	C₂×C₄	—	94	4	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
259	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
260	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
261	C₂×C₄	#41×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
263	C₂×C₄	C₂×#10625350	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	94	4	no	—
266	C₈	#13093983	1	4	yes	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
273	C₂²	#15689670	1	2	yes	—
274	C₂²	#15689924	1	2	yes	—
275	C₂²	#15689930	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	376	4	no	—
280	C₄	—	188	4	no	—
281	C₄	C₂×#13093983	1	1	yes	characteristic
282	C₄	#41×Dic₄₇	1	1	yes	characteristic
283	C₄	#15689655	1	2	yes	—
284	C₂	—	8	4	no	—
285	C₂	#20245853	1	1	yes	characteristic
286	C₂	#20252240	1	1	yes	characteristic
287	C₂	#20252272	1	1	yes	characteristic
288	C₂	#20254075	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#21284
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20246031	C₂	semidirect
3. #20244127	C₂	semidirect
4. #20244073	C₂	semidirect
5. C₄×#13093956	C₂	semidirect
6. #20243213	C₂	semidirect
7. #20243074	C₂	semidirect
25. C₄×D₈×C₄₇	C₄	semidirect
155. C₄×D₈	Dic₄₇	semidirect
173. C₄₇	#21284	semidirect

Maximal subgroups

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

subgroup	index
#20246031	2
#20244265	2
#20244127	2
#20244073	2
C₄×#13093956	2
#20243213	2
#20243074	2
C₂×C₄×D₈×C₄₇	2
#21284	47