Subgroups — #26372717

Order 12032.

488 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254027	C₂	1	2	yes	maximal, complemented
2	#20245541	C₂	1	2	yes	maximal, complemented
3	#20245526	C₂	1	2	yes	maximal, complemented
4	#20245501	C₂	1	2	yes	maximal, complemented
5	#20245479	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245474	C₂	1	1	yes	maximal, characteristic
7	#20245451	C₂	1	2	yes	maximal, complemented
8	#20245444	C₂	1	2	yes	maximal, complemented
9	C₄₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689744	—	2	2	no	—
11	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
12	C₂×#13093969	C₂²	1	2	yes	—
13	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
14	#15689528	C₂²	1	2	yes	—
15	#15689507	C₂²	1	2	yes	—
16	#15689483	—	2	2	no	—
17	#15689369	—	2	2	no	—
18	#15689364	C₂²	1	2	yes	—
19	#15689359	C₂²	1	2	yes	—
20	#15689225	—	2	2	no	—
21	C₄×Dic₁₈₈	C₂²	1	2	yes	—
22	C₄×D₃₇₆	C₂²	1	2	yes	—
23	#15688949	C₂²	1	1	yes	characteristic
24	#15688949	C₂²	1	2	yes	—
25	#15688949	C₂²	1	1	yes	characteristic
26	#15688928	C₂²	1	2	yes	—
27	#15688928	C₂²	1	2	yes	—
28	#15688925	C₂²	1	2	yes	—
29	#15688925	C₂²	1	2	yes	—
30	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#405	C₂²	1	1	yes	characteristic
33	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₂×Dic₁₈₈	C₂³	1	2	yes	—
36	C₂×#10625325	—	2	2	no	—
37	C₂×D₃₇₆	C₂³	1	2	yes	—
38	#13094024	—	2	2	no	—
39	#13094019	—	4	2	no	—
40	#13094012	—	2	2	no	—
41	#13094010	—	4	2	no	—
42	#13094006	—	4	2	no	—
43	#13094003	—	2	2	no	—
44	#13094002	—	4	2	no	—
45	#13093998	—	2	2	no	—
46	C₄×Dic₉₄	—	2	2	no	—
47	C₄×D₁₈₈	—	2	2	no	—
48	Dic₃₇₆	—	4	1	no	—
49	#13093969	—	4	2	no	—
50	D₇₅₂	—	4	1	no	—
51	#13093958	—	2	2	no	—
52	#13093958	C₂³	1	2	yes	—
53	#13093958	C₂³	1	2	yes	—
54	#13093949	—	2	2	no	—
55	#13093949	—	2	2	no	—
56	#13093946	—	2	2	no	—
57	#13093946	—	2	2	no	—
58	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	#117×C₄₇	D₄	1	1	yes	characteristic
61	M₃₂×C₄₇	—	4	1	no	—
62	C₂×C₁₆×C₄₇	—	4	1	no	—
63	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
66	#97×C₄₇	D₄	1	2	yes	—
67	C₄×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	4	2	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×C₄×D₄₇	—	4	2	no	—
72	C₂×Dic₉₄	—	2	2	no	—
73	C₂×Dic₉₄	—	4	2	no	—
74	M₁₆×C₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
82	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
83	C₄²×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²⋊Dic₄₇	—	4	2	no	—
85	(C₂×D₄₇)⋊C₄	—	4	2	no	—
86	(C₂×D₄₇)⋊C₄	—	4	2	no	—
87	(C₂×D₄₇)⋊C₄	—	4	2	no	—
88	(C₂×D₄₇)⋊C₄	—	4	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	Dic₄₇⋊C₄	—	4	2	no	—
93	Dic₄₇⋊C₄	—	4	2	no	—
94	C₄×Dic₄₇	—	4	2	no	—
95	Dic₁₈₈	—	2	2	no	—
96	Dic₁₈₈	—	2	2	no	—
97	D₃₇₆	—	2	2	no	—
98	D₃₇₆	—	2	2	no	—
99	#10625325	—	4	4	no	—
100	C₁₆×C₄₇	—	2	1	no	—
101	C₁₆×C₄₇	—	2	1	no	—
102	C₁₆×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂²⋊D₄₇	—	16	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	D₁₈₈	—	4	2	no	—
116	D₁₈₈	—	4	2	no	—
117	C₄×D₄₇	—	16	2	no	—
118	Dic₉₄	—	4	2	no	—
119	Dic₉₄	—	4	2	no	—
120	Dic₉₄	—	16	2	no	—
121	C₈×C₄₇	—	2	1	no	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
125	#33828	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	1	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
130	C₂×D₄₇	—	8	2	no	—
131	C₂×D₄₇	—	8	2	no	—
132	C₂×D₄₇	—	8	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	#537	1	1	yes	characteristic
137	C₄×C₄₇	#575	1	1	yes	characteristic
138	Dic₄₇	—	8	2	no	—
139	Dic₄₇	—	8	2	no	—
140	Dic₄₇	—	8	2	no	—
141	#3175	—	47	2	no	—
142	#2227	—	47	2	no	—
143	#2224	—	47	2	no	—
144	#2220	—	47	2	no	—
145	#2216	—	47	1	no	—
146	#2215	—	47	1	no	—
147	#2211	—	47	2	no	—
148	#2210	—	47	2	no	—
149	#2151	D₄₇	1	1	yes	characteristic, complemented
150	C₂×C₄₇	—	4	1	no	—
151	C₂×C₄₇	#3245	1	1	yes	characteristic
152	C₂×C₄₇	#3415	1	1	yes	characteristic
153	C₂×C₄₇	#3417	1	1	yes	characteristic
154	D₄₇	—	16	2	no	—
155	#542	—	94	2	no	—
156	C₂×Q₃₂	—	47	1	no	—
157	C₂×SD₃₂	—	47	2	no	—
158	C₂×D₁₆	—	47	1	no	—
159	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
160	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
161	#500	—	47	2	no	—
162	#495	—	47	2	no	—
163	#489	—	94	2	no	—
164	#468	—	94	2	no	—
165	#467	—	47	2	no	—
166	#466	—	47	2	no	—
167	#440	—	94	2	no	—
168	C₄×Q₁₆	—	47	2	no	—
169	C₄×D₈	—	47	2	no	—
170	#405	C₂×D₄₇	1	1	yes	characteristic
171	#366	—	47	1	no	—
172	#366	—	47	1	no	—
173	#366	—	47	2	no	—
174	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
175	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
176	#358	—	47	2	no	—
177	#358	—	47	2	no	—
178	#357	—	47	2	no	—
179	#357	—	47	2	no	—
180	C₄₇	#33828	1	1	yes	characteristic, complemented, Sylow
181	C₂×Q₁₆	—	47	2	no	—
182	C₂×SD₁₆	—	94	2	no	—
183	C₂×D₈	—	47	2	no	—
184	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊Q₈	—	94	2	no	—
187	#126	—	188	2	no	—
188	#124	—	94	2	no	—
189	#124	—	188	2	no	—
190	#123	—	188	2	no	—
191	C₂²⋊Q₈	—	94	2	no	—
192	C₂²⋊Q₈	—	188	2	no	—
193	#121	—	94	2	no	—
194	C₄×Q₈	—	94	2	no	—
195	C₄×D₄	—	94	2	no	—
196	#117	D₁₈₈	1	1	yes	characteristic
197	Q₃₂	—	188	1	no	—
198	SD₃₂	—	188	2	no	—
199	D₁₆	—	188	1	no	—
200	M₃₂	—	4	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
204	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
205	#107	—	47	2	no	—
206	#107	—	47	2	no	—
207	#107	—	94	2	no	—
208	#103	—	94	2	no	—
209	#103	—	94	2	no	—
210	#102	—	94	2	no	—
211	#102	—	94	2	no	—
212	#97	D₁₈₈	1	2	yes	—
213	C₄×C₈	C₂²×D₄₇	1	2	yes	—
214	C₂×Q₈	—	94	2	no	—
215	C₂×Q₈	—	188	2	no	—
216	C₂×D₄	—	94	2	no	—
217	C₂×D₄	—	188	2	no	—
218	C₂²×C₄	—	188	2	no	—
219	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
220	Q₁₆	—	94	2	no	—
221	Q₁₆	—	94	2	no	—
222	SD₁₆	—	188	4	no	—
223	D₈	—	94	2	no	—
224	D₈	—	94	2	no	—
225	M₁₆	—	4	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
229	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
231	C₄⋊C₄	—	94	2	no	—
232	C₄⋊C₄	—	94	2	no	—
233	C₄⋊C₄	—	188	2	no	—
234	C₄⋊C₄	—	188	2	no	—
235	C₄⋊C₄	—	188	2	no	—
236	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
237	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
238	C₂²⋊C₄	—	188	2	no	—
239	C₂²⋊C₄	—	188	2	no	—
240	C₂²⋊C₄	—	188	2	no	—
241	C₂²⋊C₄	—	188	2	no	—
242	C₂²⋊C₄	—	188	2	no	—
243	C₂²⋊C₄	D₃₇₆	1	2	yes	—
244	C₄²	—	188	2	no	—
245	C₄²	C₂×D₁₈₈	1	2	yes	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	188	2	no	—
250	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
251	Q₈	—	188	2	no	—
252	Q₈	—	188	2	no	—
253	Q₈	—	752	2	no	—
254	D₄	—	188	2	no	—
255	D₄	—	188	2	no	—
256	D₄	—	752	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	—	188	2	no	—
262	C₂×C₄	—	752	2	no	—
263	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
264	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
265	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	1	no	—
269	C₈	#13094069	1	1	yes	characteristic
270	C₈	#13094072	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	376	2	no	—
275	C₂²	—	376	2	no	—
276	C₂²	—	376	2	no	—
277	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	376	2	no	—
282	C₄	—	376	2	no	—
283	C₄	—	376	2	no	—
284	C₄	#15689704	1	1	yes	characteristic
285	C₄	#15689923	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	752	2	no	—
288	C₂	#20254698	1	1	yes	characteristic
289	C₂	#20256529	1	1	yes	characteristic
290	C₂	#20256541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2151	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20245474	C₂
hypercenter	#2151	D₄₇
hypocenter	C₄₇	#33828
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

Complements

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

N	H ≅ G/N	type
1. #20254027	C₂	semidirect
2. #20245541	C₂	semidirect
3. #20245526	C₂	semidirect
4. #20245501	C₂	semidirect
5. #20245479	C₂	semidirect
7. #20245451	C₂	semidirect
8. #20245444	C₂	semidirect
9. C₄₇×#2151	C₂	semidirect
149. #2151	D₄₇	semidirect
180. C₄₇	#33828	semidirect

Maximal subgroups

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

subgroup	index
#20254027	2
#20245541	2
#20245526	2
#20245501	2
#20245479	2
#20245474	2
#20245451	2
#20245444	2
C₄₇×#2151	2
#33828	47