Subgroups — #26132770

Order 12032.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256490	C₂	1	1	yes	maximal, characteristic
2	#20245153	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245017	C₂	1	1	yes	maximal, characteristic
4	#20243225	C₂	1	1	yes	maximal, characteristic
5	#20241473	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	C₂²	1	1	yes	characteristic
9	#15689541	—	2	2	no	—
10	#15689539	—	2	2	no	—
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	M₃₂×D₄₇	—	2	2	no	—
14	C₂×#13093952	C₄	1	1	yes	characteristic
15	#15689120	C₄	1	1	yes	characteristic
16	#15689078	C₂²	1	1	yes	characteristic
17	#15688943	—	2	2	no	—
18	#15688908	—	2	2	no	—
19	#15688888	C₄	1	1	yes	characteristic
20	#15688883	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×C₄⋊C₁₆	—	2	2	no	—
25	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
26	#13094030	—	2	1	no	—
27	#13094030	—	2	1	no	—
28	#13094030	D₄	1	2	yes	—
29	#13094030	Q₈	1	2	yes	—
30	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
31	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
32	C₂×#10625324	—	2	1	no	—
33	C₂×C₈×D₄₇	—	2	1	no	—
34	#13093983	C₂×C₄	1	1	yes	characteristic
35	#13093964	—	2	1	no	—
36	#13093964	—	2	1	no	—
37	#13093964	—	2	1	no	—
38	#13093964	D₄	1	2	yes	—
39	#13093963	—	2	4	no	—
40	C₂×C₄₇⋊C₁₆	—	2	1	no	—
41	C₂×C₄₇⋊C₁₆	—	2	1	no	—
42	C₁₆×D₄₇	—	2	4	no	—
43	#13093952	C₈	1	2	yes	—
44	#13093952	C₈	1	2	yes	—
45	#13093949	—	2	1	no	—
46	#13093946	—	2	1	no	—
47	#13093934	—	2	1	no	—
48	C₈×Dic₄₇	—	2	1	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₄₇	—	2	1	no	—
53	M₃₂×C₄₇	—	2	1	no	—
54	M₃₂×C₄₇	—	2	1	no	—
55	M₃₂×C₄₇	—	2	1	no	—
56	C₂×C₁₆×C₄₇	—	2	1	no	—
57	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
58	C₄×C₈×C₄₇	—	2	1	no	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₇	—	2	1	no	—
63	#10625350	—	2	1	no	—
64	#10625350	—	2	1	no	—
65	#10625350	C₄⋊C₄	1	2	yes	—
66	#10625350	C₂×C₈	1	2	yes	—
67	C₂×D₁₈₈	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₄₇	—	2	1	no	—
69	C₂×Dic₉₄	M₁₆	1	1	yes	characteristic
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
78	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
79	C₄²×C₄₇	—	2	1	no	—
80	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
82	C₂²⋊Dic₄₇	—	2	1	no	—
83	C₄⋊Dic₄₇	—	2	1	no	—
84	C₄×Dic₄₇	—	2	1	no	—
85	#10625329	—	2	1	no	—
86	#10625329	—	2	1	no	—
87	#10625329	C₄⋊C₄	1	2	yes	—
88	#10625329	C₂×C₈	1	2	yes	—
89	C₂×C₄₇⋊C₈	—	2	1	no	—
90	C₂×C₄₇⋊C₈	—	2	1	no	—
91	#10625324	—	2	2	no	—
92	#10625324	—	2	2	no	—
93	C₈×D₄₇	—	2	2	no	—
94	C₈×D₄₇	—	2	2	no	—
95	C₁₆×C₄₇	—	2	2	no	—
96	C₁₆×C₄₇	—	4	1	no	—
97	C₄₇⋊C₁₆	—	2	2	no	—
98	C₄₇⋊C₁₆	—	4	1	no	—
99	C₂³×C₄₇	#131	1	1	yes	characteristic
100	C₂²×D₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
110	C₂²⋊D₄₇	—	2	2	no	—
111	C₂²⋊D₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	1	no	—
113	C₂×Dic₄₇	—	4	1	no	—
114	D₁₈₈	—	2	1	no	—
115	D₁₈₈	C₄⋊C₈	1	2	yes	—
116	C₄×D₄₇	—	2	2	no	—
117	C₄×D₄₇	—	2	2	no	—
118	Dic₉₄	—	2	1	no	—
119	Dic₉₄	C₄⋊C₈	1	2	yes	—
120	C₈×C₄₇	—	4	1	no	—
121	C₈×C₄₇	#120	1	1	yes	characteristic
122	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₄₇	#123	1	1	yes	characteristic
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	—	2	2	no	—
127	#12918	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	#424	1	1	yes	characteristic
130	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₄₇	#435	1	1	yes	characteristic
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	8	1	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	4	1	no	—
136	C₄×C₄₇	#385	1	1	yes	characteristic
137	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
139	C₄×C₄₇	#435	1	1	yes	characteristic
140	Dic₄₇	—	2	2	no	—
141	Dic₄₇	—	8	1	no	—
142	#3407	—	47	1	no	—
143	#2153	—	47	1	no	—
144	#2150	D₄₇	1	1	yes	characteristic, complemented
145	#2117	—	47	1	no	—
146	#1764	—	47	1	no	—
147	#1378	—	47	1	no	—
148	#1376	—	47	1	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	#1850	1	1	yes	characteristic
153	D₄₇	—	4	2	no	—
154	C₂×#131	—	47	1	no	—
155	#504	—	94	2	no	—
156	#504	—	94	2	no	—
157	C₂×M₃₂	—	47	1	no	—
158	C₂×M₃₂	—	47	1	no	—
159	C₂×M₃₂	—	94	2	no	—
160	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
161	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#104	—	47	1	no	—
163	#416	—	47	1	no	—
164	#405	—	47	1	no	—
165	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
166	#364	—	94	2	no	—
167	C₄⋊C₁₆	—	2	2	no	—
168	#350	—	94	2	no	—
169	#341	—	47	1	no	—
170	#339	—	47	1	no	—
171	C₄₇	#12918	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	47	1	no	—
173	#131	—	47	2	no	—
174	#131	—	47	2	no	—
175	#131	—	94	1	no	—
176	#131	—	94	1	no	—
177	C₂×M₁₆	—	47	1	no	—
178	C₂×M₁₆	—	47	1	no	—
179	C₂×M₁₆	—	94	1	no	—
180	C₂²×C₈	—	94	1	no	—
181	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
183	#117	—	47	1	no	—
184	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	47	2	no	—
190	M₃₂	—	94	4	no	—
191	M₃₂	—	94	1	no	—
192	M₃₂	—	94	1	no	—
193	M₃₂	—	94	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	94	4	no	—
196	C₂×C₁₆	—	94	1	no	—
197	C₂×C₁₆	—	94	1	no	—
198	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
199	#104	—	47	2	no	—
200	#104	—	47	2	no	—
201	#103	—	94	1	no	—
202	#102	—	94	1	no	—
203	#97	—	94	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	94	1	no	—
207	#41	—	47	2	no	—
208	#41	—	47	2	no	—
209	#41	—	94	1	no	—
210	#41	—	94	1	no	—
211	C₂×Q₈	—	47	1	no	—
212	C₂×D₄	—	47	1	no	—
213	C₂×D₄	—	94	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	94	1	no	—
216	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
217	M₁₆	—	47	2	no	—
218	M₁₆	—	47	2	no	—
219	M₁₆	—	94	1	no	—
220	M₁₆	—	94	1	no	—
221	M₁₆	—	94	2	no	—
222	M₁₆	—	94	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	94	1	no	—
227	C₂×C₈	—	94	1	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	#10625350	1	1	yes	characteristic
231	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
232	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
233	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
234	C₂×C₈	#10625354	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₄⋊C₄	—	94	1	no	—
237	C₂²⋊C₄	—	94	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	94	1	no	—
240	C₄²	C₈×D₄₇	1	1	yes	characteristic
241	C₄²	C₈×D₄₇	1	1	yes	characteristic
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	94	2	no	—
245	C₁₆	—	188	1	no	—
246	C₂³	—	94	1	no	—
247	C₂³	#13094030	1	1	yes	characteristic
248	Q₈	—	47	2	no	—
249	Q₈	—	94	1	no	—
250	D₄	—	47	2	no	—
251	D₄	—	94	2	no	—
252	D₄	—	94	2	no	—
253	D₄	—	94	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	94	1	no	—
258	C₂×C₄	—	94	2	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	188	1	no	—
261	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₄	#13093986	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
266	C₂×C₄	#13094032	1	1	yes	characteristic
267	C₈	—	4	1	no	—
268	C₈	—	94	2	no	—
269	C₈	—	94	2	no	—
270	C₈	#13093992	1	1	yes	characteristic
271	C₈	#13094000	1	1	yes	characteristic
272	C₈	#13094002	1	1	yes	characteristic
273	C₈	#13094009	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	94	2	no	—
276	C₂²	—	376	1	no	—
277	C₂²	#15689148	1	1	yes	characteristic
278	C₂²	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
279	C₂²	#15689193	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	94	2	no	—
283	C₄	—	376	1	no	—
284	C₄	#15688994	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
286	C₄	#15689188	1	1	yes	characteristic
287	C₄	#15689192	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	188	2	no	—
292	C₂	#20243564	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13093992
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2150	C₂
socle	C₂×C₄₇	#1850
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#12918
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #20245153	C₂	semidirect
5. #20241473	C₂	semidirect
6. #20241470	C₂	semidirect
7. C₄₇×#2150	C₂	semidirect
144. #2150	D₄₇	semidirect
171. C₄₇	#12918	semidirect

Maximal subgroups

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

subgroup	index
#20256490	2
#20245153	2
#20245017	2
#20243225	2
#20241473	2
#20241470	2
C₄₇×#2150	2
#12918	47