Subgroups — #26131375

Order 12032.

410 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689539	C₂	1	1	yes	maximal, characteristic
2	#20245008	C₂	1	1	yes	maximal, characteristic
3	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243189	C₂	1	1	yes	maximal, characteristic
5	#20241475	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	C₂²	1	1	yes	characteristic
9	#15689539	—	2	1	no	—
10	#15689539	—	2	1	no	—
11	#15689539	—	2	1	no	—
12	#15689539	—	2	1	no	—
13	C₂×#13093964	C₂²	1	1	yes	characteristic
14	C₂×#13093964	C₂²	1	1	yes	characteristic
15	C₂×#13093963	C₄	1	2	yes	complemented
16	C₂×C₁₆×D₄₇	C₄	1	2	yes	complemented
17	#15689105	C₄	1	1	yes	characteristic
18	#15689096	C₄	1	1	yes	characteristic
19	#15689078	C₂²	1	1	yes	characteristic
20	#15688904	C₄	1	4	yes	complemented
21	#15688898	—	2	1	no	—
22	C₁₆×Dic₄₇	—	2	1	no	—
23	#15688895	C₄	1	1	yes	characteristic
24	#15688893	C₄	1	1	yes	characteristic
25	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#405	C₂²	1	1	yes	characteristic
28	C₄₇×#346	—	2	1	no	—
29	C₄×C₁₆×C₄₇	—	2	1	no	—
30	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
31	#13094030	—	2	1	no	—
32	#13094030	—	2	1	no	—
33	#13094030	—	2	1	no	—
34	#13094030	—	2	1	no	—
35	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
36	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
37	C₂×#10625324	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
39	#13093983	C₂×C₄	1	1	yes	characteristic
40	#13093964	—	2	1	no	—
41	#13093964	—	2	1	no	—
42	#13093964	—	2	1	no	—
43	#13093964	C₂×C₄	1	2	yes	—
44	#13093963	—	2	2	no	—
45	#13093963	—	2	2	no	—
46	C₂×C₄₇⋊C₁₆	—	2	1	no	—
47	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
48	C₁₆×D₄₇	—	2	2	no	—
49	C₁₆×D₄₇	—	2	2	no	—
50	#13093942	C₈	1	2	yes	—
51	#13093940	C₈	1	2	yes	—
52	#13093938	C₈	1	4	yes	—
53	#13093934	—	2	1	no	—
54	C₈×Dic₄₇	—	2	1	no	—
55	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₄₇	—	2	1	no	—
59	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₄₇	—	2	1	no	—
61	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
66	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
67	#97×C₄₇	—	2	1	no	—
68	C₄×C₈×C₄₇	—	2	1	no	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
71	#10625350	—	2	1	no	—
72	#10625350	—	2	1	no	—
73	#10625350	—	2	1	no	—
74	#10625350	—	2	1	no	—
75	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₄₇	C₄²	1	2	yes	—
77	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
78	M₁₆×C₄₇	—	2	1	no	—
79	M₁₆×C₄₇	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
86	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	2	1	no	—
88	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
89	C₄²×C₄₇	—	2	1	no	—
90	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₄₇	—	2	1	no	—
92	C₄×Dic₄₇	—	2	1	no	—
93	#10625329	—	2	1	no	—
94	#10625329	—	2	1	no	—
95	#10625329	—	2	1	no	—
96	#10625329	C₂×C₈	1	2	yes	—
97	C₂×C₄₇⋊C₈	—	2	1	no	—
98	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
99	#10625324	—	2	2	no	—
100	#10625324	—	2	2	no	—
101	C₈×D₄₇	—	2	2	no	—
102	C₈×D₄₇	—	2	2	no	—
103	C₁₆×C₄₇	—	2	2	no	—
104	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
105	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
106	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
107	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₄₇⋊C₁₆	—	2	2	no	—
109	C₄₇⋊C₁₆	—	2	2	no	—
110	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
111	C₂²×D₄₇	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
119	C₂²⋊D₄₇	—	2	2	no	—
120	C₂²⋊D₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	C₄×C₈	1	2	yes	—
123	D₁₈₈	—	2	1	no	—
124	D₁₈₈	—	2	1	no	—
125	C₄×D₄₇	—	2	2	no	—
126	C₄×D₄₇	—	2	2	no	—
127	Dic₉₄	—	2	1	no	—
128	Dic₉₄	—	2	1	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈×C₄₇	#117	1	1	yes	characteristic
132	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
133	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
134	C₄₇⋊C₈	—	2	2	no	—
135	C₄₇⋊C₈	—	2	2	no	—
136	#12613	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	4	1	no	—
138	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
139	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
141	C₂×D₄₇	—	2	2	no	—
142	C₂×D₄₇	—	4	2	no	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₄₇	#405	1	1	yes	characteristic
146	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
147	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
148	Dic₄₇	—	2	2	no	—
149	Dic₄₇	—	4	2	no	—
150	C₂×#504	—	47	1	no	—
151	#2115	—	47	1	no	—
152	#2108	—	47	1	no	—
153	#2108	D₄₇	1	1	yes	characteristic, complemented
154	#1757	—	47	1	no	—
155	#1378	—	47	1	no	—
156	#1378	—	47	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	2	1	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
161	D₄₇	—	4	2	no	—
162	C₂×#131	—	47	1	no	—
163	#504	—	94	1	no	—
164	#504	—	94	1	no	—
165	#504	—	94	1	no	—
166	#504	—	94	1	no	—
167	C₂×M₃₂	—	47	1	no	—
168	C₂×M₃₂	—	47	2	no	—
169	C₂×M₃₂	—	47	1	no	—
170	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×C₁₆	—	47	2	no	—
172	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
173	#413	—	47	1	no	—
174	#410	—	47	1	no	—
175	#405	—	47	1	no	—
176	#405	C₂×D₄₇	1	1	yes	characteristic
177	#349	—	47	4	no	—
178	#346	—	2	1	no	—
179	#346	—	94	1	no	—
180	C₄×C₁₆	—	2	1	no	—
181	C₄×C₁₆	—	94	1	no	—
182	#344	—	47	1	no	—
183	#344	—	47	1	no	—
184	C₄₇	#12613	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	47	1	no	—
186	#131	—	94	1	no	—
187	#131	—	94	1	no	—
188	#131	—	94	1	no	—
189	#131	—	94	1	no	—
190	C₂×M₁₆	—	47	1	no	—
191	C₂×M₁₆	—	47	2	no	—
192	C₂×M₁₆	—	47	1	no	—
193	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
194	C₂²×C₈	—	47	2	no	—
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	#117	—	47	1	no	—
197	#117	C₄×D₄₇	1	1	yes	characteristic
198	M₃₂	—	2	1	no	—
199	M₃₂	—	47	2	no	—
200	M₃₂	—	94	1	no	—
201	M₃₂	—	94	1	no	—
202	M₃₂	—	94	1	no	—
203	M₃₂	—	94	2	no	—
204	M₃₂	—	94	2	no	—
205	M₃₂	C₄×D₄₇	1	2	yes	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	47	2	no	—
208	C₂×C₁₆	—	94	1	no	—
209	C₂×C₁₆	—	94	2	no	—
210	C₂×C₁₆	—	94	2	no	—
211	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
212	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
214	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
215	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
217	#101	—	47	2	no	—
218	#100	—	47	2	no	—
219	#99	—	47	4	no	—
220	#97	—	2	1	no	—
221	#97	—	94	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	94	1	no	—
224	#41	—	94	1	no	—
225	#41	—	94	1	no	—
226	#41	—	94	1	no	—
227	#41	—	94	1	no	—
228	C₂×Q₈	—	47	1	no	—
229	C₂×D₄	—	47	2	no	—
230	C₂×D₄	—	47	1	no	—
231	C₂²×C₄	—	47	2	no	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	M₁₆	—	2	1	no	—
234	M₁₆	—	47	2	no	—
235	M₁₆	—	94	1	no	—
236	M₁₆	—	94	1	no	—
237	M₁₆	—	94	1	no	—
238	M₁₆	—	94	2	no	—
239	M₁₆	—	94	2	no	—
240	M₁₆	C₈×D₄₇	1	2	yes	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	47	2	no	—
243	C₂×C₈	—	94	1	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂×C₈	#10625350	1	1	yes	characteristic
250	C₂×C₈	#10625350	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	94	1	no	—
254	C₂²⋊C₄	—	47	2	no	—
255	C₂²⋊C₄	C₈×D₄₇	1	2	yes	—
256	C₄²	—	2	1	no	—
257	C₄²	—	94	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	94	2	no	—
260	C₁₆	—	94	2	no	—
261	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
262	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
264	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
265	C₂³	—	47	2	no	—
266	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
267	Q₈	—	94	1	no	—
268	Q₈	—	94	1	no	—
269	D₄	—	94	2	no	—
270	D₄	—	94	2	no	—
271	D₄	—	94	1	no	—
272	D₄	—	94	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	47	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
279	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
282	C₂×C₄	#13094030	1	1	yes	characteristic
283	C₂×C₄	#13094030	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	94	2	no	—
286	C₈	—	94	2	no	—
287	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
288	C₈	#13093981	1	1	yes	characteristic
289	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
290	C₈	C₄×D₁₈₈	1	1	yes	characteristic
291	C₂²	—	4	1	no	—
292	C₂²	—	94	2	no	—
293	C₂²	—	188	2	no	—
294	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
295	C₂²	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
296	C₂²	C₈×D₁₈₈	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	94	2	no	—
299	C₄	—	188	2	no	—
300	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
301	C₄	#15689077	1	1	yes	characteristic
302	C₄	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
303	C₄	C₈×D₁₈₈	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	188	2	no	—
308	C₂	C₈×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂×C₄₇	C₈×C₂²⋊C₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#12613
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #20244984	C₂	semidirect
5. #20241475	C₂	semidirect
6. #20241474	C₂	semidirect
7. C₄₇×#2108	C₂	semidirect
15. C₂×#13093963	C₄	semidirect
16. C₂×C₁₆×D₄₇	C₄	semidirect
20. #15688904	C₄	semidirect
153. #2108	D₄₇	semidirect
184. C₄₇	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689539	2
#20245008	2
#20244984	2
#20243189	2
#20241475	2
#20241474	2
C₄₇×#2108	2
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