Subgroups — #26247214

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094016	C₂	1	1	yes	maximal, characteristic
2	#20244870	C₂	1	2	yes	maximal
3	#20244869	C₂	1	2	yes	maximal
4	#20244726	C₂	1	2	yes	maximal
5	#20244723	C₂	1	2	yes	maximal
6	#20243286	C₂	1	2	yes	maximal
7	#20243286	C₂	1	2	yes	maximal
8	#20243137	C₂	1	1	yes	maximal, characteristic
9	C₄×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094016	C₂²	1	1	yes	characteristic
11	C₂×#13093957	C₂²	1	2	yes	—
12	C₂×#13093957	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	2	yes	—
14	C₂×#13093955	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	#15689021	C₂²	1	2	yes	—
18	#15689007	—	2	2	no	—
19	#15688989	C₂²	1	1	yes	characteristic
20	C₄×Dic₄₇⋊C₄	—	2	2	no	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688885	C₂²	1	2	yes	—
23	#15688885	C₂²	1	2	yes	—
24	#15688885	C₂²	1	2	yes	—
25	#15688885	C₂²	1	2	yes	—
26	#15688878	C₂²	1	2	yes	—
27	#15688878	C₂²	1	2	yes	—
28	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#389	C₂²	1	2	yes	—
30	C₄₇×#387	—	2	2	no	—
31	C₄₇×#382	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
34	C₄³×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13094016	—	2	4	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×Dic₄₇⋊C₄	—	2	2	no	—
41	C₂×Dic₄₇⋊C₄	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	#13093957	—	2	4	no	—
44	#13093957	—	2	4	no	—
45	#13093955	—	2	4	no	—
46	#13093955	—	2	4	no	—
47	#13093935	Q₈	1	4	yes	—
48	#13093935	Q₈	1	4	yes	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	#125×C₄₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	D₄	1	2	yes	—
59	C₂×C₄²×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	#95×C₄₇	—	2	2	no	—
64	#95×C₄₇	—	2	2	no	—
65	#95×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	8	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₄⋊C₄×C₄₇	#41	1	4	yes	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	4	2	no	—
84	C₄²×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	—	4	4	no	—
86	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
87	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	#41	1	4	yes	—
92	Dic₄₇⋊C₄	—	2	8	no	—
93	Dic₄₇⋊C₄	—	4	4	no	—
94	C₄×Dic₄₇	—	4	4	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	#23840	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	#525	1	1	yes	characteristic
132	C₂²×C₄₇	#525	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₄₇	#575	1	2	yes	—
135	C₂²×C₄₇	#575	1	2	yes	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	4	2	no	—
139	C₄×C₄₇	—	4	2	no	—
140	C₄×C₄₇	—	4	4	no	—
141	C₄×C₄₇	—	8	2	no	—
142	C₄×C₄₇	#533	1	2	yes	—
143	C₄×C₄₇	#533	1	2	yes	—
144	Dic₄₇	—	4	4	no	—
145	Dic₄₇	—	8	2	no	—
146	C₄×#125	—	47	1	no	—
147	C₄×#125	D₄₇	1	1	yes	characteristic
148	#2085	—	47	2	no	—
149	#2085	—	47	2	no	—
150	#2059	—	47	2	no	—
151	#2059	—	47	2	no	—
152	#1776	—	47	2	no	—
153	#1776	—	47	2	no	—
154	#1745	—	47	1	no	—
155	C₂×C₄₇	#2399	1	1	yes	characteristic
156	C₂×C₄₇	#3111	1	2	yes	—
157	C₂×C₄₇	#3111	1	2	yes	—
158	C₂×C₄₇	#3166	1	1	yes	characteristic
159	C₂×C₄₇	#3166	1	1	yes	characteristic
160	C₂×#125	—	47	1	no	—
161	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#107	—	47	2	no	—
163	C₂×#107	—	47	2	no	—
164	C₂×#106	—	47	2	no	—
165	C₂×#106	—	47	2	no	—
166	C₂×#97	—	47	1	no	—
167	C₂×#97	—	47	1	no	—
168	#389	—	47	2	no	—
169	#389	C₂×D₄₇	1	2	yes	—
170	#387	—	2	2	no	—
171	#387	—	94	2	no	—
172	#382	—	47	1	no	—
173	#382	C₂×D₄₇	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	2	no	—
175	C₄×C₄⋊C₄	—	47	2	no	—
176	C₄×C₄⋊C₄	—	94	2	no	—
177	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
178	C₄³	C₂×D₄₇	1	1	yes	characteristic
179	#340	—	47	2	no	—
180	#340	—	47	2	no	—
181	#340	—	47	2	no	—
182	#340	—	47	2	no	—
183	#336	—	47	2	no	—
184	#336	—	47	2	no	—
185	C₄₇	#23840	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	#125	—	2	4	no	—
189	#125	—	94	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	47	2	no	—
193	C₂×C₄⋊C₄	—	47	2	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	94	2	no	—
200	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	#107	—	94	4	no	—
206	#107	—	94	4	no	—
207	#106	—	94	4	no	—
208	#106	—	94	4	no	—
209	#97	—	47	4	no	—
210	#97	—	47	4	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	94	2	no	—
215	#95	—	94	2	no	—
216	#95	—	94	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	94	2	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
226	C₂×C₈	—	47	4	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	94	2	no	—
231	C₂×C₈	—	94	2	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	47	4	no	—
238	C₄⋊C₄	—	94	8	no	—
239	C₄⋊C₄	—	94	2	no	—
240	C₄⋊C₄	—	94	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	188	4	no	—
243	C₄⋊C₄	#10625350	1	4	yes	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	4	no	—
249	C₄²	—	188	4	no	—
250	C₄²	Q₈×D₄₇	1	2	yes	—
251	C₄²	Q₈×D₄₇	1	2	yes	—
252	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	94	4	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	#13094022	1	2	yes	—
274	C₂×C₄	#13094022	1	2	yes	—
275	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
277	C₂×C₄	#41×D₄₇	1	2	yes	—
278	C₂×C₄	C₂×#10625350	1	2	yes	—
279	C₈	—	94	4	no	—
280	C₈	—	94	4	no	—
281	C₂²	#15689630	1	1	yes	characteristic
282	C₂²	#15689634	1	1	yes	characteristic
283	C₂²	C₂×#13094022	1	1	yes	characteristic
284	C₂²	#15689921	1	2	yes	—
285	C₂²	#15689921	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	2	no	—
292	C₄	—	376	2	no	—
293	C₄	—	188	4	no	—
294	C₄	#15689677	1	2	yes	—
295	C₄	#15689677	1	2	yes	—
296	C₂	#20246679	1	1	yes	characteristic
297	C₂	#20253491	1	2	yes	—
298	C₂	#20253497	1	2	yes	—
299	C₂	#20253937	1	1	yes	characteristic
300	C₂	#20253941	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#125×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×#125	D₄₇
hypocenter	C₄₇	#23840
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
185. C₄₇	#23840	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094016	2
#20244870	2
#20244869	2
#20244726	2
#20244723	2
#20243286	2
#20243137	2
C₄×#125×C₄₇	2
#23840	47