Subgroups — #26104750

Order 12032.

562 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	#20243082	C₂	1	1	yes	maximal, characteristic
3	#20241210	C₂	1	4	yes	maximal, complemented
4	C₄×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂×#13094024	C₄	1	1	yes	characteristic
7	C₂×#13094003	—	2	1	no	—
8	#15689029	—	2	1	no	—
9	#15689021	—	2	1	no	—
10	#15689021	C₂²	1	2	yes	—
11	#15688993	C₄	1	1	yes	characteristic
12	#15688983	—	2	1	no	—
13	#15688983	C₄	1	2	yes	—
14	#15688878	—	2	1	no	—
15	#15688878	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	—	2	1	no	—
19	C₄₇×#336	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	4	2	no	—
23	C₂×#10625329	—	4	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13094024	—	4	2	no	—
26	#13094003	—	4	2	no	—
27	#13094003	—	4	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×Dic₄₇⋊C₄	—	2	2	no	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²⋊C₈×C₄₇	—	2	4	no	—
47	C₄×C₈×C₄₇	—	4	2	no	—
48	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	D₈	1	2	yes	—
61	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂×Dic₉₄	—	4	2	no	—
68	C₂×Dic₉₄	—	4	2	no	—
69	C₂×Dic₉₄	—	4	2	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	—	4	2	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₂²⋊Dic₄₇	—	4	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	2	no	—
80	C₄⋊Dic₄₇	—	4	2	no	—
81	C₄⋊Dic₄₇	—	4	2	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	Dic₄₇⋊C₄	—	4	2	no	—
84	Dic₄₇⋊C₄	—	4	2	no	—
85	#10625329	—	4	8	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	#102	1	2	yes	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	#99	1	2	yes	—
112	C₂×C₄×C₄₇	#101	1	2	yes	—
113	C₂×C₄×C₄₇	#102	1	2	yes	—
114	C₂×C₄×C₄₇	#104	1	2	yes	—
115	C₂×C₄×C₄₇	#104	1	2	yes	—
116	C₂×C₄×C₄₇	#131	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#137	1	2	yes	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	Dic₉₄	—	8	4	no	—
127	C₈×C₄₇	—	4	4	no	—
128	C₄₇⋊C₈	—	4	4	no	—
129	#7685	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	—	4	1	no	—
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	—	4	1	no	—
137	C₂²×C₄₇	#408	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
141	C₂²×C₄₇	#419	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	#327	1	4	yes	—
151	Dic₄₇	—	8	2	no	—
152	Dic₄₇	—	8	2	no	—
153	#2857	—	47	1	no	—
154	C₄×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
155	#1728	—	47	1	no	—
156	#1271	—	47	4	no	—
157	C₂×C₄₇	—	4	2	no	—
158	C₂×C₄₇	#1472	1	1	yes	characteristic
159	C₂×C₄₇	#1485	1	1	yes	characteristic
160	C₂×C₄₇	#1492	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#327	1	1	yes	characteristic
162	C₂×C₄₇	#1508	1	1	yes	characteristic
163	C₂×C₄₇	#1526	1	1	yes	characteristic
164	C₂×C₄₇	#1546	1	1	yes	characteristic
165	C₂²×M₁₆	—	94	1	no	—
166	C₂×C₄⋊Q₈	—	47	1	no	—
167	C₂×C₂²⋊Q₈	—	94	1	no	—
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	C₂×C₄×C₈	—	2	1	no	—
171	#391	—	94	1	no	—
172	#389	—	47	2	no	—
173	#389	—	94	1	no	—
174	#384	—	47	1	no	—
175	#380	—	47	2	no	—
176	#380	—	94	1	no	—
177	#336	—	47	2	no	—
178	#336	—	94	1	no	—
179	#336	C₂×D₄₇	1	2	yes	—
180	C₄₇	#7685	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	94	1	no	—
182	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
183	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
184	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×M₁₆	—	94	4	no	—
186	C₂×M₁₆	—	188	2	no	—
187	C₂×M₁₆	—	188	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	94	2	no	—
190	C₄⋊Q₈	—	188	2	no	—
191	C₂²⋊Q₈	—	188	2	no	—
192	C₂²⋊Q₈	—	188	2	no	—
193	C₂×C₄⋊C₄	—	47	2	no	—
194	C₂×C₄⋊C₄	—	47	2	no	—
195	C₂×C₄⋊C₄	—	94	1	no	—
196	C₂×C₄⋊C₄	—	94	1	no	—
197	C₂×C₄⋊C₄	—	94	1	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₂²⋊C₄	—	94	2	no	—
200	C₂×C₄²	—	4	2	no	—
201	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
202	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	4	no	—
206	C₄×C₈	—	4	2	no	—
207	#95	—	94	2	no	—
208	#95	—	94	2	no	—
209	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
210	C₂×Q₈	—	188	2	no	—
211	C₂×Q₈	—	188	2	no	—
212	C₂×Q₈	—	188	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	94	2	no	—
222	C₂²×C₄	—	94	2	no	—
223	C₂²×C₄	#10625325	1	2	yes	—
224	C₂²×C₄	D₃₇₆	1	2	yes	—
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	M₁₆	—	188	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	—	188	2	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	188	2	no	—
240	C₄⋊C₄	—	188	2	no	—
241	C₄⋊C₄	—	188	2	no	—
242	C₄⋊C₄	—	188	2	no	—
243	C₄⋊C₄	—	188	2	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₂²⋊C₄	—	188	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	#13093946	1	2	yes	—
257	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
258	Q₈	—	376	4	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	#13093938	1	2	yes	—
280	C₂×C₄	#13093942	1	2	yes	—
281	C₂×C₄	#13093946	1	2	yes	—
282	C₂×C₄	#13093950	1	2	yes	—
283	C₂×C₄	#13093952	1	2	yes	—
284	C₂×C₄	#13094030	1	2	yes	—
285	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
287	C₂×C₄	#13094054	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₈	—	188	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	#15689089	1	1	yes	characteristic
298	C₂²	C₂×#13093938	1	1	yes	characteristic
299	C₂²	C₂×#13093942	1	1	yes	characteristic
300	C₂²	C₂×#13093946	1	1	yes	characteristic
301	C₂²	#15689131	1	1	yes	characteristic
302	C₂²	C₂×#13093950	1	1	yes	characteristic
303	C₂²	C₂×#13093952	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	376	2	no	—
311	C₄	—	376	2	no	—
312	C₄	#15688858	1	4	yes	—
313	C₂	—	4	2	no	—
314	C₂	#20241751	1	1	yes	characteristic
315	C₂	#20241808	1	1	yes	characteristic
316	C₂	#20241840	1	1	yes	characteristic
317	C₂	C₂×#15688858	1	1	yes	characteristic
318	C₂	#20241907	1	1	yes	characteristic
319	C₂	#20241982	1	1	yes	characteristic
320	C₂	#20242059	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#7685
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20241210	C₂	semidirect
180. C₄₇	#7685	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
#20243082	2
#20241210	2
C₄×C₂²⋊C₈×C₄₇	2
#7685	47