Subgroups — #26112282

Order 12032.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic
2	#20242069	C₂	1	2	yes	maximal, complemented
3	#20241702	C₂	1	1	yes	maximal, characteristic
4	#20241246	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2628	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	—	2	1	no	—
7	#15689176	C₂²	1	2	yes	—
8	#15689150	—	2	4	no	—
9	#15689147	—	2	1	no	—
10	#15689147	C₂²	1	2	yes	—
11	#15688872	—	2	4	no	—
12	#15688849	—	2	1	no	—
13	#15688847	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#400	—	2	1	no	—
17	C₄₇×#399	—	2	1	no	—
18	C₄₇×#395	C₄	1	2	yes	—
19	C₄₇×#393	C₄	1	2	yes	—
20	C₄₇×#389	—	2	1	no	—
21	C₄₇×#382	C₂²	1	1	yes	characteristic
22	C₄₇×#380	—	2	1	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	4	2	no	—
26	C₂×#10625329	—	4	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₄₇	—	2	2	no	—
37	C₄⋊Q₈×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#95×C₄₇	—	2	2	no	—
52	#95×C₄₇	—	2	2	no	—
53	#95×C₄₇	—	2	1	no	—
54	#95×C₄₇	C₂×C₄	1	2	yes	—
55	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Q₈×C₄₇	—	2	4	no	—
57	C₂×Q₈×C₄₇	—	8	2	no	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	4	1	no	—
75	C₄⋊C₄×C₄₇	—	4	2	no	—
76	C₄⋊C₄×C₄₇	—	4	1	no	—
77	C₄⋊C₄×C₄₇	—	4	1	no	—
78	C₄⋊C₄×C₄₇	—	4	1	no	—
79	C₄⋊C₄×C₄₇	—	8	2	no	—
80	C₂²⋊C₄×C₄₇	—	2	2	no	—
81	C₂²⋊C₄×C₄₇	—	2	2	no	—
82	C₂²⋊C₄×C₄₇	—	8	2	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	—	2	1	no	—
85	C₄²×C₄₇	C₂×D₄	1	2	yes	—
86	C₄²×C₄₇	#41	1	2	yes	—
87	#10625329	—	4	4	no	—
88	#10625329	—	4	2	no	—
89	#10625329	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	4	4	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	4	1	no	—
96	C₂³×C₄₇	—	4	1	no	—
97	C₂³×C₄₇	#103	1	2	yes	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₄₇	—	8	4	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
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	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	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	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	#103	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#121	1	2	yes	—
122	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
127	C₄₇⋊C₈	—	4	4	no	—
128	C₄₇⋊C₈	—	8	2	no	—
129	#9191	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	4	2	no	—
134	C₂²×C₄₇	—	4	1	no	—
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	—	8	1	no	—
137	C₂²×C₄₇	#386	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
139	C₂²×C₄₇	#417	1	1	yes	characteristic
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	4	1	no	—
144	C₄×C₄₇	—	4	1	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	—	8	2	no	—
147	C₄×C₄₇	#450	1	1	yes	characteristic
148	C₄×C₄₇	#451	1	1	yes	characteristic
149	C₄×C₄₇	#483	1	1	yes	characteristic
150	C₄×C₄₇	#484	1	1	yes	characteristic
151	#2905	—	47	1	no	—
152	#2628	D₄₇	1	1	yes	characteristic
153	#1548	—	47	2	no	—
154	#1458	—	47	1	no	—
155	#1286	—	47	2	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	4	2	no	—
159	C₂×C₄₇	#1872	1	1	yes	characteristic
160	C₂×C₄₇	#1909	1	1	yes	characteristic
161	C₂×C₄₇	#1914	1	1	yes	characteristic
162	C₂²×M₁₆	—	94	1	no	—
163	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
165	#431	—	47	2	no	—
166	#424	—	94	4	no	—
167	#423	—	47	2	no	—
168	#423	—	94	1	no	—
169	#400	—	2	1	no	—
170	#399	—	2	1	no	—
171	#395	Dic₄₇	1	2	yes	—
172	#393	Dic₄₇	1	2	yes	—
173	#389	—	2	1	no	—
174	#382	C₂×D₄₇	1	1	yes	characteristic
175	#380	—	2	1	no	—
176	#333	—	94	4	no	—
177	#324	—	94	1	no	—
178	#323	—	94	1	no	—
179	C₄₇	#9191	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
181	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	188	2	no	—
185	C₂×M₁₆	—	188	2	no	—
186	C₂²×C₈	—	94	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	47	4	no	—
203	C₄⋊C₈	—	94	4	no	—
204	C₄⋊C₈	—	94	2	no	—
205	C₄⋊C₈	—	94	2	no	—
206	C₂²⋊C₈	—	94	2	no	—
207	C₄×C₈	—	94	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	1	no	—
211	#95	C₂×Dic₄₇	1	2	yes	—
212	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	#10625336	1	2	yes	—
222	C₂²×C₄	#10625337	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	M₁₆	—	188	4	no	—
230	M₁₆	—	188	2	no	—
231	M₁₆	—	188	2	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	188	2	no	—
234	C₂×C₈	—	188	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	8	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	D₄×D₄₇	1	2	yes	—
249	C₄²	#10625352	1	2	yes	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	#13093948	1	2	yes	—
255	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
256	Q₈	—	8	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#13093948	1	2	yes	—
276	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
277	C₂×C₄	#13093992	1	1	yes	characteristic
278	C₂×C₄	#13093996	1	2	yes	—
279	C₂×C₄	#13094005	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
281	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
282	C₂×C₄	#13094049	1	1	yes	characteristic
283	C₂×C₄	#13094054	1	1	yes	characteristic
284	C₈	—	376	2	no	—
285	C₈	—	188	4	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	#15689003	1	1	yes	characteristic
294	C₂²	C₂×#13093948	1	1	yes	characteristic
295	C₂²	#15689124	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	#15689273	1	1	yes	characteristic
304	C₄	#15689278	1	1	yes	characteristic
305	C₄	#15689451	1	1	yes	characteristic
306	C₄	#15689456	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#20243696	1	1	yes	characteristic
311	C₂	#20243911	1	1	yes	characteristic
312	C₂	#20243936	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689003
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2628	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2628	D₄₇
hypocenter	C₄₇	#9191
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2628

Complements

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

N	H ≅ G/N	type
2. #20242069	C₂	semidirect
4. #20241246	C₂	semidirect
179. C₄₇	#9191	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
#20242069	2
#20241702	2
#20241246	2
C₄₇×#2628	2
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