Subgroups — #26134779

Order 12032.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689507	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688937	C₂	1	1	yes	maximal, characteristic
4	#20245109	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245105	C₂	1	1	yes	maximal, characteristic
6	#20243267	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
9	C₂²×#10625325	—	2	1	no	—
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	C₂×#13094012	—	2	1	no	—
12	#15689507	—	2	2	no	—
13	#15689507	C₄	1	4	yes	complemented
14	C₂×#13093960	C₂²	1	1	yes	characteristic
15	#15689161	C₂²	1	1	yes	characteristic
16	#15689078	—	2	1	no	—
17	#15688940	—	2	2	no	—
18	#15688937	—	2	2	no	—
19	#15688928	—	2	2	no	—
20	#15688925	—	2	2	no	—
21	#15688883	—	2	1	no	—
22	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#364	—	2	2	no	—
26	C₂²×Dic₉₄	—	2	1	no	—
27	C₂²×D₁₈₈	—	2	1	no	—
28	C₂×Dic₁₈₈	—	2	1	no	—
29	C₂×Dic₁₈₈	—	2	1	no	—
30	C₂×Dic₁₈₈	—	2	1	no	—
31	C₂×Dic₁₈₈	—	2	1	no	—
32	C₂×Dic₁₈₈	—	2	1	no	—
33	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
34	C₂×#10625325	—	2	4	no	—
35	C₂×#10625325	—	4	1	no	—
36	C₂×#10625325	—	4	1	no	—
37	C₂×#10625325	—	4	2	no	—
38	C₂×D₃₇₆	—	2	1	no	—
39	C₂×D₃₇₆	—	2	1	no	—
40	C₂×D₃₇₆	—	2	1	no	—
41	C₂×D₃₇₆	—	2	1	no	—
42	C₂×D₃₇₆	—	2	1	no	—
43	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
44	C₂×#10625329	—	2	1	no	—
45	#13094012	—	2	4	no	—
46	#13094012	—	4	2	no	—
47	#13093983	—	2	1	no	—
48	C₂×(C₂×D₄₇)⋊C₄	—	4	2	no	—
49	#13093960	—	2	1	no	—
50	#13093960	D₄	1	2	yes	—
51	#13093958	D₄	1	2	yes	—
52	#13093956	—	2	1	no	—
53	#13093934	—	4	1	no	—
54	C₈×Dic₄₇	—	4	1	no	—
55	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
58	M₃₂×C₄₇	—	2	1	no	—
59	M₃₂×C₄₇	—	2	1	no	—
60	M₃₂×C₄₇	—	2	1	no	—
61	M₃₂×C₄₇	—	2	1	no	—
62	C₂×C₁₆×C₄₇	—	2	1	no	—
63	C₂×C₁₆×C₄₇	—	2	1	no	—
64	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₄₇	D₄	1	2	yes	—
66	C₂³×D₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	4	1	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×D₁₈₈	—	4	1	no	—
72	C₂×D₁₈₈	—	4	1	no	—
73	C₂×D₁₈₈	—	4	1	no	—
74	C₂×D₁₈₈	—	4	1	no	—
75	C₂×D₁₈₈	—	4	1	no	—
76	C₂×Dic₉₄	—	2	2	no	—
77	C₂×Dic₉₄	—	4	1	no	—
78	C₂×Dic₉₄	—	4	1	no	—
79	C₂×Dic₉₄	—	4	1	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	1	no	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
84	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
85	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
92	C₄²×C₄₇	—	2	1	no	—
93	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
94	C₂²⋊Dic₄₇	—	4	1	no	—
95	(C₂×D₄₇)⋊C₄	—	4	8	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	4	1	no	—
98	#10625329	—	2	2	no	—
99	#10625329	—	4	1	no	—
100	C₂×C₄₇⋊C₈	—	4	1	no	—
101	Dic₁₈₈	—	2	2	no	—
102	Dic₁₈₈	—	2	2	no	—
103	Dic₁₈₈	—	4	1	no	—
104	Dic₁₈₈	—	4	1	no	—
105	D₃₇₆	—	2	2	no	—
106	D₃₇₆	—	2	2	no	—
107	D₃₇₆	—	4	1	no	—
108	D₃₇₆	—	4	1	no	—
109	#10625325	—	4	4	no	—
110	#10625325	—	4	4	no	—
111	C₁₆×C₄₇	—	4	1	no	—
112	C₁₆×C₄₇	—	4	1	no	—
113	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×D₄₇	—	4	2	no	—
115	C₂²×D₄₇	—	8	4	no	—
116	C₂²×D₄₇	—	8	1	no	—
117	C₂²×D₄₇	—	8	1	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	8	1	no	—
129	C₂×Dic₄₇	—	8	1	no	—
130	D₁₈₈	—	4	2	no	—
131	D₁₈₈	—	4	2	no	—
132	D₁₈₈	—	8	1	no	—
133	D₁₈₈	—	8	1	no	—
134	Dic₉₄	—	4	2	no	—
135	Dic₉₄	—	4	2	no	—
136	Dic₉₄	—	8	1	no	—
137	Dic₉₄	—	8	1	no	—
138	C₈×C₄₇	#102	1	2	yes	—
139	C₈×C₄₇	#102	1	2	yes	—
140	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
141	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
142	C₈×C₄₇	#124	1	1	yes	characteristic
143	C₈×C₄₇	#127	1	1	yes	characteristic
144	C₄₇⋊C₈	—	4	2	no	—
145	#13260	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	#418	1	1	yes	characteristic
148	C₂²×C₄₇	#459	1	1	yes	characteristic
149	C₂²×C₄₇	#460	1	1	yes	characteristic
150	C₂×D₄₇	—	8	4	no	—
151	C₂×D₄₇	—	8	2	no	—
152	C₂×D₄₇	—	16	2	no	—
153	C₂×D₄₇	—	16	2	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	#390	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
157	C₄×C₄₇	#459	1	1	yes	characteristic
158	C₄×C₄₇	#460	1	1	yes	characteristic
159	Dic₄₇	—	8	2	no	—
160	Dic₄₇	—	8	2	no	—
161	C₂×#495	—	47	1	no	—
162	C₂×#364	D₄₇	1	1	yes	characteristic, complemented
163	C₂×#362	—	47	1	no	—
164	C₂×#361	—	47	1	no	—
165	#2141	—	47	1	no	—
166	#2140	—	47	1	no	—
167	#1772	—	47	1	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	1	no	—
170	C₂×C₄₇	#1982	1	1	yes	characteristic
171	C₂×C₄₇	#2212	1	2	yes	—
172	D₄₇	—	16	2	no	—
173	C₂²×Q₁₆	—	47	1	no	—
174	C₂²×SD₁₆	—	94	1	no	—
175	C₂²×D₈	—	47	1	no	—
176	C₂×#124	—	94	1	no	—
177	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
178	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
179	#495	—	47	4	no	—
180	#495	—	94	2	no	—
181	C₂×#108	—	47	1	no	—
182	#427	—	47	1	no	—
183	#405	—	94	1	no	—
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#364	—	2	2	no	—
186	#362	—	94	2	no	—
187	#361	—	94	2	no	—
188	#358	—	94	2	no	—
189	#357	—	94	2	no	—
190	#339	—	94	1	no	—
191	C₄₇	#13260	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	94	1	no	—
193	C₂²×D₄	—	94	1	no	—
194	C₂×Q₁₆	—	47	2	no	—
195	C₂×Q₁₆	—	94	1	no	—
196	C₂×Q₁₆	—	94	1	no	—
197	C₂×Q₁₆	—	94	1	no	—
198	C₂×Q₁₆	—	94	1	no	—
199	C₂×Q₁₆	—	94	1	no	—
200	C₂×SD₁₆	—	94	4	no	—
201	C₂×SD₁₆	—	188	2	no	—
202	C₂×SD₁₆	—	188	1	no	—
203	C₂×SD₁₆	—	188	1	no	—
204	C₂×D₈	—	47	2	no	—
205	C₂×D₈	—	94	1	no	—
206	C₂×D₈	—	94	1	no	—
207	C₂×D₈	—	94	1	no	—
208	C₂×D₈	—	94	1	no	—
209	C₂×D₈	—	94	1	no	—
210	C₂×M₁₆	—	94	1	no	—
211	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	#124	—	94	4	no	—
214	#124	—	188	2	no	—
215	#117	—	94	1	no	—
216	C₂×C₂²⋊C₄	—	188	2	no	—
217	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	#108	—	47	2	no	—
225	#108	—	94	1	no	—
226	#107	—	47	2	no	—
227	#106	—	94	1	no	—
228	#97	—	188	1	no	—
229	C₄×C₈	—	188	1	no	—
230	C₄×C₈	C₄×D₄₇	1	2	yes	—
231	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
232	C₂⁴	—	188	1	no	—
233	C₂×Q₈	—	94	2	no	—
234	C₂×Q₈	—	188	1	no	—
235	C₂×Q₈	—	188	1	no	—
236	C₂×Q₈	—	188	1	no	—
237	C₂×Q₈	—	188	1	no	—
238	C₂×Q₈	—	188	1	no	—
239	C₂×D₄	—	94	2	no	—
240	C₂×D₄	—	188	1	no	—
241	C₂×D₄	—	188	1	no	—
242	C₂×D₄	—	188	1	no	—
243	C₂×D₄	—	188	1	no	—
244	C₂×D₄	—	188	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	188	1	no	—
247	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
248	Q₁₆	—	94	2	no	—
249	Q₁₆	—	94	2	no	—
250	Q₁₆	—	188	1	no	—
251	Q₁₆	—	188	1	no	—
252	SD₁₆	—	188	4	no	—
253	SD₁₆	—	188	4	no	—
254	D₈	—	94	2	no	—
255	D₈	—	94	2	no	—
256	D₈	—	188	1	no	—
257	D₈	—	188	1	no	—
258	M₁₆	—	94	2	no	—
259	M₁₆	—	188	1	no	—
260	C₂×C₈	—	188	1	no	—
261	C₂×C₈	#10625325	1	2	yes	—
262	C₂×C₈	D₃₇₆	1	1	yes	characteristic
263	C₂×C₈	D₃₇₆	1	1	yes	characteristic
264	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
267	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	#10625354	1	1	yes	characteristic
269	C₂×C₈	#10625354	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₄⋊C₄	—	94	2	no	—
272	C₂²⋊C₄	—	188	8	no	—
273	C₂²⋊C₄	—	188	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	188	1	no	—
276	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
277	C₁₆	—	4	1	no	—
278	C₁₆	—	4	1	no	—
279	C₂³	—	376	4	no	—
280	C₂³	—	376	1	no	—
281	C₂³	—	376	1	no	—
282	C₂³	—	188	2	no	—
283	C₂³	#13094052	1	1	yes	characteristic
284	Q₈	—	376	1	no	—
285	Q₈	—	376	1	no	—
286	Q₈	—	188	2	no	—
287	Q₈	—	188	2	no	—
288	D₄	—	376	1	no	—
289	D₄	—	376	1	no	—
290	D₄	—	188	2	no	—
291	D₄	—	188	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	376	1	no	—
295	C₂×C₄	—	376	1	no	—
296	C₂×C₄	—	188	2	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#13093999	1	1	yes	characteristic
300	C₂×C₄	#13093999	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
303	C₂×C₄	#13094052	1	1	yes	characteristic
304	C₈	—	188	2	no	—
305	C₈	#13093946	1	2	yes	—
306	C₈	#13093946	1	2	yes	—
307	C₈	#13093986	1	1	yes	characteristic
308	C₈	#13093986	1	1	yes	characteristic
309	C₈	#13094013	1	1	yes	characteristic
310	C₈	#13094020	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	376	4	no	—
313	C₂²	—	376	2	no	—
314	C₂²	—	752	2	no	—
315	C₂²	—	752	2	no	—
316	C₂²	#15689129	1	1	yes	characteristic
317	C₂²	#15689319	1	1	yes	characteristic
318	C₂²	#15689324	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	376	2	no	—
321	C₄	—	376	2	no	—
322	C₄	#15689024	1	1	yes	characteristic
323	C₄	C₂×#13093946	1	1	yes	characteristic
324	C₄	#15689319	1	1	yes	characteristic
325	C₄	#15689324	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	752	2	no	—
329	C₂	#20244314	1	1	yes	characteristic
330	C₂	#20245458	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689319
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	#459
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	#13260
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#15688940	C₂	semidirect
4. #20245109	C₂	semidirect
6. #20243267	C₂	semidirect
7. C₂×C₄₇×#364	C₂	semidirect
13. #15689507	C₄	semidirect
162. C₂×#364	D₄₇	semidirect
191. C₄₇	#13260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689507	2
C₂×#15688940	2
C₂×#15688937	2
#20245109	2
#20245105	2
#20243267	2
C₂×C₄₇×#364	2
#13260	47