Subgroups — #26133673

Order 12032.

484 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688938	C₂	1	2	yes	maximal, complemented
4	#20241498	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×#13094003	—	2	1	no	—
8	C₂×#13093964	—	2	4	no	—
9	C₂×#13093964	—	2	1	no	—
10	C₂×#13093964	C₂²	1	2	yes	—
11	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
12	#15689364	—	2	1	no	—
13	#15689364	—	2	1	no	—
14	#15689364	C₄	1	2	yes	—
15	#15689364	C₄	1	2	yes	—
16	C₂×#13093958	C₂²	1	1	yes	characteristic
17	C₂×#13093949	—	2	1	no	—
18	#15688938	—	2	4	no	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#108×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#407	—	2	1	no	—
22	C₄₇×#341	—	2	1	no	—
23	C₂²×Dic₉₄	—	2	1	no	—
24	C₂×Dic₁₈₈	—	2	2	no	—
25	C₂×Dic₁₈₈	—	2	2	no	—
26	C₂×Dic₁₈₈	—	2	1	no	—
27	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
28	#13094003	—	2	2	no	—
29	#13094003	—	2	2	no	—
30	#13094003	—	4	1	no	—
31	#13094003	—	4	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×Dic₄₇⋊C₄	—	4	1	no	—
34	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
35	#13093964	—	2	4	no	—
36	#13093964	—	2	2	no	—
37	#13093964	D₄	1	4	yes	—
38	C₂×C₄₇⋊C₁₆	—	2	4	no	—
39	C₂×C₄₇⋊C₁₆	—	2	2	no	—
40	#13093958	—	2	1	no	—
41	#13093958	C₂×C₄	1	2	yes	—
42	#13093949	—	2	2	no	—
43	#13093949	—	2	2	no	—
44	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	#108×C₄₇	—	2	2	no	—
55	#108×C₄₇	—	2	2	no	—
56	C₂²⋊C₈×C₄₇	—	4	1	no	—
57	C₂²⋊C₈×C₄₇	—	4	1	no	—
58	C₂⁴×C₄₇	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	4	1	no	—
68	C₂²×Dic₄₇	—	4	1	no	—
69	C₂×Dic₉₄	—	2	2	no	—
70	C₂×Dic₉₄	—	4	2	no	—
71	C₂×Dic₉₄	—	4	2	no	—
72	C₂×Dic₉₄	—	4	1	no	—
73	M₁₆×C₄₇	—	4	2	no	—
74	M₁₆×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	1	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	#41	1	2	yes	—
89	C₂²⋊Dic₄₇	—	4	2	no	—
90	C₂²⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	4	1	no	—
93	Dic₄₇⋊C₄	—	4	2	no	—
94	Dic₄₇⋊C₄	—	4	2	no	—
95	Dic₁₈₈	—	2	4	no	—
96	Dic₁₈₈	—	4	1	no	—
97	Dic₁₈₈	—	4	1	no	—
98	C₄₇⋊C₁₆	—	2	4	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	#102	1	1	yes	characteristic
103	C₂³×C₄₇	#102	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
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	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#135	1	2	yes	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	8	2	no	—
121	C₂×Dic₄₇	—	8	2	no	—
122	Dic₉₄	—	4	2	no	—
123	Dic₉₄	—	4	2	no	—
124	Dic₉₄	—	8	2	no	—
125	C₈×C₄₇	—	2	1	no	—
126	C₈×C₄₇	—	2	1	no	—
127	C₈×C₄₇	—	8	2	no	—
128	C₈×C₄₇	C₄×D₄	1	2	yes	—
129	C₈×C₄₇	#120	1	1	yes	characteristic
130	C₈×C₄₇	#123	1	1	yes	characteristic
131	#13066	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	#362	1	2	yes	complemented
141	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
142	C₂²×C₄₇	#465	1	2	yes	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	#386	1	1	yes	characteristic
146	C₄×C₄₇	#416	1	1	yes	characteristic
147	C₄×C₄₇	#466	1	2	yes	—
148	Dic₄₇	—	8	2	no	—
149	Dic₄₇	—	8	2	no	—
150	C₂²×M₃₂	—	47	1	no	—
151	C₂×#467	—	47	1	no	—
152	C₂×#362	—	47	2	no	—
153	#1810	D₄₇	1	1	yes	characteristic
154	#1387	—	47	2	no	—
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	—	4	1	no	—
158	C₂×C₄₇	#1893	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
160	C₂×C₄₇	#2148	1	1	yes	characteristic
161	C₂²×Q₁₆	—	47	1	no	—
162	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂×C₂²⋊Q₈	—	94	1	no	—
164	C₂×M₃₂	—	47	2	no	—
165	C₂×M₃₂	—	94	4	no	—
166	C₂×M₃₂	—	94	1	no	—
167	C₂²×C₁₆	—	94	1	no	—
168	#467	—	47	2	no	—
169	#467	—	47	2	no	—
170	#467	—	94	1	no	—
171	#467	—	94	1	no	—
172	C₂×#108	C₂×D₄₇	1	2	yes	—
173	C₂×#107	—	47	1	no	—
174	C₂×#103	—	94	1	no	—
175	#407	—	2	1	no	—
176	#362	—	94	4	no	—
177	#341	—	2	1	no	—
178	C₄₇	#13066	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	94	1	no	—
180	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂×Q₁₆	—	47	2	no	—
182	C₂×Q₁₆	—	94	1	no	—
183	C₂×Q₁₆	—	94	2	no	—
184	C₂×Q₁₆	—	94	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
191	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²⋊Q₈	—	94	2	no	—
195	C₂²⋊Q₈	—	94	2	no	—
196	C₂²⋊Q₈	—	188	1	no	—
197	C₂²⋊Q₈	—	188	1	no	—
198	C₂×C₄⋊C₄	—	94	1	no	—
199	C₂×C₄⋊C₄	—	188	1	no	—
200	C₂×C₂²⋊C₄	—	188	1	no	—
201	M₃₂	—	47	4	no	—
202	M₃₂	—	94	4	no	—
203	M₃₂	—	94	2	no	—
204	C₂×C₁₆	—	94	4	no	—
205	C₂×C₁₆	—	94	2	no	—
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	#107	—	47	2	no	—
209	#107	—	94	1	no	—
210	#103	—	94	2	no	—
211	#103	—	94	2	no	—
212	C₂²⋊C₈	—	4	1	no	—
213	C₂²⋊C₈	—	4	1	no	—
214	C₂⁴	#10625336	1	1	yes	characteristic
215	C₂×Q₈	—	94	2	no	—
216	C₂×Q₈	—	188	2	no	—
217	C₂×Q₈	—	188	2	no	—
218	C₂×Q₈	—	188	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	188	1	no	—
222	C₂²×C₄	—	188	1	no	—
223	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	#10625334	1	1	yes	characteristic
226	C₂²×C₄	#10625334	1	1	yes	characteristic
227	C₂²×C₄	#10625336	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	Q₁₆	—	94	4	no	—
230	Q₁₆	—	188	1	no	—
231	Q₁₆	—	188	1	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
246	C₂×C₈	D₄×D₄₇	1	2	yes	—
247	C₂×C₈	#10625352	1	2	yes	—
248	C₄⋊C₄	—	94	2	no	—
249	C₄⋊C₄	—	188	2	no	—
250	C₄⋊C₄	—	188	2	no	—
251	C₄⋊C₄	—	188	1	no	—
252	C₂²⋊C₄	—	188	2	no	—
253	C₂²⋊C₄	—	188	2	no	—
254	C₁₆	—	94	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	#13093944	1	1	yes	characteristic
259	C₂³	#13093944	1	1	yes	characteristic
260	C₂³	C₂×#10625336	1	1	yes	characteristic
261	Q₈	—	376	2	no	—
262	Q₈	—	188	2	no	—
263	Q₈	—	188	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	376	2	no	—
272	C₂×C₄	—	376	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	#13093944	1	1	yes	characteristic
276	C₂×C₄	#13093944	1	1	yes	characteristic
277	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	#13093996	1	2	yes	—
279	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
280	C₂×C₄	#13094044	1	2	yes	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	8	2	no	—
284	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
285	C₈	#13093994	1	1	yes	characteristic
286	C₈	#13094008	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	#15688938	1	2	yes	complemented
296	C₂²	C₂×#13093944	1	1	yes	characteristic
297	C₂²	#15689354	1	2	yes	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	376	2	no	—
301	C₄	—	376	2	no	—
302	C₄	#15689000	1	1	yes	characteristic
303	C₄	#15689117	1	1	yes	characteristic
304	C₄	#15689357	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	1	no	—
308	C₂	#20243814	1	1	yes	characteristic
309	C₂	C₂×#15688938	1	1	yes	characteristic
310	C₂	#20245135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1810	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂⁴×C₄₇	SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1810	D₄₇
hypocenter	C₄₇	#13066
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. C₂×#15688938	C₂	semidirect
4. #20241498	C₂	semidirect
140. C₂²×C₄₇	#362	semidirect
178. C₄₇	#13066	semidirect
295. C₂²	#15688938	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
C₂×#15689364	2
C₂×#15688938	2
#20241498	2
C₄₇×#1810	2
#13066	47