Subgroups — #26369454

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252082	C₂	1	1	yes	maximal, characteristic
4	#20245217	C₂	1	4	yes	maximal, complemented
5	#20245216	C₂	1	4	yes	maximal, complemented
6	#20245148	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245003	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2112	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094030	C₂²	1	1	yes	characteristic
11	#15689634	C₄	1	1	yes	characteristic
12	#15689541	—	2	4	no	—
13	C₂×#13093964	—	2	4	no	—
14	C₂×#13093964	C₂²	1	2	yes	—
15	C₂×#13093964	C₂²	1	2	yes	—
16	C₂×#13093963	C₂²	1	2	yes	complemented
17	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₄₇	C₂²	1	2	yes	complemented
20	#15689194	C₄	1	4	yes	—
21	C₈×C₂²⋊D₄₇	C₂²	1	4	yes	—
22	#15689149	C₄	1	1	yes	characteristic
23	#15689089	C₄	1	1	yes	characteristic
24	#15689087	C₄	1	1	yes	characteristic
25	#15689078	C₂²	1	1	yes	characteristic
26	Dic₄₇⋊C₁₆	C₂²	1	4	yes	complemented
27	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	complemented
28	#15688898	C₂²	1	2	yes	complemented
29	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
30	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
34	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
35	#13094030	—	2	2	no	—
36	#13094030	—	2	2	no	—
37	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
38	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
39	C₂×#10625324	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₇	C₂³	1	2	yes	—
41	#13094007	C₈	1	2	yes	—
42	#13094003	C₈	1	2	yes	—
43	#13093998	C₈	1	2	yes	—
44	#13093993	C₈	1	2	yes	—
45	C₄×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
46	#13093983	C₂×C₄	1	1	yes	characteristic
47	#13093964	—	2	4	no	—
48	#13093964	—	2	4	no	—
49	#13093963	—	2	4	no	—
50	C₂×C₄₇⋊C₁₆	—	2	2	no	—
51	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
52	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
53	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₁₆×D₄₇	—	2	4	no	—
56	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
57	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
58	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
59	#13093934	C₂×C₄	1	2	yes	—
60	C₈×Dic₄₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	—	2	2	no	—
63	C₂²×C₈×C₄₇	—	2	2	no	—
64	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	M₃₂×C₄₇	—	2	2	no	—
69	M₃₂×C₄₇	—	2	2	no	—
70	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
71	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
72	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
80	C₂×C₂²⋊D₄₇	C₂×C₈	1	4	yes	—
81	#10625350	—	2	2	no	—
82	#10625350	—	2	2	no	—
83	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	#41	1	2	yes	—
94	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₄₇	C₂×C₈	1	4	yes	—
97	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
98	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
99	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
100	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
101	#10625329	—	2	2	no	—
102	#10625329	—	2	2	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
105	#10625324	—	2	4	no	—
106	C₈×D₄₇	—	2	2	no	—
107	C₈×D₄₇	—	2	2	no	—
108	C₁₆×C₄₇	—	2	2	no	—
109	C₁₆×C₄₇	—	2	2	no	—
110	C₄₇⋊C₁₆	—	2	2	no	—
111	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
112	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	—	4	2	no	—
115	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
116	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#131	1	2	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂²⋊D₄₇	—	2	4	no	—
128	C₂²⋊D₄₇	—	4	4	no	—
129	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
130	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
131	D₁₈₈	—	2	2	no	—
132	C₄×D₄₇	—	2	2	no	—
133	C₄×D₄₇	—	2	2	no	—
134	Dic₉₄	—	2	2	no	—
135	C₈×C₄₇	—	2	2	no	—
136	C₈×C₄₇	C₄×D₄	1	2	yes	—
137	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₄₇⋊C₈	—	2	2	no	—
140	C₄₇⋊C₈	—	2	2	no	—
141	#33433	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	C₈×D₄	1	2	yes	—
148	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₄₇	—	2	4	no	—
150	C₂×D₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	C₈×D₄	1	2	yes	—
153	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
155	Dic₄₇	—	2	4	no	—
156	Dic₄₇	—	2	2	no	—
157	C₂×#504	—	47	1	no	—
158	C₂²×M₃₂	—	47	1	no	—
159	#2929	—	47	1	no	—
160	#2169	—	47	4	no	—
161	#2168	—	47	4	no	—
162	#2151	—	47	1	no	—
163	#2113	—	47	1	no	—
164	#2112	D₄₇	1	1	yes	characteristic, complemented
165	#2108	—	47	1	no	—
166	C₂×C₄₇	—	2	2	no	—
167	C₂×C₄₇	—	4	2	no	—
168	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₄₇	#3407	1	2	yes	—
170	D₄₇	—	4	2	no	—
171	C₂×#131	—	47	1	no	—
172	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
173	#525	—	47	1	no	—
174	#504	—	94	4	no	—
175	C₂×M₃₂	—	47	2	no	—
176	C₂×M₃₂	—	47	2	no	—
177	C₂×M₃₂	—	47	2	no	—
178	C₂×M₃₂	—	94	4	no	—
179	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
180	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₁₆	—	47	1	no	—
182	C₂²×C₁₆	—	47	1	no	—
183	C₂²×C₁₆	—	47	2	no	—
184	#435	—	47	4	no	—
185	C₈×D₄	—	47	4	no	—
186	#424	—	47	1	no	—
187	#408	—	47	1	no	—
188	#407	—	47	1	no	—
189	#405	—	47	1	no	—
190	C₄⋊C₁₆	—	47	4	no	—
191	C₂²⋊C₁₆	—	47	4	no	—
192	C₂²⋊C₁₆	C₂×D₄₇	1	4	yes	—
193	#346	—	47	2	no	—
194	C₄×C₁₆	—	47	2	no	—
195	C₄₇	#33433	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	47	1	no	—
197	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
198	#131	—	94	2	no	—
199	#131	—	94	2	no	—
200	C₂×M₁₆	—	47	1	no	—
201	C₂×M₁₆	—	47	1	no	—
202	C₂×M₁₆	—	47	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₄₇	1	2	yes	—
208	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	#123	—	47	2	no	—
211	C₂²⋊Q₈	—	47	2	no	—
212	#121	—	47	2	no	—
213	#120	—	47	2	no	—
214	C₄×D₄	—	47	4	no	—
215	#117	—	47	1	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	2	2	no	—
218	M₃₂	—	94	4	no	—
219	M₃₂	—	94	4	no	—
220	M₃₂	—	94	4	no	—
221	C₂×C₁₆	—	47	4	no	—
222	C₂×C₁₆	—	47	1	no	—
223	C₂×C₁₆	—	47	1	no	—
224	C₂×C₁₆	—	47	2	no	—
225	C₂×C₁₆	—	94	4	no	—
226	C₂×C₁₆	—	94	2	no	—
227	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
228	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
229	C₄⋊C₈	—	47	4	no	—
230	C₂²⋊C₈	—	47	4	no	—
231	C₂²⋊C₈	—	47	4	no	—
232	#97	—	47	2	no	—
233	C₄×C₈	—	47	2	no	—
234	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
235	#41	—	94	2	no	—
236	#41	—	94	2	no	—
237	C₂×Q₈	—	47	1	no	—
238	C₂×D₄	—	47	2	no	—
239	C₂×D₄	—	47	4	no	—
240	C₂×D₄	—	47	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	C₈×D₄₇	1	2	yes	—
245	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	M₁₆	—	94	2	no	—
249	M₁₆	—	94	2	no	—
250	M₁₆	—	94	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	47	2	no	—
256	C₂×C₈	—	47	2	no	—
257	C₂×C₈	—	94	2	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
261	C₂×C₈	D₄×D₄₇	1	2	yes	—
262	C₂×C₈	#10625352	1	2	yes	—
263	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
264	C₄⋊C₄	—	47	4	no	—
265	C₄⋊C₄	—	47	2	no	—
266	C₂²⋊C₄	—	47	2	no	—
267	C₂²⋊C₄	—	47	4	no	—
268	C₂²⋊C₄	—	47	4	no	—
269	C₄²	—	47	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	47	2	no	—
273	C₁₆	—	47	2	no	—
274	C₁₆	—	94	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	47	2	no	—
278	C₂³	C₂×C₈×D₄₇	1	2	yes	—
279	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
280	Q₈	—	94	2	no	—
281	D₄	—	94	4	no	—
282	D₄	—	94	2	no	—
283	D₄	—	188	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	47	4	no	—
289	C₂×C₄	—	47	2	no	—
290	C₂×C₄	—	94	2	no	—
291	C₂×C₄	—	94	2	no	—
292	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
295	C₂×C₄	#13094032	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	94	2	no	—
299	C₈	—	94	2	no	—
300	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
301	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
302	C₈	C₂×#10625352	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	94	2	no	—
309	C₂²	—	94	4	no	—
310	C₂²	(C₂²⋊D₄₇)⋊C₈	1	2	yes	—
311	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	94	2	no	—
314	C₄	—	94	4	no	—
315	C₄	(C₂²⋊D₄₇)⋊C₈	1	2	yes	—
316	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
317	C₄	C₂×#13094032	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	188	2	no	—
321	C₂	C₂×(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
322	C₂	#20256490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2112	C₂
prime radical	#15689634	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2112	D₄₇
hypocenter	C₄₇	#33433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
1. C₂×#15689541	C₂	semidirect
2. C₂²×#13093964	C₂	semidirect
4. #20245217	C₂	semidirect
5. #20245216	C₂	semidirect
6. #20245148	C₂	semidirect
7. #20245003	C₂	semidirect
8. #20244987	C₂	semidirect
9. C₄₇×#2112	C₂	semidirect
16. C₂×#13093963	C₂²	semidirect
19. C₂×C₁₆×D₄₇	C₂²	semidirect
26. Dic₄₇⋊C₁₆	C₂²	semidirect
27. (C₂×D₄₇)⋊C₁₆	C₂²	semidirect
28. #15688898	C₂²	semidirect
29. C₁₆×Dic₄₇	C₂²	semidirect
51. C₂×C₄₇⋊C₁₆	D₄	semidirect
164. #2112	D₄₇	semidirect
195. C₄₇	#33433	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689541	2
C₂²×#13093964	2
#20252082	2
#20245217	2
#20245216	2
#20245148	2
#20245003	2
#20244987	2
C₄₇×#2112	2
#33433	47