Subgroups — #26369463

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂×#15689540	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252098	C₂	1	1	yes	maximal, characteristic
4	#20245217	C₂	1	4	yes	maximal, complemented
5	(C₂²⋊D₄₇)⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	#20245143	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245001	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094030	C₂²	1	1	yes	characteristic
11	#15689669	C₄	1	1	yes	characteristic
12	#15689541	—	2	4	no	—
13	#15689540	—	2	4	no	—
14	C₂×#13093964	C₂²	1	1	yes	characteristic
15	C₂×#13093964	C₂²	1	2	yes	—
16	C₂×#13093963	C₂²	1	2	yes	—
17	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
20	#15689199	C₄	1	4	yes	—
21	C₈×C₂²⋊D₄₇	C₂²	1	4	yes	—
22	#15689146	C₄	1	1	yes	characteristic
23	#15689091	C₄	1	1	yes	characteristic
24	#15689090	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
26	Dic₄₇⋊C₁₆	C₂²	1	4	yes	—
27	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
28	C₁₆×Dic₄₇	C₂²	1	2	yes	—
29	C₁₆×Dic₄₇	C₂²	1	2	yes	—
30	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×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×#10625324	C₂×C₄	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₄₇	C₂³	1	2	yes	—
41	#13094022	C₈	1	1	yes	characteristic
42	#13094020	C₈	1	1	yes	characteristic
43	#13094013	C₈	1	1	yes	characteristic
44	#13094011	C₈	1	1	yes	characteristic
45	#13093996	C₈	1	4	yes	—
46	C₄×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
48	#13093964	—	2	2	no	—
49	#13093964	—	2	4	no	—
50	#13093963	—	2	4	no	—
51	C₂×C₄₇⋊C₁₆	—	2	4	no	—
52	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
53	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₁₆×D₄₇	—	2	4	no	—
57	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
58	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
59	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
60	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
61	C₈×Dic₄₇	C₂³	1	2	yes	—
62	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
63	#131×C₄₇	—	2	2	no	—
64	#131×C₄₇	—	2	2	no	—
65	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	M₃₂×C₄₇	—	2	2	no	—
70	C₂×C₁₆×C₄₇	—	2	2	no	—
71	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
72	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
73	#41×C₄₇	—	2	2	no	—
74	#41×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
77	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₄₇	C₂×C₈	1	4	yes	—
81	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
82	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
83	#10625350	—	2	2	no	—
84	#10625350	—	2	2	no	—
85	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
88	M₁₆×C₄₇	—	2	2	no	—
89	M₁₆×C₄₇	—	2	4	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₄₇	#41	1	2	yes	—
96	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	C₂×C₈	1	4	yes	—
98	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
99	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
100	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
103	#10625329	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
107	#10625324	—	2	4	no	—
108	C₈×D₄₇	—	2	2	no	—
109	C₈×D₄₇	—	2	2	no	—
110	C₁₆×C₄₇	—	2	2	no	—
111	C₁₆×C₄₇	—	2	2	no	—
112	C₄₇⋊C₁₆	—	2	2	no	—
113	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
114	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
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	Q₈×C₄₇	—	2	2	no	—
119	D₄×C₄₇	—	2	4	no	—
120	D₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#131	1	2	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₄₇	—	2	4	no	—
129	C₂²⋊D₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
132	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
133	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
134	D₁₈₈	—	2	2	no	—
135	C₄×D₄₇	—	2	2	no	—
136	C₄×D₄₇	—	2	2	no	—
137	Dic₉₄	—	2	2	no	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	C₄×D₄	1	2	yes	—
140	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₄₇⋊C₈	—	2	2	no	—
143	C₄₇⋊C₈	—	2	2	no	—
144	#33434	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	4	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	C₈×D₄	1	2	yes	—
149	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₄₇	—	2	4	no	—
151	C₂×D₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	C₈×D₄	1	2	yes	—
154	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
156	Dic₄₇	—	2	2	no	—
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	#504	1	4	yes	complemented
159	C₂×#504	—	47	1	no	—
160	C₂×#504	—	47	1	no	—
161	#2931	—	47	1	no	—
162	#2169	—	47	4	no	—
163	D₄×C₁₆	—	47	4	no	—
164	#2150	—	47	1	no	—
165	#2113	—	47	1	no	—
166	#2113	D₄₇	1	1	yes	characteristic, complemented
167	C₂×C₄×C₁₆	—	47	1	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	—	4	2	no	—
170	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#504	1	2	yes	—
172	D₄₇	—	4	2	no	—
173	C₂×#131	—	47	1	no	—
174	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
175	#532	—	47	1	no	—
176	#504	—	94	4	no	—
177	#504	—	94	4	no	—
178	C₂×M₃₂	—	47	1	no	—
179	C₂×M₃₂	—	47	2	no	—
180	C₂×M₃₂	—	47	2	no	—
181	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
182	C₂²×C₁₆	—	47	1	no	—
183	C₂²×C₁₆	—	47	2	no	—
184	C₂²×C₁₆	—	47	2	no	—
185	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
186	#436	—	47	4	no	—
187	C₈×D₄	—	47	4	no	—
188	#423	—	47	1	no	—
189	#408	—	47	1	no	—
190	#408	—	47	1	no	—
191	C₂×C₄×C₈	—	47	1	no	—
192	C₄⋊C₁₆	—	47	4	no	—
193	C₂²⋊C₁₆	—	47	4	no	—
194	C₂²⋊C₁₆	C₂×D₄₇	1	4	yes	—
195	C₄×C₁₆	—	47	2	no	—
196	C₄×C₁₆	—	47	2	no	—
197	C₄₇	#33434	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	47	1	no	—
199	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	94	2	no	—
203	#131	—	94	2	no	—
204	C₂×M₁₆	—	47	1	no	—
205	C₂×M₁₆	—	47	2	no	—
206	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
207	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	—	47	2	no	—
210	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
211	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₄⋊Q₈	—	47	1	no	—
213	#127	—	47	1	no	—
214	#124	—	47	1	no	—
215	#124	—	47	1	no	—
216	#121	—	47	4	no	—
217	C₄×D₄	—	47	4	no	—
218	C₂×C₄²	—	47	1	no	—
219	M₃₂	—	2	2	no	—
220	M₃₂	—	94	4	no	—
221	M₃₂	—	94	4	no	—
222	M₃₂	—	94	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	C₂×C₁₆	—	47	4	no	—
225	C₂×C₁₆	—	47	1	no	—
226	C₂×C₁₆	—	47	1	no	—
227	C₂×C₁₆	—	47	2	no	—
228	C₂×C₁₆	—	94	4	no	—
229	C₂×C₁₆	—	94	4	no	—
230	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
231	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
232	C₄⋊C₈	—	47	4	no	—
233	C₂²⋊C₈	—	47	4	no	—
234	C₂²⋊C₈	—	47	4	no	—
235	C₄×C₈	—	47	2	no	—
236	C₄×C₈	—	47	2	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	94	2	no	—
240	#41	—	94	2	no	—
241	C₂×Q₈	—	47	1	no	—
242	C₂×Q₈	C₈×D₄₇	1	1	yes	characteristic
243	C₂×D₄	—	47	4	no	—
244	C₂×D₄	—	47	2	no	—
245	C₂×D₄	—	47	1	no	—
246	C₂×D₄	C₈×D₄₇	1	2	yes	—
247	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	—	47	2	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	94	2	no	—
255	M₁₆	—	94	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	47	2	no	—
259	C₂×C₈	—	47	2	no	—
260	C₂×C₈	—	94	2	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
265	C₂×C₈	D₄×D₄₇	1	2	yes	—
266	C₂×C₈	#10625352	1	2	yes	—
267	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
268	C₄⋊C₄	—	47	4	no	—
269	C₂²⋊C₄	—	47	4	no	—
270	C₂²⋊C₄	—	47	4	no	—
271	C₄²	—	47	1	no	—
272	C₄²	—	47	1	no	—
273	C₄²	—	47	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	47	2	no	—
277	C₁₆	—	47	2	no	—
278	C₁₆	—	94	2	no	—
279	C₂³	—	47	2	no	—
280	C₂³	C₂×C₈×D₄₇	1	2	yes	—
281	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	94	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	94	8	no	—
287	D₄	—	94	4	no	—
288	D₄	—	94	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	47	2	no	—
292	C₂×C₄	—	47	2	no	—
293	C₂×C₄	—	47	2	no	—
294	C₂×C₄	—	94	4	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	94	2	no	—
297	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
298	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
300	C₂×C₄	#13094032	1	2	yes	—
301	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	94	2	no	—
304	C₈	—	94	2	no	—
305	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
306	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
307	C₈	C₂×#10625352	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	94	2	no	—
312	C₂²	—	94	4	no	—
313	C₂²	(C₂²⋊D₄₇)⋊C₈	1	2	yes	—
314	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	47	4	no	—
317	C₄	—	94	2	no	—
318	C₄	—	94	2	no	—
319	C₄	(C₂²⋊D₄₇)⋊C₈	1	2	yes	—
320	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
321	C₄	C₂×#13094032	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	188	2	no	—
325	C₂	C₂×(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
326	C₂	#504×D₄₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2113	C₂
prime radical	#15689669	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2113	D₄₇
hypocenter	C₄₇	#33434
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. C₂×#15689541	C₂	semidirect
2. C₂×#15689540	C₂	semidirect
4. #20245217	C₂	semidirect
5. (C₂²⋊D₄₇)⋊C₁₆	C₂	semidirect
6. #20245143	C₂	semidirect
7. #20245001	C₂	semidirect
8. C₂×C₁₆×Dic₄₇	C₂	semidirect
9. C₄₇×#2113	C₂	semidirect
52. C₂×C₄₇⋊C₁₆	D₄	semidirect
158. Dic₄₇	#504	semidirect
166. #2113	D₄₇	semidirect
197. C₄₇	#33434	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
C₂×#15689540	2
#20252098	2
#20245217	2
(C₂²⋊D₄₇)⋊C₁₆	2
#20245143	2
#20245001	2
C₂×C₁₆×Dic₄₇	2
C₄₇×#2113	2
#33434	47