Subgroups — #26109421

Order 12032.

450 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689200	C₂	1	1	yes	maximal, characteristic
2	#20243559	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241814	C₂	1	1	yes	maximal, characteristic
5	#20241812	C₂	1	1	yes	maximal, characteristic
6	#20241285	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	—	2	1	no	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689200	—	2	1	no	—
11	#15689200	—	2	1	no	—
12	#15689200	C₄	1	2	yes	—
13	#15689200	C₄	1	2	yes	—
14	#15689176	C₂²	1	1	yes	characteristic
15	C₂×#13093959	—	2	1	no	—
16	C₂×#13093959	—	2	1	no	—
17	#15689147	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15688875	—	2	1	no	—
21	#15688873	—	2	1	no	—
22	#15688852	—	2	2	no	—
23	#15688850	—	2	2	no	—
24	C₄₇×#431	C₂²	1	1	yes	characteristic
25	C₄₇×#423	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×D₁₈₈	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	1	no	—
29	C₂×#10625329	—	2	1	no	—
30	C₂×#10625324	—	2	2	no	—
31	C₂×#10625324	—	2	2	no	—
32	C₂×#10625324	—	4	2	no	—
33	C₂×#10625324	—	4	2	no	—
34	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
36	C₄×D₁₈₈	—	2	1	no	—
37	C₄×D₁₈₈	—	2	1	no	—
38	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
39	C₄×D₁₈₈	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	Q₈	1	1	yes	characteristic
41	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
42	#13093959	—	4	2	no	—
43	#13093959	—	4	2	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
48	(C₂×D₄₇)⋊C₈	—	2	2	no	—
49	(C₂×D₄₇)⋊C₈	—	2	2	no	—
50	C₄×C₄₇⋊C₈	—	2	1	no	—
51	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
54	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₄₇	—	2	1	no	—
56	C₂²×C₈×C₄₇	—	2	1	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	C₄⋊C₈×C₄₇	—	2	1	no	—
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₄₇	—	2	1	no	—
64	C₄×C₈×C₄₇	—	2	1	no	—
65	C₂³×D₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	1	no	—
70	C₂×D₁₈₈	—	2	1	no	—
71	C₂×D₁₈₈	—	4	2	no	—
72	C₂×D₁₈₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₄₇	—	2	2	no	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	4	1	no	—
76	C₂×C₄×D₄₇	—	4	1	no	—
77	C₂×C₄×D₄₇	—	4	1	no	—
78	C₂×C₄×D₄₇	—	4	1	no	—
79	M₁₆×C₄₇	—	4	1	no	—
80	M₁₆×C₄₇	—	4	1	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	—	4	1	no	—
85	C₂×C₈×C₄₇	—	4	1	no	—
86	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	#41	1	1	yes	characteristic
90	(C₂×D₄₇)⋊C₄	—	2	2	no	—
91	(C₂×D₄₇)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	1	no	—
93	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
94	#10625329	—	4	1	no	—
95	#10625329	—	4	1	no	—
96	C₂×C₄₇⋊C₈	—	2	1	no	—
97	C₂×C₄₇⋊C₈	—	2	1	no	—
98	C₂×C₄₇⋊C₈	—	4	1	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
105	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
106	#10625324	—	4	4	no	—
107	#10625324	—	4	4	no	—
108	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×D₄₇	—	2	2	no	—
110	C₂²×D₄₇	—	4	2	no	—
111	C₂²×D₄₇	—	4	2	no	—
112	C₂²×D₄₇	—	4	1	no	—
113	C₂²×D₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	4	1	no	—
128	C₂×Dic₄₇	—	4	1	no	—
129	D₁₈₈	—	2	4	no	—
130	D₁₈₈	—	4	2	no	—
131	C₄×D₄₇	—	4	2	no	—
132	C₄×D₄₇	—	4	2	no	—
133	C₄×D₄₇	—	4	2	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₈×C₄₇	—	4	2	no	—
136	C₈×C₄₇	—	8	1	no	—
137	C₈×C₄₇	—	8	1	no	—
138	C₄₇⋊C₈	—	2	1	no	—
139	C₄₇⋊C₈	—	2	1	no	—
140	C₄₇⋊C₈	—	8	1	no	—
141	C₄₇⋊C₈	—	8	1	no	—
142	C₄₇⋊C₈	#102	1	2	yes	—
143	C₄₇⋊C₈	#102	1	2	yes	—
144	#8662	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	#380	1	1	yes	characteristic
148	C₂²×C₄₇	#474	1	1	yes	characteristic
149	C₂²×C₄₇	#476	1	1	yes	characteristic
150	C₂×D₄₇	—	4	2	no	—
151	C₂×D₄₇	—	4	2	no	—
152	C₂×D₄₇	—	4	2	no	—
153	C₂×D₄₇	—	8	2	no	—
154	C₂×D₄₇	—	8	2	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
159	C₄×C₄₇	#417	1	1	yes	characteristic
160	C₄×C₄₇	#430	1	2	yes	—
161	C₄×C₄₇	#459	1	1	yes	characteristic
162	C₄×C₄₇	#461	1	1	yes	characteristic
163	Dic₄₇	—	4	2	no	—
164	C₂×#436	—	47	1	no	—
165	#1848	—	47	1	no	—
166	#1564	—	47	1	no	—
167	#1486	—	47	1	no	—
168	#1486	—	47	1	no	—
169	#1301	—	47	1	no	—
170	#1301	D₄₇	1	1	yes	characteristic, complemented
171	C₂×C₄₇	—	2	1	no	—
172	C₂×C₄₇	—	2	1	no	—
173	C₂×C₄₇	#1802	1	1	yes	characteristic
174	C₂×C₄₇	#1811	1	1	yes	characteristic
175	C₂×C₄₇	#1814	1	1	yes	characteristic
176	D₄₇	—	8	2	no	—
177	C₂²×M₁₆	—	94	1	no	—
178	C₂×C₄×D₄	—	47	1	no	—
179	#436	—	47	2	no	—
180	#436	—	47	2	no	—
181	#436	—	94	1	no	—
182	#436	—	94	1	no	—
183	#431	—	47	1	no	—
184	#431	C₂×D₄₇	1	1	yes	characteristic
185	C₂×#108	—	94	1	no	—
186	C₂×#108	—	94	1	no	—
187	#423	—	47	1	no	—
188	#423	C₂×D₄₇	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₂²⋊C₈	—	94	1	no	—
191	C₂×C₄×C₈	—	47	1	no	—
192	#335	—	94	1	no	—
193	#334	—	94	1	no	—
194	D₄⋊C₈	—	94	2	no	—
195	D₄⋊C₈	—	94	2	no	—
196	C₄₇	#8662	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	47	1	no	—
198	C₂³×C₄	—	94	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	94	2	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂×M₁₆	—	94	1	no	—
203	C₂×M₁₆	—	188	2	no	—
204	C₂×M₁₆	—	188	2	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₄×D₄	—	47	2	no	—
209	C₄×D₄	—	47	2	no	—
210	C₄×D₄	—	94	1	no	—
211	C₄×D₄	—	94	1	no	—
212	C₂×C₄⋊C₄	—	47	1	no	—
213	C₂×C₂²⋊C₄	—	94	1	no	—
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	#108	—	188	2	no	—
216	#108	—	188	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	94	1	no	—
222	C₄⋊C₈	—	94	1	no	—
223	C₄⋊C₈	—	94	1	no	—
224	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	94	2	no	—
227	C₂²⋊C₈	—	94	2	no	—
228	C₂²⋊C₈	—	94	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	47	1	no	—
231	C₄×C₈	—	47	1	no	—
232	C₄×C₈	—	47	1	no	—
233	C₄×C₈	—	47	1	no	—
234	C₄×C₈	—	94	1	no	—
235	C₂⁴	—	94	1	no	—
236	C₂×D₄	—	47	2	no	—
237	C₂×D₄	—	94	1	no	—
238	C₂×D₄	—	188	2	no	—
239	C₂²×C₄	—	94	1	no	—
240	C₂²×C₄	—	94	2	no	—
241	C₂²×C₄	—	94	2	no	—
242	C₂²×C₄	—	188	1	no	—
243	C₂²×C₄	—	188	1	no	—
244	C₂²×C₄	—	188	1	no	—
245	C₂²×C₄	—	188	1	no	—
246	C₂²×C₄	#10625350	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	188	4	no	—
252	M₁₆	—	188	4	no	—
253	M₁₆	—	188	1	no	—
254	M₁₆	—	188	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	47	1	no	—
261	C₂×C₈	—	47	1	no	—
262	C₂×C₈	—	47	2	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	47	1	no	—
265	C₂×C₈	—	47	1	no	—
266	C₂×C₈	—	94	1	no	—
267	C₂×C₈	—	94	1	no	—
268	C₂×C₈	—	188	1	no	—
269	C₂×C₈	—	188	1	no	—
270	C₄⋊C₄	—	47	2	no	—
271	C₄⋊C₄	—	94	1	no	—
272	C₂²⋊C₄	—	94	2	no	—
273	C₂²⋊C₄	—	94	2	no	—
274	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₄²	#10625350	1	1	yes	characteristic
276	C₄²	D₄×D₄₇	1	1	yes	characteristic
277	C₄²	D₄×D₄₇	1	1	yes	characteristic
278	C₂³	—	94	2	no	—
279	C₂³	—	188	2	no	—
280	C₂³	—	188	2	no	—
281	C₂³	—	188	1	no	—
282	C₂³	—	188	1	no	—
283	C₂³	#13094002	1	1	yes	characteristic
284	D₄	—	94	4	no	—
285	D₄	—	188	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	188	1	no	—
290	C₂×C₄	—	188	1	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	#13093981	1	1	yes	characteristic
298	C₂×C₄	#13093997	1	1	yes	characteristic
299	C₂×C₄	#13093997	1	1	yes	characteristic
300	C₂×C₄	#13094002	1	1	yes	characteristic
301	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
303	C₂×C₄	#13094051	1	1	yes	characteristic
304	C₂×C₄	#13094056	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	—	8	1	no	—
308	C₈	—	47	2	no	—
309	C₈	—	47	2	no	—
310	C₈	—	94	1	no	—
311	C₈	—	94	1	no	—
312	C₈	—	376	1	no	—
313	C₈	—	376	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	376	2	no	—
317	C₂²	—	376	2	no	—
318	C₂²	—	188	2	no	—
319	C₂²	—	188	2	no	—
320	C₂²	—	188	2	no	—
321	C₂²	#15688982	1	1	yes	characteristic
322	C₂²	#15689402	1	1	yes	characteristic
323	C₂²	#15689412	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	188	2	no	—
328	C₄	#102×D₄₇	1	1	yes	characteristic
329	C₄	#15689123	1	1	yes	characteristic
330	C₄	#15689170	1	2	yes	—
331	C₄	#15689317	1	1	yes	characteristic
332	C₄	#15689327	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	376	2	no	—
336	C₂	#20243387	1	1	yes	characteristic
337	C₂	#20243429	1	1	yes	characteristic
338	C₂	#20243441	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093997
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1301	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂²×D₁₈₈	Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1301	D₄₇
hypocenter	C₄₇	#8662
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #20243559	C₂	semidirect
3. #20242134	C₂	semidirect
6. #20241285	C₂	semidirect
7. C₄₇×#1301	C₂	semidirect
170. #1301	D₄₇	semidirect
196. C₄₇	#8662	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689200	2
#20243559	2
#20242134	2
#20241814	2
#20241812	2
#20241285	2
C₄₇×#1301	2
#8662	47