Subgroups — #26112301

Order 12032.

524 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20249154	C₂	1	1	yes	maximal, characteristic
2	#20242061	C₂	1	2	yes	maximal, complemented
3	#20241702	C₂	1	1	yes	maximal, characteristic
4	#20241248	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13094003	—	2	1	no	—
8	#15689176	C₂²	1	2	yes	—
9	#15689053	—	2	1	no	—
10	#15689053	C₄	1	2	yes	—
11	#15689048	—	2	1	no	—
12	#15689048	C₄	1	2	yes	—
13	#15689029	—	2	1	no	—
14	#15688989	C₂²	1	1	yes	characteristic
15	#15688870	—	2	4	no	—
16	#15688849	—	2	1	no	—
17	#15688847	—	2	1	no	—
18	C₂²×M₁₆×C₄₇	—	2	1	no	—
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#424	—	2	4	no	—
21	C₄₇×#423	—	2	1	no	—
22	C₄₇×#423	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	—	2	1	no	—
24	#13094024	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094003	—	4	2	no	—
27	#13094003	—	4	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
31	C₂×Dic₄₇⋊C₄	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄×C₄₇⋊C₈	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₄₇⋊#95	—	2	1	no	—
39	C₄₇⋊#95	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	4	2	no	—
44	C₂×M₁₆×C₄₇	—	4	2	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	#117×C₄₇	—	2	2	no	—
47	#117×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	—	2	4	no	—
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
57	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	D₈	1	2	yes	—
66	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂×Dic₉₄	—	4	1	no	—
73	C₂×Dic₉₄	—	4	1	no	—
74	C₂×Dic₉₄	—	4	1	no	—
75	C₂×Dic₉₄	—	4	1	no	—
76	C₂×Dic₉₄	—	4	1	no	—
77	C₂×Dic₉₄	—	4	1	no	—
78	M₁₆×C₄₇	—	4	4	no	—
79	M₁₆×C₄₇	—	4	2	no	—
80	M₁₆×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	4	4	no	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₂²⋊C₄×C₄₇	—	2	2	no	—
86	C₂²⋊C₄×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	1	no	—
88	C₄²×C₄₇	—	2	1	no	—
89	C₄²×C₄₇	C₂×D₄	1	2	yes	—
90	C₄²×C₄₇	#41	1	2	yes	—
91	C₂²⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	1	no	—
95	C₄⋊Dic₄₇	—	4	1	no	—
96	Dic₄₇⋊C₄	—	4	4	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	4	1	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	#102	1	2	yes	—
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	#102	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#121	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	Dic₉₄	—	8	1	no	—
132	Dic₉₄	—	8	1	no	—
133	Dic₉₄	—	8	1	no	—
134	Dic₉₄	—	8	1	no	—
135	C₈×C₄₇	—	4	4	no	—
136	C₄₇⋊C₈	—	8	2	no	—
137	#9195	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	4	2	no	—
142	C₂²×C₄₇	—	4	1	no	—
143	C₂²×C₄₇	—	4	1	no	—
144	C₂²×C₄₇	—	8	1	no	—
145	C₂²×C₄₇	#386	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
147	C₂²×C₄₇	#417	1	1	yes	characteristic
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	4	1	no	—
152	C₄×C₄₇	—	4	1	no	—
153	C₄×C₄₇	#447	1	1	yes	characteristic
154	C₄×C₄₇	#448	1	1	yes	characteristic
155	C₄×C₄₇	#480	1	1	yes	characteristic
156	C₄×C₄₇	#481	1	1	yes	characteristic
157	Dic₄₇	—	8	2	no	—
158	Dic₄₇	—	8	2	no	—
159	#2905	D₄₇	1	1	yes	characteristic
160	#2621	—	47	1	no	—
161	#1546	—	47	2	no	—
162	#1458	—	47	1	no	—
163	#1286	—	47	2	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	4	2	no	—
167	C₂×C₄₇	#1874	1	1	yes	characteristic
168	C₂×C₄₇	#1909	1	1	yes	characteristic
169	C₂×C₄₇	#1915	1	1	yes	characteristic
170	C₂²×M₁₆	—	2	1	no	—
171	C₂×C₄⋊Q₈	—	47	1	no	—
172	C₂×C₂²⋊Q₈	—	94	1	no	—
173	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
174	#431	—	47	2	no	—
175	#424	—	2	4	no	—
176	#423	—	2	1	no	—
177	#423	C₂×D₄₇	1	2	yes	—
178	#398	—	47	2	no	—
179	#398	—	94	1	no	—
180	#397	—	47	2	no	—
181	#397	—	94	1	no	—
182	#391	—	94	1	no	—
183	#382	—	47	1	no	—
184	#332	—	94	4	no	—
185	#324	—	94	1	no	—
186	#323	—	94	1	no	—
187	C₄₇	#9195	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	94	1	no	—
189	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	4	2	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₄⋊Q₈	—	94	2	no	—
196	C₄⋊Q₈	—	94	2	no	—
197	C₂²⋊Q₈	—	188	2	no	—
198	C₂²⋊Q₈	—	188	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	C₂×C₄⋊C₄	—	47	2	no	—
202	C₂×C₄⋊C₄	—	94	2	no	—
203	C₂×C₄⋊C₄	—	94	1	no	—
204	C₂×C₄⋊C₄	—	94	1	no	—
205	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	94	2	no	—
208	C₂×C₂²⋊C₄	C₄×D₄₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	D₁₈₈	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	C₄⋊C₈	—	2	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	94	2	no	—
215	C₄⋊C₈	C₂²⋊D₄₇	1	4	yes	—
216	C₂²⋊C₈	—	94	2	no	—
217	C₄×C₈	—	94	2	no	—
218	#95	—	47	2	no	—
219	#95	—	94	2	no	—
220	#95	—	94	2	no	—
221	#95	—	94	1	no	—
222	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
223	C₂×Q₈	—	188	1	no	—
224	C₂×Q₈	—	188	1	no	—
225	C₂×Q₈	—	188	1	no	—
226	C₂×Q₈	—	188	1	no	—
227	C₂×Q₈	—	188	1	no	—
228	C₂×Q₈	—	188	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	94	2	no	—
234	C₂²×C₄	—	94	2	no	—
235	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	#10625334	1	2	yes	—
239	C₂²×C₄	#10625336	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	M₁₆	—	4	4	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	188	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	188	4	no	—
253	C₄⋊C₄	—	188	2	no	—
254	C₄⋊C₄	—	188	1	no	—
255	C₄⋊C₄	—	188	1	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	188	4	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	#10625350	1	2	yes	—
262	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	#13093944	1	2	yes	—
268	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	Q₈	—	376	1	no	—
270	Q₈	—	376	1	no	—
271	Q₈	—	376	1	no	—
272	Q₈	—	376	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	#13093944	1	2	yes	—
292	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
293	C₂×C₄	#13093993	1	1	yes	characteristic
294	C₂×C₄	#13093998	1	2	yes	—
295	C₂×C₄	#13094007	1	1	yes	characteristic
296	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
297	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
298	C₂×C₄	#13094050	1	1	yes	characteristic
299	C₂×C₄	#13094055	1	1	yes	characteristic
300	C₈	—	4	4	no	—
301	C₈	—	376	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	#15689002	1	1	yes	characteristic
310	C₂²	C₂×#13093944	1	1	yes	characteristic
311	C₂²	#15689122	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	376	2	no	—
318	C₄	—	376	2	no	—
319	C₄	#15689257	1	1	yes	characteristic
320	C₄	#15689262	1	1	yes	characteristic
321	C₄	#15689436	1	1	yes	characteristic
322	C₄	#15689441	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	4	2	no	—
326	C₂	#20243705	1	1	yes	characteristic
327	C₂	#20243910	1	1	yes	characteristic
328	C₂	#20243940	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689002
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2905	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#9195
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #20242061	C₂	semidirect
4. #20241248	C₂	semidirect
187. C₄₇	#9195	semidirect

Maximal subgroups

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

subgroup	index
#20249154	2
#20242061	2
#20241702	2
#20241248	2
C₄₇×#2905	2
#9195	47