Subgroups — #26104948

Order 12032.

494 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243456	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	#20241211	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#391	—	2	1	no	—
15	C₄₇×#389	—	2	1	no	—
16	C₄₇×#389	C₂²	1	2	yes	—
17	C₄₇×#384	C₄	1	1	yes	characteristic
18	C₄₇×#380	—	2	1	no	—
19	C₄₇×#380	C₄	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
25	C₂²×Q₈×C₄₇	—	2	1	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₇	—	4	2	no	—
30	C₂²⋊Q₈×C₄₇	—	4	2	no	—
31	C₂²⋊Q₈×C₄₇	—	4	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	4	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#95×C₄₇	—	2	2	no	—
45	#95×C₄₇	—	2	2	no	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	—	4	2	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₄⋊C₄×C₄₇	—	4	2	no	—
71	C₄⋊C₄×C₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	4	2	no	—
73	C₄⋊C₄×C₄₇	—	4	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	4	4	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂³×C₄₇	—	2	1	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	#102	1	2	yes	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₄₇	—	8	4	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	#99	1	2	yes	—
118	C₂×C₄×C₄₇	#101	1	2	yes	—
119	C₂×C₄×C₄₇	#102	1	2	yes	—
120	C₂×C₄×C₄₇	#131	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#137	1	2	yes	—
124	C₄₇⋊C₈	—	8	2	no	—
125	C₄₇⋊C₈	—	8	2	no	—
126	#7728	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	4	1	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	#408	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
138	C₂²×C₄₇	#419	1	1	yes	characteristic
139	C₂²×C₄₇	#421	1	1	yes	characteristic
140	C₂²×C₄₇	#421	1	1	yes	characteristic
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	8	2	no	—
149	C₄×C₄₇	—	8	2	no	—
150	#2857	D₄₇	1	1	yes	characteristic
151	#1817	—	47	1	no	—
152	#1797	—	47	1	no	—
153	#1271	—	47	4	no	—
154	C₂×C₄₇	—	4	2	no	—
155	C₂×C₄₇	#1472	1	1	yes	characteristic
156	C₂×C₄₇	#1480	1	1	yes	characteristic
157	C₂×C₄₇	#1497	1	1	yes	characteristic
158	C₂×C₄₇	#1511	1	1	yes	characteristic
159	C₂×C₄₇	#1519	1	1	yes	characteristic
160	C₂×C₄₇	#1537	1	1	yes	characteristic
161	C₂×C₄₇	#1546	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
163	C₂×C₂²⋊Q₈	—	2	1	no	—
164	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	94	1	no	—
166	C₂×C₂²⋊C₈	—	94	1	no	—
167	C₂×C₂²⋊C₈	—	94	1	no	—
168	#391	—	2	1	no	—
169	#389	—	2	1	no	—
170	#389	C₂×D₄₇	1	2	yes	—
171	#384	Dic₄₇	1	1	yes	characteristic
172	#380	—	2	1	no	—
173	#380	Dic₄₇	1	2	yes	—
174	#336	—	47	2	no	—
175	#336	—	47	2	no	—
176	#336	—	94	1	no	—
177	C₄₇	#7728	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	2	1	no	—
179	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂²×C₈	—	94	2	no	—
183	C₂²×C₈	—	94	2	no	—
184	C₄⋊Q₈	—	4	2	no	—
185	C₂²⋊Q₈	—	4	2	no	—
186	C₂²⋊Q₈	—	4	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₄²	—	4	2	no	—
195	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₄⋊C₈	—	188	2	no	—
200	C₂²⋊C₈	—	188	2	no	—
201	C₂²⋊C₈	—	188	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	#10625334	1	2	yes	—
219	C₂²×C₄	#10625335	1	2	yes	—
220	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂×C₈	—	188	2	no	—
227	C₂×C₈	—	188	2	no	—
228	C₂×C₈	—	188	2	no	—
229	C₂×C₈	—	188	2	no	—
230	C₂×C₈	—	188	2	no	—
231	C₂×C₈	—	188	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	#13093945	1	2	yes	—
251	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
252	Q₈	—	8	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#13093939	1	2	yes	—
276	C₂×C₄	#13093943	1	2	yes	—
277	C₂×C₄	#13093945	1	2	yes	—
278	C₂×C₄	#13094031	1	2	yes	—
279	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
281	C₂×C₄	#13094055	1	2	yes	—
282	C₈	—	376	2	no	—
283	C₈	—	376	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	#15689090	1	1	yes	characteristic
292	C₂²	C₂×#13093939	1	1	yes	characteristic
293	C₂²	C₂×#13093943	1	1	yes	characteristic
294	C₂²	C₂×#13093945	1	1	yes	characteristic
295	C₂²	#15689132	1	1	yes	characteristic
296	C₂²	#15689140	1	1	yes	characteristic
297	C₂²	#15689140	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#20241752	1	1	yes	characteristic
309	C₂	#20241787	1	1	yes	characteristic
310	C₂	#20241861	1	1	yes	characteristic
311	C₂	#20241921	1	1	yes	characteristic
312	C₂	#20241956	1	1	yes	characteristic
313	C₂	#20242026	1	1	yes	characteristic
314	C₂	#20242060	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2857	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2857	D₄₇
hypocenter	C₄₇	#7728
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. #20241211	C₂	semidirect
177. C₄₇	#7728	semidirect

Maximal subgroups

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

subgroup	index
#20243456	2
#20243368	2
#20241211	2
C₄₇×#2857	2
#7728	47