Subgroups — #26104973

Order 12032.

366 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	#20241214	C₂	1	2	yes	maximal
4	#20241211	C₂	1	2	yes	maximal
5	C₄₇×#2863	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	C₂²	1	2	yes	—
11	#15688878	C₂²	1	2	yes	—
12	C₄₇×#391	C₄	1	1	yes	characteristic
13	C₄₇×#391	C₄	1	1	yes	characteristic
14	C₄₇×#391	C₂²	1	1	yes	characteristic
15	C₄₇×#389	C₄	1	1	yes	characteristic
16	C₄₇×#389	C₄	1	1	yes	characteristic
17	C₄₇×#389	C₂²	1	1	yes	characteristic
18	C₄₇×#387	—	2	1	no	—
19	C₄₇×#387	—	2	1	no	—
20	C₄₇×#387	—	2	1	no	—
21	C₄₇×#383	—	2	1	no	—
22	C₄³×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	#95×C₄₇	—	2	1	no	—
42	#95×C₄₇	—	2	1	no	—
43	#95×C₄₇	—	2	1	no	—
44	#95×C₄₇	—	2	1	no	—
45	#95×C₄₇	—	2	1	no	—
46	#95×C₄₇	—	2	1	no	—
47	#95×C₄₇	—	2	1	no	—
48	#95×C₄₇	—	2	1	no	—
49	#95×C₄₇	—	2	1	no	—
50	C₂×Q₈×C₄₇	—	2	2	no	—
51	C₂×Q₈×C₄₇	—	4	1	no	—
52	C₂×Q₈×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	4	2	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	4	1	no	—
68	C₄⋊C₄×C₄₇	—	4	1	no	—
69	C₄⋊C₄×C₄₇	—	4	1	no	—
70	C₄⋊C₄×C₄₇	—	4	1	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	1	no	—
73	C₄²×C₄₇	—	2	1	no	—
74	C₄²×C₄₇	—	4	1	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₄²×C₄₇	—	4	1	no	—
78	C₄²×C₄₇	SD₁₆	1	2	yes	—
79	C₄²×C₄₇	Q₁₆	1	2	yes	—
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	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₄₇	—	8	2	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	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	—	4	1	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	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	#103	1	2	yes	—
117	C₂×C₄×C₄₇	#103	1	2	yes	—
118	C₂×C₄×C₄₇	#131	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
123	C₄₇⋊C₈	—	8	2	no	—
124	C₄₇⋊C₈	—	8	2	no	—
125	#7733	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	#408	1	1	yes	characteristic
127	C₂²×C₄₇	#410	1	1	yes	characteristic
128	C₂²×C₄₇	#413	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
130	C₂²×C₄₇	#417	1	1	yes	characteristic
131	C₂²×C₄₇	#421	1	1	yes	characteristic
132	C₂²×C₄₇	#421	1	1	yes	characteristic
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	4	1	no	—
141	C₄×C₄₇	—	4	1	no	—
142	C₄×C₄₇	—	8	1	no	—
143	C₄×C₄₇	—	8	1	no	—
144	C₄×C₄₇	—	8	1	no	—
145	C₄×C₄₇	—	8	1	no	—
146	#2863	D₄₇	1	1	yes	characteristic
147	#1831	—	47	1	no	—
148	#1831	—	47	1	no	—
149	#1272	—	47	2	no	—
150	#1271	—	47	2	no	—
151	C₂×C₄₇	#1473	1	1	yes	characteristic
152	C₂×C₄₇	#1481	1	1	yes	characteristic
153	C₂×C₄₇	#1495	1	1	yes	characteristic
154	C₂×C₄₇	#1512	1	1	yes	characteristic
155	C₂×C₄₇	#1518	1	1	yes	characteristic
156	C₂×C₄₇	#1535	1	1	yes	characteristic
157	C₂×C₄₇	#1550	1	1	yes	characteristic
158	C₂×C₄⋊C₈	—	94	1	no	—
159	C₂×C₄⋊C₈	—	94	1	no	—
160	#391	Dic₄₇	1	1	yes	characteristic
161	#391	Dic₄₇	1	1	yes	characteristic
162	#391	C₂×D₄₇	1	1	yes	characteristic
163	#389	Dic₄₇	1	1	yes	characteristic
164	#389	Dic₄₇	1	1	yes	characteristic
165	#389	C₂×D₄₇	1	1	yes	characteristic
166	#387	—	2	1	no	—
167	#387	—	2	1	no	—
168	#387	—	2	1	no	—
169	#383	—	2	1	no	—
170	C₄³	C₂×D₄₇	1	1	yes	characteristic
171	#336	—	47	2	no	—
172	#336	—	47	2	no	—
173	#336	—	94	1	no	—
174	#336	—	94	1	no	—
175	C₄₇	#7733	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₈	—	94	2	no	—
178	C₂²×C₈	—	94	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	188	2	no	—
193	C₄⋊C₈	—	188	2	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₂×C₈	—	188	2	no	—
220	C₂×C₈	—	188	2	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	#10625336	1	2	yes	—
238	C₄²	#10625337	1	2	yes	—
239	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
240	Q₈	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	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	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	#13093948	1	2	yes	—
270	C₂×C₄	#13093948	1	2	yes	—
271	C₂×C₄	#13094031	1	2	yes	—
272	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
273	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
274	C₂×C₄	#13094049	1	1	yes	characteristic
275	C₂×C₄	#13094054	1	1	yes	characteristic
276	C₈	—	376	2	no	—
277	C₈	—	376	2	no	—
278	C₂²	#15689090	1	1	yes	characteristic
279	C₂²	#15689097	1	1	yes	characteristic
280	C₂²	#15689107	1	1	yes	characteristic
281	C₂²	C₂×#13093948	1	1	yes	characteristic
282	C₂²	#15689124	1	1	yes	characteristic
283	C₂²	#15689140	1	1	yes	characteristic
284	C₂²	#15689140	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	8	1	no	—
295	C₄	—	8	1	no	—
296	C₄	—	8	1	no	—
297	C₄	—	8	1	no	—
298	C₂	#20241757	1	1	yes	characteristic
299	C₂	#20241791	1	1	yes	characteristic
300	C₂	#20241853	1	1	yes	characteristic
301	C₂	#20241926	1	1	yes	characteristic
302	C₂	#20241952	1	1	yes	characteristic
303	C₂	#20242019	1	1	yes	characteristic
304	C₂	#20242078	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×#2863	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2863	D₄₇
hypocenter	C₄₇	#7733
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2863

Complements

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

N	H ≅ G/N	type
175. C₄₇	#7733	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
#20241214	2
#20241211	2
C₄₇×#2863	2
#7733	47