Subgroups — #26109699

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
6	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
7	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	#15688993	—	2	1	no	—
11	#15688993	—	2	1	no	—
12	#15688993	C₄	1	2	yes	—
13	#15688993	C₂²	1	2	yes	—
14	#15688878	—	2	1	no	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
18	C₄₇×#336	—	2	1	no	—
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	4	4	no	—
27	C₂×C₄⋊Dic₄₇	—	4	4	no	—
28	C₂×C₄⋊Dic₄₇	—	4	4	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	—	2	4	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	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	C₄⋊C₈×C₄₇	—	2	4	no	—
42	C₂²⋊C₈×C₄₇	—	4	2	no	—
43	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	C₂²×Dic₄₇	—	4	4	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	#41	1	4	yes	—
72	C₄⋊Dic₄₇	—	4	4	no	—
73	C₄⋊Dic₄₇	—	4	4	no	—
74	C₄⋊Dic₄₇	—	4	4	no	—
75	C₄⋊Dic₄₇	—	4	4	no	—
76	C₄⋊Dic₄₇	—	4	4	no	—
77	C₄⋊Dic₄₇	—	4	4	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂³×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	#100	1	2	yes	—
86	C₂³×C₄₇	#103	1	2	yes	—
87	C₂³×C₄₇	#103	1	2	yes	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	8	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	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	#101	1	2	yes	—
103	C₂×C₄×C₄₇	#103	1	2	yes	—
104	C₂×C₄×C₄₇	#103	1	2	yes	—
105	C₂×C₄×C₄₇	#117	1	2	yes	—
106	C₂×C₄×C₄₇	#123	1	2	yes	—
107	C₂×C₄×C₄₇	#123	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	8	8	no	—
118	C₈×C₄₇	—	8	2	no	—
119	C₄₇⋊C₈	—	8	2	no	—
120	#8725	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	#333	1	4	yes	—
128	C₂²×C₄₇	#381	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
133	C₂²×C₄₇	#418	1	1	yes	characteristic
134	C₂²×C₄₇	#418	1	1	yes	characteristic
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	#481	1	2	yes	—
140	C₄×C₄₇	#481	1	2	yes	—
141	C₄×C₄₇	#484	1	2	yes	—
142	C₄×C₄₇	#484	1	2	yes	—
143	Dic₄₇	—	8	4	no	—
144	C₂×#384	—	47	1	no	—
145	#1797	—	47	1	no	—
146	#1797	D₄₇	1	1	yes	characteristic
147	#1296	—	47	4	no	—
148	C₂×C₄₇	—	2	4	no	—
149	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
150	C₂×C₄₇	#1549	1	1	yes	characteristic
151	C₂×C₄₇	#1549	1	1	yes	characteristic
152	C₂×C₄₇	#1555	1	1	yes	characteristic
153	C₂×C₄₇	#1821	1	1	yes	characteristic
154	C₂×C₄₇	#1824	1	1	yes	characteristic
155	C₂×C₄₇	#1824	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	—	47	2	no	—
157	C₂²×C₄⋊C₄	—	94	1	no	—
158	C₂²×C₄⋊C₄	—	94	1	no	—
159	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
162	C₂×C₂²⋊C₈	—	2	1	no	—
163	C₂×C₂²⋊C₈	—	94	1	no	—
164	#384	—	47	2	no	—
165	#384	—	47	2	no	—
166	#384	—	94	1	no	—
167	#384	—	94	1	no	—
168	#336	—	2	1	no	—
169	#336	—	94	1	no	—
170	C₄₇	#8725	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	94	2	no	—
172	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
173	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
174	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	94	2	no	—
177	C₂×C₄⋊C₄	—	47	4	no	—
178	C₂×C₄⋊C₄	—	94	2	no	—
179	C₂×C₄⋊C₄	—	94	2	no	—
180	C₂×C₄⋊C₄	—	94	2	no	—
181	C₂×C₄⋊C₄	—	94	2	no	—
182	C₂×C₄⋊C₄	—	94	2	no	—
183	C₂×C₄⋊C₄	—	188	4	no	—
184	C₂×C₄⋊C₄	—	188	4	no	—
185	C₂×C₄⋊C₄	—	188	4	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
188	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	94	4	no	—
193	C₂²⋊C₈	—	4	2	no	—
194	C₂²⋊C₈	—	188	2	no	—
195	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	94	4	no	—
204	C₂²×C₄	—	188	4	no	—
205	C₂²×C₄	—	188	4	no	—
206	C₂²×C₄	—	188	4	no	—
207	C₂²×C₄	#10625325	1	2	yes	—
208	C₂²×C₄	Dic₁₈₈	1	2	yes	—
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	#10625335	1	2	yes	—
213	C₂²×C₄	#10625337	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	188	2	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₄⋊C₄	—	188	4	no	—
224	C₄⋊C₄	—	188	4	no	—
225	C₄⋊C₄	—	188	4	no	—
226	C₄⋊C₄	—	188	4	no	—
227	C₄⋊C₄	—	188	4	no	—
228	C₄⋊C₄	—	188	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	#10625352	1	4	yes	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	#13093940	1	2	yes	—
238	C₂³	#13093947	1	2	yes	—
239	C₂³	#13093949	1	2	yes	—
240	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	—	2	8	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	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	376	8	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	#13093942	1	2	yes	—
259	C₂×C₄	#13093947	1	2	yes	—
260	C₂×C₄	#13093949	1	2	yes	—
261	C₂×C₄	#13093980	1	2	yes	—
262	C₂×C₄	#13094005	1	2	yes	—
263	C₂×C₄	#13094008	1	2	yes	—
264	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
265	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
266	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
268	C₂×C₄	#13094049	1	2	yes	—
269	C₂×C₄	#13094052	1	2	yes	—
270	C₈	—	8	2	no	—
271	C₈	—	376	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	#15688871	1	4	yes	—
279	C₂²	#15688984	1	1	yes	characteristic
280	C₂²	C₂×#13093940	1	1	yes	characteristic
281	C₂²	C₂×#13093942	1	1	yes	characteristic
282	C₂²	C₂×#13093947	1	1	yes	characteristic
283	C₂²	C₂×#13093949	1	1	yes	characteristic
284	C₂²	#15689127	1	1	yes	characteristic
285	C₂²	#15689129	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	376	4	no	—
291	C₄	#15689439	1	2	yes	—
292	C₄	#15689442	1	2	yes	—
293	C₄	#15689456	1	2	yes	—
294	C₄	#15689459	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	C₂×#15688871	1	1	yes	characteristic
297	C₂	#20242072	1	1	yes	characteristic
298	C₂	#20242074	1	1	yes	characteristic
299	C₂	#20242101	1	1	yes	characteristic
300	C₂	#20243466	1	1	yes	characteristic
301	C₂	#20243476	1	1	yes	characteristic
302	C₂	#20243479	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#8725
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. #20241272	C₂	semidirect
170. C₄₇	#8725	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688993	2
#20243368	2
#20241272	2
C₄₇×#1797	2
#8725	47