Subgroups — #26124668

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688894	C₂	1	2	yes	maximal
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	#20243086	C₂	1	2	yes	maximal
4	#20243085	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15688894	—	2	4	no	—
10	#15688890	C₄	1	4	yes	—
11	#15688878	—	2	2	no	—
12	#15688877	—	2	4	no	—
13	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#424	—	2	4	no	—
16	C₄₇×#423	—	2	2	no	—
17	C₂×#10625329	—	2	1	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	D₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	D₄	1	4	yes	—
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₄⋊C₈×C₄₇	—	2	8	no	—
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	#41	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	2	yes	—
58	C₂²×C₄×C₄₇	#41	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	2	4	no	—
61	M₁₆×C₄₇	—	4	2	no	—
62	M₁₆×C₄₇	—	4	2	no	—
63	M₁₆×C₄₇	—	4	2	no	—
64	M₁₆×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₂²⋊C₄×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₂²⋊Dic₄₇	—	2	4	no	—
73	C₂²⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	4	2	no	—
76	C₄×Dic₄₇	—	4	2	no	—
77	C₄×Dic₄₇	—	4	2	no	—
78	#10625329	—	2	4	no	—
79	#10625329	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	2	2	no	—
83	C₂×C₄₇⋊C₈	—	4	1	no	—
84	C₂×C₄₇⋊C₈	—	4	1	no	—
85	C₂×C₄₇⋊C₈	—	4	1	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	1	no	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	#117	1	2	yes	—
91	C₂³×C₄₇	C₄×D₄	1	2	yes	—
92	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#121	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	#123	1	2	yes	—
103	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#123	1	2	yes	—
105	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#124	1	2	yes	—
108	C₂×C₄×C₄₇	#125	1	2	yes	—
109	C₂×C₄×C₄₇	#125	1	2	yes	—
110	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#126	1	2	yes	—
112	C₂×C₄×C₄₇	#126	1	2	yes	—
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₄₇	—	4	8	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	C₈×C₄₇	—	4	4	no	—
117	C₄₇⋊C₈	—	4	2	no	—
118	C₄₇⋊C₈	—	4	2	no	—
119	#11288	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	4	2	no	—
123	C₂²×C₄₇	—	4	1	no	—
124	C₂²×C₄₇	—	4	1	no	—
125	C₂²×C₄₇	#382	1	1	yes	characteristic
126	C₂²×C₄₇	#385	1	1	yes	characteristic
127	C₂²×C₄₇	#387	1	2	yes	—
128	C₂²×C₄₇	#388	1	2	yes	—
129	C₂²×C₄₇	#389	1	1	yes	characteristic
130	C₄×C₄₇	—	4	4	no	—
131	C₄×C₄₇	#393	1	1	yes	characteristic
132	C₄×C₄₇	#394	1	1	yes	characteristic
133	C₄×C₄₇	#397	1	1	yes	characteristic
134	C₄×C₄₇	#398	1	2	yes	—
135	C₄×C₄₇	#400	1	1	yes	characteristic
136	C₄×C₄₇	#400	1	2	yes	—
137	Dic₄₇	—	8	4	no	—
138	#2905	D₄₇	1	1	yes	characteristic
139	C₂×#344	—	47	2	no	—
140	#1740	—	47	1	no	—
141	#1729	—	47	2	no	—
142	#1729	—	47	1	no	—
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	—	2	1	no	—
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	#1442	1	1	yes	characteristic
148	C₂×C₄₇	#1868	1	2	yes	—
149	C₂²×M₁₆	—	47	1	no	—
150	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
151	C₂³×C₈	—	47	1	no	—
152	C₂×#117	—	47	2	no	—
153	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
154	#424	—	2	4	no	—
155	#423	—	2	2	no	—
156	#344	—	94	4	no	—
157	#342	—	47	4	no	—
158	#336	—	94	4	no	—
159	#336	—	94	2	no	—
160	C₄₇	#11288	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	47	2	no	—
169	C₂×M₁₆	—	94	1	no	—
170	C₂×M₁₆	—	94	2	no	—
171	C₂×M₁₆	—	94	2	no	—
172	C₂²×C₈	—	47	2	no	—
173	C₂²×C₈	—	94	2	no	—
174	C₂²×C₈	—	94	1	no	—
175	C₂²×C₈	—	94	1	no	—
176	C₂²×C₈	—	94	2	no	—
177	C₂²×C₈	—	94	1	no	—
178	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	94	2	no	—
182	#117	—	94	2	no	—
183	#117	—	94	2	no	—
184	#117	—	94	2	no	—
185	C₂×C₄⋊C₄	—	94	2	no	—
186	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
187	C₂×C₂²⋊C₄	—	47	4	no	—
188	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
189	C₂×C₄²	—	94	2	no	—
190	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
191	C₄⋊C₈	—	2	8	no	—
192	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
195	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
196	C₂²×C₄	#10625350	1	2	yes	—
197	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	#10625352	1	1	yes	characteristic
199	C₂²×C₄	#10625352	1	2	yes	—
200	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	#10625354	1	2	yes	—
202	C₂²×C₄	#10625354	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	94	4	no	—
210	M₁₆	—	188	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	94	2	no	—
215	C₂×C₈	—	94	4	no	—
216	C₂×C₈	—	94	2	no	—
217	C₂×C₈	—	188	1	no	—
218	C₂×C₈	—	188	1	no	—
219	C₂×C₈	—	188	1	no	—
220	C₂×C₈	—	188	2	no	—
221	C₂×C₈	—	188	1	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	188	2	no	—
224	C₄⋊C₄	—	188	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	94	4	no	—
227	C₂²⋊C₄	—	94	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	188	2	no	—
231	C₄²	—	188	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
234	C₂³	#13093982	1	2	yes	—
235	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
236	C₂³	#13093986	1	1	yes	characteristic
237	C₂³	#13093989	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	#13093993	1	1	yes	characteristic
243	C₂×C₄	#13093995	1	1	yes	characteristic
244	C₂×C₄	#13093996	1	2	yes	—
245	C₂×C₄	#13094000	1	2	yes	—
246	C₂×C₄	#13094000	1	1	yes	characteristic
247	C₂×C₄	#13094003	1	2	yes	—
248	C₂×C₄	#13094005	1	2	yes	—
249	C₂×C₄	#13094007	1	1	yes	characteristic
250	C₂×C₄	#13094009	1	2	yes	—
251	C₂×C₄	#13094011	1	1	yes	characteristic
252	C₂×C₄	#13094013	1	1	yes	characteristic
253	C₂×C₄	#13094013	1	2	yes	—
254	C₂×C₄	#13094014	1	2	yes	—
255	C₂×C₄	#13094015	1	2	yes	—
256	C₂×C₄	#13094015	1	1	yes	characteristic
257	C₂×C₄	#13094019	1	2	yes	—
258	C₂×C₄	#13094019	1	2	yes	—
259	C₂×C₄	#13094023	1	1	yes	characteristic
260	C₈	—	4	4	no	—
261	C₈	—	188	2	no	—
262	C₈	—	188	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#15688988	1	1	yes	characteristic
269	C₂²	#15688996	1	1	yes	characteristic
270	C₂²	#15689004	1	2	yes	—
271	C₂²	#15689013	1	2	yes	—
272	C₂²	#15689019	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	376	4	no	—
275	C₄	#15689034	1	1	yes	characteristic
276	C₄	#15689037	1	1	yes	characteristic
277	C₄	#15689049	1	1	yes	characteristic
278	C₄	#15689056	1	2	yes	—
279	C₄	#15689061	1	1	yes	characteristic
280	C₄	#15689062	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	#20241647	1	1	yes	characteristic
286	C₂	#20243674	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093993
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2905	C₂
socle	C₂²×C₄₇	#385
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#11288
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
160. C₄₇	#11288	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688894	2
#20243121	2
#20243086	2
#20243085	2
C₄₇×#2905	2
#11288	47