Subgroups — #26125051

Order 12032.

774 conjugacy classes of subgroups.

The subgroup lattice has 258 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	C₂×#15688891	C₂	1	4	yes	maximal
3	#20243082	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₇	C₄	1	4	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	#15688993	—	2	1	no	—
9	#15688993	—	2	1	no	—
10	#15688993	—	2	2	no	—
11	#15688891	—	2	8	no	—
12	#15688878	—	2	1	no	—
13	#15688878	—	2	1	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	D₄	1	4	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	4	8	no	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₄₇	—	2	2	no	—
33	C₂×M₁₆×C₄₇	—	2	4	no	—
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	—	2	4	no	—
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	4	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	2	yes	—
52	C₂²×C₄×C₄₇	#41	1	4	yes	—
53	C₂²×C₄×C₄₇	#41	1	2	yes	—
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	—	4	8	no	—
57	M₁₆×C₄₇	—	2	8	no	—
58	M₁₆×C₄₇	—	4	4	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	—	2	4	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄⋊Dic₄₇	—	4	8	no	—
66	C₄⋊Dic₄₇	—	4	8	no	—
67	C₄⋊Dic₄₇	—	4	4	no	—
68	C₄⋊Dic₄₇	—	4	2	no	—
69	C₄⋊Dic₄₇	—	4	2	no	—
70	#10625329	—	2	8	no	—
71	#10625329	—	4	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂³×C₄₇	—	2	2	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₄₇	#117	1	2	yes	—
79	C₂³×C₄₇	C₄×D₄	1	2	yes	—
80	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₄₇	—	2	16	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	#100	1	4	yes	—
87	C₂×C₄×C₄₇	#101	1	4	yes	—
88	C₂×C₄×C₄₇	#121	1	2	yes	—
89	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
90	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₄₇	#123	1	2	yes	—
92	C₂×C₄×C₄₇	#123	1	4	yes	—
93	C₂×C₄×C₄₇	#123	1	2	yes	—
94	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#125	1	4	yes	—
97	C₂×C₄×C₄₇	#126	1	4	yes	—
98	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×Dic₄₇	—	4	4	no	—
101	C₂×Dic₄₇	—	4	8	no	—
102	C₂×Dic₄₇	—	8	8	no	—
103	C₈×C₄₇	—	4	4	no	—
104	C₄₇⋊C₈	—	4	4	no	—
105	#11354	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	—	2	2	no	—
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	4	no	—
110	C₂²×C₄₇	#381	1	1	yes	characteristic
111	C₂²×C₄₇	#387	1	4	yes	—
112	C₂²×C₄₇	#390	1	1	yes	characteristic
113	C₂²×C₄₇	#391	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#100	1	2	yes	—
115	C₂²×C₄₇	C₂×#101	1	2	yes	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	#397	1	2	yes	—
119	C₄×C₄₇	#398	1	2	yes	—
120	C₄×C₄₇	#399	1	2	yes	—
121	C₄×C₄₇	#400	1	2	yes	—
122	Dic₄₇	—	8	4	no	—
123	C₂×#384	—	47	1	no	—
124	C₂×#343	—	47	4	no	—
125	#1728	—	47	1	no	—
126	#1728	D₄₇	1	1	yes	characteristic
127	C₂×C₄₇	—	2	4	no	—
128	C₂×C₄₇	#1445	1	1	yes	characteristic
129	C₂×C₄₇	#1821	1	2	yes	—
130	C₂×C₄₇	#1965	1	1	yes	characteristic
131	C₂×C₄₇	#1966	1	1	yes	characteristic
132	C₂×C₄₇	#1997	1	2	yes	—
133	C₂²×M₁₆	—	47	2	no	—
134	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
135	C₂²×C₄⋊C₄	—	47	2	no	—
136	C₂²×C₄⋊C₄	—	47	4	no	—
137	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
138	#384	—	94	1	no	—
139	#384	—	94	1	no	—
140	#384	—	94	2	no	—
141	#343	—	94	8	no	—
142	#336	—	2	2	no	—
143	#336	—	94	1	no	—
144	#336	—	94	1	no	—
145	C₄₇	#11354	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	—	47	4	no	—
147	C₂³×C₄	C₄×D₄₇	1	2	yes	—
148	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	47	4	no	—
153	C₂×M₁₆	—	94	2	no	—
154	C₂×M₁₆	—	94	4	no	—
155	C₂×M₁₆	—	94	4	no	—
156	C₂×M₁₆	Dic₉₄	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	94	2	no	—
159	C₂×C₄⋊C₄	—	94	4	no	—
160	C₂×C₄⋊C₄	—	94	4	no	—
161	C₂×C₄⋊C₄	—	94	2	no	—
162	C₂×C₄⋊C₄	—	94	4	no	—
163	C₂×C₄⋊C₄	—	94	4	no	—
164	C₂×C₄⋊C₄	—	94	2	no	—
165	C₂×C₄⋊C₄	—	188	8	no	—
166	C₂×C₄²	—	2	4	no	—
167	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
168	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
170	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	94	8	no	—
175	C₂²×C₄	—	94	4	no	—
176	C₂²×C₄	—	188	8	no	—
177	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
178	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
179	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
180	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
181	C₂²×C₄	#10625350	1	2	yes	—
182	C₂²×C₄	D₄×D₄₇	1	2	yes	—
183	C₂²×C₄	#10625352	1	4	yes	—
184	C₂²×C₄	#10625354	1	2	yes	—
185	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
186	M₁₆	—	2	8	no	—
187	M₁₆	—	4	4	no	—
188	M₁₆	—	94	8	no	—
189	M₁₆	—	188	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	94	4	no	—
194	C₂×C₈	—	188	2	no	—
195	C₂×C₈	—	188	2	no	—
196	C₄⋊C₄	—	188	8	no	—
197	C₄⋊C₄	—	188	8	no	—
198	C₄⋊C₄	—	188	4	no	—
199	C₄⋊C₄	—	188	2	no	—
200	C₄⋊C₄	—	188	2	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
207	C₂³	#13093980	1	2	yes	—
208	C₂³	#13093986	1	2	yes	—
209	C₂³	C₄×Dic₉₄	1	2	yes	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	376	8	no	—
216	C₂×C₄	—	188	4	no	—
217	C₂×C₄	—	188	8	no	—
218	C₂×C₄	#13093940	1	4	yes	—
219	C₂×C₄	#13093942	1	4	yes	—
220	C₂×C₄	#13093999	1	2	yes	—
221	C₂×C₄	#13094001	1	4	yes	—
222	C₂×C₄	#13094003	1	2	yes	—
223	C₂×C₄	#13094005	1	2	yes	—
224	C₂×C₄	#13094006	1	4	yes	—
225	C₂×C₄	#13094008	1	2	yes	—
226	C₂×C₄	#13094012	1	1	yes	characteristic
227	C₂×C₄	#13094013	1	1	yes	characteristic
228	C₂×C₄	#13094015	1	4	yes	—
229	C₂×C₄	#13094019	1	4	yes	—
230	C₂×C₄	#13094020	1	1	yes	characteristic
231	C₂×C₄	#13094024	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	188	4	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	#15688984	1	1	yes	characteristic
239	C₂²	#15689006	1	4	yes	—
240	C₂²	#15689024	1	1	yes	characteristic
241	C₂²	#15689029	1	1	yes	characteristic
242	C₂²	C₂×#13093940	1	2	yes	—
243	C₂²	C₂×#13093942	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	376	4	no	—
247	C₄	#15689050	1	2	yes	—
248	C₄	#15689055	1	2	yes	—
249	C₄	#15689059	1	2	yes	—
250	C₄	#15689062	1	2	yes	—
251	C₂	—	2	4	no	—
252	C₂	#20241657	1	1	yes	characteristic
253	C₂	#20243466	1	2	yes	—
254	C₂	#20244227	1	1	yes	characteristic
255	C₂	#20244231	1	1	yes	characteristic
256	C₂	#20244377	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#11354
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
145. C₄₇	#11354	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
C₂×#15688891	2
#20243082	2
C₄₇×#1728	2
#11354	47