Subgroups — #26125193

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094016	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	C₂×#13094016	—	2	8	no	—
7	C₂×#13094016	—	2	2	no	—
8	C₂×#13094016	—	2	2	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×Dic₄₇⋊C₄	C₄	1	4	yes	—
11	#15688891	—	2	8	no	—
12	#15688878	—	2	2	no	—
13	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#336	—	2	2	no	—
16	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
17	C₂×#10625329	—	2	2	no	—
18	C₂×#10625329	—	2	4	no	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	Q₈	1	4	yes	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094016	—	2	16	no	—
23	#13094016	—	2	16	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	4	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	D₄	1	8	yes	—
28	C₂×Dic₄₇⋊C₄	—	2	8	no	—
29	C₂×Dic₄₇⋊C₄	—	2	8	no	—
30	C₂×Dic₄₇⋊C₄	—	2	8	no	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₄₇	—	2	2	no	—
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	—	2	4	no	—
36	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	—	2	4	no	—
39	C₂×C₄²×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	—	4	4	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	2	yes	—
52	C₂²×C₄×C₄₇	#41	1	2	yes	—
53	C₂²×Dic₄₇	—	2	4	no	—
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	—	4	8	no	—
56	C₂²×Dic₄₇	C₂²⋊C₄	1	8	yes	—
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	4	no	—
64	C₄⋊Dic₄₇	—	2	8	no	—
65	C₄⋊Dic₄₇	—	2	8	no	—
66	Dic₄₇⋊C₄	—	2	32	no	—
67	#10625329	—	2	8	no	—
68	#10625329	—	4	4	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂³×C₄₇	—	2	4	no	—
73	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
76	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	—	4	4	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	#120	1	2	yes	—
82	C₂×C₄×C₄₇	#120	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
85	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
87	C₂×C₄×C₄₇	#124	1	4	yes	—
88	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
92	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×Dic₄₇	—	2	8	no	—
94	C₂×Dic₄₇	—	4	8	no	—
95	C₂×Dic₄₇	—	4	8	no	—
96	C₂×Dic₄₇	—	4	8	no	—
97	C₈×C₄₇	—	4	4	no	—
98	C₄₇⋊C₈	—	4	4	no	—
99	#11383	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	—	2	4	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	#379	1	1	yes	characteristic
105	C₂²×C₄₇	#391	1	4	yes	—
106	C₂²×C₄₇	#391	1	1	yes	characteristic
107	C₂²×C₄₇	#391	1	1	yes	characteristic
108	C₂²×C₄₇	#413	1	4	yes	—
109	C₄×C₄₇	—	4	4	no	—
110	C₄×C₄₇	#393	1	2	yes	—
111	C₄×C₄₇	#393	1	2	yes	—
112	C₄×C₄₇	#395	1	2	yes	—
113	C₄×C₄₇	#395	1	2	yes	—
114	Dic₄₇	—	4	8	no	—
115	C₂²×#125	—	47	1	no	—
116	C₂×#343	—	47	4	no	—
117	#1728	—	47	1	no	—
118	#1728	D₄₇	1	1	yes	characteristic
119	C₂×C₄₇	—	2	4	no	—
120	C₂×C₄₇	#1446	1	1	yes	characteristic
121	C₂×C₄₇	#1785	1	2	yes	—
122	C₂×C₄₇	#1998	1	2	yes	—
123	C₂×C₄₇	#1998	1	2	yes	—
124	C₂²×M₁₆	—	47	2	no	—
125	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
126	C₂×#125	—	94	8	no	—
127	C₂×#125	—	94	2	no	—
128	C₂×#125	—	94	2	no	—
129	C₂²×C₄⋊C₄	—	47	2	no	—
130	C₂²×C₄⋊C₄	—	47	4	no	—
131	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
132	#343	—	94	8	no	—
133	#336	—	2	2	no	—
134	#336	—	94	2	no	—
135	C₄₇	#11383	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	—	47	4	no	—
137	C₂³×C₄	C₄×D₄₇	1	2	yes	—
138	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	47	4	no	—
143	C₂×M₁₆	—	94	2	no	—
144	C₂×M₁₆	—	94	4	no	—
145	C₂×M₁₆	—	94	4	no	—
146	C₂×M₁₆	Dic₉₄	1	4	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	94	2	no	—
149	#125	—	94	16	no	—
150	#125	—	94	16	no	—
151	C₂×C₄⋊C₄	—	47	8	no	—
152	C₂×C₄⋊C₄	—	94	8	no	—
153	C₂×C₄⋊C₄	—	94	8	no	—
154	C₂×C₄⋊C₄	—	94	8	no	—
155	C₂×C₄⋊C₄	—	94	2	no	—
156	C₂×C₄⋊C₄	—	94	4	no	—
157	C₂×C₄⋊C₄	—	94	2	no	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	D₁₈₈	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
161	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	47	8	no	—
166	C₂²×C₄	—	94	4	no	—
167	C₂²×C₄	—	94	4	no	—
168	C₂²×C₄	—	188	8	no	—
169	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
170	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
171	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
172	C₂²×C₄	#10625350	1	2	yes	—
173	C₂²×C₄	D₄×D₄₇	1	4	yes	—
174	C₂²×C₄	#10625352	1	2	yes	—
175	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
176	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
177	M₁₆	—	2	8	no	—
178	M₁₆	—	4	4	no	—
179	M₁₆	—	94	8	no	—
180	M₁₆	—	188	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	94	4	no	—
185	C₂×C₈	—	188	2	no	—
186	C₂×C₈	—	188	2	no	—
187	C₄⋊C₄	—	94	32	no	—
188	C₄⋊C₄	—	94	8	no	—
189	C₄⋊C₄	—	94	8	no	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₂²⋊C₄×D₄₇	1	2	yes	—
194	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
195	C₂³	#13093989	1	2	yes	—
196	C₂³	C₄×Dic₉₄	1	2	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	94	8	no	—
202	C₂×C₄	—	188	8	no	—
203	C₂×C₄	—	188	8	no	—
204	C₂×C₄	—	188	8	no	—
205	C₂×C₄	#13093992	1	2	yes	—
206	C₂×C₄	#13093994	1	2	yes	—
207	C₂×C₄	#13094001	1	4	yes	—
208	C₂×C₄	#13094002	1	4	yes	—
209	C₂×C₄	#13094003	1	2	yes	—
210	C₂×C₄	#13094004	1	2	yes	—
211	C₂×C₄	#13094010	1	4	yes	—
212	C₂×C₄	#13094011	1	1	yes	characteristic
213	C₂×C₄	#13094012	1	1	yes	characteristic
214	C₂×C₄	#13094022	1	1	yes	characteristic
215	C₂×C₄	#13094023	1	4	yes	—
216	C₂×C₄	#13094024	1	1	yes	characteristic
217	C₈	—	4	4	no	—
218	C₈	—	188	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	#15688978	1	1	yes	characteristic
224	C₂²	#15689027	1	4	yes	—
225	C₂²	#15689028	1	1	yes	characteristic
226	C₂²	#15689029	1	1	yes	characteristic
227	C₂²	#15689105	1	4	yes	—
228	C₄	—	4	4	no	—
229	C₄	—	188	8	no	—
230	C₄	#15689035	1	2	yes	—
231	C₄	#15689036	1	2	yes	—
232	C₄	#15689043	1	2	yes	—
233	C₄	#15689044	1	2	yes	—
234	C₂	—	2	4	no	—
235	C₂	#20241658	1	1	yes	characteristic
236	C₂	#20243314	1	2	yes	—
237	C₂	#20244380	1	2	yes	—
238	C₂	#20244381	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	#11383
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
135. C₄₇	#11383	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13094016	2
C₂×#15688891	2
#20243082	2
C₄₇×#1728	2
#11383	47