Subgroups — #26121190

Order 12032.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244144	C₂	1	2	yes	maximal
2	#20244106	C₂	1	1	yes	maximal, characteristic
3	#20241271	C₂	1	2	yes	maximal
4	#20241223	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093957	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₄₇⋊C₈	—	2	1	no	—
10	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	#15688885	—	2	2	no	—
12	#15688877	—	2	1	no	—
13	C₂×#107×C₄₇	—	2	1	no	—
14	C₂×#107×C₄₇	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#384	C₂²	1	2	yes	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	—	2	1	no	—
22	#13093957	—	4	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
26	Dic₄₇⋊C₈	—	4	2	no	—
27	C₄₇⋊#95	—	2	1	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#107×C₄₇	—	4	2	no	—
35	#107×C₄₇	—	4	2	no	—
36	C₄×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
41	C₂²×Dic₄₇	—	2	1	no	—
42	C₂²×Dic₄₇	—	2	1	no	—
43	C₂×C₈×C₄₇	—	2	1	no	—
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	D₈	1	2	yes	—
48	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
49	C₄⋊C₄×C₄₇	—	4	2	no	—
50	C₄⋊C₄×C₄₇	—	4	2	no	—
51	C₄⋊C₄×C₄₇	—	4	2	no	—
52	C₄⋊C₄×C₄₇	—	4	2	no	—
53	C₄⋊C₄×C₄₇	—	8	2	no	—
54	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
55	C₄²×C₄₇	#41	1	2	yes	—
56	C₄⋊Dic₄₇	—	2	1	no	—
57	C₄⋊Dic₄₇	—	2	2	no	—
58	C₄⋊Dic₄₇	Q₁₆	1	2	yes	—
59	C₄×Dic₄₇	—	4	1	no	—
60	C₄×Dic₄₇	—	4	1	no	—
61	C₂×C₄₇⋊C₈	—	2	4	no	—
62	C₂×C₄₇⋊C₈	—	4	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	1	no	—
65	C₂×C₄₇⋊C₈	—	4	1	no	—
66	C₂×C₄₇⋊C₈	—	4	1	no	—
67	C₂³×C₄₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	—	8	2	no	—
72	C₂×C₄×C₄₇	—	8	2	no	—
73	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#125	1	2	yes	—
76	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
81	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
84	C₂×Dic₄₇	—	2	2	no	—
85	C₂×Dic₄₇	—	4	1	no	—
86	C₂×Dic₄₇	—	4	1	no	—
87	C₂×Dic₄₇	—	4	1	no	—
88	C₂×Dic₄₇	—	4	1	no	—
89	C₂×Dic₄₇	—	4	1	no	—
90	C₈×C₄₇	—	2	2	no	—
91	C₈×C₄₇	—	2	2	no	—
92	C₄₇⋊C₈	—	4	4	no	—
93	C₄₇⋊C₈	—	8	1	no	—
94	#10650	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	#398	1	1	yes	characteristic
96	C₂²×C₄₇	#459	1	1	yes	characteristic
97	C₂²×C₄₇	#462	1	1	yes	characteristic
98	C₂²×C₄₇	#467	1	1	yes	characteristic
99	C₂²×C₄₇	#469	1	1	yes	characteristic
100	C₂²×C₄₇	#472	1	1	yes	characteristic
101	C₂²×C₄₇	#473	1	1	yes	characteristic
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	16	2	no	—
104	C₄×C₄₇	#477	1	2	yes	—
105	C₄×C₄₇	#478	1	2	yes	—
106	C₄×C₄₇	#480	1	1	yes	characteristic
107	C₄×C₄₇	#484	1	1	yes	characteristic
108	C₄×C₄₇	#500	1	1	yes	characteristic
109	C₄×C₄₇	#501	1	1	yes	characteristic
110	Dic₄₇	—	4	2	no	—
111	Dic₄₇	—	8	1	no	—
112	#1947	—	47	2	no	—
113	#1939	—	47	1	no	—
114	#1845	D₄₇	1	1	yes	characteristic
115	#1296	—	47	2	no	—
116	#1277	—	47	1	no	—
117	C₂×C₄₇	#1669	1	1	yes	characteristic
118	C₂×C₄₇	#1676	1	1	yes	characteristic
119	C₂×C₄₇	#1687	1	1	yes	characteristic
120	C₂×C₄₇	#1694	1	1	yes	characteristic
121	C₂×C₄₇	#2083	1	1	yes	characteristic
122	C₂×C₄₇	#2087	1	1	yes	characteristic
123	C₂×C₄₇	#2089	1	1	yes	characteristic
124	C₂×#107	—	2	1	no	—
125	C₂×#107	—	2	1	no	—
126	C₂×#107	—	94	2	no	—
127	C₂×C₄⋊C₈	—	47	2	no	—
128	C₂×C₄⋊C₈	—	47	1	no	—
129	C₂×C₄⋊C₈	—	94	1	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	#384	C₂×D₄₇	1	2	yes	—
132	C₄×C₄⋊C₄	—	47	1	no	—
133	#340	—	94	2	no	—
134	#336	—	94	1	no	—
135	C₄₇	#10650	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	94	2	no	—
137	C₂²×C₈	—	94	1	no	—
138	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	47	1	no	—
144	C₂×C₄⋊C₄	—	47	1	no	—
145	C₂×C₄²	—	94	1	no	—
146	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	#107	—	4	2	no	—
148	#107	—	4	2	no	—
149	#107	—	188	4	no	—
150	C₄⋊C₈	—	47	4	no	—
151	C₄⋊C₈	—	94	2	no	—
152	C₄⋊C₈	—	94	2	no	—
153	C₄⋊C₈	—	188	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	94	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	94	1	no	—
158	C₂²×C₄	—	94	1	no	—
159	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
160	C₂²×C₄	#10625352	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	94	4	no	—
167	C₂×C₈	—	188	1	no	—
168	C₂×C₈	—	188	1	no	—
169	C₂×C₈	—	188	1	no	—
170	C₂×C₈	—	188	2	no	—
171	C₂×C₈	—	188	2	no	—
172	C₂×C₈	#10625334	1	2	yes	—
173	C₂×C₈	#10625337	1	2	yes	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	8	2	no	—
179	C₄⋊C₄	—	47	2	no	—
180	C₄⋊C₄	—	94	1	no	—
181	C₄⋊C₄	—	94	2	no	—
182	C₄²	—	188	1	no	—
183	C₄²	—	188	1	no	—
184	C₄²	#10625350	1	2	yes	—
185	C₄²	Q₈×D₄₇	1	2	yes	—
186	C₂³	#13093995	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	94	2	no	—
193	C₂×C₄	—	188	1	no	—
194	C₂×C₄	—	188	1	no	—
195	C₂×C₄	—	188	1	no	—
196	C₂×C₄	—	188	1	no	—
197	C₂×C₄	—	188	1	no	—
198	C₂×C₄	#13094002	1	2	yes	—
199	C₂×C₄	#13094007	1	1	yes	characteristic
200	C₂×C₄	#13094014	1	2	yes	—
201	C₂×C₄	#13094022	1	1	yes	characteristic
202	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
204	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
205	C₂×C₄	#13094047	1	1	yes	characteristic
206	C₂×C₄	#13094048	1	2	yes	—
207	C₂×C₄	#13094050	1	1	yes	characteristic
208	C₂×C₄	#13094055	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	376	1	no	—
212	C₈	—	188	4	no	—
213	C₂²	#15689054	1	1	yes	characteristic
214	C₂²	#15689315	1	1	yes	characteristic
215	C₂²	#15689330	1	1	yes	characteristic
216	C₂²	#15689361	1	1	yes	characteristic
217	C₂²	#15689373	1	1	yes	characteristic
218	C₂²	#15689390	1	1	yes	characteristic
219	C₂²	#15689395	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	16	2	no	—
222	C₄	—	376	1	no	—
223	C₄	—	188	2	no	—
224	C₄	#15689417	1	2	yes	—
225	C₄	#15689426	1	2	yes	—
226	C₄	#15689436	1	1	yes	characteristic
227	C₄	#15689458	1	1	yes	characteristic
228	C₄	#15689526	1	1	yes	characteristic
229	C₄	#15689529	1	1	yes	characteristic
230	C₂	#20242750	1	1	yes	characteristic
231	C₂	#20242789	1	1	yes	characteristic
232	C₂	#20242864	1	1	yes	characteristic
233	C₂	#20242903	1	1	yes	characteristic
234	C₂	#20244860	1	1	yes	characteristic
235	C₂	#20244882	1	1	yes	characteristic
236	C₂	#20244894	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	#121
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#10650
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
135. C₄₇	#10650	semidirect

Maximal subgroups

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

subgroup	index
#20244144	2
#20244106	2
#20241271	2
#20241223	2
C₄₇×#1845	2
#10650	47