Subgroups — #26121184

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	#20244146	C₂	1	2	yes	maximal
2	#20243548	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	2	yes	maximal
4	#20241223	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	—	2	1	no	—
7	C₂×#13093958	—	2	1	no	—
8	C₂×#13093957	—	2	2	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#15688993	C₂²	1	2	yes	—
11	#15688885	—	2	2	no	—
12	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	1	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	#13093958	—	4	2	no	—
22	#13093958	—	4	2	no	—
23	#13093957	—	4	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
26	C₂²×C₈×C₄₇	—	2	1	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	—	2	1	no	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	—	2	2	no	—
34	C₄⋊C₈×C₄₇	—	4	2	no	—
35	C₄×C₈×C₄₇	—	2	2	no	—
36	#95×C₄₇	—	2	1	no	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
42	C₂²×Dic₄₇	—	4	2	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	—	4	1	no	—
48	C₂×C₈×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	—	4	1	no	—
50	C₂×C₈×C₄₇	D₈	1	2	yes	—
51	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
52	C₄⋊C₄×C₄₇	—	2	1	no	—
53	C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
55	C₄²×C₄₇	—	4	1	no	—
56	C₄²×C₄₇	—	4	1	no	—
57	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
58	C₄²×C₄₇	#41	1	2	yes	—
59	C₄⋊Dic₄₇	—	4	2	no	—
60	C₄⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	4	2	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	8	2	no	—
64	C₂×C₄₇⋊C₈	—	2	4	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	—	4	2	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	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	1	no	—
72	C₂×C₄×C₄₇	—	4	1	no	—
73	C₂×C₄×C₄₇	—	4	1	no	—
74	C₂×C₄×C₄₇	—	4	1	no	—
75	C₂×C₄×C₄₇	—	4	1	no	—
76	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#125	1	2	yes	—
79	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
84	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#137	1	2	yes	—
87	C₂×Dic₄₇	—	8	2	no	—
88	C₂×Dic₄₇	—	8	2	no	—
89	C₂×Dic₄₇	—	8	2	no	—
90	C₈×C₄₇	—	2	2	no	—
91	C₈×C₄₇	—	2	2	no	—
92	C₈×C₄₇	—	8	1	no	—
93	C₄₇⋊C₈	—	4	4	no	—
94	#10649	—	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	#466	1	1	yes	characteristic
99	C₂²×C₄₇	#470	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	—	4	2	no	—
104	C₄×C₄₇	—	8	1	no	—
105	C₄×C₄₇	#474	1	2	yes	—
106	C₄×C₄₇	#479	1	2	yes	—
107	C₄×C₄₇	#480	1	1	yes	characteristic
108	C₄×C₄₇	#484	1	1	yes	characteristic
109	C₄×C₄₇	#500	1	1	yes	characteristic
110	C₄×C₄₇	#501	1	1	yes	characteristic
111	Dic₄₇	—	16	2	no	—
112	#1947	—	47	2	no	—
113	#1939	D₄₇	1	1	yes	characteristic
114	#1845	—	47	1	no	—
115	#1296	—	47	2	no	—
116	#1277	—	47	1	no	—
117	C₂×C₄₇	#1667	1	1	yes	characteristic
118	C₂×C₄₇	#1674	1	1	yes	characteristic
119	C₂×C₄₇	#1689	1	1	yes	characteristic
120	C₂×C₄₇	#1696	1	1	yes	characteristic
121	C₂×C₄₇	#2083	1	1	yes	characteristic
122	C₂×C₄₇	#2086	1	1	yes	characteristic
123	C₂×C₄₇	#2089	1	1	yes	characteristic
124	C₂×#107	—	94	1	no	—
125	C₂×#107	—	94	1	no	—
126	C₂×#107	—	94	2	no	—
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	—	47	2	no	—
129	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	#384	—	47	2	no	—
132	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
133	#340	—	94	2	no	—
134	#336	—	2	1	no	—
135	C₄₇	#10649	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	94	2	no	—
138	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	94	2	no	—
141	C₂×C₄⋊C₄	—	94	2	no	—
142	C₂×C₄⋊C₄	—	94	2	no	—
143	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
144	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	#107	—	188	2	no	—
148	#107	—	188	2	no	—
149	#107	—	188	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	4	2	no	—
152	C₄⋊C₈	—	47	4	no	—
153	C₄⋊C₈	—	94	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	188	2	no	—
159	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
160	C₂²×C₄	#10625350	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	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	94	4	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	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	376	2	no	—
177	C₄⋊C₄	—	188	2	no	—
178	C₄⋊C₄	—	188	2	no	—
179	C₄⋊C₄	—	188	2	no	—
180	C₄⋊C₄	—	188	2	no	—
181	C₄⋊C₄	D₃₇₆	1	2	yes	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	#10625352	1	2	yes	—
185	C₄²	Q₈×D₄₇	1	2	yes	—
186	C₂³	#13093998	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	376	2	no	—
196	C₂×C₄	—	376	2	no	—
197	C₂×C₄	—	376	2	no	—
198	C₂×C₄	#13094004	1	2	yes	—
199	C₂×C₄	#13094005	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	C₂×D₃₇₆	1	1	yes	characteristic
204	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
205	C₂×C₄	#13094046	1	1	yes	characteristic
206	C₂×C₄	#13094049	1	1	yes	characteristic
207	C₂×C₄	#13094054	1	1	yes	characteristic
208	C₂×C₄	#13094057	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	8	1	no	—
212	C₈	—	188	4	no	—
213	C₂²	#15689053	1	1	yes	characteristic
214	C₂²	#15689318	1	1	yes	characteristic
215	C₂²	#15689333	1	1	yes	characteristic
216	C₂²	#15689359	1	1	yes	characteristic
217	C₂²	#15689381	1	1	yes	characteristic
218	C₂²	#15689393	1	1	yes	characteristic
219	C₂²	#15689398	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	752	2	no	—
224	C₄	#15689404	1	2	yes	—
225	C₄	#15689431	1	2	yes	—
226	C₄	#15689434	1	1	yes	characteristic
227	C₄	#15689456	1	1	yes	characteristic
228	C₄	#15689526	1	1	yes	characteristic
229	C₄	#15689533	1	1	yes	characteristic
230	C₂	#20242743	1	1	yes	characteristic
231	C₂	#20242782	1	1	yes	characteristic
232	C₂	#20242877	1	1	yes	characteristic
233	C₂	#20242916	1	1	yes	characteristic
234	C₂	#20244859	1	1	yes	characteristic
235	C₂	#20244876	1	1	yes	characteristic
236	C₂	#20244893	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1939	C₂
socle, prime radical	C₂³×C₄₇	#121
hypercenter	#1939	D₄₇
hypocenter	C₄₇	#10649
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
135. C₄₇	#10649	semidirect

Maximal subgroups

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

subgroup	index
#20244146	2
#20243548	2
#20241272	2
#20241223	2
C₄₇×#1939	2
#10649	47