Subgroups — #26134880

Order 12032.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#20243548	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal
4	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093958	—	2	1	no	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	#15688993	—	2	1	no	—
9	C₄×C₄₇⋊C₁₆	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
11	C₄₇×#500	—	2	2	no	—
12	C₄₇×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₄₇	—	2	1	no	—
14	C₂×#107×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄⋊Dic₄₇	—	4	1	no	—
18	C₂×C₄⋊Dic₄₇	—	4	1	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	complemented
22	#13093958	—	2	4	no	—
23	#13093958	—	4	2	no	—
24	C₂²×Q₈×C₄₇	—	4	1	no	—
25	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₇	—	2	4	no	—
28	C₄⋊Q₈×C₄₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	#107×C₄₇	—	2	4	no	—
33	#107×C₄₇	—	2	2	no	—
34	#107×C₄₇	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₂×Q₈×C₄₇	—	4	4	no	—
38	C₂×Q₈×C₄₇	—	8	2	no	—
39	C₂×Q₈×C₄₇	—	8	2	no	—
40	C₂²×C₄×C₄₇	—	4	2	no	—
41	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
48	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₄₇	#41	1	2	yes	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	—	4	4	no	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₄₇	SD₁₆	1	2	yes	—
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₄⋊Dic₄₇	—	4	2	no	—
59	C₄⋊Dic₄₇	—	8	2	no	—
60	C₄⋊Dic₄₇	—	8	2	no	—
61	C₄₇⋊C₁₆	—	2	4	no	—
62	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
63	Q₈×C₄₇	—	8	8	no	—
64	C₂×C₄×C₄₇	—	4	4	no	—
65	C₂×C₄×C₄₇	—	8	2	no	—
66	C₂×C₄×C₄₇	—	8	2	no	—
67	C₂×C₄×C₄₇	#102	1	2	yes	—
68	C₂×C₄×C₄₇	#102	1	2	yes	—
69	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
70	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
71	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
72	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₄₇	—	8	2	no	—
80	C₂×Dic₄₇	—	8	2	no	—
81	C₂×Dic₄₇	—	8	2	no	—
82	C₈×C₄₇	#103	1	4	yes	—
83	C₈×C₄₇	C₄×Q₈	1	2	yes	—
84	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
85	#13282	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
87	C₂²×C₄₇	#486	1	1	yes	characteristic
88	C₂²×C₄₇	#500	1	1	yes	characteristic
89	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
90	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
91	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
92	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
93	C₄×C₄₇	—	8	4	no	—
94	C₄×C₄₇	#357	1	4	yes	—
95	C₄×C₄₇	#358	1	4	yes	—
96	C₄×C₄₇	#391	1	1	yes	characteristic
97	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
98	C₄×C₄₇	#487	1	1	yes	characteristic
99	C₄×C₄₇	#498	1	1	yes	characteristic
100	Dic₄₇	—	16	2	no	—
101	C₂×#500	D₄₇	1	1	yes	characteristic
102	C₂×C₄×C₁₆	—	47	1	no	—
103	#1845	—	47	1	no	—
104	#1380	—	47	4	no	—
105	C₂×C₄₇	#1990	1	1	yes	characteristic
106	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
107	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
108	C₂×C₄₇	#2240	1	1	yes	characteristic
109	C₂×C₄₇	#2241	1	1	yes	characteristic
110	C₂×C₄₇	#2252	1	1	yes	characteristic
111	C₂×C₄₇	#2254	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂²×C₁₆	—	47	2	no	—
114	#500	—	2	2	no	—
115	#500	Dic₄₇	1	4	yes	complemented
116	C₂×#107	—	2	1	no	—
117	C₂×#107	—	47	2	no	—
118	C₂×#107	—	94	1	no	—
119	C₂×#107	C₂×D₄₇	1	2	yes	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	#384	—	94	1	no	—
122	C₄×C₁₆	—	94	2	no	—
123	C₄₇	#13282	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	4	1	no	—
125	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	4	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	94	2	no	—
132	C₂×C₄⋊C₄	—	188	1	no	—
133	C₂×C₄⋊C₄	—	188	1	no	—
134	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
135	C₂×C₁₆	—	47	4	no	—
136	C₂×C₁₆	—	94	2	no	—
137	C₂×C₁₆	—	94	2	no	—
138	#107	—	2	4	no	—
139	#107	—	2	2	no	—
140	#107	—	94	4	no	—
141	#107	—	188	2	no	—
142	#107	C₂×Dic₄₇	1	4	yes	—
143	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
144	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂×Q₈	—	8	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	188	2	no	—
150	C₂²×C₄	#10625334	1	1	yes	characteristic
151	C₂²×C₄	#10625334	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂×C₈	#10625336	1	2	yes	—
154	C₂×C₈	#10625337	1	2	yes	—
155	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
156	C₂×C₈	Q₈×D₄₇	1	2	yes	—
157	C₂×C₈	#10625354	1	2	yes	—
158	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	376	2	no	—
164	C₄⋊C₄	—	376	2	no	—
165	C₄⋊C₄	—	188	2	no	—
166	C₄⋊C₄	—	188	2	no	—
167	C₄²	#10625335	1	2	yes	—
168	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
169	C₁₆	—	94	4	no	—
170	C₂³	C₂×#10625334	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	376	2	no	—
176	C₂×C₄	—	376	2	no	—
177	C₂×C₄	—	376	2	no	—
178	C₂×C₄	#13093945	1	2	yes	—
179	C₂×C₄	#13093945	1	2	yes	—
180	C₂×C₄	#13093965	1	2	yes	—
181	C₂×C₄	#13093967	1	2	yes	—
182	C₂×C₄	#13093968	1	2	yes	—
183	C₂×C₄	#13093970	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
185	C₂×C₄	#13094013	1	1	yes	characteristic
186	C₂×C₄	#13094022	1	1	yes	characteristic
187	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
188	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
189	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
190	C₈	#13093948	1	4	yes	—
191	C₈	Q₈×Dic₄₇	1	2	yes	—
192	C₈	#13094000	1	2	yes	—
193	C₂²	C₂×#13093945	1	1	yes	characteristic
194	C₂²	#15689471	1	1	yes	characteristic
195	C₂²	#15689526	1	1	yes	characteristic
196	C₂²	C₂×#13093965	1	1	yes	characteristic
197	C₂²	C₂×#13093967	1	1	yes	characteristic
198	C₂²	C₂×#13093968	1	1	yes	characteristic
199	C₂²	C₂×#13093970	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	752	2	no	—
202	C₄	#15688924	1	4	yes	—
203	C₄	#15688927	1	4	yes	—
204	C₄	#15689030	1	1	yes	characteristic
205	C₄	C₂×#13093948	1	1	yes	characteristic
206	C₄	#15689475	1	1	yes	characteristic
207	C₄	#15689519	1	1	yes	characteristic
208	C₂	#20244350	1	1	yes	characteristic
209	C₂	C₂×#15688924	1	1	yes	characteristic
210	C₂	C₂×#15688927	1	1	yes	characteristic
211	C₂	#20245613	1	1	yes	characteristic
212	C₂	#20245617	1	1	yes	characteristic
213	C₂	#20245661	1	1	yes	characteristic
214	C₂	#20245668	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#13282
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₄₇×#500	C₄	semidirect
21. C₂×C₄₇⋊C₁₆	Q₈	semidirect
115. #500	Dic₄₇	semidirect
123. C₄₇	#13282	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
#20243548	2
#20241479	2
C₂×C₄₇×#500	2
#13282	47