Finite Group Databasebeta

← #18005051

Subgroups — #18005051

Order 4736.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924569C211yesmaximal, characteristic, complemented
2C2×#11924567C211yesmaximal, characteristic, complemented
3C2×#11924566C211yesmaximal, characteristic, complemented
4#14558079C211yesmaximal, characteristic
5C2×#11924534C211yesmaximal, characteristic, complemented
6#14557926C212yesmaximal, complemented
7#14557924C212yesmaximal, complemented
8#14557911C212yesmaximal, complemented
9#14557904C212yesmaximal, complemented
10M16×Dic37C211yesmaximal, characteristic, complemented
11C2×#137×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×Dic74C2211yescharacteristic
14#11924569—24no—
15#11924567—24no—
16#11924566—24no—
17C2×Dic148C2212yes—
18C2×#10596942C2212yes—
19C2×#10596942C2212yes—
20C2×#10596930C2212yes—
21C2×#10596940C2212yes—
22C2×#10596941C2212yes—
23C2×#10596934C2211yescharacteristic
24#11924534—22no—
25#11924534C2212yes—
26#11924534C2212yes—
27#11924534C2211yescharacteristic
28#11924532C2211yescharacteristic
29#11924525C2211yescharacteristic
30#11924523C2211yescharacteristic
31#11924508—22no—
32C4×C22⋊D37—22no—
33C2×#10596936—22no—
34C2×C4×Dic37C2211yescharacteristic
35#11924446C2212yes—
36C8×Dic37C2212yes—
37C2×#41×C37C2211yescharacteristic
38C22×Q8×C37C2211yescharacteristic
39#137×C37—24no—
40C2×Q16×C37C2212yes—
41C2×SD16×C37C2212yes—
42C2×M16×C37C2211yescharacteristic
43#41×C37—22no—
44#41×C37—22no—
45C2×Q8×C37—21no—
46C2×Q8×C37—22no—
47C2×Q8×C37—22no—
48C2×Q8×C37C2311yescharacteristic
49C2×Q8×C37C2312yes—
50C2×D4×C37—21no—
51C2×D4×C37C2311yescharacteristic
52C22×C4×C37—21no—
53C22×C4×C37—21no—
54C22×C4×C37C2311yescharacteristic
55C2×C22⋊D37—21no—
56C2×C22⋊D37—22no—
57C22×Dic37—21no—
58C22×Dic37D412yes—
59#10596962—22no—
60#10596962—22no—
61C2×D148C2311yescharacteristic
62C2×C4×D37—21no—
63C2×Dic74—22no—
64C2×Dic74—22no—
65C2×Dic74—21no—
66C2×Dic74C2311yescharacteristic
67C2×Dic74C2312yes—
68Q16×C37—24no—
69SD16×C37—24no—
70M16×C37D414yescomplemented
71C2×C8×C37C2312yes—
72C22⋊Dic37—22no—
73#10596942—24no—
74#10596942—24no—
75#10596941—24no—
76#10596940—24no—
77#10596938—22no—
78#10596936—24no—
79#10596936—24no—
80#10596936—22no—
81C4×Dic37D412yes—
82C4×Dic37C2311yescharacteristic
83C4×Dic37C2311yescharacteristic
84#10596934D414yescomplemented
85C2×#64436C2312yes—
86Dic148—24no—
87#10596930—24no—
88C23×C37—21no—
89C23×C37C2×D411yescharacteristic
90C22×D37—21no—
91Q8×C37—22no—
92Q8×C37—21no—
93Q8×C37—22no—
94Q8×C37—21no—
95Q8×C37—42no—
96D4×C37—21no—
97D4×C37—21no—
98D4×C37—42no—
99C2×C4×C37—22no—
100C2×C4×C37—21no—
101C2×C4×C37—42no—
102C2×C4×C37—42no—
103C2×C4×C37C2×D411yescharacteristic
104C2×C4×C37C2×D412yes—
105C2×C4×C37C2×D412yes—
106C2×C4×C37C2411yescharacteristic
107C22⋊D37—42no—
108C22⋊D37—44no—
109C2×Dic37—24no—
110C2×Dic37—22no—
111C2×Dic37—21no—
112C2×Dic37—42no—
113C2×Dic37C2×D412yes—
114C2×Dic37C2×D412yes—
115D148—21no—
116D148—21no—
117C4×D37—42no—
118Dic74—22no—
119Dic74—21no—
120Dic74—22no—
121Dic74—21no—
122Dic74—42no—
123C8×C37C2×D414yes—
124#64436C2×D414yes—
125C22×C37—22no—
126C22×C37—41no—
127C22×C37—42no—
128C22×C37#12712yes—
129C22×C37C22×D411yescharacteristic
130C2×D37—41no—
131C2×D37—42no—
132C4×C37—41no—
133C4×C37—42no—
134C4×C37#12712yes—
135C4×C37C22×D411yescharacteristic
136C4×C37C22×D411yescharacteristic
137Dic37—22no—
138Dic37—42no—
139Dic37—41no—
140Dic37#13714yescomplemented
141#3152—371nomaximal, Sylow
142C2×C37—22no—
143C2×C37—81no—
144C2×C37C2×#12711yescharacteristic
145C2×C37C2×#13712yes—
146D37—81no—
147C2×#137—371no—
148C2×#137—371no—
149C2×#137—371no—
150C2×#137D3711yescharacteristic, complemented
151#532—371no—
152C2×C4⋊Q8—371no—
153#497—372no—
154#497—372no—
155#494—372no—
156#492—372no—
157C4×M16—371no—
158C37#315211yescharacteristic, complemented, Sylow
159C2×#41—371no—
160C2×#41C2×D3711yescharacteristic
161C22×Q8—371no—
162C22×Q8C2×D3711yescharacteristic
163#137—24no—
164#137—744no—
165#137—744no—
166#137—744no—
167C2×Q16—372no—
168C2×Q16—372no—
169C2×Q16—372no—
170C2×Q16C2×D3712yes—
171C2×SD16—372no—
172C2×SD16—372no—
173C2×SD16—372no—
174C2×SD16C2×D3712yes—
175C2×M16—371no—
176C2×M16C2×D3711yescharacteristic
177C4⋊Q8—372no—
178C4⋊Q8—372no—
179C4⋊Q8—371no—
180C4⋊Q8—742no—
181#127—371no—
182#124—371no—
183#124—371no—
184#121—742no—
185C4×D4—742no—
186C2×C4⋊C4—742no—
187C2×C42—371no—
188#97—372no—
189C4×C8—372no—
190#41—22no—
191#41—22no—
192#41—742no—
193#41—742no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—371no—
198C2×Q8—372no—
199C2×Q8—741no—
200C2×Q8—742no—
201C2×Q8—742no—
202C2×Q8C22×D3711yescharacteristic
203C2×Q8C22×D3712yes—
204C2×D4—21no—
205C2×D4—371no—
206C2×D4—742no—
207C2×D4—741no—
208C2×D4C22×D3711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—372no—
212C22×C4—741no—
213C22×C4—741no—
214C22×C4C22×D3711yescharacteristic
215Q16—24no—
216Q16—744no—
217Q16—744no—
218Q16—744no—
219SD16—24no—
220SD16—744no—
221SD16—744no—
222SD16—744no—
223M16—374no—
224M16C22⋊D3714yescomplemented
225C2×C8—372no—
226C2×C8C22×D3712yes—
227C4⋊C4—744no—
228C4⋊C4—744no—
229C4⋊C4—742no—
230C22⋊C4—742no—
231C22⋊C4—742no—
232C42—372no—
233C42—371no—
234C42—371no—
235C23—21no—
236C23—741no—
237C23D4×D3711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—742no—
244Q8—741no—
245Q8—742no—
246Q8—741no—
247Q8—1482no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—741no—
252D4—741no—
253D4—1484no—
254D4—1482no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—372no—
260C2×C4—372no—
261C2×C4—744no—
262C2×C4—741no—
263C2×C4—742no—
264C2×C4—1482no—
265C2×C4—1482no—
266C2×C4D4×D3711yescharacteristic
267C2×C4D4×D3712yes—
268C2×C4C2×C22⋊D3712yes—
269C2×C4C23×D3711yescharacteristic
270C8—374no—
271C8C2×C22⋊D3714yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—1482no—
276C22—1481no—
277C22#1192453212yes—
278C22C2×D4×D3711yescharacteristic
279C4—41no—
280C4—42no—
281C4—374no—
282C4—742no—
283C4—1481no—
284C4—1482no—
285C4#1192453212yes—
286C4C2×D4×D3711yescharacteristic
287C4C22×C22⋊D3711yescharacteristic
288C2—22no—
289C2—81no—
290C2—2961no—
291C2C2×#1192453211yescharacteristic
292C2#137×D3712yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×#137×C37C2
socleC22×C37C22×D4
prime radical#14558079C2
hypercenterC2×#137D37
hypocenterC37#3152
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#11924569C2semidirect
2. C2×#11924567C2semidirect
3. C2×#11924566C2semidirect
5. C2×#11924534C2semidirect
6. #14557926C2semidirect
7. #14557924C2semidirect
8. #14557911C2semidirect
9. #14557904C2semidirect
10. M16×Dic37C2semidirect
11. C2×#137×C37C2semidirect
70. M16×C37D4semidirect
84. #10596934D4semidirect
140. Dic37#137semidirect
150. C2×#137D37semidirect
158. C37#3152semidirect
224. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245692
C2×#119245672
C2×#119245662
#145580792
C2×#119245342
#145579262
#145579242
#145579112
#145579042
M16×Dic372
C2×#137×C372
#315237