Finite Group Databasebeta

← #18005367

Subgroups — #18005367

Order 4736.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D37C212yesmaximal
2#14557870C212yesmaximal, complemented
3#14557867C212yesmaximal, complemented
4#14557774C211yesmaximal, characteristic, complemented
5#14557772C212yesmaximal, complemented
6#14557690C212yesmaximal, complemented
7#14557687C212yesmaximal, complemented
8C2×#11924448C211yesmaximal, characteristic
9C37×#467C211yesmaximal, characteristic, complemented
10C2×Q8×D37C2212yes—
11C22×C4×D37C2211yescharacteristic
12C2×Dic148C2211yescharacteristic, complemented
13C2×#10596942C2212yescomplemented
14C22×#64436C2211yescharacteristic
15#11924516C2212yes—
16#11924515C2212yes—
17#11924514—22no—
18#11924513—22no—
19#11924512C2212yes—
20C4⋊C4×D37—22no—
21C4⋊C4×D37C2212yes—
22C22⋊C4×D37—22no—
23#11924470C2211yescharacteristic, complemented
24#11924469C2212yescomplemented
25#11924461C2212yescomplemented
26#11924460C2212yescomplemented
27#11924459C2212yescomplemented
28#11924448C2212yes—
29#11924448C2212yes—
30C2×Q16×C37C2211yescharacteristic, complemented
31C22×C8×C37C2211yescharacteristic
32C22⋊Q8×C37C2212yes—
33#107×C37C2211yescharacteristic, complemented
34#103×C37C2212yescomplemented
35C23×D37D411yescharacteristic
36C2×Q8×C37C2312yes—
37C22×C4×C37C2311yescharacteristic
38C22×Dic37D411yescharacteristic
39Q8×D37—42no—
40Q8×D37—42no—
41C2×C4×D37—22no—
42C2×C4×D37—22no—
43C2×C4×D37—41no—
44C2×C4×D37—41no—
45C2×C4×D37D412yes—
46C2×C4×D37C2312yes—
47C2×Dic74—22no—
48C2×Dic74C2312yes—
49Q16×C37—41no—
50C2×C8×C37—41no—
51C2×C8×C37C2311yescharacteristic
52C2×C8×C37C2311yescharacteristic
53C4⋊C4×C37—22no—
54C4⋊C4×C37C2312yes—
55C22⋊C4×C37—22no—
56C22⋊Dic37—22no—
57#10596942—42no—
58#10596938—22no—
59#10596938—22no—
60C4⋊Dic37—22no—
61C4⋊Dic37C2312yes—
62#10596936—22no—
63#10596936—22no—
64#10596936—22no—
65C2×#64436—41no—
66C2×#64436C2312yes—
67Dic148—41no—
68C23×C37C2×D411yescharacteristic
69C22×D37—41no—
70C22×D37—41no—
71C22×D37Q1614yescomplemented
72C22×D37C2×D412yes—
73Q8×C37—22no—
74Q8×C37—22no—
75C2×C4×C37—21no—
76C2×C4×C37—21no—
77C2×C4×C37—22no—
78C2×C4×C37—22no—
79C2×C4×C37C2×D411yescharacteristic
80C2×C4×C37C2411yescharacteristic
81C2×Dic37—22no—
82C2×Dic37—22no—
83C2×Dic37—41no—
84C2×Dic37C2×D412yes—
85C4×D37—22no—
86C4×D37—22no—
87C4×D37—42no—
88C4×D37—44no—
89C4×D37—44no—
90Dic74—22no—
91Dic74—22no—
92Dic74—44no—
93C8×C37—21no—
94C8×C37—21no—
95#64436—22no—
96C22×C37—21no—
97C22×C37—21no—
98C22×C37C2×Q1612yes—
99C22×C37C22×D411yescharacteristic
100C2×D37—22no—
101C2×D37—22no—
102C2×D37—44no—
103C2×D37C2×Q1614yes—
104C4×C37—21no—
105C4×C37—42no—
106C4×C37—42no—
107C4×C37H(5,𝔽2)11yescharacteristic
108C4×C37H(5,𝔽2)11yescharacteristic
109Dic37—22no—
110Dic37—42no—
111Dic37—42no—
112#3189—371nomaximal, Sylow
113C2×C37—22no—
114C2×C37#53411yescharacteristic
115C2×C37C22×Q1611yescharacteristic
116C2×C37#57511yescharacteristic
117D37—24no—
118C2×C22⋊Q8—372no—
119#484—372no—
120#484—372no—
121#467—371no—
122#467—372no—
123#467D3711yescharacteristic, complemented
124#451—372no—
125#451—372no—
126C2×C22⋊C8—371no—
127C37#318911yescharacteristic, complemented, Sylow
128C22×Q8—372no—
129C23×C4—371no—
130C2×Q16—372no—
131C2×Q16—371no—
132C2×Q16C2×D3711yescharacteristic, complemented
133C22×C8—371no—
134C22×C8C2×D3711yescharacteristic
135C22⋊Q8—372no—
136C22⋊Q8—372no—
137C22⋊Q8—372no—
138C22⋊Q8—742no—
139C22⋊Q8—742no—
140C22⋊Q8C2×D3712yes—
141C2×C4⋊C4—372no—
142C2×C4⋊C4—742no—
143C2×C22⋊C4—742no—
144#107—371no—
145#107—372no—
146#107C2×D3711yescharacteristic, complemented
147#103—372no—
148#103—372no—
149#103—372no—
150#103C2×D3712yescomplemented
151C22⋊C8—372no—
152C22⋊C8—372no—
153C24—371no—
154C2×Q8—372no—
155C2×Q8—742no—
156C2×Q8—1482no—
157C2×Q8—1482no—
158C2×Q8C22×D3712yes—
159C22×C4—371no—
160C22×C4—372no—
161C22×C4—372no—
162C22×C4—742no—
163C22×C4—742no—
164C22×C4—1481no—
165C22×C4—1481no—
166C22×C4C22×D3711yescharacteristic
167Q16—41no—
168Q16—1481no—
169Q16—1482no—
170C2×C8—41no—
171C2×C8—372no—
172C2×C8—1481no—
173C2×C8C22×D3711yescharacteristic
174C2×C8C22×D3711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—372no—
177C4⋊C4—742no—
178C4⋊C4—742no—
179C4⋊C4—742no—
180C4⋊C4—742no—
181C4⋊C4C22×D3712yes—
182C22⋊C4—22no—
183C22⋊C4—742no—
184C22⋊C4—742no—
185C22⋊C4—742no—
186C23—372no—
187C23—374no—
188C23—1481no—
189C23—1481no—
190C23D4×D3711yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—742no—
194Q8—742no—
195Q8—1484no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—372no—
201C2×C4—742no—
202C2×C4—742no—
203C2×C4—742no—
204C2×C4—742no—
205C2×C4—1482no—
206C2×C4—1484no—
207C2×C4—1484no—
208C2×C4—1481no—
209C2×C4D4×D3711yescharacteristic
210C2×C4C23×D3711yescharacteristic
211C8—21no—
212C8—21no—
213C8—742no—
214C22—21no—
215C22—21no—
216C22—374no—
217C22—742no—
218C22—742no—
219C22—1484no—
220C22Q16×D3712yes—
221C22C2×D4×D3711yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—742no—
226C4—1482no—
227C4—1482no—
228C4#1192458111yescharacteristic
229C4#1192458211yescharacteristic
230C2—22no—
231C2—744no—
232C2#1455809111yescharacteristic
233C2C2×Q16×D3711yescharacteristic
234C2#1455833011yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#467C2
socleC22×C37C22×D4
prime radicalC23×D37D4
hypercenter#467D37
hypocenterC37#3189
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #14557870C2semidirect
3. #14557867C2semidirect
4. #14557774C2semidirect
5. #14557772C2semidirect
6. #14557690C2semidirect
7. #14557687C2semidirect
9. C37×#467C2semidirect
12. C2×Dic148C22semidirect
13. C2×#10596942C22semidirect
23. #11924470C22semidirect
24. #11924469C22semidirect
25. #11924461C22semidirect
26. #11924460C22semidirect
27. #11924459C22semidirect
30. C2×Q16×C37C22semidirect
33. #107×C37C22semidirect
34. #103×C37C22semidirect
71. C22×D37Q16semidirect
123. #467D37semidirect
127. C37#3189semidirect
132. C2×Q16C2×D37semidirect
146. #107C2×D37semidirect
150. #103C2×D37semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D372
#145578702
#145578672
#145577742
#145577722
#145576902
#145576872
C2×#119244482
C37×#4672
#318937