Finite Group Databasebeta

← #16829887

Subgroups — #16829887

Order 3712.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#14271714C211yesmaximal, characteristic
2C4⋊Q8×D29C211yesmaximal, characteristic, complemented
3#14271524C214yesmaximal, complemented
4#14271513C211yesmaximal, characteristic, complemented
5#14271511C211yesmaximal, characteristic, complemented
6#14271510C211yesmaximal, characteristic, complemented
7#14271163C214yesmaximal, complemented
8C4×#90772C211yesmaximal, characteristic, complemented
9C29×#487C211yesmaximal, characteristic, complemented
10C2×#90809C2212yes—
11C2×Q8×D29—21no—
12C29×C4⋊Q8C2211yescharacteristic
13C29×C4⋊Q8C2211yescharacteristic
14#11745309C2212yes—
15#11745306—21no—
16C2×#90772C2212yes—
17#11745257—21no—
18#11745257—21no—
19#11745255—21no—
20#11745255—21no—
21C4⋊C4×D29—21no—
22C4⋊C4×D29C2212yes—
23#11745250—21no—
24#11745250C2212yes—
25#11745236C2211yescharacteristic
26C42×D29C2211yescharacteristic
27#11745231C2211yescharacteristic
28C29×#103C2214yes—
29C4×C8×C29C2211yescharacteristic
30#11745183C2214yes—
31#11745169C2214yes—
32#11745163C2212yes—
33#11745157C2214yes—
34C4×#6597C2211yescharacteristic
35C2×Q8×C29—21no—
36C2×Q8×C29C2312yes—
37#90809—28no—
38Q8×D29—44no—
39C2×D116—22no—
40C2×D116—21no—
41C2×D116—21no—
42C2×D116C2312yes—
43C2×C4×D29—22no—
44C2×C4×D29—22no—
45C2×C4×D29D411yescharacteristic
46C2×C4×D29D411yescharacteristic
47C2×C4×D29C2311yescharacteristic
48C2×Dic58—21no—
49C2×Dic58—22no—
50C2×C8×C29C2312yes—
51C4⋊C4×C29—21no—
52C4⋊C4×C29—21no—
53C4⋊C4×C29—21no—
54C4⋊C4×C29C2312yes—
55C42×C29C2311yescharacteristic
56#90781—22no—
57#90781—22no—
58C4⋊Dic29—21no—
59C4⋊Dic29C2312yes—
60#90779—22no—
61#90779—22no—
62C4×Dic29D411yescharacteristic
63C4×Dic29D411yescharacteristic
64C4×Dic29C2311yescharacteristic
65C2×#6597C2312yes—
66#90772—24no—
67C22×D29—22no—
68C22×D29C2×D411yescharacteristic
69Q8×C29—24no—
70Q8×C29—42no—
71C2×C4×C29—22no—
72C2×C4×C29—22no—
73C2×C4×C29C2×D411yescharacteristic
74C2×C4×C29C2×D411yescharacteristic
75C2×C4×C29C2411yescharacteristic
76C2×Dic29—22no—
77C2×Dic29C2×D411yescharacteristic
78C2×Dic29C2×D411yescharacteristic
79C2×Dic29C2×D411yescharacteristic
80D116—24no—
81D116—44no—
82D116—42no—
83D116—42no—
84C4×D29—22no—
85C4×D29—22no—
86C4×D29—44no—
87C4×D29—44no—
88C4×D29#4114yes—
89Dic58—44no—
90Dic58—42no—
91C8×C29—22no—
92#6597—22no—
93#3141—291nomaximal, Sylow
94C22×C29C22×D411yescharacteristic
95C2×D29—21no—
96C2×D29—21no—
97C2×D29—42no—
98C2×D29—42no—
99C2×D29—42no—
100C2×D29#12412yes—
101C4×C29—42no—
102C4×C29—42no—
103C4×C29AGL(1,ℤ/8)12yes—
104C4×C29#13712yes—
105C4×C29C2×#4112yes—
106Dic29—21no—
107Dic29—21no—
108Dic29—42no—
109Dic29#12412yes—
110#532—291no—
111C2×C4⋊Q8—291no—
112#489—294no—
113#487—291no—
114#487D2911yescharacteristic, complemented
115#486—291no—
116#486—291no—
117#417—294no—
118C4×M16—291no—
119C2×C29C2×#12411yescharacteristic
120C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C29C2×#13711yescharacteristic
122D29—22no—
123D29—82no—
124C2×#41—292no—
125C22×Q8—581no—
126C2×M16—292no—
127C4⋊Q8—292no—
128C4⋊Q8—291no—
129C4⋊Q8—581no—
130C4⋊Q8—581no—
131C4⋊Q8C2×D2911yescharacteristic
132C4⋊Q8C2×D2911yescharacteristic
133#127—291no—
134#124—292no—
135#121—581no—
136#121—581no—
137C4×D4—581no—
138C4×D4—581no—
139C2×C4⋊C4—292no—
140C2×C4⋊C4—581no—
141C2×C42—291no—
142#103—294no—
143#103C2×D2914yes—
144#102—294no—
145#102—294no—
146#97—292no—
147C4×C8—291no—
148C4×C8C2×D2911yescharacteristic
149C29#314111yescharacteristic, complemented, Sylow
150#41—588no—
151C2×Q8—21no—
152C2×Q8—582no—
153C2×Q8—581no—
154C2×Q8—1164no—
155C2×Q8C22×D2912yes—
156C2×D4—292no—
157C2×D4—582no—
158C2×D4—581no—
159C2×D4—581no—
160C22×C4—291no—
161C22×C4—291no—
162C22×C4—291no—
163C22×C4—582no—
164C22×C4—582no—
165M16—584no—
166C2×C8—292no—
167C2×C8C22×D2912yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—292no—
172C4⋊C4—582no—
173C4⋊C4—582no—
174C4⋊C4—581no—
175C4⋊C4C22×D2912yes—
176C22⋊C4—582no—
177C22⋊C4—582no—
178C42—291no—
179C42—291no—
180C42—291no—
181C42C22×D2911yescharacteristic
182C23—291no—
183C23—582no—
184Q8—24no—
185Q8—42no—
186Q8—1164no—
187Q8—1162no—
188D4—584no—
189D4—1164no—
190D4—1162no—
191D4—1162no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—291no—
195C2×C4—294no—
196C2×C4—291no—
197C2×C4—291no—
198C2×C4—582no—
199C2×C4—582no—
200C2×C4—582no—
201C2×C4—1164no—
202C2×C4—1164no—
203C2×C4D4×D2911yescharacteristic
204C2×C4D4×D2911yescharacteristic
205C2×C4C23×D2911yescharacteristic
206C8—22no—
207C8—582no—
208C22—292no—
209C22—581no—
210C22—581no—
211C22—1162no—
212C22—1162no—
213C22—1162no—
214C22C2×D4×D2911yescharacteristic
215C4—42no—
216C4—42no—
217C4—292no—
218C4—581no—
219C4—581no—
220C4—1162no—
221C4#1174527612yes—
222C4#1174528012yes—
223C4#41×D2912yes—
224C2—582no—
225C2—2322no—
226C2#124×D2911yescharacteristic
227C2C2×#1174527611yescharacteristic
228C2C2×#1174528011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#487C2
socleC22×C29C22×D4
prime radical#14271714C2
hypercenter#487D29
hypocenterC29#3141
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D29C2semidirect
3. #14271524C2semidirect
4. #14271513C2semidirect
5. #14271511C2semidirect
6. #14271510C2semidirect
7. #14271163C2semidirect
8. C4×#90772C2semidirect
9. C29×#487C2semidirect
114. #487D29semidirect
149. C29#3141semidirect

Maximal subgroups

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

subgroupindex
#142717142
C4⋊Q8×D292
#142715242
#142715132
#142715112
#142715102
#142711632
C4×#907722
C29×#4872
#314129