Finite Group Databasebeta

← #16819703

Subgroups — #16819703

Order 3712.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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#14271637C211yesmaximal, characteristic
2#14271206C211yesmaximal, characteristic, complemented
3#14271205C211yesmaximal, characteristic, complemented
4C2×#11745171C211yesmaximal, characteristic
5#14271065C211yesmaximal, characteristic
6#14270923C211yesmaximal, characteristic, complemented
7C29×#339C211yesmaximal, characteristic, complemented
8C22×D116C2211yescharacteristic
9C2×M16×C29C2211yescharacteristic
10C29×#117C2211yescharacteristic
11C2×C42×C29C2211yescharacteristic
12C2×#90781—22no—
13#11745288C2211yescharacteristic
14C2×C4⋊Dic29C2211yescharacteristic
15C2×#90777C2211yescharacteristic
16#11745255—21no—
17#11745255—21no—
18C22⋊C4×D29—22no—
19C4×D116—21no—
20C4×D116—21no—
21#11745171C414yescomplemented
22#11745166—21no—
23#11745165—21no—
24#11745156—21no—
25#11745155—21no—
26C23×D29D412yes—
27C22×C4×C29—21no—
28C22×C4×C29C2311yescharacteristic
29C22×Dic29—21no—
30C2×D116—21no—
31C2×D116—21no—
32C2×D116—21no—
33C2×D116—21no—
34C2×D116C2×C412yes—
35C2×D116Q812yes—
36C2×C4×D29—22no—
37C2×C4×D29—22no—
38M16×C29—21no—
39M16×C29C2×C412yescomplemented
40C2×C8×C29—21no—
41C4⋊C4×C29D411yescharacteristic
42C4⋊C4×C29D411yescharacteristic
43C22⋊C4×C29—21no—
44C42×C29—21no—
45C42×C29—21no—
46C42×C29—21no—
47C22⋊Dic29—21no—
48#90781—22no—
49#90781—22no—
50#90781—44no—
51C4⋊Dic29—21no—
52C4⋊Dic29—21no—
53C4⋊Dic29D411yescharacteristic
54C4⋊Dic29D411yescharacteristic
55C4×Dic29—21no—
56#90777—21no—
57#90777C2×C412yescomplemented
58C2×#6597—21no—
59C23×C29C2×D411yescharacteristic
60C22×D29—22no—
61C22×D29—22no—
62C22×D29—42no—
63C22×D29—42no—
64C22×D29C4⋊C414yes—
65C2×C4×C29—22no—
66C2×C4×C29—21no—
67C2×C4×C29—21no—
68C2×C4×C29—41no—
69C2×C4×C29—41no—
70C2×C4×C29C22×C411yescharacteristic
71C2×C4×C29C2×D411yescharacteristic
72C2×C4×C29C2×D411yescharacteristic
73C2×C4×C29C2×Q811yescharacteristic
74C2×C4×C29#4111yescharacteristic
75C2×C4×C29#4111yescharacteristic
76C2×Dic29—21no—
77C2×Dic29—21no—
78C2×Dic29—42no—
79C2×Dic29—41no—
80D116—22no—
81D116—22no—
82D116—22no—
83D116—22no—
84C4×D29—82no—
85C4×D29—82no—
86C8×C29—22no—
87#6597—22no—
88#1856—291nomaximal, Sylow
89C22×C29—21no—
90C22×C29—21no—
91C22×C29C2×C4⋊C411yescharacteristic
92C22×C29#12011yescharacteristic
93C22×C29#12311yescharacteristic
94C2×D29—24no—
95C2×D29—42no—
96C2×D29—42no—
97C2×D29—44no—
98C2×D29—82no—
99C4×C29—42no—
100C4×C29—41no—
101C4×C29—41no—
102C4×C29C4×D411yescharacteristic
103C4×C29C4×D411yescharacteristic
104C4×C29C22⋊Q811yescharacteristic
105C4×C29C22⋊Q811yescharacteristic
106Dic29—41no—
107Dic29—41no—
108Dic29—81no—
109#518—291no—
110#428—291no—
111#428—291no—
112C2×#100—291no—
113#390—291no—
114#339—291no—
115#339D2911yescharacteristic, complemented
116C2×C29—21no—
117C2×C29—21no—
118C2×C29#38511yescharacteristic
119C2×C29#45311yescharacteristic
120C2×C29#45511yescharacteristic
121D29—44no—
122C22×D4—291no—
123C2×M16—291no—
124C2×M16C2×D2911yescharacteristic
125C4×D4—581no—
126C4×D4—581no—
127C4×D4—581no—
128C4×D4—581no—
129#117—291no—
130#117C2×D2911yescharacteristic
131C2×C4⋊C4—291no—
132C2×C22⋊C4—582no—
133C2×C22⋊C4—582no—
134C2×C42C2×D2911yescharacteristic
135#107—581no—
136#107—581no—
137#106—581no—
138#106—581no—
139#100—294no—
140C29#185611yescharacteristic, complemented, Sylow
141C24—292no—
142C2×D4—292no—
143C2×D4—292no—
144C2×D4—581no—
145C2×D4—581no—
146C2×D4—581no—
147C2×D4—581no—
148C22×C4—21no—
149C22×C4—582no—
150C22×C4—582no—
151C22×C4—581no—
152C22×C4C22×D2911yescharacteristic
153M16—21no—
154M16—292no—
155M16—581no—
156M16C4×D2912yescomplemented
157C2×C8—21no—
158C2×C8—581no—
159C4⋊C4—291no—
160C4⋊C4—291no—
161C4⋊C4—581no—
162C4⋊C4—581no—
163C4⋊C4D11611yescharacteristic
164C4⋊C4D11611yescharacteristic
165C22⋊C4—21no—
166C22⋊C4—582no—
167C22⋊C4—582no—
168C22⋊C4—581no—
169C22⋊C4—1164no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—581no—
174C23—294no—
175C23—582no—
176C23—582no—
177C23—1162no—
178C23—1162no—
179C23D4×D2911yescharacteristic
180D4—582no—
181D4—582no—
182D4—582no—
183D4—582no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—581no—
190C2×C4—581no—
191C2×C4—1162no—
192C2×C4—1161no—
193C2×C4—2322no—
194C2×C4—2322no—
195C2×C4C2×C4×D2911yescharacteristic
196C2×C4C2×D11611yescharacteristic
197C2×C4#9080511yescharacteristic
198C2×C4D4×D2911yescharacteristic
199C2×C4#9080711yescharacteristic
200C2×C4Q8×D2911yescharacteristic
201C8—22no—
202C8—582no—
203C22—21no—
204C22—21no—
205C22—584no—
206C22—1164no—
207C22—1162no—
208C22—1162no—
209C22—2322no—
210C22#1174524411yescharacteristic
211C22#1174524511yescharacteristic
212C22C4⋊C4×D2911yescharacteristic
213C4—42no—
214C4—41no—
215C4—41no—
216C4—1161no—
217C4—1161no—
218C4—2321no—
219C4C4×D11611yescharacteristic
220C4#1174524311yescharacteristic
221C4#1174525811yescharacteristic
222C4#1174525911yescharacteristic
223C2—21no—
224C2—21no—
225C2—1164no—
226C2#1427103711yescharacteristic
227C2#1427133211yescharacteristic
228C2#1427134211yescharacteristic
subgroupG/H
centerC22#11745244
commutatorC2×C4×C29C22×C4
FrattiniC22×C4C22×D29
FittingC29×#339C2
socleC22×C29#120
prime radicalC22×D116C22
cosocleC22×C4×C29C23
hypercenter#339D29
hypocenterC29#1856
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #14271206C2semidirect
3. #14271205C2semidirect
6. #14270923C2semidirect
7. C29×#339C2semidirect
21. #11745171C4semidirect
39. M16×C29C2×C4semidirect
57. #90777C2×C4semidirect
115. #339D29semidirect
140. C29#1856semidirect
156. M16C4×D29semidirect

Maximal subgroups

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

subgroupindex
#142716372
#142712062
#142712052
C2×#117451712
#142710652
#142709232
C29×#3392
#185629