Finite Group Databasebeta

← #16835471

Subgroups — #16835471

Order 3712.

408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 90 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×#90800C211yesmaximal, characteristic, complemented
2C2×#11745219C212yesmaximal, complemented
3C2×C22⋊#6599C211yesmaximal, characteristic, complemented
4#14272029C212yesmaximal, complemented
5C2×#11745243C211yesmaximal, characteristic
6C22×C22⋊D29C411yescharacteristic
7C23×Dic29C2211yescharacteristic
8C22×C4×D29C2211yescharacteristic
9C2×#90800—22no—
10C2×#90800—24no—
11C22×#6599C2212yes—
12C22×#6599C2212yes—
13C2×C22⋊C4×C29C411yescharacteristic
14C2×#90781C411yescharacteristic
15C2×#90779C411yescharacteristic
16C2×C4×Dic29C2211yescharacteristic
17#11745243—22no—
18#11745243—22no—
19C22⋊#6599—22no—
20#11745219—24no—
21C24×C29C2×C411yescharacteristic
22C23×D29C2×C411yescharacteristic
23C22×C4×C29C2×C411yescharacteristic
24C22×C4×C29C2×C411yescharacteristic
25C2×C22⋊D29—22no—
26C2×C22⋊D29—22no—
27C2×C22⋊D29—22no—
28C22×Dic29—22no—
29C22×Dic29—22no—
30C22×Dic29C2×C411yescharacteristic
31C22×Dic29C2×C411yescharacteristic
32C22×Dic29D412yes—
33C22×Dic29D412yes—
34C22×Dic29C2311yescharacteristic
35C2×C4×D29—22no—
36C2×C4×D29—21no—
37C2×C4×D29—21no—
38C2×C4×D29—22no—
39#90800—28no—
40C2×#6599—22no—
41C2×#6599—22no—
42C2×#6599—22no—
43C2×#6599—22no—
44C2×#6599—22no—
45C2×#6599D414yescomplemented
46C22⋊C4×C29—22no—
47#90781—22no—
48#90779—22no—
49C4×Dic29—21no—
50C4×Dic29—21no—
51C23×C29—22no—
52C23×C29—21no—
53C23×C29—21no—
54C23×C29—22no—
55C23×C29C22⋊C412yes—
56C23×C29C22×C411yescharacteristic
57C22×D29—21no—
58C22×D29—22no—
59C22×D29—21no—
60C22×D29—22no—
61C22×D29M1612yescomplemented
62C2×C4×C29—21no—
63C2×C4×C29—21no—
64C2×C4×C29—22no—
65C2×C4×C29—22no—
66C22⋊D29—28no—
67C2×Dic29—21no—
68C2×Dic29—22no—
69C2×Dic29—21no—
70C2×Dic29—22no—
71C2×Dic29—24no—
72C2×Dic29—22no—
73C2×Dic29C22⋊C412yes—
74C2×Dic29M1612yescomplemented
75C2×Dic29C2×D412yes—
76C2×Dic29C2×D411yescharacteristic
77C2×Dic29C2×D411yescharacteristic
78C2×Dic29#4112yes—
79C4×D29—41no—
80C4×D29—42no—
81C4×D29—41no—
82#6599—24no—
83#6599—42no—
84#1870—291nomaximal, Sylow
85C22×C29—22no—
86C22×C29—22no—
87C22×C29—22no—
88C22×C29—42no—
89C22×C29—42no—
90C22×C29C2×C22⋊C411yescharacteristic
91C22×C29C4×D412yes—
92C22×C29C2×M1611yescharacteristic
93C22×C29C2×M1611yescharacteristic
94C22×C29#13112yes—
95C2×D29—22no—
96C2×D29—24no—
97C2×D29—41no—
98C2×D29—41no—
99C2×D29—42no—
100C4×C29—41no—
101C4×C29—41no—
102Dic29—22no—
103Dic29—22no—
104Dic29#12011yescharacteristic
105Dic29#12112yes—
106Dic29#12311yescharacteristic
107C22×M16—291no—
108C2×C4×D4—291no—
109C2×C22⋊C8—291no—
110C2×C22⋊C8—292no—
111#336—292no—
112C2×C29—42no—
113C2×C29#38611yescharacteristic
114C2×C29#40711yescharacteristic
115C2×C29#40811yescharacteristic
116C2×C29#43512yes—
117C2×C29#43612yes—
118D29—42no—
119C22×D4—291no—
120C23×C4—291no—
121C23×C4—291no—
122C2×M16—584no—
123C2×M16—582no—
124C22×C8—292no—
125C22×C8—292no—
126C4×D4—582no—
127C4×D4—582no—
128C2×C4⋊C4—291no—
129C2×C22⋊C4—291no—
130C2×C22⋊C4#118111yescharacteristic
131C2×C42—291no—
132C22⋊C8—584no—
133C22⋊C8—582no—
134C29#187011yescharacteristic, complemented, Sylow
135C24—291no—
136C24C2×#118111yescharacteristic
137C2×D4—582no—
138C2×D4—582no—
139C2×D4—582no—
140C22×C4—291no—
141C22×C4—292no—
142C22×C4—292no—
143C22×C4—291no—
144C22×C4—291no—
145C22×C4—582no—
146C22×C4—581no—
147C22×C4—581no—
148C22×C4—582no—
149C22×C4—582no—
150C22×C4—582no—
151C22×C4C2×#118111yescharacteristic
152C22×C4C2×#118111yescharacteristic
153M16—588no—
154C2×C8—294no—
155C2×C8—582no—
156C2×C8—582no—
157C2×C8—582no—
158C2×C8—582no—
159C2×C8—582no—
160C4⋊C4—582no—
161C22⋊C4—22no—
162C22⋊C4—582no—
163C42—581no—
164C42—581no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—22no—
169C23—292no—
170C23—581no—
171C23—582no—
172C23—582no—
173C23—581no—
174C23C22⋊#118112yes—
175C23C22×#118111yescharacteristic
176D4—588no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—292no—
182C2×C4—292no—
183C2×C4—291no—
184C2×C4—291no—
185C2×C4—292no—
186C2×C4—292no—
187C2×C4—582no—
188C2×C4—584no—
189C2×C4—582no—
190C2×C4—581no—
191C2×C4—582no—
192C2×C4—581no—
193C2×C4—1161no—
194C2×C4—1162no—
195C2×C4—1161no—
196C8—584no—
197C8—1162no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—582no—
204C22—584no—
205C22—1161no—
206C22—1161no—
207C22—1162no—
208C22D29⋊M1611yescharacteristic
209C22D29⋊M1611yescharacteristic
210C22#1174534612yes—
211C22D4×#118112yes—
212C22C2×C22⋊#118111yescharacteristic
213C4—41no—
214C4—41no—
215C4—291no—
216C4—292no—
217C4—291no—
218C4—582no—
219C4—582no—
220C2—42no—
221C2—1162no—
222C2#1427211111yescharacteristic
223C2#1427213511yescharacteristic
224C2#1427213611yescharacteristic
225C2#1427219512yes—
226C2#1427219612yes—
subgroupG/H
center, FrattiniC23C22×#1181
commutator, socleC23×C29C22×C4
FittingC2×C22⋊C4×C29C4
prime radicalC22×C22⋊D29C4
cosocleC22×Dic29C23
hypercenterC2×C22⋊C4#1181
hypocenterC29#1870
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#90800C2semidirect
2. C2×#11745219C2semidirect
3. C2×C22⋊#6599C2semidirect
4. #14272029C2semidirect
45. C2×#6599D4semidirect
61. C22×D29M16semidirect
74. C2×Dic29M16semidirect
134. C29#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#908002
C2×#117452192
C2×C22⋊#65992
#142720292
C2×#117452432
#187029