Finite Group Databasebeta

← #16835497

Subgroups — #16835497

Order 3712.

360 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 68 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#14272230C211yesmaximal, characteristic
2#14272156C212yesmaximal, complemented
3D29⋊#100C211yesmaximal, characteristic
4#14272034C212yesmaximal, complemented
5#124×D29C211yesmaximal, characteristic
6C2×Q8×D29C2211yescharacteristic
7C2×D4×D29C2211yescharacteristic
8C2×C22⋊#1181—22no—
9D4×#1181—24no—
10#11745345C2212yes—
11D29⋊M16C2212yes—
12C29×#124C411yescharacteristic, complemented
13#11745309C411yescharacteristic, complemented
14#11745298C411yescharacteristic, complemented
15#11745247—22no—
16C22⋊C4×D29—22no—
17#11745237C411yescharacteristic, complemented
18C42×D29C2211yescharacteristic
19#11745229—22no—
20#11745225—24no—
21C23×D29D412yes—
22C22×#1181—24no—
23C2×Q8×C29C2×C411yescharacteristic, complemented
24C2×D4×C29C2×C411yescharacteristic
25C2×C22⋊D29D412yes—
26Q8×D29—22no—
27Q8×D29—22no—
28D4×D29—22no—
29D4×D29—22no—
30C2×D116C2×C411yescharacteristic
31C2×C4×D29—21no—
32C2×C4×D29—21no—
33C2×C4×D29C2311yescharacteristic
34C2×Dic58—21no—
35C2×Dic58C2×C411yescharacteristic, complemented
36C22⋊#1181—24no—
37C22⋊#1181—22no—
38#90800—22no—
39C4⋊#1181D414yes—
40C4×#1181—22no—
41#90796—22no—
42D29⋊C8—22no—
43C22⋊C4×C29—22no—
44C42×C29C2×C411yescharacteristic, complemented
45C22⋊Dic29—22no—
46#90781—22no—
47#90781—22no—
48C4×Dic29—21no—
49C4×Dic29C2×C411yescharacteristic, complemented
50C23×C29C22⋊C412yes—
51C22×D29—22no—
52C22×D29—22no—
53C22×D29—44no—
54C22×D29C22⋊C412yes—
55C22×D29C2×D411yescharacteristic
56C2×#1181—24no—
57C2×#1181—84no—
58Q8×C29—22no—
59D4×C29—22no—
60C2×C4×C29—21no—
61C2×C4×C29—21no—
62C2×C4×C29C22×C411yescharacteristic
63C22⋊D29—24no—
64C2×Dic29—21no—
65C2×Dic29—21no—
66C2×Dic29C2×D411yescharacteristic
67D116—22no—
68C4×D29—42no—
69C4×D29—41no—
70C4×D29—41no—
71C4×D29C2×D412yes—
72C4×D29#4112yes—
73Dic58—22no—
74Dic58—42no—
75#6599—42no—
76#1885—291nomaximal, Sylow
77C22×C29—22no—
78C22×C29—42no—
79C22×C29C2×C22⋊C411yescharacteristic
80C2×D29—22no—
81C2×D29—22no—
82C2×D29—44no—
83C2×D29—42no—
84C2×D29—82no—
85C2×D29#12011yescharacteristic
86C2×D29#12311yescharacteristic
87#1181—44no—
88C4×C29—41no—
89C4×C29—41no—
90C4×C29C4×D412yes—
91Dic29—41no—
92Dic29—41no—
93Dic29#12112yes—
94C2×#124—291no—
95#518—291no—
96#419—292no—
97C2×#100—291no—
98#339—292no—
99C2×C29—21no—
100C2×C29—42no—
101C2×C29#38611yescharacteristic
102D29—21no—
103D29—42no—
104D29#45512yes—
105C22×Q8—291no—
106C22×D4—291no—
107C2×M16—292no—
108#124—291no—
109#124—291no—
110#124—291no—
111#124—582no—
112#124#118111yescharacteristic, complemented
113C4×D4—584no—
114#117—292no—
115C2×C22⋊C4—582no—
116C2×C22⋊C4—582no—
117C2×C42—291no—
118#103—584no—
119#100—582no—
120C29#188511yescharacteristic, complemented, Sylow
121C24—292no—
122C2×Q8—291no—
123C2×Q8—581no—
124C2×Q8—582no—
125C2×Q8—582no—
126C2×Q8C2×#118111yescharacteristic, complemented
127C2×D4—292no—
128C2×D4—291no—
129C2×D4—582no—
130C2×D4—582no—
131C2×D4C2×#118111yescharacteristic
132C22×C4—291no—
133C22×C4—581no—
134C22×C4—581no—
135C22×C4—584no—
136M16—582no—
137M16—582no—
138C2×C8—582no—
139C4⋊C4—294no—
140C22⋊C4—22no—
141C22⋊C4—582no—
142C22⋊C4—582no—
143C22⋊C4—582no—
144C22⋊C4—584no—
145C22⋊C4—582no—
146C42—291no—
147C42—581no—
148C42—582no—
149C42C2×#118111yescharacteristic, complemented
150C23—291no—
151C23—292no—
152C23—582no—
153C23—582no—
154C23—1164no—
155C23C22⋊#118112yes—
156Q8—22no—
157Q8—582no—
158Q8—1162no—
159D4—22no—
160D4—584no—
161D4—582no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—291no—
165C2×C4—292no—
166C2×C4—292no—
167C2×C4—584no—
168C2×C4—581no—
169C2×C4—581no—
170C2×C4—1161no—
171C2×C4—1161no—
172C2×C4—1162no—
173C2×C4—2324no—
174C2×C4C22×#118111yescharacteristic
175C8—1162no—
176C22—22no—
177C22—42no—
178C22—291no—
179C22—291no—
180C22—582no—
181C22—582no—
182C22—1162no—
183C22—1164no—
184C22—2322no—
185C22C2×C22⋊#118111yescharacteristic
186C4—41no—
187C4—41no—
188C4—292no—
189C4—1161no—
190C4—1164no—
191C4—1161no—
192C4D4×#118112yes—
193C2—21no—
194C2—42no—
195C2—292no—
196C2—581no—
197C2—1162no—
198C2#1427211111yescharacteristic
subgroupG/H
centerC2#14272111
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC29×#124C4
socleC2×C29#386
prime radicalC2×D4×D29C22
cosocleC2×C4×D29C23
hypercenter#124#1181
hypocenterC29#1885
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #14272156C2semidirect
4. #14272034C2semidirect
12. C29×#124C4semidirect
13. #11745309C4semidirect
14. #11745298C4semidirect
17. #11745237C4semidirect
23. C2×Q8×C29C2×C4semidirect
35. C2×Dic58C2×C4semidirect
44. C42×C29C2×C4semidirect
49. C4×Dic29C2×C4semidirect
112. #124#1181semidirect
120. C29#1885semidirect
126. C2×Q8C2×#1181semidirect
149. C42C2×#1181semidirect

Maximal subgroups

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

subgroupindex
#142722302
#142721562
D29⋊#1002
#142720342
#124×D292
#188529