Finite Group Databasebeta

← #16834892

Subgroups — #16834892

Order 3712.

294 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 64 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#14272134C211yesmaximal, characteristic
2C4×C22⋊#1181C211yesmaximal, characteristic
3#14272017C214yesmaximal, complemented
4C2×#11745308C211yesmaximal, characteristic
5C22×Q8×C29C411yescharacteristic, complemented
6C22×C4×D29C2211yescharacteristic
7C2×C22⋊#1181—21no—
8C2×C4×#1181—21no—
9C2×#90796—21no—
10#11745308—21no—
11#11745308—21no—
12#11745308C412yescomplemented
13#11745308C2212yes—
14C2×#90781—21no—
15C2×C4⋊Dic29C411yescharacteristic, complemented
16C2×#90779—21no—
17C22⋊#6599—21no—
18(C2×#1181)⋊C4C2212yes—
19#11745219C2212yes—
20C23×D29D411yescharacteristic
21C22×#1181—22no—
22C2×Q8×C29—21no—
23C2×Q8×C29—42no—
24C2×Q8×C29C2×C412yes—
25C22×C4×C29—21no—
26C22×C4×C29C2×C411yescharacteristic
27C22×Dic29—21no—
28C22×Dic29D411yescharacteristic
29C2×C4×D29—42no—
30C2×C4×D29C2×C411yescharacteristic
31C2×C4×D29D411yescharacteristic
32C2×C4×D29D411yescharacteristic
33C2×C4×D29C2311yescharacteristic
34C22⋊#1181—24no—
35C2×#6599—22no—
36C4×#1181—42no—
37#90796—42no—
38#90781—22no—
39#90781—22no—
40C4⋊Dic29—21no—
41C4⋊Dic29C2×C412yes—
42#90779—22no—
43#90779—22no—
44C23×C29C22⋊C411yescharacteristic
45C22×D29—22no—
46C22×D29—22no—
47C22×D29—41no—
48C22×D29C22⋊C411yescharacteristic
49C22×D29C2×D411yescharacteristic
50C2×#1181—24no—
51C2×#1181—42no—
52C2×#1181—42no—
53Q8×C29—42no—
54Q8×C29—42no—
55C2×C4×C29—22no—
56C2×C4×C29—22no—
57C2×C4×C29—42no—
58C2×C4×C29C22⋊C411yescharacteristic
59C2×C4×C29C22×C411yescharacteristic
60C2×Dic29—22no—
61C2×Dic29—41no—
62C2×Dic29—42no—
63C2×Dic29C22⋊C411yescharacteristic
64C2×Dic29C2×D411yescharacteristic
65C4×D29—22no—
66C4×D29—22no—
67C4×D29—41no—
68C4×D29—41no—
69#6599—42no—
70#1508—291nomaximal, Sylow
71C22×C29—21no—
72C22×C29—21no—
73C22×C29#10412yes—
74C22×C29C2×C22⋊C411yescharacteristic
75C2×D29—21no—
76C2×D29—21no—
77C2×D29—21no—
78C2×D29—21no—
79C2×D29—21no—
80C2×D29—44no—
81C2×D29#13712yes—
82#1181—44no—
83C4×C29—21no—
84C4×C29—42no—
85C4×C29#9912yes—
86Dic29—21no—
87Dic29—21no—
88Dic29—42no—
89C2×C22⋊Q8—291no—
90#407—291no—
91C4×C22⋊C4—291no—
92#328—294no—
93C2×C29—22no—
94C2×C29C2×#9911yescharacteristic
95C2×C29#41911yescharacteristic
96C2×C29C2×#10411yescharacteristic
97D29—22no—
98D29—41no—
99C22×Q8#118111yescharacteristic, complemented
100C23×C4—291no—
101C2×M16—581no—
102C22⋊Q8—292no—
103C22⋊Q8—292no—
104C22⋊Q8—581no—
105C22⋊Q8—581no—
106C2×C4⋊C4—291no—
107C2×C4⋊C4—581no—
108C2×C22⋊C4—581no—
109C2×C22⋊C4—581no—
110C2×C42—581no—
111C22⋊C8—292no—
112C22⋊C8—581no—
113#95—292no—
114C29#150811yescharacteristic, complemented, Sylow
115C24—291no—
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×#118112yes—
119C22×C4—21no—
120C22×C4—291no—
121C22×C4—291no—
122C22×C4—291no—
123C22×C4—291no—
124C22×C4—291no—
125C22×C4—582no—
126C22×C4—581no—
127C22×C4—1162no—
128C22×C4C2×#118111yescharacteristic
129M16—1162no—
130C2×C8—582no—
131C4⋊C4—292no—
132C4⋊C4—582no—
133C4⋊C4—582no—
134C4⋊C4—581no—
135C22⋊C4—582no—
136C22⋊C4—582no—
137C22⋊C4—584no—
138C42—1162no—
139C23—291no—
140C23—291no—
141C23—582no—
142C23—582no—
143C23—1161no—
144C23C22⋊#118111yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—291no—
151C2×C4—291no—
152C2×C4—582no—
153C2×C4—582no—
154C2×C4—584no—
155C2×C4—582no—
156C2×C4—1161no—
157C2×C4—1161no—
158C2×C4—1161no—
159C2×C4—1162no—
160C2×C4—1162no—
161C2×C4—1162no—
162C2×C4C22⋊#118111yescharacteristic
163C2×C4C22×#118111yescharacteristic
164C8—1162no—
165C22—21no—
166C22—21no—
167C22—292no—
168C22—581no—
169C22—581no—
170C22—581no—
171C22—581no—
172C22—581no—
173C22—1164no—
174C22#1174522612yes—
175C22C2×C22⋊#118111yescharacteristic
176C4—21no—
177C4—42no—
178C4—581no—
179C4—581no—
180C4—1164no—
181C4—1162no—
182C4#1174522712yes—
183C2—22no—
184C2—582no—
185C2—1161no—
186C2C2×#1174522711yescharacteristic
187C2#1427215611yescharacteristic
188C2C2×#1174522611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#1181
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC22×Q8×C29C4
socleC22×C29C2×C22⋊C4
prime radicalC23×D29D4
cosocleC2×C4×D29C23
hypercenterC22×Q8#1181
hypocenterC29#1508
LCSG ▷ C2×C4×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #14272017C2semidirect
5. C22×Q8×C29C4semidirect
12. #11745308C4semidirect
15. C2×C4⋊Dic29C4semidirect
99. C22×Q8#1181semidirect
114. C29#1508semidirect

Maximal subgroups

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

subgroupindex
#142721342
C4×C22⋊#11812
#142720172
C2×#117453082
#150829