Finite Group Databasebeta

← #37535405

Subgroups — #37535405

Order 23168.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#26041830C211yesmaximal, characteristic
2#26041262C212yesmaximal, complemented
3#26041078C211yesmaximal, characteristic, complemented
4#26041075C212yesmaximal, complemented
5#356×C181C211yesmaximal, characteristic, complemented
6C2×#15668450C411yescharacteristic
7C2×#15668450C2211yescharacteristic
8M16×D181—22no—
9#19153813C411yescharacteristic
10#19153805C411yescharacteristic
11#19153805C2211yescharacteristic
12#19153791—22no—
13#19153778—22no—
14C42×D181C411yescharacteristic
15#19153735C2212yes—
16#19153732—22no—
17C4⋊Q8×C181C2211yescharacteristic
18#101×C181C2212yes—
19#15668450—41no—
20#15668450—41no—
21#15668450—41no—
22#15668450—41no—
23C2×D724—22no—
24C2×D724—21no—
25C2×D724C2×C411yescharacteristic
26C2×D724C2×C411yescharacteristic
27C2×D724D412yes—
28C2×C4×D181—21no—
29C2×C4×D181—21no—
30C2×C4×D181C2×C411yescharacteristic
31C2×C4×D181D412yes—
32#15668433—22no—
33#15668432—42no—
34C8×D181—42no—
35#15668426—22no—
36#15668426—22no—
37C4×Dic181—21no—
38C4×Dic181C2×C411yescharacteristic
39C2×Q8×C181C2×C411yescharacteristic
40C2×Q8×C181C2311yescharacteristic
41M16×C181—22no—
42C4⋊C4×C181—22no—
43C42×C181C2×C411yescharacteristic
44C22×D181—21no—
45C22×D181C22⋊C412yes—
46C22×D181C22⋊C411yescharacteristic
47D724—41no—
48D724—42no—
49D724—41no—
50D724—44no—
51D724—42no—
52C2×Dic181—21no—
53C2×Dic181C22⋊C411yescharacteristic
54C4×D181—22no—
55C4×D181—22no—
56C4×D181—21no—
57C4×D181—21no—
58C4×D181—42no—
59C4×D181—41no—
60C4×D181—41no—
61#13089678—42no—
62Q8×C181—41no—
63Q8×C181—41no—
64C2×C4×C181—21no—
65C2×C4×C181—21no—
66C2×C4×C181C22×C411yescharacteristic
67C2×C4×C181C2×D412yes—
68C8×C181—42no—
69C22×C181C2×C22⋊C411yescharacteristic
70C2×D181—21no—
71C2×D181—22no—
72C2×D181—41no—
73C2×D181—42no—
74C2×D181—42no—
75C2×D181—41no—
76C4×C181—22no—
77C4×C181—22no—
78C4×C181—21no—
79C4×C181—41no—
80Dic181—41no—
81Dic181#9911yescharacteristic
82Dic181#9911yescharacteristic
83C2×C181—21no—
84C2×C181C2×#9911yescharacteristic
85D181—41no—
86D181—42no—
87D181—81no—
88C181#213211yescharacteristic, complemented, Sylow
89#2132—1811nomaximal, Sylow
90#532—1811no—
91#413—1812no—
92#356—1811no—
93#356D18111yescharacteristic, complemented
94#355—1812no—
95C2×#41—1811no—
96C2×#41—1811no—
97C2×M16—3622no—
98C4⋊Q8C2×D18111yescharacteristic
99#127—1811no—
100#124—1811no—
101#124—1811no—
102#121—3622no—
103C4×D4—3622no—
104C2×C42—1811no—
105#101—1812no—
106#101C2×D18112yes—
107#100—3622no—
108#41—7241no—
109#41—7241no—
110#41—7241no—
111#41—7241no—
112C2×Q8C4×D18111yescharacteristic
113C2×Q8C22×D18111yescharacteristic
114C2×D4—3622no—
115C2×D4—3621no—
116C2×D4—1811no—
117C2×D4—1811no—
118C2×D4—1812no—
119C22×C4—3621no—
120C22×C4—3621no—
121C22×C4—1811no—
122C22×C4—1812no—
123M16—22no—
124M16—3622no—
125M16—7242no—
126C2×C8—7242no—
127C4⋊C4—22no—
128C22⋊C4—3622no—
129C22⋊C4—3622no—
130C42—3621no—
131C42—1811no—
132C42C4×D18111yescharacteristic
133C23—3621no—
134C23—1812no—
135C23—1811no—
136Q8—41no—
137Q8—41no—
138D4—7242no—
139D4—7241no—
140D4—7241no—
141D4—7244no—
142D4—7242no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—3622no—
146C2×C4—3622no—
147C2×C4—3621no—
148C2×C4—3621no—
149C2×C4—3621no—
150C2×C4—1811no—
151C2×C4—7242no—
152C2×C4—7241no—
153C2×C4—7241no—
154C2×C4C2×C4×D18111yescharacteristic
155C2×C4D4×D18112yes—
156C8—42no—
157C8—7242no—
158C22—3621no—
159C22—3622no—
160C22—7242no—
161C22—7241no—
162C22—7242no—
163C22—7241no—
164C22C22⋊C4×D18111yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—1811no—
170C4—1811no—
171C4—7241no—
172C2—21no—
173C2—14481no—
174C2—7241no—
175C2—7242no—
176C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC2×C4×C181C22×C4
FrattiniC2×Q8C22×D181
Fitting#356×C181C2
socleC2×C181C2×#99
prime radical#26041830C2
cosocleC2×Q8×C181C23
hypercenter#356D181
hypocenterC181#2132
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #26041262C2semidirect
3. #26041078C2semidirect
4. #26041075C2semidirect
5. #356×C181C2semidirect
88. C181#2132semidirect
93. #356D181semidirect

Maximal subgroups

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

subgroupindex
#260418302
#260412622
#260410782
#260410752
#356×C1812
#2132181