Finite Group Databasebeta

← #37544853

Subgroups — #37544853

Order 23168.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041836C211yesmaximal, characteristic, complemented
2#26041682C211yesmaximal, characteristic, complemented
3#26041675C211yesmaximal, characteristic, complemented
4#26041625C211yesmaximal, characteristic, complemented
5#26041625C211yesmaximal, characteristic, complemented
6#26041600C214yesmaximal, complemented
7#26041579C214yesmaximal, complemented
8#26041332C211yesmaximal, characteristic, complemented
9#532×C181C211yesmaximal, characteristic
10#19153816C2211yescharacteristic
11#19153816C2211yescharacteristic
12#19153808C2212yes—
13#19153795—21no—
14#19153795—21no—
15C4×Dic362—21no—
16C4×Dic362—21no—
17#19153775C2212yes—
18#19153749C2214yes—
19#19153747C2214yes—
20C4⋊#13089678C2212yes—
21#19153737C2214yes—
22#19153737C2214yes—
23C22⋊#13089678C2212yes—
24C4×#13089678C2211yescharacteristic
25C4×#13089678C2211yescharacteristic
26C2×#41×C181—21no—
27C4⋊Q8×C181C2211yescharacteristic
28#127×C181C2211yescharacteristic
29#124×C181—21no—
30#121×C181—21no—
31#121×C181C2212yes—
32C4×D4×C181—21no—
33C4×D4×C181C2212yes—
34C2×C42×C181C2211yescharacteristic
35C2×Dic362—21no—
36C2×Dic362—21no—
37C2×#13089678C2312yes—
38C2×#13089678C2312yes—
39C4⋊Dic181—21no—
40C4⋊Dic181—21no—
41C4⋊Dic181C2312yes—
42C4⋊Dic181C2312yes—
43#15668428—22no—
44#15668428—22no—
45C22⋊Dic181—22no—
46C4×Dic181—22no—
47#41×C181—44no—
48C2×Q8×C181—21no—
49C2×D4×C181—22no—
50C2×D4×C181—21no—
51C2×D4×C181—21no—
52C2×D4×C181C2312yes—
53C22×C4×C181—22no—
54C22×C4×C181D411yescharacteristic
55C22×C4×C181D411yescharacteristic
56C22×C4×C181C2311yescharacteristic
57C4⋊C4×C181—21no—
58C4⋊C4×C181C2312yes—
59C22⋊C4×C181—22no—
60C22⋊C4×C181—22no—
61C42×C181D411yescharacteristic
62C42×C181D411yescharacteristic
63C42×C181C2311yescharacteristic
64C42×C181C2311yescharacteristic
65Dic362—42no—
66Dic362—42no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69#13089678—22no—
70#13089678—22no—
71C23×C181—22no—
72C23×C181C2×D411yescharacteristic
73Q8×C181—42no—
74D4×C181—24no—
75D4×C181—44no—
76D4×C181—42no—
77D4×C181—42no—
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—44no—
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2411yescharacteristic
89C22×C181—21no—
90C22×C181—21no—
91C22×C181—21no—
92C22×C181—42no—
93C22×C181—42no—
94C22×C181—42no—
95C22×C181C22×D411yescharacteristic
96C4×C181—21no—
97C4×C181—21no—
98C4×C181—21no—
99C4×C181—42no—
100C4×C181#13512yes—
101C4×C181AGL(1,ℤ/8)12yes—
102C4×C1812−1+412yes—
103Dic181—42no—
104Dic181—42no—
105C2×C181—41no—
106C2×C181—82no—
107C2×C181#53611yescharacteristic
108C2×C181C2×#13511yescharacteristic
109C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
110C181#322811yescharacteristic, complemented, Sylow
111#3228—1811nomaximal, Sylow
112#533—1811no—
113#532D18111yescharacteristic
114#500—1811no—
115#498—1811no—
116#486—1811no—
117#486—1811no—
118#482—1814no—
119#478—1814no—
120#431—1811no—
121C2×#41—21no—
122C4⋊Q8—1811no—
123C4⋊Q8—1811no—
124C4⋊Q8C2×D18111yescharacteristic
125#127C2×D18111yescharacteristic
126#125—1812no—
127#124—21no—
128C22⋊Q8—3621no—
129C22⋊Q8—3621no—
130#121—21no—
131#121C2×D18112yes—
132C4×Q8—3621no—
133C4×Q8—3621no—
134C4×D4—21no—
135C4×D4C2×D18112yes—
136#117—1812no—
137C2×C42C2×D18111yescharacteristic
138#107—1814no—
139#106—1814no—
140C4⋊C8—1812no—
141#102—1814no—
142#102—1814no—
143C22⋊C8—1812no—
144C4×C8—1811no—
145C4×C8—1811no—
146#41—44no—
147C2×Q8—21no—
148C2×Q8—3621no—
149C2×Q8—3621no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D18112yes—
154C22×C4—22no—
155C22×C4C22⋊D18111yescharacteristic
156C22×C4C22⋊D18111yescharacteristic
157C22×C4C22×D18111yescharacteristic
158C2×C8—1812no—
159C2×C8—1812no—
160C4⋊C4—21no—
161C4⋊C4—3622no—
162C4⋊C4—3622no—
163C4⋊C4—3621no—
164C4⋊C4—3621no—
165C4⋊C4—1812no—
166C4⋊C4—1812no—
167C4⋊C4C22×D18112yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—3622no—
171C42—3622no—
172C42C22⋊D18111yescharacteristic
173C42C22⋊D18111yescharacteristic
174C42C22×D18111yescharacteristic
175C42C22×D18111yescharacteristic
176C23—22no—
177C23C2×C22⋊D18111yescharacteristic
178Q8—42no—
179Q8—7242no—
180Q8—7242no—
181D4—24no—
182D4—44no—
183D4—42no—
184D4—42no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—3622no—
191C2×C4—3622no—
192C2×C4C2×C22⋊D18111yescharacteristic
193C2×C4C2×C22⋊D18111yescharacteristic
194C2×C4C2×C22⋊D18111yescharacteristic
195C2×C4C2×C22⋊D18111yescharacteristic
196C2×C4C2×C22⋊D18111yescharacteristic
197C2×C4C23×D18111yescharacteristic
198C8—3622no—
199C8—3622no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C22×C22⋊D18111yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—7242no—
212C4—7242no—
213C4#1915383612yes—
214C4#1915384212yes—
215C4#1915386312yes—
216C2—41no—
217C2—82no—
218C2#2604185011yescharacteristic
219C2C2×#1915383611yescharacteristic
220C2C2×#1915384211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#532×C181C2
socleC22×C181C22×D4
hypercenter#532D181
hypocenterC181#3228
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
2. #26041682C2semidirect
3. #26041675C2semidirect
4. #26041625C2semidirect
5. #26041625C2semidirect
6. #26041600C2semidirect
7. #26041579C2semidirect
8. #26041332C2semidirect
110. C181#3228semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260416822
#260416752
#260416252
#260416002
#260415792
#260413322
#532×C1812
#3228181