Finite Group Databasebeta

← #37532794

Subgroups — #37532794

Order 23168.

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

normal subgroups only

This group has 236 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#26041990C211yesmaximal, characteristic, complemented
2#26041653C211yesmaximal, characteristic, complemented
3#26041480C211yesmaximal, characteristic, complemented
4#26041475C211yesmaximal, characteristic
5#26041009C211yesmaximal, characteristic, complemented
6#26041006C211yesmaximal, characteristic, complemented
7#333×C181C211yesmaximal, characteristic, complemented
8C2×#15668438—21no—
9C2×#15668436—22no—
10C2×D1448—21no—
11#19153813C2211yescharacteristic
12#19153807—21no—
13#19153796—21no—
14#19153792—22no—
15C4×Dic362C2211yescharacteristic
16C4×D724C2211yescharacteristic
17#19153778—21no—
18C4⋊C4×D181—21no—
19#19153738—21no—
20#19153738—21no—
21C4×#13089678C2211yescharacteristic
22C4⋊Q8×C181C2211yescharacteristic
23C4⋊C8×C181C2211yescharacteristic
24C4⋊C8×C181C2211yescharacteristic
25C2×Dic362D411yescharacteristic
26C2×D724—21no—
27C2×D724—21no—
28C2×D724—21no—
29C2×D724D411yescharacteristic
30C2×C4×D181—22no—
31C2×C4×D181—21no—
32#15668438—42no—
33#15668436—44no—
34D1448—42no—
35C2×#13089678—21no—
36C4⋊Dic181—22no—
37C4⋊Dic181D411yescharacteristic
38C4⋊Dic181D411yescharacteristic
39#15668428—22no—
40#15668428—21no—
41#15668426—22no—
42#15668426—21no—
43C4×Dic181—21no—
44C2×Q8×C181D412yes—
45C2×C8×C181—21no—
46C2×C8×C181—21no—
47C4⋊C4×C181—21no—
48C4⋊C4×C181—21no—
49C42×C181C2311yescharacteristic
50C22×D181—21no—
51C22×D181—41no—
52Dic362—21no—
53Dic362—21no—
54D724—21no—
55D724—42no—
56D724—41no—
57D724—41no—
58D724—41no—
59D724—41no—
60D724SD1612yescomplemented
61C2×Dic181—21no—
62C2×Dic181—21no—
63C2×Dic181—21no—
64C4×D181—41no—
65C4×D181—41no—
66C4×D181—82no—
67#13089678—22no—
68Q8×C181—42no—
69C2×C4×C181—22no—
70C2×C4×C181C2×D411yescharacteristic
71C2×C4×C181C2×D411yescharacteristic
72C2×C4×C181C2×D411yescharacteristic
73C8×C181—41no—
74C8×C181—41no—
75C22×C181#12011yescharacteristic
76C2×D181—22no—
77C2×D181—41no—
78C2×D181—41no—
79C2×D181—81no—
80C2×D181—81no—
81C2×D181—81no—
82C4×C181—21no—
83C4×C181—42no—
84C4×C181C2×SD1611yescharacteristic
85C4×C181#13511yescharacteristic
86C4×C181AGL(1,ℤ/8)11yescharacteristic
87C4×C181#13711yescharacteristic
88Dic181—41no—
89Dic181—41no—
90Dic181—41no—
91C2×C181#44811yescharacteristic
92C2×C181#44911yescharacteristic
93C2×C181#45311yescharacteristic
94D181—42no—
95D181—161no—
96C181#162211yescharacteristic, complemented, Sylow
97#1622—1811nomaximal, Sylow
98#554—1811no—
99#492—1811no—
100#460—1811no—
101#459—1811no—
102#333D18111yescharacteristic, complemented
103Q8⋊C8—1811no—
104D4⋊C8—1811no—
105C2×SD16—3621no—
106C2×SD16—3622no—
107C2×D8—3621no—
108C4⋊Q8C2×D18111yescharacteristic
109#127—1811no—
110#125—3621no—
111C22⋊Q8—3621no—
112C22⋊Q8—3622no—
113C4×Q8—1811no—
114C4×D4—3621no—
115C4×D4—1811no—
116C2×C4⋊C4—3621no—
117C4⋊C8C2×D18111yescharacteristic
118C4⋊C8C2×D18111yescharacteristic
119#102—3621no—
120#102—3621no—
121C4×C8—1811no—
122C2×Q8—1811no—
123C2×Q8D72412yes—
124C2×D4—3621no—
125C2×D4—3621no—
126C2×D4—3621no—
127C2×D4—1811no—
128C22×C4—3622no—
129C22×C4—3621no—
130SD16—7242no—
131SD16—7244no—
132D8—7242no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3621no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—3622no—
139C4⋊C4—3622no—
140C4⋊C4—3621no—
141C4⋊C4—1811no—
142C4⋊C4—1811no—
143C22⋊C4—3622no—
144C22⋊C4—3621no—
145C42—3621no—
146C42C22×D18111yescharacteristic
147C23—3621no—
148C23—7241no—
149Q8—42no—
150Q8—3621no—
151Q8—3621no—
152D4—3621no—
153D4—1812no—
154D4—7242no—
155D4—7241no—
156D4—7241no—
157D4—7241no—
158D4—7241no—
159C2×C4—22no—
160C2×C4—3621no—
161C2×C4—3621no—
162C2×C4—3621no—
163C2×C4—14482no—
164C2×C4—7241no—
165C2×C4—7241no—
166C2×C4D4×D18111yescharacteristic
167C2×C4D4×D18111yescharacteristic
168C2×C4C2×D72411yescharacteristic
169C8—41no—
170C8—41no—
171C8—3622no—
172C22—3622no—
173C22—14481no—
174C22—14481no—
175C22—14481no—
176C22—7241no—
177C22—7241no—
178C22#1915378611yescharacteristic
179C4—21no—
180C4—42no—
181C4—7241no—
182C4—7241no—
183C4—7241no—
184C4SD16×D18111yescharacteristic
185C4#1915383911yescharacteristic
186C4#1915384311yescharacteristic
187C4#1915384811yescharacteristic
188C2—28961no—
189C2—7242no—
190C2#2604141911yescharacteristic
191C2#2604142711yescharacteristic
192C2#2604144911yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#333×C181C2
socleC22×C181#120
prime radical#26041475C2
hypercenter#333D181
hypocenterC181#1622
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #26041990C2semidirect
2. #26041653C2semidirect
3. #26041480C2semidirect
5. #26041009C2semidirect
6. #26041006C2semidirect
7. #333×C181C2semidirect
60. D724SD16semidirect
96. C181#1622semidirect
102. #333D181semidirect

Maximal subgroups

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

subgroupindex
#260419902
#260416532
#260414802
#260414752
#260410092
#260410062
#333×C1812
#1622181