Finite Group Databasebeta

← #37533039

Subgroups — #37533039

Order 23168.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#26042023C211yesmaximal, characteristic
2#26041688C211yesmaximal, characteristic, complemented
3#26041674C211yesmaximal, characteristic, complemented
4#26041639C211yesmaximal, characteristic, complemented
5#26041020C211yesmaximal, characteristic, complemented
6#26041018C211yesmaximal, characteristic, complemented
7#329×C181C211yesmaximal, characteristic, complemented
8C2×#15668439—21no—
9C2×#15668437—21no—
10#19153816C2211yescharacteristic
11#19153808C2211yescharacteristic
12#19153804C2211yescharacteristic
13#19153800—21no—
14#19153798—21no—
15#19153793—21no—
16#19153792—21no—
17#19153789—21no—
18#19153788—21no—
19#19153750—21no—
20#19153748—21no—
21#19153741—21no—
22#19153738—21no—
23#19153727C2211yescharacteristic
24#124×C181C2211yescharacteristic
25#97×C181C2211yescharacteristic
26#97×C181C2211yescharacteristic
27C2×Dic362—21no—
28C2×Dic362D411yescharacteristic
29C2×D724D411yescharacteristic
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C22×Dic181—21no—
33C2×C4×D181—21no—
34#15668439—42no—
35#15668437—42no—
36C2×#13089678—21no—
37C4⋊Dic181—21no—
38C4⋊Dic181—21no—
39C4⋊Dic181—21no—
40C4⋊Dic181D411yescharacteristic
41C4⋊Dic181D411yescharacteristic
42#15668428—21no—
43#15668428—21no—
44#15668428—21no—
45C22⋊Dic181—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—21no—
49C2×Q8×C181D411yescharacteristic
50C2×D4×C181D411yescharacteristic
51C2×C8×C181—21no—
52C2×C8×C181—21no—
53C22⋊C4×C181—21no—
54C22⋊C4×C181—21no—
55C42×C181C2311yescharacteristic
56C22×D181—21no—
57Dic362—41no—
58Dic362—41no—
59Dic362—41no—
60D724—41no—
61C22⋊D181—81no—
62C22⋊D181—81no—
63C2×Dic181—21no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—41no—
67C2×Dic181—81no—
68C4×D181—81no—
69#13089678—22no—
70C23×C181—21no—
71Q8×C181—41no—
72D4×C181—41no—
73C2×C4×C181—21no—
74C2×C4×C181C2×D411yescharacteristic
75C2×C4×C181C2×D411yescharacteristic
76C2×C4×C181C2×D411yescharacteristic
77C8×C181—22no—
78C8×C181—41no—
79C22×C181—41no—
80C22×C181—41no—
81C22×C181—41no—
82C22×C181#12011yescharacteristic
83C2×D181—41no—
84C2×D181—41no—
85C2×D181—41no—
86C4×C181—21no—
87C4×C181—21no—
88C4×C181—21no—
89C4×C181—41no—
90Dic181—41no—
91Dic181—41no—
92Dic181—41no—
93Dic181—81no—
94C2×C181—81no—
95C2×C181#45311yescharacteristic
96C2×C181#45511yescharacteristic
97C2×C181#45611yescharacteristic
98D181—81no—
99C181#166111yescharacteristic, complemented, Sylow
100#1661—1811nomaximal, Sylow
101#562—1811no—
102#501—1811no—
103#497—1811no—
104#489—1811no—
105#330—1811no—
106#329—1811no—
107#329D18111yescharacteristic, complemented
108C2×Q16—3621no—
109C2×SD16—3621no—
110C4⋊Q8—1811no—
111#125—1811no—
112#124—1811no—
113#124C2×D18111yescharacteristic
114#123—3621no—
115#123—3621no—
116C22⋊Q8—3621no—
117C22⋊Q8—3621no—
118#121—3621no—
119#121—3621no—
120#107—3621no—
121#106—3621no—
122#103—3621no—
123#102—3621no—
124#97—1811no—
125#97C2×D18111yescharacteristic
126#97C2×D18111yescharacteristic
127C2×Q8—3621no—
128C2×Q8—1811no—
129C2×Q8D72411yescharacteristic
130C2×D4—3621no—
131C2×D4—3621no—
132C2×D4—1811no—
133C2×D4D72411yescharacteristic
134C22×C4—3621no—
135C22×C4—3621no—
136Q16—7242no—
137SD16—7242no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3621no—
141C4⋊C4—3621no—
142C4⋊C4—3621no—
143C4⋊C4—3621no—
144C4⋊C4—3621no—
145C4⋊C4—3621no—
146C4⋊C4—3621no—
147C4⋊C4—1811no—
148C4⋊C4—1811no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—3621no—
152C22⋊C4—3621no—
153C22⋊C4—3621no—
154C22⋊C4—3621no—
155C42C22×D18111yescharacteristic
156C23—21no—
157C23—3621no—
158Q8—41no—
159Q8—7241no—
160Q8—7241no—
161Q8—7241no—
162D4—41no—
163D4—14481no—
164D4—14481no—
165D4—7241no—
166C2×C4—21no—
167C2×C4—3621no—
168C2×C4—3621no—
169C2×C4—3621no—
170C2×C4—14481no—
171C2×C4—14481no—
172C2×C4—7241no—
173C2×C4D4×D18111yescharacteristic
174C2×C4D4×D18111yescharacteristic
175C2×C4C2×D72411yescharacteristic
176C8—22no—
177C8—41no—
178C8—3622no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—7241no—
183C22—7241no—
184C22—7241no—
185C22#1915378611yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—14481no—
191C4—7241no—
192C4—7241no—
193C4—7241no—
194C2—81no—
195C2—14481no—
196C2#2604144911yescharacteristic
197C2#2604145811yescharacteristic
198C2#2604146311yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#329×C181C2
socleC22×C181#120
prime radical#26042023C2
hypercenter#329D181
hypocenterC181#1661
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26041688C2semidirect
3. #26041674C2semidirect
4. #26041639C2semidirect
5. #26041020C2semidirect
6. #26041018C2semidirect
7. #329×C181C2semidirect
99. C181#1661semidirect
107. #329D181semidirect

Maximal subgroups

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

subgroupindex
#260420232
#260416882
#260416742
#260416392
#260410202
#260410182
#329×C1812
#1661181