Finite Group Databasebeta

← #37508318

Subgroups — #37508318

Order 22912.

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#26039183C211yesmaximal, characteristic
2#26038848C211yesmaximal, characteristic, complemented
3#26038834C211yesmaximal, characteristic, complemented
4#26038799C211yesmaximal, characteristic, complemented
5#26038180C211yesmaximal, characteristic, complemented
6#26038178C211yesmaximal, characteristic, complemented
7#329×C179C211yesmaximal, characteristic, complemented
8C2×#15668249—21no—
9C2×#15668247—21no—
10#19153113C2211yescharacteristic
11#19153105C2211yescharacteristic
12#19153101C2211yescharacteristic
13#19153097—21no—
14#19153095—21no—
15#19153090—21no—
16#19153089—21no—
17#19153086—21no—
18#19153085—21no—
19#19153047—21no—
20#19153045—21no—
21#19153038—21no—
22#19153035—21no—
23#19153024C2211yescharacteristic
24#124×C179C2211yescharacteristic
25#97×C179C2211yescharacteristic
26#97×C179C2211yescharacteristic
27C2×Dic358—21no—
28C2×Dic358D411yescharacteristic
29C2×D716D411yescharacteristic
30C2×C22⋊D179—21no—
31C2×C22⋊D179—21no—
32C22×Dic179—21no—
33C2×C4×D179—21no—
34#15668249—42no—
35#15668247—42no—
36C2×C179⋊C8—21no—
37C4⋊Dic179—21no—
38C4⋊Dic179—21no—
39C4⋊Dic179—21no—
40C4⋊Dic179D411yescharacteristic
41C4⋊Dic179D411yescharacteristic
42Dic179⋊C4—21no—
43Dic179⋊C4—21no—
44Dic179⋊C4—21no—
45C22⋊Dic179—21no—
46(C2×D179)⋊C4—21no—
47(C2×D179)⋊C4—21no—
48(C2×D179)⋊C4—21no—
49C2×Q8×C179D411yescharacteristic
50C2×D4×C179D411yescharacteristic
51C2×C8×C179—21no—
52C2×C8×C179—21no—
53C22⋊C4×C179—21no—
54C22⋊C4×C179—21no—
55C42×C179C2311yescharacteristic
56C22×D179—21no—
57Dic358—41no—
58Dic358—41no—
59Dic358—41no—
60D716—41no—
61C22⋊D179—81no—
62C22⋊D179—81no—
63C2×Dic179—21no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—41no—
67C2×Dic179—81no—
68C4×D179—81no—
69C179⋊C8—22no—
70C23×C179—21no—
71Q8×C179—41no—
72D4×C179—41no—
73C2×C4×C179—21no—
74C2×C4×C179C2×D411yescharacteristic
75C2×C4×C179C2×D411yescharacteristic
76C2×C4×C179C2×D411yescharacteristic
77C8×C179—22no—
78C8×C179—41no—
79C22×C179—41no—
80C22×C179—41no—
81C22×C179—41no—
82C22×C179#12011yescharacteristic
83C2×D179—41no—
84C2×D179—41no—
85C2×D179—41no—
86C4×C179—21no—
87C4×C179—21no—
88C4×C179—21no—
89C4×C179—41no—
90Dic179—41no—
91Dic179—41no—
92Dic179—41no—
93Dic179—81no—
94C2×C179—81no—
95C2×C179#45311yescharacteristic
96C2×C179#45511yescharacteristic
97C2×C179#45611yescharacteristic
98D179—81no—
99C179#166111yescharacteristic, complemented, Sylow
100#1661—1791nomaximal, Sylow
101#562—1791no—
102#501—1791no—
103#497—1791no—
104#489—1791no—
105#330—1791no—
106#329—1791no—
107#329D17911yescharacteristic, complemented
108C2×Q16—3581no—
109C2×SD16—3581no—
110C4⋊Q8—1791no—
111#125—1791no—
112#124—1791no—
113#124C2×D17911yescharacteristic
114#123—3581no—
115#123—3581no—
116C22⋊Q8—3581no—
117C22⋊Q8—3581no—
118#121—3581no—
119#121—3581no—
120#107—3581no—
121#106—3581no—
122#103—3581no—
123#102—3581no—
124#97—1791no—
125#97C2×D17911yescharacteristic
126#97C2×D17911yescharacteristic
127C2×Q8—3581no—
128C2×Q8—1791no—
129C2×Q8D71611yescharacteristic
130C2×D4—3581no—
131C2×D4—3581no—
132C2×D4—1791no—
133C2×D4D71611yescharacteristic
134C22×C4—3581no—
135C22×C4—3581no—
136Q16—7162no—
137SD16—7162no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3581no—
141C4⋊C4—3581no—
142C4⋊C4—3581no—
143C4⋊C4—3581no—
144C4⋊C4—3581no—
145C4⋊C4—3581no—
146C4⋊C4—3581no—
147C4⋊C4—1791no—
148C4⋊C4—1791no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—3581no—
152C22⋊C4—3581no—
153C22⋊C4—3581no—
154C22⋊C4—3581no—
155C42C22×D17911yescharacteristic
156C23—21no—
157C23—3581no—
158Q8—41no—
159Q8—7161no—
160Q8—7161no—
161Q8—7161no—
162D4—41no—
163D4—14321no—
164D4—14321no—
165D4—7161no—
166C2×C4—21no—
167C2×C4—3581no—
168C2×C4—3581no—
169C2×C4—3581no—
170C2×C4—14321no—
171C2×C4—14321no—
172C2×C4—7161no—
173C2×C4D4×D17911yescharacteristic
174C2×C4D4×D17911yescharacteristic
175C2×C4C2×D71611yescharacteristic
176C8—22no—
177C8—41no—
178C8—3582no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—7161no—
183C22—7161no—
184C22—7161no—
185C22#1915308311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—14321no—
191C4—7161no—
192C4—7161no—
193C4—7161no—
194C2—81no—
195C2—14321no—
196C2#2603860911yescharacteristic
197C2#2603861811yescharacteristic
198C2#2603862311yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#329×C179C2
socleC22×C179#120
prime radical#26039183C2
hypercenter#329D179
hypocenterC179#1661
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26038848C2semidirect
3. #26038834C2semidirect
4. #26038799C2semidirect
5. #26038180C2semidirect
6. #26038178C2semidirect
7. #329×C179C2semidirect
99. C179#1661semidirect
107. #329D179semidirect

Maximal subgroups

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

subgroupindex
#260391832
#260388482
#260388342
#260387992
#260381802
#260381782
#329×C1792
#1661179