Finite Group Databasebeta

← #37510067

Subgroups — #37510067

Order 22912.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#26039239C211yesmaximal, characteristic, complemented
2#26038950C212yesmaximal, complemented
3#26038455C212yesmaximal, complemented
4#26038196C211yesmaximal, characteristic, complemented
5#407×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2212yes—
7#19153146—22no—
8#19153141—22no—
9#19153135—22no—
10#19153096—22no—
11#19153092—22no—
12#19153087—22no—
13#19153082—22no—
14C4×C22⋊D179—24no—
15#19153072C2212yes—
16#19153039—24no—
17C23×C4×C179C2211yescharacteristic
18C2×M16×C179C2211yescharacteristic
19C2×M16×C179C2211yescharacteristic
20C22⋊C8×C179—22no—
21#15668259—22no—
22#15668259—22no—
23#15668259—22no—
24#15668259—22no—
25C2×Dic358D412yescomplemented
26C2×D716D412yescomplemented
27C2×C22⋊D179—24no—
28C2×C22⋊D179—22no—
29C2×C4×D179—22no—
30Dic716—22no—
31#15668248—24no—
32D1432—22no—
33C4⋊Dic179—22no—
34Dic179⋊C4—24no—
35C22⋊Dic179—24no—
36C22⋊Dic179—22no—
37(C2×D179)⋊C4—24no—
38C4×Dic179D414yescomplemented
39C24×C179D411yescharacteristic
40C22×C4×C179—21no—
41C22×C4×C179—22no—
42C22×C4×C179—22no—
43C22×C4×C179—21no—
44C22×C4×C179D411yescharacteristic
45C22×C4×C179C2311yescharacteristic
46M16×C179—22no—
47M16×C179—22no—
48C2×C8×C179—21no—
49C2×C8×C179D412yescomplemented
50C22×D179—22no—
51Dic358—22no—
52Dic358—22no—
53D716—22no—
54D716—22no—
55C22⋊D179—48no—
56C22⋊D179—42no—
57C22⋊D179—42no—
58C2×Dic179—24no—
59C2×Dic179—22no—
60C4×D179—42no—
61C4×D179—42no—
62C23×C179—22no—
63C23×C179—21no—
64C23×C179—42no—
65C23×C179C2×D411yescharacteristic
66C2×C4×C179—22no—
67C2×C4×C179—21no—
68C2×C4×C179—41no—
69C2×C4×C179—42no—
70C2×C4×C179—41no—
71C2×C4×C179C2×D412yes—
72C2×C4×C179C2×D412yes—
73C2×C4×C179C2×D411yescharacteristic
74C2×C4×C179#4111yescharacteristic
75C8×C179—22no—
76C8×C179—41no—
77C22×C179—22no—
78C22×C179—24no—
79C22×C179—41no—
80C22×C179—42no—
81C22×C179—41no—
82C22×C179—41no—
83C22×C179#12012yes—
84C22×C179#12111yescharacteristic
85C2×D179—44no—
86C2×D179—42no—
87C4×C179—41no—
88C4×C179#12011yescharacteristic
89C4×C179#12112yes—
90C4×C179#12711yescharacteristic
91Dic179—44no—
92Dic179—42no—
93C2×C179—22no—
94C2×C179—21no—
95C2×C179—42no—
96C2×C179#39211yescharacteristic
97D179—82no—
98C179#201011yescharacteristic, complemented, Sylow
99#2010—1791nomaximal, Sylow
100#575—1791no—
101#525—1792no—
102#421—1792no—
103#407D17911yescharacteristic, complemented
104H(3,ℤ/4)—1791no—
105C2×#41—1792no—
106C23×C4C2×D17911yescharacteristic
107#137—3582no—
108AGL(1,ℤ/8)—3582no—
109#135—3582no—
110C2×M16C2×D17911yescharacteristic
111C2×M16C2×D17911yescharacteristic
112#123—3582no—
113C22⋊Q8—3582no—
114#121—3582no—
115#120—3582no—
116C4×D4—3584no—
117#117—1792no—
118#104—3584no—
119C22⋊C8—22no—
120C24D71611yescharacteristic
121#41—3582no—
122#41—3582no—
123#41—3582no—
124#41—3582no—
125C2×Q8—1792no—
126C2×D4—3584no—
127C2×D4—3582no—
128C2×D4—1792no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—3582no—
134C22×C4D71611yescharacteristic
135C22×C4C22×D17911yescharacteristic
136Q16—3582no—
137SD16—3584no—
138D8—3582no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8C22⋊D17912yescomplemented
143C4⋊C4—3584no—
144C4⋊C4—3582no—
145C22⋊C4—3584no—
146C22⋊C4—3584no—
147C22⋊C4—3582no—
148C42—1794no—
149C23—22no—
150C23—21no—
151C23—42no—
152C23—3582no—
153C23C2×D71611yescharacteristic
154Q8—3582no—
155Q8—3582no—
156D4—3582no—
157D4—3582no—
158D4—7168no—
159D4—7162no—
160D4—7162no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—3584no—
167C2×C4—3582no—
168C2×C4—7162no—
169C2×C4—7162no—
170C2×C4D4×D17912yes—
171C2×C4D4×D17912yes—
172C2×C4C2×C22⋊D17911yescharacteristic
173C2×C4#1566825911yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—7164no—
183C22—7162no—
184C22#1915308312yes—
185C22#1915308711yescharacteristic
186C4—41no—
187C4—7164no—
188C4—7162no—
189C4#1915308111yescharacteristic
190C4#1915308612yes—
191C4#1915310911yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—14322no—
196C2#2603834811yescharacteristic
subgroupG/H
centerC4#19153109
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#407×C179C2
socleC2×C179#392
hypercenter#407D179
hypocenterC179#2010
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #26039239C2semidirect
2. #26038950C2semidirect
3. #26038455C2semidirect
4. #26038196C2semidirect
5. #407×C179C2semidirect
25. C2×Dic358D4semidirect
26. C2×D716D4semidirect
38. C4×Dic179D4semidirect
49. C2×C8×C179D4semidirect
98. C179#2010semidirect
103. #407D179semidirect
142. C2×C8C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260392392
#260389502
#260384552
#260381962
#407×C1792
#2010179