Finite Group Databasebeta

← #37508199

Subgroups — #37508199

Order 22912.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#26039111C211yesmaximal, characteristic
2#26038787C211yesmaximal, characteristic, complemented
3#26038641C211yesmaximal, characteristic, complemented
4#26038182C211yesmaximal, characteristic, complemented
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#475×C179C211yesmaximal, characteristic, complemented
8C2×#15668260—21no—
9C2×#15668246—21no—
10#19153110—21no—
11#19153110C2211yescharacteristic
12#19153088—21no—
13#19153088—22no—
14C4×D716—21no—
15C4×D716C2211yescharacteristic
16C4⋊(C179⋊C8)C2211yescharacteristic
17#19153037—21no—
18#19153033—22no—
19C4×C179⋊C8C2211yescharacteristic
20C4⋊Q8×C179C2211yescharacteristic
21C4×Q8×C179C2211yescharacteristic
22#106×C179—21no—
23C4⋊C8×C179C2211yescharacteristic
24#103×C179—21no—
25#15668260—42no—
26#15668260—42no—
27C2×D716—22no—
28C2×D716—22no—
29C2×D716—21no—
30C2×D716—21no—
31C2×D716D411yescharacteristic
32C2×D716D412yes—
33C2×C4×D179—22no—
34C2×C4×D179—21no—
35#15668246—42no—
36C2×C179⋊C8—21no—
37C2×C179⋊C8—21no—
38C4⋊Dic179D411yescharacteristic
39(C2×D179)⋊C4—22no—
40(C2×D179)⋊C4—21no—
41C2×Q8×C179—21no—
42C2×Q8×C179D411yescharacteristic
43C2×C8×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179D411yescharacteristic
48C42×C179—21no—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51C22×D179—22no—
52D716—21no—
53D716—21no—
54D716—42no—
55D716—44no—
56D716—41no—
57D716—41no—
58D716—41no—
59D716—41no—
60D716—82no—
61C2×Dic179—21no—
62C4×D179—44no—
63C4×D179—41no—
64C4×D179—41no—
65C179⋊C8—41no—
66C179⋊C8—41no—
67Q8×C179—21no—
68Q8×C179—41no—
69Q8×C179—41no—
70Q8×C179SD1612yescomplemented
71C2×C4×C179—21no—
72C2×C4×C179—21no—
73C2×C4×C179—41no—
74C2×C4×C179C2×D411yescharacteristic
75C2×C4×C179C2×D411yescharacteristic
76C2×C4×C179C2×D411yescharacteristic
77C8×C179—41no—
78C22×C179#12011yescharacteristic
79C2×D179—41no—
80C2×D179—42no—
81C2×D179—42no—
82C2×D179—41no—
83C2×D179—42no—
84C2×D179—41no—
85C4×C179—22no—
86C4×C179—21no—
87C4×C179—41no—
88C4×C179—81no—
89C4×C179C2×SD1611yescharacteristic
90C4×C179#13511yescharacteristic
91C4×C179AGL(1,ℤ/8)11yescharacteristic
92C4×C179AGL(1,ℤ/8)11yescharacteristic
93Dic179—41no—
94C2×C179#44911yescharacteristic
95C2×C179#45011yescharacteristic
96C2×C179#45511yescharacteristic
97D179—81no—
98D179—82no—
99C179#164211yescharacteristic, complemented, Sylow
100#1642—1791nomaximal, Sylow
101#550—1791no—
102#486—1791no—
103#475D17911yescharacteristic, complemented
104#461—1791no—
105#331—1791no—
106Q8⋊C8—1791no—
107D4⋊C8—1791no—
108C2×#41—3581no—
109C2×SD16—3581no—
110C4⋊Q8C2×D17911yescharacteristic
111#127—3581no—
112#127—1791no—
113#121—3582no—
114#121—3581no—
115C4×Q8C2×D17911yescharacteristic
116C4×D4—3581no—
117C4×D4—1791no—
118#106—21no—
119C4⋊C8—1791no—
120C4⋊C8C2×D17911yescharacteristic
121#103—21no—
122#103—3581no—
123#102—3582no—
124C4×C8—1791no—
125#41—7162no—
126#41—7162no—
127C2×Q8—21no—
128C2×Q8C22⋊D17911yescharacteristic
129C2×D4—3582no—
130C2×D4—3582no—
131C2×D4—3581no—
132C2×D4—3581no—
133C2×D4—1791no—
134C2×D4—1792no—
135C22×C4—3582no—
136C22×C4—3581no—
137SD16—7162no—
138C2×C8—21no—
139C2×C8—3581no—
140C2×C8—3581no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1791no—
145C4⋊C4C22⋊D17911yescharacteristic
146C22⋊C4—3582no—
147C22⋊C4—3581no—
148C42—21no—
149C42C22×D17911yescharacteristic
150C23—3581no—
151C23—3582no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1566824612yescomplemented
156D4—3581no—
157D4—3581no—
158D4—14322no—
159D4—7162no—
160D4—7164no—
161D4—7161no—
162D4—7161no—
163D4—7161no—
164D4—7161no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—3581no—
169C2×C4—7164no—
170C2×C4—7161no—
171C2×C4—7161no—
172C2×C4D4×D17911yescharacteristic
173C2×C4D4×D17911yescharacteristic
174C2×C4C2×C22⋊D17911yescharacteristic
175C8—41no—
176C8—7161no—
177C8—7161no—
178C22—7161no—
179C22—7162no—
180C22—7162no—
181C22—7161no—
182C22—7161no—
183C22—7162no—
184C22#1915308111yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—7161no—
190C4C2×#1566824611yescharacteristic
191C4#1915313411yescharacteristic
192C4#1915313811yescharacteristic
193C4#1915314011yescharacteristic
194C2—14321no—
195C2—14322no—
196C2#2603858311yescharacteristic
197C2#2603858911yescharacteristic
198C2#2603861511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#475×C179C2
socleC22×C179#120
prime radical#26039111C2
hypercenter#475D179
hypocenterC179#1642
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #26038787C2semidirect
3. #26038641C2semidirect
4. #26038182C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
6. #26038166C2semidirect
7. #475×C179C2semidirect
70. Q8×C179SD16semidirect
99. C179#1642semidirect
103. #475D179semidirect
155. Q8#15668246semidirect

Maximal subgroups

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

subgroupindex
#260391112
#260387872
#260386412
#260381822
Q8⋊(C179⋊C8)2
#260381662
#475×C1792
#1642179