Finite Group Databasebeta

← #37508109

Subgroups — #37508109

Order 22912.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#26039162C211yesmaximal, characteristic, complemented
2#26038785C211yesmaximal, characteristic, complemented
3#26038631C211yesmaximal, characteristic
4#26038184C211yesmaximal, characteristic, complemented
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#460×C179C211yesmaximal, characteristic, complemented
8C2×#15668244—21no—
9#19153113C2211yescharacteristic
10#19153108—22no—
11#19153101—21no—
12#19153089—21no—
13C4×Dic358—21no—
14C4×D716C2211yescharacteristic
15#19153071—21no—
16C4⋊(C179⋊C8)C2211yescharacteristic
17#19153034—22no—
18#19153034—21no—
19C4×C179⋊C8C2211yescharacteristic
20C2×SD16×C179—21no—
21#127×C179C2211yescharacteristic
22C4×Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24#102×C179—21no—
25C2×Dic358—21no—
26C2×D716D411yescharacteristic
27C2×C4×D179—21no—
28#15668244—42no—
29C2×C179⋊C8—21no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—21no—
32C4⋊Dic179D411yescharacteristic
33C4⋊Dic179D412yes—
34Dic179⋊C4—22no—
35(C2×D179)⋊C4—22no—
36(C2×D179)⋊C4—22no—
37(C2×D179)⋊C4—21no—
38C4×Dic179—22no—
39C2×Q8×C179D411yescharacteristic
40C2×D4×C179—21no—
41C2×D4×C179—21no—
42C2×D4×C179—21no—
43SD16×C179—42no—
44C2×C8×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179D411yescharacteristic
47C42×C179—21no—
48C42×C179C2311yescharacteristic
49C22×D179—21no—
50Dic358—41no—
51Dic358—41no—
52D716—21no—
53D716—21no—
54C2×Dic179—21no—
55C2×Dic179—22no—
56C4×D179—41no—
57C4×D179—41no—
58C179⋊C8—41no—
59C179⋊C8—41no—
60C23×C179—41no—
61Q8×C179—21no—
62Q8×C179—21no—
63D4×C179—42no—
64D4×C179—41no—
65D4×C179—41no—
66D4×C179—41no—
67D4×C179—41no—
68C2×C4×C179—21no—
69C2×C4×C179—21no—
70C2×C4×C179C2×D411yescharacteristic
71C2×C4×C179C2×D411yescharacteristic
72C2×C4×C179C2×D411yescharacteristic
73C8×C179—41no—
74C22×C179—81no—
75C22×C179—81no—
76C22×C179—81no—
77C22×C179#12011yescharacteristic
78C2×D179—41no—
79C2×D179—41no—
80C2×D179—41no—
81C4×C179—21no—
82C4×C179—41no—
83C4×C179—41no—
84C4×C179#13511yescharacteristic
85C4×C179#13511yescharacteristic
86C4×C179AGL(1,ℤ/8)11yescharacteristic
87C4×C179AGL(1,ℤ/8)11yescharacteristic
88Dic179—41no—
89Dic179—42no—
90C2×C179—161no—
91C2×C179#44911yescharacteristic
92C2×C179#44911yescharacteristic
93C2×C179#45411yescharacteristic
94D179—81no—
95C179#162811yescharacteristic, complemented, Sylow
96#1628—1791nomaximal, Sylow
97#556—1791no—
98#486—1791no—
99#460D17911yescharacteristic, complemented
100#459—1791no—
101#332—1791no—
102Q8⋊C8—1791no—
103D4⋊C8—1791no—
104C2×SD16—21no—
105C2×D8—3581no—
106C4⋊Q8—1791no—
107#127C2×D17911yescharacteristic
108#126—3582no—
109#124—3581no—
110C22⋊Q8—3581no—
111C4×Q8—3581no—
112C4×Q8C2×D17911yescharacteristic
113C4×D4—1791no—
114#117—3581no—
115C4⋊C8—1791no—
116C4⋊C8C2×D17911yescharacteristic
117#102—21no—
118#102—3581no—
119#102—3582no—
120C4×C8—1791no—
121C2×Q8—3581no—
122C2×Q8C22⋊D17911yescharacteristic
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1791no—
127C22×C4—3581no—
128SD16—42no—
129D8—7162no—
130C2×C8—21no—
131C2×C8—3581no—
132C2×C8—3581no—
133C4⋊C4—21no—
134C4⋊C4—3582no—
135C4⋊C4—3581no—
136C4⋊C4—1791no—
137C4⋊C4—1792no—
138C4⋊C4C22⋊D17911yescharacteristic
139C22⋊C4—3582no—
140C22⋊C4—3582no—
141C22⋊C4—3581no—
142C42—21no—
143C42—3582no—
144C42C22×D17911yescharacteristic
145C23—41no—
146C23—3581no—
147Q8—21no—
148Q8—21no—
149Q8—7161no—
150Q8—7161no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3581no—
157D4—3581no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—3581no—
161C2×C4—3582no—
162C2×C4—7161no—
163C2×C4—7161no—
164C2×C4D4×D17911yescharacteristic
165C2×C4D4×D17911yescharacteristic
166C2×C4C2×C22⋊D17911yescharacteristic
167C8—41no—
168C8—7161no—
169C8—7161no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—7161no—
174C22—7161no—
175C22—7161no—
176C22#1915308111yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—7161no—
181C4—7162no—
182C4#1915313311yescharacteristic
183C4#1915313411yescharacteristic
184C4#1915313911yescharacteristic
185C4#1915314011yescharacteristic
186C2—161no—
187C2—14321no—
188C2#2603858211yescharacteristic
189C2#2603858311yescharacteristic
190C2#2603861011yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#460×C179C2
socleC22×C179#120
prime radical#26038631C2
hypercenter#460D179
hypocenterC179#1628
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #26039162C2semidirect
2. #26038785C2semidirect
4. #26038184C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
6. #26038166C2semidirect
7. #460×C179C2semidirect
95. C179#1628semidirect
99. #460D179semidirect

Maximal subgroups

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

subgroupindex
#260391622
#260387852
#260386312
#260381842
Q8⋊(C179⋊C8)2
#260381662
#460×C1792
#1628179