Finite Group Databasebeta

← #37508201

Subgroups — #37508201

Order 22912.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039109C211yesmaximal, characteristic
2#26038785C211yesmaximal, characteristic, complemented
3#26038725C211yesmaximal, characteristic, complemented
4#26038645C211yesmaximal, characteristic, complemented
5#26038169C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#331×C179C211yesmaximal, characteristic, complemented
8C2×#15668258—21no—
9C2×#15668248—21no—
10#19153113C2211yescharacteristic
11#19153109—21no—
12#19153085—22no—
13#19153084—21no—
14C4×Dic358C2211yescharacteristic
15C4×D716C2211yescharacteristic
16D4×Dic179—21no—
17#19153045—21no—
18#19153038—21no—
19#19153038—21no—
20#19153034—22no—
21C4×C179⋊C8C2211yescharacteristic
22#127×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25#15668258—42no—
26#15668258—42no—
27C2×Dic358—21no—
28C2×Dic358D411yescharacteristic
29C2×D716D411yescharacteristic
30C2×C22⋊D179—22no—
31C2×C22⋊D179—22no—
32C22×Dic179—22no—
33C2×C4×D179—21no—
34#15668248—42no—
35C2×C179⋊C8—21no—
36C4⋊Dic179—21no—
37C4⋊Dic179—21no—
38C4⋊Dic179D411yescharacteristic
39C4⋊Dic179D411yescharacteristic
40Dic179⋊C4—21no—
41C22⋊Dic179—22no—
42(C2×D179)⋊C4—21no—
43C4×Dic179—21no—
44C2×D4×C179—21no—
45C2×D4×C179—21no—
46C2×D4×C179D412yes—
47C2×C8×C179—21no—
48C2×C8×C179—21no—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51Dic358—21no—
52Dic358—41no—
53Dic358—41no—
54Dic358SD1612yescomplemented
55D716—21no—
56D716—21no—
57C22⋊D179—44no—
58C22⋊D179—82no—
59C2×Dic179—21no—
60C2×Dic179—21no—
61C2×Dic179—21no—
62C2×Dic179—44no—
63C2×Dic179—41no—
64C4×D179—41no—
65C4×D179—41no—
66C179⋊C8—41no—
67C23×C179—22no—
68D4×C179—42no—
69D4×C179—41no—
70D4×C179—41no—
71D4×C179—41no—
72D4×C179—41no—
73C2×C4×C179C2×D411yescharacteristic
74C2×C4×C179C2×D411yescharacteristic
75C2×C4×C179C2×D411yescharacteristic
76C8×C179—41no—
77C8×C179—41no—
78C22×C179—42no—
79C22×C179—42no—
80C22×C179—42no—
81C22×C179#12011yescharacteristic
82C2×D179—41no—
83C2×D179—41no—
84C2×D179—41no—
85C4×C179—21no—
86C4×C179C2×SD1611yescharacteristic
87C4×C179#13511yescharacteristic
88C4×C179AGL(1,ℤ/8)11yescharacteristic
89C4×C179AGL(1,ℤ/8)11yescharacteristic
90Dic179—22no—
91Dic179—41no—
92Dic179—41no—
93Dic179—81no—
94C2×C179—82no—
95C2×C179#44911yescharacteristic
96C2×C179#45011yescharacteristic
97C2×C179#45511yescharacteristic
98D179—81no—
99C179#164211yescharacteristic, complemented, Sylow
100#1642—1791nomaximal, Sylow
101#550—1791no—
102#486—1791no—
103#475—1791no—
104#461—1791no—
105#331D17911yescharacteristic, complemented
106Q8⋊C8—1791no—
107D4⋊C8—1791no—
108C2×#41—3581no—
109C2×SD16—3581no—
110C4⋊Q8—1791no—
111#127—3581no—
112#127C2×D17911yescharacteristic
113#121—3582no—
114#121—3581no—
115C4×Q8—1791no—
116C4×D4—3581no—
117C4×D4—1791no—
118#106—3581no—
119C4⋊C8C2×D17911yescharacteristic
120C4⋊C8C2×D17911yescharacteristic
121#103—3581no—
122#103—3581no—
123#102—3582no—
124C4×C8—1791no—
125#41—7162no—
126#41—7162no—
127C2×Q8—3581no—
128C2×Q8—1791no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3582no—
132C2×D4—3582no—
133C2×D4—1791no—
134C2×D4D71612yes—
135C22×C4—3582no—
136C22×C4—3581no—
137SD16—7162no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3581no—
141C4⋊C4—3581no—
142C4⋊C4—3581no—
143C4⋊C4—3581no—
144C4⋊C4—1791no—
145C4⋊C4—1791no—
146C22⋊C4—3582no—
147C22⋊C4—3581no—
148C42—3581no—
149C42C22×D17911yescharacteristic
150C23—22no—
151C23—3581no—
152Q8—3581no—
153Q8—1792no—
154Q8—7161no—
155Q8—7161no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—3581no—
162D4—3581no—
163D4—14322no—
164D4—7164no—
165C2×C4—3581no—
166C2×C4—3581no—
167C2×C4—3581no—
168C2×C4—7164no—
169C2×C4—7161no—
170C2×C4—7161no—
171C2×C4—7161no—
172C2×C4D4×D17911yescharacteristic
173C2×C4D4×D17911yescharacteristic
174C2×C4C2×D71611yescharacteristic
175C8—41no—
176C8—41no—
177C8—7161no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—7161no—
182C22—7161no—
183C22—7161no—
184C22#1915308311yescharacteristic
185C4—21no—
186C4—3582no—
187C4—14321no—
188C4—7161no—
189C4—7161no—
190C4SD16×D17911yescharacteristic
191C4#1915313411yescharacteristic
192C4#1915313711yescharacteristic
193C4#1915313711yescharacteristic
194C2—82no—
195C2—14321no—
196C2#2603858511yescharacteristic
197C2#2603859111yescharacteristic
198C2#2603861911yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#331×C179C2
socleC22×C179#120
prime radical#26039109C2
hypercenter#331D179
hypocenterC179#1642
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26038785C2semidirect
3. #26038725C2semidirect
4. #26038645C2semidirect
5. #26038169C2semidirect
6. #26038166C2semidirect
7. #331×C179C2semidirect
54. Dic358SD16semidirect
99. C179#1642semidirect
105. #331D179semidirect

Maximal subgroups

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

subgroupindex
#260391092
#260387852
#260387252
#260386452
#260381692
#260381662
#331×C1792
#1642179