Finite Group Databasebeta

← #37508227

Subgroups — #37508227

Order 22912.

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

normal subgroups only

This group has 212 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#26039154C211yesmaximal, characteristic
2#26038791C211yesmaximal, characteristic, complemented
3#26038641C211yesmaximal, characteristic, complemented
4#26038187C211yesmaximal, characteristic, complemented
5D4⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#474×C179C211yesmaximal, characteristic, complemented
8C2×#15668246—21no—
9#19153113C2211yescharacteristic
10#19153099—22no—
11#19153087—21no—
12#19153084—21no—
13C4×D716C2211yescharacteristic
14#19153072—21no—
15#19153071—21no—
16C4⋊(C179⋊C8)C2211yescharacteristic
17#19153037—21no—
18#19153036—22no—
19C4×C179⋊C8C2211yescharacteristic
20C4⋊Q8×C179C2211yescharacteristic
21C4×D4×C179C2211yescharacteristic
22#107×C179—21no—
23C4⋊C8×C179C2211yescharacteristic
24#102×C179—21no—
25C2×Dic358D412yes—
26C2×D716D411yescharacteristic
27C2×C22⋊D179—22no—
28C2×C4×D179—21no—
29#15668246—42no—
30C2×C179⋊C8—21no—
31C2×C179⋊C8—21no—
32C4⋊Dic179—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179D411yescharacteristic
35C22⋊Dic179—22no—
36(C2×D179)⋊C4—22no—
37(C2×D179)⋊C4—21no—
38C4×Dic179—22no—
39C2×Q8×C179—21no—
40C2×D4×C179D411yescharacteristic
41C22×C4×C179—21no—
42C2×C8×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179D411yescharacteristic
46C22⋊C4×C179—21no—
47C42×C179C2311yescharacteristic
48C22×D179—21no—
49Dic358—42no—
50D716—21no—
51D716—21no—
52C22⋊D179—82no—
53C2×Dic179—21no—
54C2×Dic179—22no—
55C4×D179—41no—
56C4×D179—41no—
57C179⋊C8—41no—
58C179⋊C8—41no—
59C23×C179—21no—
60Q8×C179—41no—
61Q8×C179—41no—
62D4×C179—21no—
63D4×C179—21no—
64C2×C4×C179—21no—
65C2×C4×C179—41no—
66C2×C4×C179—41no—
67C2×C4×C179—41no—
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×D411yescharacteristic
71C8×C179—41no—
72C22×C179—41no—
73C22×C179—41no—
74C22×C179—41no—
75C22×C179#12011yescharacteristic
76C2×D179—41no—
77C2×D179—41no—
78C2×D179—41no—
79C4×C179—21no—
80C4×C179—41no—
81C4×C179—81no—
82C4×C179#13511yescharacteristic
83C4×C179#13511yescharacteristic
84C4×C179AGL(1,ℤ/8)11yescharacteristic
85C4×C179#13711yescharacteristic
86Dic179—41no—
87Dic179—42no—
88C2×C179—81no—
89C2×C179#44911yescharacteristic
90C2×C179#45211yescharacteristic
91C2×C179#45511yescharacteristic
92D179—81no—
93C179#164611yescharacteristic, complemented, Sylow
94#1646—1791nomaximal, Sylow
95#555—1791no—
96#487—1791no—
97#474D17911yescharacteristic, complemented
98#461—1791no—
99#333—1791no—
100D4⋊C8—1791no—
101D4⋊C8—1791no—
102C2×SD16—3581no—
103C4⋊Q8—1791no—
104C4⋊Q8C2×D17911yescharacteristic
105#124—3582no—
106#121—3581no—
107#121—3581no—
108C4×D4—1791no—
109C4×D4C2×D17911yescharacteristic
110#117—3581no—
111#117—3581no—
112#107—21no—
113C4⋊C8—1791no—
114C4⋊C8C2×D17911yescharacteristic
115#103—3582no—
116#103—3581no—
117#102—21no—
118C4×C8—1791no—
119C2×Q8—21no—
120C2×Q8—1792no—
121C2×D4—3582no—
122C2×D4—1791no—
123C2×D4C22⋊D17911yescharacteristic
124C22×C4—21no—
125C22×C4—3581no—
126SD16—7162no—
127C2×C8—21no—
128C2×C8—3581no—
129C2×C8—3581no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3581no—
133C4⋊C4—3581no—
134C4⋊C4—1791no—
135C4⋊C4C22⋊D17911yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3582no—
138C22⋊C4—3582no—
139C22⋊C4—3581no—
140C42—3582no—
141C42C22×D17911yescharacteristic
142C23—21no—
143C23—3581no—
144Q8—41no—
145Q8—41no—
146Q8—7162no—
147D4—21no—
148D4—21no—
149D4—3581no—
150D4—3581no—
151D4—14322no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3581no—
157C2×C4—3582no—
158C2×C4—7161no—
159C2×C4—7161no—
160C2×C4D4×D17911yescharacteristic
161C2×C4D4×D17911yescharacteristic
162C2×C4C2×C22⋊D17911yescharacteristic
163C8—41no—
164C8—7161no—
165C8—7161no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—7161no—
170C22—7161no—
171C22—7161no—
172C22#1915308111yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—7161no—
177C4—7162no—
178C4#1915313211yescharacteristic
179C4#1915313311yescharacteristic
180C4#1915313711yescharacteristic
181C4#1915314411yescharacteristic
182C2—81no—
183C2—14321no—
184C2#2603858111yescharacteristic
185C2#2603859911yescharacteristic
186C2#2603861511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#474×C179C2
socleC22×C179#120
prime radical#26039154C2
hypercenter#474D179
hypocenterC179#1646
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #26038791C2semidirect
3. #26038641C2semidirect
4. #26038187C2semidirect
5. D4⋊(C179⋊C8)C2semidirect
6. #26038166C2semidirect
7. #474×C179C2semidirect
93. C179#1646semidirect
97. #474D179semidirect

Maximal subgroups

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

subgroupindex
#260391542
#260387912
#260386412
#260381872
D4⋊(C179⋊C8)2
#260381662
#474×C1792
#1646179