Finite Group Databasebeta

← #37508019

Subgroups — #37508019

Order 22912.

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

normal subgroups only

This group has 288 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#26039001C211yesmaximal, characteristic
2#26038757C212yesmaximal, complemented
3#26038172C212yesmaximal, complemented
4#26038163C211yesmaximal, characteristic, complemented
5#399×C179C211yesmaximal, characteristic, complemented
6C22×C22⋊D179—21no—
7#19153097—22no—
8#19153087—21no—
9#19153087C2212yes—
10#19153083—22no—
11#19153044—22no—
12#19153033—22no—
13C22⋊(C179⋊C8)C2212yescomplemented
14C2×C4⋊C4×C179—21no—
15C2×C4⋊C4×C179C2211yescharacteristic, complemented
16C2×C22⋊C4×C179—21no—
17C2×C22⋊C4×C179C2211yescharacteristic
18#95×C179C2211yescharacteristic, complemented
19C23×D179—21no—
20C2×D716—21no—
21C2×D716D412yes—
22C2×C22⋊D179—22no—
23C2×C22⋊D179—22no—
24C2×C22⋊D179—42no—
25C2×C22⋊D179—42no—
26C22×Dic179—21no—
27C2×C179⋊C8—22no—
28C4⋊Dic179—21no—
29C4⋊Dic179D412yes—
30(C2×D179)⋊C4—22no—
31(C2×D179)⋊C4—22no—
32C24×C179D411yescharacteristic
33C22×C4×C179—21no—
34C22×C4×C179—21no—
35C22×C4×C179D411yescharacteristic
36C22×C4×C179C2311yescharacteristic
37C4⋊C4×C179—22no—
38C4⋊C4×C179—42no—
39C22⋊C4×C179—22no—
40C22⋊C4×C179—42no—
41C22×D179—22no—
42C22×D179—42no—
43C22×D179—81no—
44C22×D179—81no—
45D716—42no—
46D716—81no—
47C22⋊D179—44no—
48C22⋊D179—42no—
49C22⋊D179—42no—
50C2×Dic179—22no—
51C2×Dic179—42no—
52C179⋊C8—42no—
53C23×C179—21no—
54C23×C179—21no—
55C23×C179—41no—
56C23×C179—41no—
57C23×C179SD1612yes—
58C23×C179C2×D411yescharacteristic
59C2×C4×C179—22no—
60C2×C4×C179—21no—
61C2×C4×C179—41no—
62C2×C4×C179—41no—
63C2×C4×C179—41no—
64C2×C4×C179—41no—
65C2×C4×C179—41no—
66C2×C4×C179—41no—
67C2×C4×C179—41no—
68C2×C4×C179C2×D412yes—
69C22×C179—21no—
70C22×C179—21no—
71C22×C179—22no—
72C22×C179—42no—
73C22×C179—41no—
74C22×C179—41no—
75C22×C179—81no—
76C22×C179#12011yescharacteristic
77C22×C179C2×SD1611yescharacteristic
78C22×C179AGL(1,ℤ/8)11yescharacteristic
79C2×D179—42no—
80C2×D179—42no—
81C2×D179—42no—
82C2×D179—84no—
83C4×C179—22no—
84C4×C179—42no—
85C4×C179—41no—
86C4×C179—81no—
87Dic179—42no—
88C2×C179—21no—
89C2×C179—21no—
90C2×C179—42no—
91C2×C179#45011yescharacteristic
92C2×C179#45311yescharacteristic
93C2×C179#45811yescharacteristic
94D179—82no—
95C179#161311yescharacteristic, complemented, Sylow
96#1613—1791nomaximal, Sylow
97#534—1791no—
98#481—1792no—
99#399D17911yescharacteristic, complemented
100#327—1792no—
101#323—1791no—
102C22×D4—3581no—
103#123—3582no—
104#121—3581no—
105#121—1792no—
106#120—3582no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D17911yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D17911yescharacteristic
111#106—3582no—
112#102—3582no—
113C22⋊C8—1792no—
114#95C2×D17911yescharacteristic, complemented
115C24—3581no—
116C24C22⋊D17911yescharacteristic
117C2×D4—3581no—
118C2×D4—3582no—
119C2×D4—3582no—
120C2×D4—1792no—
121C2×D4—7162no—
122C2×D4—7162no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—3581no—
126C22×C4C22⋊D17911yescharacteristic
127C22×C4C22×D17911yescharacteristic
128C2×C8—3582no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—3581no—
132C4⋊C4—1792no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—3582no—
136C22⋊C4—3582no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—3582no—
142C23—14321no—
143C23—14321no—
144C23—7162no—
145C23#1566824612yes—
146C23C2×C22⋊D17911yescharacteristic
147D4—14321no—
148D4—7162no—
149D4—7164no—
150D4—7162no—
151D4—7162no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3582no—
162C2×C4—7162no—
163C2×C4D4×D17912yes—
164C8—7162no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—14324no—
173C22—7162no—
174C22—7162no—
175C22—7162no—
176C22#1915308111yescharacteristic
177C22C2×#1566824611yescharacteristic
178C22#1915313811yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—7162no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—14322no—
188C2#2603858911yescharacteristic
189C2#2603860611yescharacteristic
190C2#2603862811yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#399×C179C2
socleC22×C179#120
prime radical#26039001C2
hypercenter#399D179
hypocenterC179#1613
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #26038757C2semidirect
3. #26038172C2semidirect
4. #26038163C2semidirect
5. #399×C179C2semidirect
13. C22⋊(C179⋊C8)C22semidirect
15. C2×C4⋊C4×C179C22semidirect
18. #95×C179C22semidirect
95. C179#1613semidirect
99. #399D179semidirect
108. C2×C4⋊C4C2×D179semidirect
114. #95C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260390012
#260387572
#260381722
#260381632
#399×C1792
#1613179