Finite Group Databasebeta

← #37507956

Subgroups — #37507956

Order 22912.

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

normal subgroups only

This group has 256 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#26039065C211yesmaximal, characteristic
2#26038595C212yesmaximal, complemented
3#26038175C212yesmaximal, complemented
4#26038163C211yesmaximal, characteristic
5#393×C179C211yesmaximal, characteristic
6C2×#15668249—22no—
7#19153092—21no—
8#19153092C2212yes—
9#19153090—22no—
10#19153090—22no—
11C2×C22⋊Dic179—21no—
12#19153037—22no—
13C22⋊(C179⋊C8)C2212yes—
14C22×Q8×C179C2211yescharacteristic
15C2×C22⋊C4×C179—21no—
16C2×C22⋊C4×C179C2211yescharacteristic
17#95×C179—21no—
18#95×C179C2211yescharacteristic
19C2×Dic358—21no—
20C2×Dic358D412yes—
21C22×Dic179—21no—
22C22×Dic179—21no—
23#15668249—44no—
24C2×C179⋊C8—22no—
25C4⋊Dic179—21no—
26C4⋊Dic179D412yes—
27Dic179⋊C4—22no—
28Dic179⋊C4—22no—
29C22⋊Dic179—22no—
30C22⋊Dic179—22no—
31C24×C179D411yescharacteristic
32C2×Q8×C179—22no—
33C2×Q8×C179—42no—
34C22×C4×C179—21no—
35C22×C4×C179—21no—
36C22×C4×C179D411yescharacteristic
37C22×C4×C179C2311yescharacteristic
38C22⋊C4×C179—22no—
39C22⋊C4×C179—42no—
40Dic358—42no—
41Dic358—81no—
42C2×Dic179—22no—
43C2×Dic179—22no—
44C2×Dic179—81no—
45C2×Dic179—81no—
46C179⋊C8—42no—
47C23×C179—21no—
48C23×C179—21no—
49C23×C179—41no—
50C23×C179—41no—
51C23×C179Q1612yescomplemented
52C23×C179C2×D411yescharacteristic
53Q8×C179—24no—
54Q8×C179—42no—
55C2×C4×C179—22no—
56C2×C4×C179—21no—
57C2×C4×C179—41no—
58C2×C4×C179—41no—
59C2×C4×C179—41no—
60C2×C4×C179—41no—
61C2×C4×C179—41no—
62C2×C4×C179—41no—
63C2×C4×C179—41no—
64C2×C4×C179C2×D412yes—
65C22×C179—21no—
66C22×C179—21no—
67C22×C179—22no—
68C22×C179—42no—
69C22×C179—41no—
70C22×C179—41no—
71C22×C179—81no—
72C22×C179#12011yescharacteristic
73C22×C179C2×Q1611yescharacteristic
74C22×C179#13711yescharacteristic
75C4×C179—22no—
76C4×C179—41no—
77C4×C179—42no—
78C4×C179—81no—
79Dic179—42no—
80Dic179—42no—
81C2×C179—21no—
82C2×C179—21no—
83C2×C179—42no—
84C2×C179#45111yescharacteristic
85C2×C179#45511yescharacteristic
86C2×C179B(4,𝔽2)11yescharacteristic
87C179#160211yescharacteristic, complemented, Sylow
88#1602—1791nomaximal, Sylow
89#543—1791no—
90#451—1792no—
91#393D17911yescharacteristic
92#328—1792no—
93#323—1791no—
94C22×Q8C2×D17911yescharacteristic
95C2×Q16—3582no—
96C22⋊Q8—3582no—
97C22⋊Q8—3582no—
98C22⋊Q8—3581no—
99C22⋊Q8—1792no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—3581no—
102C2×C22⋊C4C2×D17911yescharacteristic
103#103—3582no—
104C22⋊C8—1792no—
105#95—21no—
106#95C2×D17911yescharacteristic
107C24C22⋊D17911yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—3581no—
111C2×Q8—1792no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—3581no—
115C22×C4—3581no—
116C22×C4C22⋊D17911yescharacteristic
117C22×C4C22×D17911yescharacteristic
118Q16—7164no—
119C2×C8—3582no—
120C4⋊C4—3582no—
121C4⋊C4—3581no—
122C4⋊C4—3582no—
123C4⋊C4—1792no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4—3582no—
127C22⋊C4—3582no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1566824912yescomplemented
133C23C2×C22⋊D17911yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—14321no—
137Q8—7162no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—3582no—
148C2×C4—3582no—
149C2×C4—14321no—
150C2×C4—14321no—
151C2×C4D4×D17912yes—
152C8—7162no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1915308111yescharacteristic
161C22C2×#1566824911yescharacteristic
162C22#1915314311yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4—7162no—
168C4—7162no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2603859411yescharacteristic
173C2#2603861611yescharacteristic
174C2#2603862511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#393×C179C2
socleC22×C179#120
prime radicalC24×C179D4
hypercenter#393D179
hypocenterC179#1602
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #26038595C2semidirect
3. #26038175C2semidirect
51. C23×C179Q16semidirect
87. C179#1602semidirect
132. C23#15668249semidirect

Maximal subgroups

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

subgroupindex
#260390652
#260385952
#260381752
#260381632
#393×C1792
#1602179