Finite Group Databasebeta

← #37508173

Subgroups — #37508173

Order 22912.

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

normal subgroups only

This group has 194 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#26039163C211yesmaximal, characteristic
2#26038791C211yesmaximal, characteristic, complemented
3#26038735C211yesmaximal, characteristic, complemented
4#26038730C211yesmaximal, characteristic
5#26038169C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#332×C179C211yesmaximal, characteristic, complemented
8#19153113C2211yescharacteristic
9#19153108—22no—
10#19153102—21no—
11#19153093—21no—
12Q8×Dic179—21no—
13C4×Dic358C2211yescharacteristic
14C4×D716C2211yescharacteristic
15#19153071—21no—
16#19153047—21no—
17#19153045—21no—
18#19153038—21no—
19#19153036—22no—
20#19153035—21no—
21C4×C179⋊C8C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25C2×Dic358—21no—
26C2×Dic358D411yescharacteristic
27C2×D716D411yescharacteristic
28C2×C4×D179—21no—
29C2×C179⋊C8—21no—
30C4⋊Dic179—22no—
31C4⋊Dic179—21no—
32C4⋊Dic179—21no—
33C4⋊Dic179D411yescharacteristic
34C4⋊Dic179D411yescharacteristic
35Dic179⋊C4—21no—
36(C2×D179)⋊C4—22no—
37(C2×D179)⋊C4—22no—
38(C2×D179)⋊C4—21no—
39C4×Dic179—22no—
40C4×Dic179—21no—
41C2×Q8×C179—21no—
42C2×C8×C179—21no—
43C2×C8×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179D412yes—
46C42×C179C2311yescharacteristic
47C22×D179—21no—
48Dic358—21no—
49Dic358—21no—
50Dic358—42no—
51D716—21no—
52D716—21no—
53C2×Dic179—21no—
54C2×Dic179—21no—
55C2×Dic179—21no—
56C2×Dic179—41no—
57C4×D179—41no—
58C4×D179—41no—
59C179⋊C8—41no—
60Q8×C179—41no—
61Q8×C179—41no—
62C2×C4×C179—22no—
63C2×C4×C179C2×D411yescharacteristic
64C2×C4×C179C2×D411yescharacteristic
65C2×C4×C179C2×D411yescharacteristic
66C8×C179—41no—
67C8×C179—41no—
68C22×C179#12011yescharacteristic
69C2×D179—41no—
70C2×D179—41no—
71C2×D179—41no—
72C4×C179—21no—
73C4×C179—42no—
74C4×C179#13511yescharacteristic
75C4×C179#13511yescharacteristic
76C4×C179#13711yescharacteristic
77C4×C179#13711yescharacteristic
78Dic179—41no—
79Dic179—41no—
80Dic179—41no—
81Dic179—81no—
82C2×C179#45211yescharacteristic
83C2×C179#45211yescharacteristic
84C2×C179#45411yescharacteristic
85D179—81no—
86C179#163811yescharacteristic, complemented, Sylow
87#1638—1791nomaximal, Sylow
88#556—1791no—
89#487—1791no—
90#477—1791no—
91#476—1791no—
92#332D17911yescharacteristic, complemented
93Q8⋊C8—1791no—
94D4⋊C8—1791no—
95C4⋊Q8—1791no—
96C4⋊Q8C2×D17911yescharacteristic
97#126—3582no—
98#124—3581no—
99C22⋊Q8—3581no—
100C4×Q8—3581no—
101C4×Q8—1791no—
102C4×D4—1791no—
103#117—3581no—
104#107—3581no—
105#106—3581no—
106C4⋊C8C2×D17911yescharacteristic
107C4⋊C8C2×D17911yescharacteristic
108#103—3581no—
109#103—3582no—
110#102—3581no—
111C4×C8—1791no—
112C2×Q8—21no—
113C2×Q8—3581no—
114C2×Q8—1791no—
115C2×D4—1791no—
116C22×C4—3581no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3581no—
120C4⋊C4—21no—
121C4⋊C4—3582no—
122C4⋊C4—3581no—
123C4⋊C4—3581no—
124C4⋊C4—3581no—
125C4⋊C4—1791no—
126C4⋊C4—1791no—
127C4⋊C4D71612yes—
128C22⋊C4—3582no—
129C22⋊C4—3582no—
130C22⋊C4—3581no—
131C42—3582no—
132C42—3581no—
133C42C22×D17911yescharacteristic
134C23—3581no—
135Q8—41no—
136Q8—41no—
137Q8—3581no—
138Q8—3581no—
139Q8—7162no—
140D4—3581no—
141D4—3581no—
142C2×C4—22no—
143C2×C4—3581no—
144C2×C4—3581no—
145C2×C4—3581no—
146C2×C4—7161no—
147C2×C4—7161no—
148C2×C4—7161no—
149C2×C4D4×D17911yescharacteristic
150C2×C4D4×D17911yescharacteristic
151C2×C4C2×D71611yescharacteristic
152C8—41no—
153C8—41no—
154C8—7161no—
155C22—7161no—
156C22—7161no—
157C22—7161no—
158C22#1915308311yescharacteristic
159C4—21no—
160C4—42no—
161C4—14321no—
162C4—7161no—
163C4—7161no—
164C4—7161no—
165C4#1915313211yescharacteristic
166C4#1915313411yescharacteristic
167C4#1915314211yescharacteristic
168C4#1915314511yescharacteristic
169C2—14321no—
170C2#2603860211yescharacteristic
171C2#2603860411yescharacteristic
172C2#2603861411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#332×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#332D179
hypocenterC179#1638
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #26038791C2semidirect
3. #26038735C2semidirect
5. #26038169C2semidirect
7. #332×C179C2semidirect
86. C179#1638semidirect
92. #332D179semidirect

Maximal subgroups

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

subgroupindex
#260391632
#260387912
#260387352
#260387302
#260381692
#260381662
#332×C1792
#1638179