Finite Group Databasebeta

← #37508271

Subgroups — #37508271

Order 22912.

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

normal subgroups only

This group has 200 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#26038835C211yesmaximal, characteristic, complemented
3#26038720C211yesmaximal, characteristic
4#26038650C211yesmaximal, characteristic, complemented
5#26038169C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#332×C179C211yesmaximal, characteristic, complemented
8C2×Dic716—21no—
9#19153113C2211yescharacteristic
10#19153108—22no—
11#19153102—21no—
12#19153093—21no—
13Q8×Dic179—21no—
14C4×Dic358C2211yescharacteristic
15C4×D716C2211yescharacteristic
16#19153071—21no—
17#19153047—21no—
18#19153044—22no—
19#19153038—21no—
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—
29Dic716—42no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—22no—
32C4⋊Dic179—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179D411yescharacteristic
35C4⋊Dic179D411yescharacteristic
36Dic179⋊C4—21no—
37(C2×D179)⋊C4—22no—
38(C2×D179)⋊C4—22no—
39(C2×D179)⋊C4—21no—
40C4×Dic179—22no—
41C4×Dic179—21no—
42C2×Q8×C179—21no—
43C2×C8×C179—21no—
44C2×C8×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179D412yes—
47C42×C179C2311yescharacteristic
48C22×D179—21no—
49Dic358—21no—
50Dic358—21no—
51Dic358—41no—
52Dic358—41no—
53D716—21no—
54D716—21no—
55C2×Dic179—21no—
56C2×Dic179—21no—
57C2×Dic179—21no—
58C2×Dic179—41no—
59C4×D179—41no—
60C4×D179—41no—
61C179⋊C8—22no—
62Q8×C179—41no—
63Q8×C179—41no—
64C2×C4×C179—22no—
65C2×C4×C179C2×D411yescharacteristic
66C2×C4×C179C2×D411yescharacteristic
67C2×C4×C179C2×D411yescharacteristic
68C8×C179—41no—
69C8×C179—41no—
70C22×C179#12011yescharacteristic
71C2×D179—41no—
72C2×D179—41no—
73C2×D179—41no—
74C4×C179—21no—
75C4×C179—42no—
76C4×C179#13511yescharacteristic
77C4×C179#13511yescharacteristic
78C4×C179AGL(1,ℤ/8)11yescharacteristic
79C4×C179#13711yescharacteristic
80Dic179—41no—
81Dic179—41no—
82Dic179—41no—
83Dic179—81no—
84C2×C179#44911yescharacteristic
85C2×C179#45211yescharacteristic
86C2×C179#45611yescharacteristic
87D179—81no—
88C179#165211yescharacteristic, complemented, Sylow
89#1652—1791nomaximal, Sylow
90#556—1791no—
91#498—1791no—
92#474—1791no—
93#462—1791no—
94#332D17911yescharacteristic, complemented
95Q8⋊C8—1791no—
96D4⋊C8—1791no—
97C2×Q16—3581no—
98C4⋊Q8—1791no—
99C4⋊Q8C2×D17911yescharacteristic
100#126—3582no—
101#124—3581no—
102C22⋊Q8—3581no—
103C4×Q8—3581no—
104C4×Q8—1791no—
105C4×D4—1791no—
106#117—3581no—
107#107—3581no—
108#106—3582no—
109C4⋊C8C2×D17911yescharacteristic
110C4⋊C8C2×D17911yescharacteristic
111#103—3581no—
112#102—3581no—
113C4×C8—1791no—
114C2×Q8—21no—
115C2×Q8—3581no—
116C2×Q8—1791no—
117C2×D4—1791no—
118C22×C4—3581no—
119Q16—7162no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3581no—
123C4⋊C4—21no—
124C4⋊C4—3582no—
125C4⋊C4—3581no—
126C4⋊C4—3581no—
127C4⋊C4—3581no—
128C4⋊C4—1791no—
129C4⋊C4—1791no—
130C4⋊C4D71612yes—
131C22⋊C4—3582no—
132C22⋊C4—3582no—
133C22⋊C4—3581no—
134C42—3582no—
135C42—3581no—
136C42C22×D17911yescharacteristic
137C23—3581no—
138Q8—41no—
139Q8—41no—
140Q8—3581no—
141Q8—3581no—
142Q8—7161no—
143Q8—7161no—
144D4—3581no—
145D4—3581no—
146C2×C4—22no—
147C2×C4—3581no—
148C2×C4—3581no—
149C2×C4—3581no—
150C2×C4—7161no—
151C2×C4—7161no—
152C2×C4—7161no—
153C2×C4D4×D17911yescharacteristic
154C2×C4D4×D17911yescharacteristic
155C2×C4C2×D71611yescharacteristic
156C8—41no—
157C8—41no—
158C8—3582no—
159C22—7161no—
160C22—7161no—
161C22—7161no—
162C22#1915308311yescharacteristic
163C4—21no—
164C4—42no—
165C4—14321no—
166C4—7161no—
167C4—7161no—
168C4—7161no—
169C4#1915313211yescharacteristic
170C4#1915313611yescharacteristic
171C4#1915314011yescharacteristic
172C4#1915314211yescharacteristic
173C2—14321no—
174C2#2603858711yescharacteristic
175C2#2603860211yescharacteristic
176C2#2603862411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#332×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#332D179
hypocenterC179#1652
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. #26038835C2semidirect
4. #26038650C2semidirect
5. #26038169C2semidirect
7. #332×C179C2semidirect
88. C179#1652semidirect
94. #332D179semidirect

Maximal subgroups

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

subgroupindex
#260391632
#260388352
#260387202
#260386502
#260381692
#260381662
#332×C1792
#1652179