Finite Group Databasebeta

← #37508187

Subgroups — #37508187

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#26039175C211yesmaximal, characteristic
2#26038785C211yesmaximal, characteristic
3#26038725C211yesmaximal, characteristic, complemented
4#26038650C211yesmaximal, characteristic, complemented
5#26038169C211yesmaximal, characteristic, complemented
6#26038169C211yesmaximal, characteristic, complemented
7#331×C179C211yesmaximal, characteristic
8C2×Dic716—21no—
9#19153113C2211yescharacteristic
10#19153100—21no—
11#19153100—21no—
12#19153094—22no—
13C4×Dic358C2211yescharacteristic
14C4×Dic358C2211yescharacteristic
15D4×Dic179—21no—
16D4×Dic179—21no—
17#19153045—21no—
18#19153038—21no—
19#19153038—21no—
20#19153034—22no—
21C4×C179⋊C8C2211yescharacteristic
22#127×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25C2×Dic358—21no—
26C2×Dic358D411yescharacteristic
27C2×Dic358D411yescharacteristic
28C22×Dic179—22no—
29Dic716—42no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—21no—
32C4⋊Dic179—21no—
33C4⋊Dic179D411yescharacteristic
34C4⋊Dic179D411yescharacteristic
35Dic179⋊C4—21no—
36Dic179⋊C4—21no—
37C22⋊Dic179—22no—
38C22⋊Dic179—22no—
39C22⋊Dic179—22no—
40C4×Dic179—21no—
41C4×Dic179—21no—
42C2×D4×C179—21no—
43C2×D4×C179—21no—
44C2×D4×C179D412yes—
45C2×C8×C179—21no—
46C2×C8×C179—21no—
47C42×C179C2311yescharacteristic
48Dic358—21no—
49Dic358—21no—
50Dic358—21no—
51Dic358—21no—
52Dic358—41no—
53Dic358—41no—
54C2×Dic179—21no—
55C2×Dic179—21no—
56C2×Dic179—21no—
57C2×Dic179—21no—
58C2×Dic179—41no—
59C2×Dic179—82no—
60C179⋊C8—41no—
61C23×C179—22no—
62D4×C179—42no—
63D4×C179—41no—
64D4×C179—41no—
65D4×C179—41no—
66D4×C179—41no—
67C2×C4×C179C2×D411yescharacteristic
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×D411yescharacteristic
70C8×C179—41no—
71C8×C179—41no—
72C22×C179—42no—
73C22×C179—42no—
74C22×C179—42no—
75C22×C179#12011yescharacteristic
76C4×C179—21no—
77C4×C179#13511yescharacteristic
78C4×C179#13511yescharacteristic
79C4×C179AGL(1,ℤ/8)11yescharacteristic
80C4×C179#13711yescharacteristic
81Dic179—41no—
82Dic179—41no—
83Dic179—41no—
84Dic179—41no—
85Dic179—81no—
86C2×C179—82no—
87C2×C179#44911yescharacteristic
88C2×C179#45211yescharacteristic
89C2×C179#45511yescharacteristic
90C179#164011yescharacteristic, complemented, Sylow
91#1640—1791nomaximal, Sylow
92#559—1791no—
93#486—1791no—
94#475—1791no—
95#462—1791no—
96#331D17911yescharacteristic
97Q8⋊C8—1791no—
98Q8⋊C8—1791no—
99C2×Q16—3581no—
100C4⋊Q8—1791no—
101#127C2×D17911yescharacteristic
102#124—3581no—
103#124—3581no—
104#123—3582no—
105C4×Q8—1791no—
106C4×Q8—1791no—
107C4×D4—3581no—
108C4×D4—3581no—
109#106—3581no—
110C4⋊C8C2×D17911yescharacteristic
111C4⋊C8C2×D17911yescharacteristic
112#103—3581no—
113#103—3581no—
114#102—3582no—
115C4×C8—1791no—
116C2×Q8—3581no—
117C2×Q8—1791no—
118C2×Q8—1791no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D71612yes—
122C22×C4—3582no—
123Q16—7162no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—3581no—
127C4⋊C4—3581no—
128C4⋊C4—3581no—
129C4⋊C4—3581no—
130C4⋊C4—3581no—
131C4⋊C4—1791no—
132C4⋊C4—1791no—
133C22⋊C4—3582no—
134C22⋊C4—3582no—
135C22⋊C4—3582no—
136C42—3581no—
137C42—3581no—
138C42C22×D17911yescharacteristic
139C23—22no—
140Q8—3581no—
141Q8—3581no—
142Q8—3581no—
143Q8—3581no—
144Q8—7161no—
145Q8—7161no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—3581no—
152C2×C4—3581no—
153C2×C4—3581no—
154C2×C4—3581no—
155C2×C4—14322no—
156C2×C4—7161no—
157C2×C4D4×D17911yescharacteristic
158C2×C4D4×D17911yescharacteristic
159C2×C4C2×D71611yescharacteristic
160C8—41no—
161C8—41no—
162C8—7161no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1915308311yescharacteristic
167C4—21no—
168C4—14321no—
169C4—7161no—
170C4—7161no—
171C4—7161no—
172C4—7161no—
173C4#1915313211yescharacteristic
174C4#1915313411yescharacteristic
175C4#1915313711yescharacteristic
176C4#1915314211yescharacteristic
177C2—82no—
178C2#2603858511yescharacteristic
179C2#2603860211yescharacteristic
180C2#2603861911yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#331×C179C2
socleC22×C179#120
prime radical#127×C179C22
hypercenter#331D179
hypocenterC179#1640
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #26038725C2semidirect
4. #26038650C2semidirect
5. #26038169C2semidirect
6. #26038169C2semidirect
90. C179#1640semidirect

Maximal subgroups

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

subgroupindex
#260391752
#260387852
#260387252
#260386502
#260381692
#331×C1792
#1640179