Finite Group Databasebeta

← #37508184

Subgroups — #37508184

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#26039174C211yesmaximal, characteristic
2#26038787C211yesmaximal, characteristic
3#26038721C211yesmaximal, characteristic, complemented
4#26038182C211yesmaximal, characteristic
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038169C211yesmaximal, characteristic, complemented
7#462×C179C211yesmaximal, characteristic, complemented
8#19153110C2211yescharacteristic
9#19153102—21no—
10#19153101—21no—
11#19153098—22no—
12C4×Dic358C2211yescharacteristic
13C4×D716—21no—
14#19153075—21no—
15#19153044—21no—
16C4⋊(C179⋊C8)C2211yescharacteristic
17#19153036—21no—
18#19153033—22no—
19C4×C179⋊C8C2211yescharacteristic
20C2×Q16×C179—21no—
21C4⋊Q8×C179C2211yescharacteristic
22C4×Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24#103×C179—21no—
25C2×Dic358D411yescharacteristic
26C2×D716—21no—
27C2×D716—21no—
28C2×D716D412yes—
29C2×C4×D179—22no—
30C2×C179⋊C8—21no—
31C2×C179⋊C8—21no—
32C4⋊Dic179D411yescharacteristic
33Dic179⋊C4—21no—
34(C2×D179)⋊C4—22no—
35(C2×D179)⋊C4—22no—
36(C2×D179)⋊C4—22no—
37C4×Dic179—21no—
38C2×Q8×C179—21no—
39C2×Q8×C179D411yescharacteristic
40Q16×C179—42no—
41C2×C8×C179—21no—
42C4⋊C4×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179D411yescharacteristic
46C42×C179—21no—
47C42×C179C2311yescharacteristic
48C22×D179—22no—
49Dic358—21no—
50Dic358—21no—
51D716—42no—
52D716—41no—
53D716—41no—
54D716—41no—
55D716—41no—
56C2×Dic179—21no—
57C2×Dic179—21no—
58C4×D179—82no—
59C179⋊C8—41no—
60C179⋊C8—41no—
61Q8×C179—21no—
62Q8×C179—21no—
63Q8×C179—41no—
64Q8×C179—41no—
65C2×C4×C179—21no—
66C2×C4×C179—21no—
67C2×C4×C179—41no—
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×D411yescharacteristic
71C8×C179—41no—
72C22×C179#12011yescharacteristic
73C2×D179—42no—
74C2×D179—42no—
75C2×D179—42no—
76C4×C179—21no—
77C4×C179—41no—
78C4×C179—41no—
79C4×C179—81no—
80C4×C179#13511yescharacteristic
81C4×C179#13511yescharacteristic
82C4×C179AGL(1,ℤ/8)11yescharacteristic
83C4×C179#13711yescharacteristic
84Dic179—41no—
85Dic179—41no—
86C2×C179#44911yescharacteristic
87C2×C179#45211yescharacteristic
88C2×C179#45511yescharacteristic
89D179—82no—
90C179#164011yescharacteristic, complemented, Sylow
91#1640—1791nomaximal, Sylow
92#559—1791no—
93#486—1791no—
94#475—1791no—
95#462D17911yescharacteristic, complemented
96#331—1791no—
97Q8⋊C8—1791no—
98Q8⋊C8—1791no—
99C2×Q16—21no—
100C4⋊Q8C2×D17911yescharacteristic
101#127—1791no—
102#124—3581no—
103#124—3581no—
104#123—3582no—
105C4×Q8—1791no—
106C4×Q8C2×D17911yescharacteristic
107C4×D4—3581no—
108C4×D4—3581no—
109#106—3581no—
110C4⋊C8—1791no—
111C4⋊C8C2×D17911yescharacteristic
112#103—21no—
113#103—3581no—
114#102—3582no—
115C4×C8—1791no—
116C2×Q8—21no—
117C2×Q8—1791no—
118C2×Q8C22⋊D17911yescharacteristic
119C2×D4—3581no—
120C2×D4—3581no—
121C2×D4—1792no—
122C22×C4—3582no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—3581no—
126C2×C8—3581no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—3581no—
131C4⋊C4—1791no—
132C4⋊C4C22⋊D17911yescharacteristic
133C22⋊C4—3582no—
134C22⋊C4—3582no—
135C22⋊C4—3582no—
136C42—21no—
137C42—3581no—
138C42C22×D17911yescharacteristic
139C23—3582no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—3581no—
145Q8—3581no—
146D4—7162no—
147D4—7161no—
148D4—7161no—
149D4—7161no—
150D4—7161no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—3581no—
155C2×C4—3581no—
156C2×C4—14322no—
157C2×C4D4×D17911yescharacteristic
158C2×C4D4×D17911yescharacteristic
159C2×C4C2×C22⋊D17911yescharacteristic
160C8—41no—
161C8—7161no—
162C8—7161no—
163C22—7162no—
164C22—7162no—
165C22—7162no—
166C22#1915308111yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—7161no—
172C4—7161no—
173C4#1915313311yescharacteristic
174C4#1915313611yescharacteristic
175C4#1915313911yescharacteristic
176C4#1915314511yescharacteristic
177C2—14322no—
178C2#2603858211yescharacteristic
179C2#2603860011yescharacteristic
180C2#2603861511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#462×C179C2
socleC22×C179#120
prime radical#19153110C22
hypercenter#462D179
hypocenterC179#1640
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #26038721C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
6. #26038169C2semidirect
7. #462×C179C2semidirect
90. C179#1640semidirect
95. #462D179semidirect

Maximal subgroups

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

subgroupindex
#260391742
#260387872
#260387212
#260381822
Q8⋊(C179⋊C8)2
#260381692
#462×C1792
#1640179