Finite Group Databasebeta

← #37508262

Subgroups — #37508262

Order 22912.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D716C211yesmaximal, characteristic
2#26038835C211yesmaximal, characteristic, complemented
3#26038641C211yesmaximal, characteristic
4#26038186C211yesmaximal, characteristic, complemented
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#475×C179C211yesmaximal, characteristic, complemented
8C2×Q8×D179—21no—
9C2×#15668246—21no—
10#19153113—21no—
11#19153113C2211yescharacteristic
12#19153093—21no—
13#19153093—22no—
14C4×D716—21no—
15C4×D716C2211yescharacteristic
16#19153044—22no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18#19153037—21no—
19C4×C179⋊C8C2211yescharacteristic
20C4⋊Q8×C179C2211yescharacteristic
21C4×Q8×C179C2211yescharacteristic
22#106×C179—21no—
23C4⋊C8×C179C2211yescharacteristic
24#103×C179—21no—
25C2×Dic358—22no—
26C2×Dic358—21no—
27Q8×D179—42no—
28Q8×D179—42no—
29C2×D716D411yescharacteristic
30C2×C4×D179—22no—
31C2×C4×D179—21no—
32#15668246—42no—
33C2×C179⋊C8—21no—
34C2×C179⋊C8—21no—
35C4⋊Dic179—22no—
36C4⋊Dic179—21no—
37C4⋊Dic179D411yescharacteristic
38C4⋊Dic179D412yes—
39(C2×D179)⋊C4—22no—
40(C2×D179)⋊C4—21no—
41C2×Q8×C179—21no—
42C2×Q8×C179D411yescharacteristic
43C2×C8×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179D411yescharacteristic
48C42×C179—21no—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51Dic358—44no—
52Dic358—41no—
53Dic358—41no—
54D716—21no—
55D716SD1612yes—
56C2×Dic179—22no—
57C2×Dic179—21no—
58C4×D179—44no—
59C4×D179—41no—
60C4×D179—41no—
61C179⋊C8—22no—
62C179⋊C8—41no—
63Q8×C179—21no—
64Q8×C179—41no—
65Q8×C179—41no—
66Q8×C179SD1612yescomplemented
67C2×C4×C179—21no—
68C2×C4×C179—21no—
69C2×C4×C179—41no—
70C2×C4×C179C2×D411yescharacteristic
71C2×C4×C179C2×D411yescharacteristic
72C2×C4×C179C2×D411yescharacteristic
73C8×C179—41no—
74C22×C179#12011yescharacteristic
75C2×D179—22no—
76C2×D179—41no—
77C2×D179—41no—
78C4×C179—22no—
79C4×C179—21no—
80C4×C179—41no—
81C4×C179—81no—
82C4×C179C2×SD1611yescharacteristic
83C4×C179C2×SD1611yescharacteristic
84C4×C179AGL(1,ℤ/8)11yescharacteristic
85C4×C179#13711yescharacteristic
86Dic179—42no—
87Dic179—41no—
88C2×C179#44811yescharacteristic
89C2×C179#45011yescharacteristic
90C2×C179#45611yescharacteristic
91D179—42no—
92C179#165111yescharacteristic, complemented, Sylow
93#1651—1791nomaximal, Sylow
94D4×Q8—1791no—
95#498—1791no—
96#475D17911yescharacteristic, complemented
97#461—1791no—
98#332—1791no—
99Q8⋊C8—1791no—
100D4⋊C8—1791no—
101C22×Q8—3581no—
102C2×SD16—3581no—
103C4⋊Q8—3581no—
104C4⋊Q8—1791no—
105C4⋊Q8C2×D17911yescharacteristic
106C22⋊Q8—3582no—
107C22⋊Q8—3581no—
108C4×Q8C2×D17911yescharacteristic
109C4×D4—3581no—
110C4×D4—1791no—
111#106—21no—
112#106—3582no—
113C4⋊C8—1791no—
114C4⋊C8C2×D17911yescharacteristic
115#103—21no—
116#103—3581no—
117C4×C8—1791no—
118C2×Q8—21no—
119C2×Q8—3582no—
120C2×Q8—3581no—
121C2×Q8—7162no—
122C2×Q8—7162no—
123C2×Q8C22⋊D17911yescharacteristic
124C2×D4—1791no—
125C22×C4—3582no—
126C22×C4—3581no—
127SD16—7162no—
128C2×C8—21no—
129C2×C8—3581no—
130C2×C8—3581no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3582no—
135C4⋊C4—3581no—
136C4⋊C4—1791no—
137C4⋊C4—1792no—
138C4⋊C4C22⋊D17911yescharacteristic
139C22⋊C4—3582no—
140C22⋊C4—3581no—
141C42—21no—
142C42C22×D17911yescharacteristic
143C23—3581no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—7164no—
148Q8—7161no—
149Q8—7161no—
150Q8#1566824612yescomplemented
151D4—3581no—
152D4—1792no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—3582no—
157C2×C4—3581no—
158C2×C4—7164no—
159C2×C4—7161no—
160C2×C4—7161no—
161C2×C4D4×D17911yescharacteristic
162C2×C4D4×D17911yescharacteristic
163C2×C4C2×C22⋊D17911yescharacteristic
164C8—41no—
165C8—3582no—
166C8—7161no—
167C22—3582no—
168C22—7161no—
169C22—7161no—
170C22#1915308111yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—7162no—
176C4—7161no—
177C4SD16×D17911yescharacteristic
178C4C2×#1566824611yescharacteristic
179C4#1915313811yescharacteristic
180C4#1915314211yescharacteristic
181C2—7162no—
182C2#2603857611yescharacteristic
183C2#2603858911yescharacteristic
184C2#2603862011yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#475×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#475D179
hypocenterC179#1651
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #26038835C2semidirect
4. #26038186C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
7. #475×C179C2semidirect
66. Q8×C179SD16semidirect
92. C179#1651semidirect
96. #475D179semidirect
150. Q8#15668246semidirect

Maximal subgroups

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

subgroupindex
Q8×D7162
#260388352
#260386412
#260381862
Q8⋊(C179⋊C8)2
#260381662
#475×C1792
#1651179