Finite Group Databasebeta

← #37510691

Subgroups — #37510691

Order 22912.

262 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 65 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#26039003C211yesmaximal, characteristic
2#26038423C211yesmaximal, characteristic, complemented
3#26038235C214yesmaximal, complemented
4#413×C179C211yesmaximal, characteristic, complemented
5C22×D716C411yescharacteristic
6C2×#15668243—21no—
7#19153109C412yes—
8#19153102C2212yes—
9#19153087—21no—
10#19153087—21no—
11#19153081—21no—
12#19153081—21no—
13#19153072C411yescharacteristic
14#19153032C2212yescomplemented
15#19153030—21no—
16C2×#41×C179C2211yescharacteristic
17C2×M16×C179—21no—
18#101×C179C2212yescomplemented
19#100×C179—21no—
20C23×D179—21no—
21C2×D716—21no—
22C2×D716—41no—
23C2×D716—41no—
24C2×D716C2×C412yes—
25C2×C22⋊D179—22no—
26C2×C22⋊D179—22no—
27#15668243—22no—
28#15668243—41no—
29C2×C179⋊C8—41no—
30C4⋊Dic179—21no—
31C22⋊Dic179—21no—
32(C2×D179)⋊C4—22no—
33(C2×D179)⋊C4—22no—
34C4×Dic179C2×C412yes—
35#41×C179—41no—
36#41×C179—41no—
37C2×Q8×C179C2311yescharacteristic
38C2×D4×C179C2×C411yescharacteristic
39C2×D4×C179D411yescharacteristic
40C2×D4×C179D411yescharacteristic
41C22×C4×C179C2×C411yescharacteristic
42C22×C4×C179D411yescharacteristic
43C22×C4×C179D411yescharacteristic
44M16×C179—22no—
45M16×C179—41no—
46C2×C8×C179—41no—
47C22×D179—22no—
48C22×D179—42no—
49C22×D179—82no—
50D716—42no—
51D716—41no—
52D716—41no—
53C22⋊D179—82no—
54C22⋊D179—82no—
55C2×Dic179—22no—
56C179⋊C8—42no—
57C23×C179C22⋊C411yescharacteristic
58C23×C179C22⋊C411yescharacteristic
59C23×C179C22⋊C411yescharacteristic
60Q8×C179—41no—
61D4×C179—41no—
62D4×C179—41no—
63D4×C179—41no—
64C2×C4×C179—21no—
65C2×C4×C179—21no—
66C2×C4×C179—21no—
67C2×C4×C179—21no—
68C2×C4×C179—21no—
69C2×C4×C179—21no—
70C2×C4×C179C22⋊C411yescharacteristic
71C2×C4×C179C22×C411yescharacteristic
72C2×C4×C179C2×D411yescharacteristic
73C2×C4×C179C2×D411yescharacteristic
74C8×C179—42no—
75C22×C179—21no—
76C22×C179—21no—
77C22×C179—21no—
78C22×C179—41no—
79C22×C179—41no—
80C22×C179—41no—
81C22×C179C2×C22⋊C411yescharacteristic
82C2×D179—42no—
83C2×D179—42no—
84C2×D179—42no—
85C2×D179—84no—
86C4×C179—21no—
87C4×C179—21no—
88C4×C179—21no—
89C4×C179#9911yescharacteristic
90C4×C179#9911yescharacteristic
91Dic179—42no—
92C2×C179—21no—
93C2×C179—41no—
94C2×C179—41no—
95C2×C179—41no—
96C2×C179C2×#9911yescharacteristic
97D179—82no—
98C179#213411yescharacteristic, complemented, Sylow
99#2134—1791nomaximal, Sylow
100#535—1791no—
101#413—1791no—
102#413D17911yescharacteristic, complemented
103#355—1794no—
104C2×#41C2×D17911yescharacteristic
105C22×D4—1791no—
106C2×M16—21no—
107C2×M16—3581no—
108#127—1792no—
109#124—1792no—
110#121—3581no—
111#121—3581no—
112#120—3581no—
113#120—3581no—
114#117—1791no—
115#101—1792no—
116#101C2×D17912yescomplemented
117#100—21no—
118#100—3581no—
119C24—3581no—
120#41—41no—
121#41—41no—
122C2×Q8C22×D17911yescharacteristic
123C2×D4—3582no—
124C2×D4—3582no—
125C2×D4—3581no—
126C2×D4—1792no—
127C2×D4—7161no—
128C2×D4—7161no—
129C2×D4C4×D17911yescharacteristic
130C2×D4C22⋊D17911yescharacteristic
131C2×D4D71611yescharacteristic
132C22×C4C4×D17911yescharacteristic
133C22×C4C22⋊D17911yescharacteristic
134C22×C4D71611yescharacteristic
135M16—22no—
136M16—41no—
137M16—3582no—
138M16—7161no—
139C2×C8—41no—
140C2×C8—7161no—
141C4⋊C4—3581no—
142C22⋊C4—3581no—
143C22⋊C4—3582no—
144C22⋊C4—3582no—
145C42—1792no—
146C23—3582no—
147C23—14322no—
148C23—7162no—
149C23(C2×D179)⋊C411yescharacteristic
150C23(C2×D179)⋊C411yescharacteristic
151C23(C2×D179)⋊C411yescharacteristic
152Q8—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—14322no—
157D4—14322no—
158D4—7162no—
159D4—7161no—
160D4—7161no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—3582no—
168C2×C4(C2×D179)⋊C411yescharacteristic
169C2×C4C2×C4×D17911yescharacteristic
170C2×C4C2×C22⋊D17911yescharacteristic
171C2×C4C2×D71611yescharacteristic
172C8—42no—
173C8—7162no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—14324no—
181C22—7162no—
182C22—7162no—
183C22—7162no—
184C22C2×(C2×D179)⋊C411yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—7162no—
189C4#1915302711yescharacteristic
190C4#1915302711yescharacteristic
191C2—21no—
192C2—41no—
193C2—41no—
194C2—41no—
195C2—14322no—
196C2C2×#1915302711yescharacteristic
subgroupG/H
centerC2C2×#19153027
commutatorC2×C4×C179C22×C4
FrattiniC2×Q8C22×D179
Fitting#413×C179C2
socleC2×C179C2×#99
prime radical#26039003C2
cosocleC2×Q8×C179C23
hypercenter#413D179
hypocenterC179#2134
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #26038423C2semidirect
3. #26038235C2semidirect
4. #413×C179C2semidirect
14. #19153032C22semidirect
18. #101×C179C22semidirect
98. C179#2134semidirect
102. #413D179semidirect
116. #101C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260390032
#260384232
#260382352
#413×C1792
#2134179