Finite Group Databasebeta

← #37510651

Subgroups — #37510651

Order 22912.

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

normal subgroups only

This group has 266 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#26038413C211yesmaximal, characteristic, complemented
3#26038227C214yesmaximal, complemented
4#413×C179C211yesmaximal, characteristic, complemented
5C22×D716C411yescharacteristic, complemented
6#19153109—21no—
7#19153102—21no—
8#19153087—21no—
9#19153087—21no—
10#19153081C412yescomplemented
11#19153081C2212yes—
12#19153072—21no—
13#19153072C411yescharacteristic, complemented
14#19153028—21no—
15#19153028C2212yes—
16C2×#41×C179C2211yescharacteristic
17C2×M16×C179—21no—
18#101×C179—21no—
19#100×C179C2212yes—
20C23×D179C2×C412yes—
21C2×D716—21no—
22C2×D716—21no—
23C2×D716—41no—
24C2×D716—41no—
25C2×C22⋊D179—22no—
26C2×C22⋊D179—22no—
27C4⋊Dic179—21no—
28C4⋊Dic179—41no—
29C22⋊Dic179—22no—
30C22⋊Dic179C2×C412yes—
31(C2×D179)⋊C4—22no—
32(C2×D179)⋊C4—22no—
33C4×Dic179—21no—
34C4×Dic179—41no—
35#41×C179—41no—
36#41×C179—41no—
37C2×Q8×C179D411yescharacteristic
38C2×D4×C179C2×C411yescharacteristic
39C2×D4×C179D411yescharacteristic
40C2×D4×C179C2311yescharacteristic
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—
56C2×Dic179—42no—
57C23×C179C22⋊C411yescharacteristic
58C23×C179C22×C411yescharacteristic
59C23×C179C2×D411yescharacteristic
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×C179C22⋊C411yescharacteristic
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—
92Dic179—82no—
93C2×C179—21no—
94C2×C179—41no—
95C2×C179—41no—
96C2×C179—41no—
97C2×C179C2×#9911yescharacteristic
98D179—82no—
99C179#212211yescharacteristic, complemented, Sylow
100#2122—1791nomaximal, Sylow
101#535—1791no—
102#413D17911yescharacteristic, complemented
103#410—1791no—
104AΣL(1,𝔽16)—1794no—
105C2×#41C2×D17911yescharacteristic
106C22×D4—1791no—
107C2×M16—21no—
108#127—3581no—
109#124—3581no—
110#121—3581no—
111#121—3581no—
112#120—1792no—
113#120—1792no—
114#117—3581no—
115#117—1791no—
116#101—21no—
117#100C2×D17912yes—
118#99—3581no—
119#99—1792no—
120C24—1792no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D17911yescharacteristic
124C2×D4—3582no—
125C2×D4—3582no—
126C2×D4—3581no—
127C2×D4—3581no—
128C2×D4—7161no—
129C2×D4—7161no—
130C2×D4C4×D17911yescharacteristic
131C2×D4D71611yescharacteristic
132C2×D4C22×D17911yescharacteristic
133C22×C4C4×D17911yescharacteristic
134C22×C4C22⋊D17911yescharacteristic
135C22×C4D71611yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—3581no—
140C4⋊C4—7161no—
141C22⋊C4—3582no—
142C22⋊C4—3582no—
143C22⋊C4—3582no—
144C22⋊C4—1792no—
145C42—3581no—
146C42—7161no—
147C23—3582no—
148C23—14322no—
149C23—7162no—
150C23(C2×D179)⋊C411yescharacteristic
151C23C2×C4×D17911yescharacteristic
152C23C2×D71611yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—14322no—
158D4—14322no—
159D4—7162no—
160D4—7161no—
161D4—7161no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—3582no—
169C2×C4—7162no—
170C2×C4(C2×D179)⋊C411yescharacteristic
171C2×C4(C2×D179)⋊C411yescharacteristic
172C2×C4(C2×D179)⋊C411yescharacteristic
173C2×C4C2×C22⋊D17911yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—14324no—
182C22—7162no—
183C22—7162no—
184C22—7162no—
185C22C2×(C2×D179)⋊C411yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—14322no—
190C4—7162no—
191C4#1915302711yescharacteristic
192C4#1915302711yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—14322no—
198C2C2×#1915302711yescharacteristic
subgroupG/H
centerC2C2×#19153027
commutatorC23×C179C22×C4
FrattiniC2×D4C22×D179
Fitting#413×C179C2
socleC2×C179C2×#99
prime radical#26039003C2
cosocleC2×D4×C179C23
hypercenter#413D179
hypocenterC179#2122
LCSG ▷ C23×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. #26038413C2semidirect
3. #26038227C2semidirect
4. #413×C179C2semidirect
5. C22×D716C4semidirect
10. #19153081C4semidirect
13. #19153072C4semidirect
99. C179#2122semidirect
102. #413D179semidirect

Maximal subgroups

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

subgroupindex
#260390032
#260384132
#260382272
#413×C1792
#2122179