Finite Group Databasebeta

← #37508232

Subgroups — #37508232

Order 22912.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039092C211yesmaximal, characteristic
2#26038716C211yesmaximal, characteristic, complemented
3#26038641C211yesmaximal, characteristic, complemented
4#26038185C211yesmaximal, characteristic, complemented
5#26038168C211yesmaximal, characteristic, complemented
6#26038168C211yesmaximal, characteristic, complemented
7#500×C179C211yesmaximal, characteristic, complemented
8C2×#15668260—21no—
9C2×#15668246—21no—
10#19153098—22no—
11#19153089—21no—
12#19153088—21no—
13C4×D716C2211yescharacteristic
14C4×D716C2211yescharacteristic
15C4⋊C4×D179—21no—
16#19153046—21no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18C4⋊(C179⋊C8)C2211yescharacteristic
19#19153037—21no—
20#19153033—21no—
21C4⋊Q8×C179C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23#107×C179—22no—
24C4×C8×C179C2211yescharacteristic
25#15668260—42no—
26#15668260—42no—
27C2×D716—22no—
28C2×D716D411yescharacteristic
29C2×D716D411yescharacteristic
30C2×C4×D179—22no—
31C2×C4×D179—21no—
32C2×C4×D179—21no—
33#15668246—42no—
34C2×C179⋊C8—21no—
35C2×C179⋊C8—21no—
36C4⋊Dic179D411yescharacteristic
37C4⋊Dic179D411yescharacteristic
38Dic179⋊C4—22no—
39(C2×D179)⋊C4—22no—
40(C2×D179)⋊C4—21no—
41(C2×D179)⋊C4—21no—
42C2×Q8×C179—21no—
43C2×Q8×C179—21no—
44C2×C8×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179—21no—
48C4⋊C4×C179D412yes—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51C22×D179—21no—
52D716—21no—
53D716—21no—
54D716—21no—
55D716—44no—
56D716D812yes—
57C2×Dic179—21no—
58C2×Dic179—21no—
59C4×D179—44no—
60C4×D179—41no—
61C4×D179—41no—
62C4×D179—41no—
63C4×D179—41no—
64C179⋊C8—41no—
65C179⋊C8—41no—
66Q8×C179—41no—
67Q8×C179—41no—
68Q8×C179—41no—
69Q8×C179—41no—
70C2×C4×C179—22no—
71C2×C4×C179—41no—
72C2×C4×C179C2×D411yescharacteristic
73C2×C4×C179C2×D411yescharacteristic
74C2×C4×C179C2×D411yescharacteristic
75C8×C179—22no—
76C22×C179#12011yescharacteristic
77C2×D179—22no—
78C2×D179—41no—
79C2×D179—41no—
80C2×D179—41no—
81C2×D179—41no—
82C2×D179—41no—
83C4×C179—21no—
84C4×C179—42no—
85C4×C179—81no—
86C4×C179C2×D811yescharacteristic
87C4×C179#13511yescharacteristic
88C4×C179AGL(1,ℤ/8)11yescharacteristic
89C4×C179#13711yescharacteristic
90Dic179—41no—
91Dic179—41no—
92C2×C179#44711yescharacteristic
93C2×C179#45211yescharacteristic
94C2×C179#45611yescharacteristic
95D179—42no—
96D179—81no—
97C179#164711yescharacteristic, complemented, Sylow
98#1647—1791nomaximal, Sylow
99#547—1791no—
100#500D17911yescharacteristic, complemented
101#474—1791no—
102#461—1791no—
103#332—1791no—
104D4⋊C8—1791no—
105D4⋊C8—1791no—
106C2×#41—3581no—
107C2×SD16—3581no—
108C4⋊Q8C2×D17911yescharacteristic
109C4⋊Q8C2×D17911yescharacteristic
110#123—3582no—
111C22⋊Q8—3581no—
112#121—3581no—
113C4×D4—1791no—
114C4×D4—1791no—
115C2×C4⋊C4—3581no—
116#107—22no—
117#107—3581no—
118C4⋊C8—1791no—
119C4⋊C8—1791no—
120#103—3581no—
121#102—3581no—
122C4×C8C2×D17911yescharacteristic
123#41—7162no—
124#41—7162no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—3582no—
128C2×D4—1791no—
129C2×D4—1791no—
130C22×C4—3582no—
131C22×C4—3581no—
132C22×C4—3581no—
133SD16—7162no—
134C2×C8—21no—
135C2×C8—3581no—
136C2×C8—3581no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—3582no—
141C4⋊C4—1791no—
142C4⋊C4—1791no—
143C4⋊C4C22⋊D17912yes—
144C22⋊C4—3582no—
145C22⋊C4—3581no—
146C22⋊C4—3581no—
147C42C22×D17911yescharacteristic
148C23—3581no—
149C23—3581no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—3581no—
155D4—3581no—
156D4—3581no—
157D4—1792no—
158D4—7164no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—3581no—
162C2×C4—3581no—
163C2×C4—7164no—
164C2×C4—7161no—
165C2×C4—7161no—
166C2×C4—7161no—
167C2×C4—7161no—
168C2×C4D4×D17911yescharacteristic
169C2×C4D4×D17911yescharacteristic
170C2×C4C2×C22⋊D17911yescharacteristic
171C8—22no—
172C8—7161no—
173C8—7161no—
174C22—3582no—
175C22—7161no—
176C22—7161no—
177C22—7161no—
178C22—7161no—
179C22—7161no—
180C22#1915308111yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—7161no—
185C4—7161no—
186C4D8×D17911yescharacteristic
187C4#1915313611yescharacteristic
188C4#1915313711yescharacteristic
189C4#1915314511yescharacteristic
190C2—14321no—
191C2—7162no—
192C2#2603857111yescharacteristic
193C2#2603860011yescharacteristic
194C2#2603862111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#500×C179C2
socleC22×C179#120
prime radical#26039092C2
hypercenter#500D179
hypocenterC179#1647
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #26038716C2semidirect
3. #26038641C2semidirect
4. #26038185C2semidirect
5. #26038168C2semidirect
6. #26038168C2semidirect
7. #500×C179C2semidirect
97. C179#1647semidirect
100. #500D179semidirect

Maximal subgroups

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

subgroupindex
#260390922
#260387162
#260386412
#260381852
#260381682
#500×C1792
#1647179