Finite Group Databasebeta

← #37508322

Subgroups — #37508322

Order 22912.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#26039183C211yesmaximal, characteristic
2#26038807C211yesmaximal, characteristic, complemented
3#26038801C211yesmaximal, characteristic, complemented
4#26038180C211yesmaximal, characteristic, complemented
5#26038179C211yesmaximal, characteristic, complemented
6#26038178C211yesmaximal, characteristic, complemented
7#489×C179C211yesmaximal, characteristic, complemented
8#19153105C2211yescharacteristic
9#19153101C2211yescharacteristic
10#19153097—21no—
11#19153095—21no—
12#19153090—21no—
13#19153089—21no—
14#19153086—21no—
15#19153085—21no—
16#19153037—21no—
17#19153037—21no—
18#19153036—21no—
19#19153033—21no—
20#19153024C2211yescharacteristic
21#19153024C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23#124×C179C2211yescharacteristic
24#103×C179—21no—
25#102×C179—21no—
26#97×C179C2211yescharacteristic
27C2×Dic358D411yescharacteristic
28C2×D716D411yescharacteristic
29C2×C22⋊D179—21no—
30C2×C22⋊D179—21no—
31C22×Dic179—21no—
32C2×C4×D179—21no—
33C2×C179⋊C8—21no—
34C2×C179⋊C8—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179D411yescharacteristic
37C4⋊Dic179D411yescharacteristic
38Dic179⋊C4—21no—
39Dic179⋊C4—21no—
40Dic179⋊C4—21no—
41C22⋊Dic179—21no—
42(C2×D179)⋊C4—21no—
43(C2×D179)⋊C4—21no—
44(C2×D179)⋊C4—21no—
45C2×Q8×C179—21no—
46C2×Q8×C179D411yescharacteristic
47C2×D4×C179D411yescharacteristic
48C2×C8×C179—21no—
49C4⋊C4×C179—21no—
50C4⋊C4×C179—21no—
51C22⋊C4×C179—21no—
52C22⋊C4×C179—21no—
53C42×C179C2311yescharacteristic
54C22×D179—21no—
55Dic358—41no—
56D716—41no—
57C22⋊D179—81no—
58C22⋊D179—81no—
59C2×Dic179—21no—
60C2×Dic179—21no—
61C2×Dic179—21no—
62C2×Dic179—81no—
63C4×D179—81no—
64C179⋊C8—41no—
65C179⋊C8—41no—
66C23×C179—21no—
67Q8×C179—41no—
68Q8×C179—42no—
69D4×C179—41no—
70C2×C4×C179—21no—
71C2×C4×C179—41no—
72C2×C4×C179C2×D411yescharacteristic
73C2×C4×C179C2×D411yescharacteristic
74C2×C4×C179C2×D411yescharacteristic
75C8×C179—41no—
76C22×C179—41no—
77C22×C179—41no—
78C22×C179—41no—
79C22×C179#12011yescharacteristic
80C2×D179—41no—
81C2×D179—41no—
82C2×D179—41no—
83C4×C179—21no—
84C4×C179—21no—
85C4×C179—21no—
86C4×C179—41no—
87C4×C179—81no—
88Dic179—41no—
89Dic179—41no—
90Dic179—41no—
91C2×C179—81no—
92C2×C179#45411yescharacteristic
93C2×C179#45511yescharacteristic
94C2×C179#45511yescharacteristic
95D179—81no—
96C179#166211yescharacteristic, complemented, Sylow
97#1662—1791nomaximal, Sylow
98#562—1791no—
99#491—1791no—
100#489—1791no—
101#489D17911yescharacteristic, complemented
102#330—1791no—
103#329—1791no—
104#329—1791no—
105C4⋊Q8C2×D17911yescharacteristic
106#125—1791no—
107#124—1791no—
108#124C2×D17911yescharacteristic
109#123—3581no—
110#123—3581no—
111C22⋊Q8—3581no—
112C22⋊Q8—3581no—
113#121—3581no—
114#121—3581no—
115#103—21no—
116#103—3581no—
117#103—3581no—
118#103—3581no—
119#102—21no—
120#102—3581no—
121#97—1791no—
122#97—1791no—
123#97C2×D17911yescharacteristic
124C2×Q8—21no—
125C2×Q8—1791no—
126C2×Q8C22⋊D17911yescharacteristic
127C2×D4—3581no—
128C2×D4—3581no—
129C2×D4—1791no—
130C2×D4C22⋊D17911yescharacteristic
131C22×C4—3581no—
132C22×C4—3581no—
133C2×C8—21no—
134C2×C8—3581no—
135C2×C8—3581no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—3581no—
139C4⋊C4—3581no—
140C4⋊C4—3581no—
141C4⋊C4—3581no—
142C4⋊C4—1791no—
143C4⋊C4—1791no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—3581no—
147C22⋊C4—3581no—
148C22⋊C4—3581no—
149C22⋊C4—3581no—
150C42C22×D17911yescharacteristic
151C23—21no—
152C23—3581no—
153Q8—41no—
154Q8—42no—
155Q8—7161no—
156D4—41no—
157D4—14321no—
158D4—14321no—
159D4—7161no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—3581no—
163C2×C4—3581no—
164C2×C4—3581no—
165C2×C4—14321no—
166C2×C4—14321no—
167C2×C4D4×D17911yescharacteristic
168C2×C4D4×D17911yescharacteristic
169C2×C4C2×C22⋊D17911yescharacteristic
170C8—41no—
171C8—7161no—
172C8—7161no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—7161no—
177C22—7161no—
178C22—7161no—
179C22#1915308111yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—81no—
185C4—7161no—
186C4—7161no—
187C4—7161no—
188C2—81no—
189C2—14321no—
190C2#2603861011yescharacteristic
191C2#2603861511yescharacteristic
192C2#2603861611yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#489×C179C2
socleC22×C179#120
prime radical#26039183C2
hypercenter#489D179
hypocenterC179#1662
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #26038807C2semidirect
3. #26038801C2semidirect
4. #26038180C2semidirect
5. #26038179C2semidirect
6. #26038178C2semidirect
7. #489×C179C2semidirect
96. C179#1662semidirect
101. #489D179semidirect

Maximal subgroups

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

subgroupindex
#260391832
#260388072
#260388012
#260381802
#260381792
#260381782
#489×C1792
#1662179