Finite Group Databasebeta

← #36352514

Subgroups — #36352514

Order 22144.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012053C211yesmaximal, characteristic
2#26011630C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#393×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173—21no—
7C2×#14583813—22no—
8#19149663—22no—
9#19149653—21no—
10#19149653C2212yes—
11#19149649—22no—
12#19149603—22no—
13C22⋊#13063436C2212yescomplemented
14C22×Q8×C173C2211yescharacteristic, complemented
15C2×C22⋊C4×C173—21no—
16C2×C22⋊C4×C173C2211yescharacteristic
17#95×C173—21no—
18#95×C173C2211yescharacteristic, complemented
19C23×D173—21no—
20C2×D692—21no—
21C2×D692D412yes—
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—42no—
25C2×C22⋊D173—42no—
26C22×Dic173—21no—
27#14583813—44no—
28C2×#13063436—22no—
29C4⋊Dic173—21no—
30C4⋊Dic173D412yes—
31#14583803—22no—
32#14583803—22no—
33C24×C173D411yescharacteristic
34C2×Q8×C173—22no—
35C2×Q8×C173—42no—
36C22×C4×C173—21no—
37C22×C4×C173—21no—
38C22×C4×C173D411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C22⋊C4×C173—22no—
41C22⋊C4×C173—42no—
42C22×D173—22no—
43C22×D173—42no—
44C22×D173—81no—
45C22×D173—81no—
46D692—42no—
47D692—81no—
48C22⋊D173—44no—
49C22⋊D173—42no—
50C22⋊D173—42no—
51C2×Dic173—22no—
52C2×Dic173—42no—
53#13063436—42no—
54C23×C173—21no—
55C23×C173—21no—
56C23×C173—41no—
57C23×C173—41no—
58C23×C173SD1612yescomplemented
59C23×C173C2×D411yescharacteristic
60Q8×C173—24no—
61Q8×C173—42no—
62C2×C4×C173—22no—
63C2×C4×C173—21no—
64C2×C4×C173—41no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173—41no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173C2×D412yes—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173—22no—
75C22×C173—42no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—81no—
79C22×C173#12011yescharacteristic
80C22×C173C2×SD1611yescharacteristic
81C22×C173AGL(1,ℤ/8)11yescharacteristic
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—84no—
86C4×C173—22no—
87C4×C173—41no—
88C4×C173—42no—
89C4×C173—81no—
90Dic173—42no—
91C2×C173—21no—
92C2×C173—21no—
93C2×C173—42no—
94C2×C173#45011yescharacteristic
95C2×C173#45511yescharacteristic
96C2×C173B(4,𝔽2)11yescharacteristic
97D173—82no—
98C173#160111yescharacteristic, complemented, Sylow
99#1601—1731nomaximal, Sylow
100#534—1731no—
101#448—1732no—
102#393D17311yescharacteristic, complemented
103#327—1732no—
104#323—1731no—
105C22×Q8C2×D17311yescharacteristic, complemented
106C22×D4—3461no—
107C2×SD16—3462no—
108#123—3462no—
109#121—3461no—
110#121—1732no—
111#120—3462no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×D17311yescharacteristic
114#103—3462no—
115C22⋊C8—1732no—
116#95—21no—
117#95C2×D17311yescharacteristic, complemented
118C24—3461no—
119C24C22⋊D17311yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—3461no—
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—1732no—
126C2×D4—6922no—
127C2×D4—6922no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—3461no—
131C22×C4C22⋊D17311yescharacteristic
132C22×C4C22×D17311yescharacteristic
133SD16—6924no—
134C2×C8—3462no—
135C4⋊C4—3461no—
136C4⋊C4—1732no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—3462no—
140C22⋊C4—3462no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—3462no—
146C23—13841no—
147C23—13841no—
148C23—6922no—
149C23#1458381312yescomplemented
150C23C2×C22⋊D17311yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—13841no—
154D4—6922no—
155D4—6924no—
156D4—6922no—
157D4—6922no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3462no—
168C2×C4—6922no—
169C2×C4D4×D17312yes—
170C8—6922no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—13844no—
179C22—6922no—
180C22—6922no—
181C22—6922no—
182C22#1914964711yescharacteristic
183C22C2×#1458381311yescharacteristic
184C22#1914970411yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4—6922no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—13842no—
194C2#2601164111yescharacteristic
195C2#2601166811yescharacteristic
196C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#393×C173C2
socleC22×C173#120
prime radical#26012053C2
hypercenter#393D173
hypocenterC173#1601
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #26011630C2semidirect
3. #26011224C2semidirect
4. #26011215C2semidirect
5. #393×C173C2semidirect
13. C22⋊#13063436C22semidirect
14. C22×Q8×C173C22semidirect
18. #95×C173C22semidirect
58. C23×C173SD16semidirect
98. C173#1601semidirect
102. #393D173semidirect
105. C22×Q8C2×D173semidirect
117. #95C2×D173semidirect
149. C23#14583813semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260116302
#260112242
#260112152
#393×C1732
#1601173