Finite Group Databasebeta

← #36352582

Subgroups — #36352582

Order 22144.

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

normal subgroups only

This group has 288 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#26011809C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#399×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173—21no—
7#19149663—22no—
8#19149653—21no—
9#19149653C2212yes—
10#19149649—22no—
11#19149610—22no—
12#19149599—22no—
13C22⋊#13063436C2212yescomplemented
14C2×C4⋊C4×C173—21no—
15C2×C4⋊C4×C173C2211yescharacteristic, complemented
16C2×C22⋊C4×C173—21no—
17C2×C22⋊C4×C173C2211yescharacteristic
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—
27C2×#13063436—22no—
28C4⋊Dic173—21no—
29C4⋊Dic173D412yes—
30#14583803—22no—
31#14583803—22no—
32C24×C173D411yescharacteristic
33C22×C4×C173—21no—
34C22×C4×C173—21no—
35C22×C4×C173D411yescharacteristic
36C22×C4×C173C2311yescharacteristic
37C4⋊C4×C173—22no—
38C4⋊C4×C173—42no—
39C22⋊C4×C173—22no—
40C22⋊C4×C173—42no—
41C22×D173—22no—
42C22×D173—42no—
43C22×D173—81no—
44C22×D173—81no—
45D692—42no—
46D692—81no—
47C22⋊D173—44no—
48C22⋊D173—42no—
49C22⋊D173—42no—
50C2×Dic173—22no—
51C2×Dic173—42no—
52#13063436—42no—
53C23×C173—21no—
54C23×C173—21no—
55C23×C173—41no—
56C23×C173—41no—
57C23×C173SD1612yes—
58C23×C173C2×D411yescharacteristic
59C2×C4×C173—22no—
60C2×C4×C173—21no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173—41no—
64C2×C4×C173—41no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D412yes—
69C22×C173—21no—
70C22×C173—21no—
71C22×C173—22no—
72C22×C173—42no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173—81no—
76C22×C173#12011yescharacteristic
77C22×C173C2×SD1611yescharacteristic
78C22×C173AGL(1,ℤ/8)11yescharacteristic
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—42no—
82C2×D173—84no—
83C4×C173—22no—
84C4×C173—42no—
85C4×C173—41no—
86C4×C173—81no—
87Dic173—42no—
88C2×C173—21no—
89C2×C173—21no—
90C2×C173—42no—
91C2×C173#45011yescharacteristic
92C2×C173#45311yescharacteristic
93C2×C173#45811yescharacteristic
94D173—82no—
95C173#161311yescharacteristic, complemented, Sylow
96#1613—1731nomaximal, Sylow
97#534—1731no—
98#481—1732no—
99#399D17311yescharacteristic, complemented
100#327—1732no—
101#323—1731no—
102C22×D4—3461no—
103#123—3462no—
104#121—3461no—
105#121—1732no—
106#120—3462no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D17311yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D17311yescharacteristic
111#106—3462no—
112#102—3462no—
113C22⋊C8—1732no—
114#95C2×D17311yescharacteristic, complemented
115C24—3461no—
116C24C22⋊D17311yescharacteristic
117C2×D4—3461no—
118C2×D4—3462no—
119C2×D4—3462no—
120C2×D4—1732no—
121C2×D4—6922no—
122C2×D4—6922no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—3461no—
126C22×C4C22⋊D17311yescharacteristic
127C22×C4C22×D17311yescharacteristic
128C2×C8—3462no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—3461no—
132C4⋊C4—1732no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—3462no—
136C22⋊C4—3462no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—3462no—
142C23—13841no—
143C23—13841no—
144C23—6922no—
145C23#1458381312yes—
146C23C2×C22⋊D17311yescharacteristic
147D4—13841no—
148D4—6922no—
149D4—6924no—
150D4—6922no—
151D4—6922no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3462no—
162C2×C4—6922no—
163C2×C4D4×D17312yes—
164C8—6922no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—13844no—
173C22—6922no—
174C22—6922no—
175C22—6922no—
176C22#1914964711yescharacteristic
177C22C2×#1458381311yescharacteristic
178C22#1914970411yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—6922no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—13842no—
188C2#2601164111yescharacteristic
189C2#2601165811yescharacteristic
190C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#399×C173C2
socleC22×C173#120
prime radical#26012053C2
hypercenter#399D173
hypocenterC173#1613
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #26011809C2semidirect
3. #26011224C2semidirect
4. #26011215C2semidirect
5. #399×C173C2semidirect
13. C22⋊#13063436C22semidirect
15. C2×C4⋊C4×C173C22semidirect
18. #95×C173C22semidirect
95. C173#1613semidirect
99. #399D173semidirect
108. C2×C4⋊C4C2×D173semidirect
114. #95C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260118092
#260112242
#260112152
#399×C1732
#1613173