Finite Group Databasebeta

← #36352549

Subgroups — #36352549

Order 22144.

280 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 57 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#26011802C212yesmaximal, complemented
2#26011416C211yesmaximal, characteristic, complemented
3#26011225C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#534×C173C211yesmaximal, characteristic
6C2×C4⋊Dic173C2211yescharacteristic, complemented
7C2×C22⋊Dic173—21no—
8#19149612—22no—
9#19149600—22no—
10C22⋊#13063436C2212yescomplemented
11#19149586—21no—
12#19149586C2211yescharacteristic, complemented
13C22×D4×C173—21no—
14#123×C173—22no—
15#121×C173—21no—
16#121×C173C2212yes—
17#120×C173—22no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C22×Dic173—21no—
20C22×Dic173—21no—
21C2×#13063436—22no—
22C4⋊Dic173—22no—
23C22⋊Dic173—42no—
24C24×C173—21no—
25C24×C173D411yescharacteristic
26C2×D4×C173—22no—
27C2×D4×C173—22no—
28C2×D4×C173—21no—
29C2×D4×C173—42no—
30C2×D4×C173—42no—
31C2×D4×C173D412yes—
32C22×C4×C173—21no—
33C22×C4×C173D411yescharacteristic
34C22×C4×C173C2311yescharacteristic
35C4⋊C4×C173—21no—
36C4⋊C4×C173D412yes—
37C22⋊C4×C173—22no—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—22no—
40C2×Dic173—41no—
41C2×Dic173—41no—
42C2×Dic173—41no—
43C2×Dic173—41no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46#13063436—42no—
47C23×C173—21no—
48C23×C173—21no—
49C23×C173—22no—
50C23×C173—41no—
51C23×C173—41no—
52C23×C173—42no—
53C23×C173—81no—
54C23×C173—81no—
55C23×C173D812yes—
56C23×C173C2×D411yescharacteristic
57D4×C173—42no—
58D4×C173—44no—
59D4×C173—42no—
60D4×C173—42no—
61D4×C173—81no—
62C2×C4×C173—21no—
63C2×C4×C173—22no—
64C2×C4×C173—41no—
65C2×C4×C173—42no—
66C2×C4×C173—41no—
67C2×C4×C173C2×D412yes—
68C22×C173—21no—
69C22×C173—21no—
70C22×C173—22no—
71C22×C173—42no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—42no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173—81no—
78C22×C173—84no—
79C22×C173#12011yescharacteristic
80C22×C173C2×D811yescharacteristic
81C22×C173AGL(1,ℤ/8)11yescharacteristic
82C4×C173—22no—
83C4×C173—42no—
84C4×C173—41no—
85Dic173—81no—
86Dic173—81no—
87C2×C173—21no—
88C2×C173—21no—
89C2×C173—42no—
90C2×C173—82no—
91C2×C173#44711yescharacteristic
92C2×C173#45511yescharacteristic
93C2×C173#45811yescharacteristic
94C173#160711yescharacteristic, complemented, Sylow
95#1607—1731nomaximal, Sylow
96#534D17311yescharacteristic
97#480—1732no—
98#397—1731no—
99#327—1732no—
100#323—1731no—
101C22×D4—21no—
102#123—22no—
103#121—21no—
104#121C2×D17312yes—
105#120—22no—
106C2×C4⋊C4—1731no—
107C2×C22⋊C4—3461no—
108C2×C22⋊C4C2×D17311yescharacteristic
109#107—3462no—
110#102—3462no—
111C22⋊C8—1732no—
112#95—3461no—
113#95—1731no—
114C24—21no—
115C24C22⋊D17311yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4C22⋊D17312yes—
122C22×C4—21no—
123C22×C4—3461no—
124C22×C4—3461no—
125C22×C4C22⋊D17311yescharacteristic
126C22×C4C22×D17311yescharacteristic
127C2×C8—3462no—
128C4⋊C4—21no—
129C4⋊C4—3462no—
130C4⋊C4C22⋊D17312yes—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—6922no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—81no—
142C23—81no—
143C23#1458381112yes—
144C23C2×C22⋊D17311yescharacteristic
145D4—42no—
146D4—44no—
147D4—42no—
148D4—42no—
149D4—81no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—6921no—
156C2×C4—6921no—
157C2×C4—6921no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4—6921no—
161C2×C4C2×C22⋊D17312yes—
162C8—6922no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—84no—
174C22#1914964811yescharacteristic
175C22C2×#1458381111yescharacteristic
176C22#1914970511yescharacteristic
177C4—22no—
178C4—41no—
179C4—42no—
180C4—13841no—
181C4—13841no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#2601162511yescharacteristic
187C2#2601166911yescharacteristic
188C2#2601168111yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting, prime radical#534×C173C2
socleC22×C173#120
hypercenter#534D173
hypocenterC173#1607
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #26011802C2semidirect
2. #26011416C2semidirect
3. #26011225C2semidirect
4. #26011215C2semidirect
6. C2×C4⋊Dic173C22semidirect
10. C22⋊#13063436C22semidirect
12. #19149586C22semidirect
94. C173#1607semidirect

Maximal subgroups

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

subgroupindex
#260118022
#260114162
#260112252
#260112152
#534×C1732
#1607173