Finite Group Databasebeta

← #36369688

Subgroups — #36369688

Order 22144.

384 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C4⋊#13063443C211yesmaximal, characteristic, complemented
2C2×#19149736C212yesmaximal, complemented
3C2×C22⋊#13063443C212yesmaximal, complemented
4C2×#19149734C211yesmaximal, characteristic, complemented
5C2×#19149639C211yesmaximal, characteristic
6C22×#13063443C2212yes—
7C22×#13063443C2212yes—
8C4⋊#13063443—22no—
9#19149736—24no—
10C22⋊#13063443—24no—
11#19149734—22no—
12C22×C22⋊D173C411yescharacteristic
13C23×Dic173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15#19149639—22no—
16#19149639—22no—
17C2×#14583805C411yescharacteristic
18C2×#14583803C411yescharacteristic
19C2×C4×Dic173C2211yescharacteristic
20C2×C22⋊C4×C173C411yescharacteristic
21C2×#13063443—22no—
22C2×#13063443—22no—
23C2×#13063443—22no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×#13063443D414yescomplemented
27C23×D173C2×C411yescharacteristic
28C2×C22⋊D173—22no—
29C2×C22⋊D173—22no—
30C2×C22⋊D173—22no—
31C22×Dic173—21no—
32C22×Dic173—21no—
33C22×Dic173—22no—
34C22×Dic173—22no—
35C22×Dic173C2×C411yescharacteristic
36C22×Dic173C2×C411yescharacteristic
37C22×Dic173C2311yescharacteristic
38C2×C4×D173—21no—
39C2×C4×D173—21no—
40C2×C4×D173—22no—
41C2×C4×D173—22no—
42#14583805—22no—
43#14583803—22no—
44C4×Dic173D412yes—
45C4×Dic173D412yes—
46C24×C173C2×C411yescharacteristic
47C22×C4×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C22⋊C4×C173—22no—
50#13063443—24no—
51#13063443—42no—
52C22×D173—21no—
53C22×D173—22no—
54C22×D173—21no—
55C22×D173—22no—
56C22×D173M1612yes—
57C22⋊D173—28no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—22no—
62C2×Dic173—42no—
63C2×Dic173—41no—
64C2×Dic173—41no—
65C2×Dic173C22⋊C412yes—
66C2×Dic173C2×D411yescharacteristic
67C2×Dic173C2×D412yes—
68C2×Dic173C2×D411yescharacteristic
69C2×Dic173#4112yes—
70C4×D173—42no—
71C4×D173—41no—
72C4×D173—41no—
73C23×C173—21no—
74C23×C173—22no—
75C23×C173—21no—
76C23×C173—22no—
77C23×C173M1612yes—
78C23×C173C22×C411yescharacteristic
79C2×C4×C173—22no—
80C2×C4×C173—21no—
81C2×C4×C173—22no—
82C2×C4×C173C22⋊C412yes—
83C22×C173—22no—
84C22×C173—24no—
85C22×C173—41no—
86C22×C173—41no—
87C22×C173—42no—
88C22×C173C2×C22⋊C411yescharacteristic
89C22×C173C4×D412yes—
90C22×C173C2×M1611yescharacteristic
91C22×C173C2×M1611yescharacteristic
92C22×C173#13112yes—
93C2×D173—22no—
94C2×D173—24no—
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—42no—
98C4×C173—22no—
99C4×C173—41no—
100Dic173—22no—
101Dic173—41no—
102Dic173#12112yes—
103Dic173#12411yescharacteristic
104Dic173#12711yescharacteristic
105C2×C173—42no—
106C2×C173#39011yescharacteristic
107C2×C173#40711yescharacteristic
108C2×C173#40811yescharacteristic
109C2×C173#43512yes—
110C2×C173#43512yes—
111D173—42no—
112C173#195511yescharacteristic, complemented, Sylow
113#1955—1731nomaximal, Sylow
114C2×C4×D4—1731no—
115C2×C4⋊C8—1731no—
116C2×C22⋊C8—1732no—
117C2×C22⋊C8—1732no—
118C2×#97—1731no—
119C22×D4—1731no—
120C23×C4—1731no—
121C23×C4—1731no—
122C22×C8—1732no—
123C22×C8—1732no—
124C4×D4—3462no—
125C4×D4—3462no—
126C2×C4⋊C4—1731no—
127C2×C22⋊C4—1731no—
128C2×C22⋊C4#1062192411yescharacteristic
129C2×C42—1731no—
130C4⋊C8—3462no—
131C22⋊C8—3464no—
132C22⋊C8—3464no—
133#97—3462no—
134C24—1731no—
135C24C2×#1062192411yescharacteristic
136C2×D4—3462no—
137C2×D4—3462no—
138C2×D4—3462no—
139C22×C4—3462no—
140C22×C4—3461no—
141C22×C4—3461no—
142C22×C4—3462no—
143C22×C4—3462no—
144C22×C4—3461no—
145C22×C4—3461no—
146C22×C4—3462no—
147C22×C4—1731no—
148C22×C4—1731no—
149C22×C4—1731no—
150C22×C4C2×#1062192411yescharacteristic
151C22×C4C2×#1062192411yescharacteristic
152C2×C8—3462no—
153C2×C8—3462no—
154C2×C8—3462no—
155C2×C8—3462no—
156C2×C8—3462no—
157C2×C8—1734no—
158C4⋊C4—3462no—
159C22⋊C4—22no—
160C22⋊C4—3462no—
161C42—1732no—
162C42—1732no—
163C23—21no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—3461no—
168C23—3462no—
169C23—3461no—
170C23—3462no—
171C23—1732no—
172C23#1458383612yes—
173C23C22×#1062192411yescharacteristic
174D4—3468no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—3461no—
179C2×C4—3461no—
180C2×C4—3461no—
181C2×C4—3462no—
182C2×C4—1732no—
183C2×C4—1731no—
184C2×C4—1731no—
185C2×C4—1732no—
186C2×C4—1732no—
187C2×C4—6922no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4—6922no—
191C2×C4—6921no—
192C2×C4—6921no—
193C2×C4C22⋊#1062192412yes—
194C8—3464no—
195C8—6922no—
196C22—22no—
197C22—24no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—3462no—
202C22—3464no—
203C22—6921no—
204C22—6921no—
205C22—6922no—
206C22C2×C22⋊#1062192411yescharacteristic
207C22D4×#1062192412yes—
208C22D173⋊M1611yescharacteristic
209C22C2×#1458383611yescharacteristic
210C22#1914976612yes—
211C4—22no—
212C4—41no—
213C4—3462no—
214C4—1732no—
215C4—1731no—
216C4—1731no—
217C4—6921no—
218C2—42no—
219C2—6922no—
220C2#2601244311yescharacteristic
221C2#2601246211yescharacteristic
222C2#2601246411yescharacteristic
223C2#2601252012yes—
224C2#2601252312yes—
subgroupG/H
center, FrattiniC23C22×#10621924
commutator, socleC23×C173C22×C4
FittingC2×C22⋊C4×C173C4
prime radicalC22×C22⋊D173C4
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1955
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#13063443C2semidirect
2. C2×#19149736C2semidirect
3. C2×C22⋊#13063443C2semidirect
4. C2×#19149734C2semidirect
26. C2×#13063443D4semidirect
112. C173#1955semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#130634432
C2×#191497362
C2×C22⋊#130634432
C2×#191497342
C2×#191496392
#1955173