Finite Group Databasebeta

← #36369023

Subgroups — #36369023

Order 22144.

264 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 58 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#26012514C211yesmaximal, characteristic, complemented
2C4×#14583836C211yesmaximal, characteristic, complemented
3#26012347C212yesmaximal, complemented
4#26012346C212yesmaximal, complemented
5#124×D173C211yesmaximal, characteristic
6C2×#14583836—21no—
7#19149746—21no—
8#19149736—21no—
9#19149734C2212yescomplemented
10#19149734C2212yescomplemented
11C4×#13063443—21no—
12C2×Q8×D173C411yescharacteristic
13C2×D4×D173C411yescharacteristic
14#19149668C2211yescharacteristic
15#19149667C411yescharacteristic
16#19149666C2211yescharacteristic
17#19149665—21no—
18#19149665—21no—
19C22⋊C4×D173—21no—
20C22⋊C4×D173—21no—
21C42×D173C2211yescharacteristic
22#124×C173C411yescharacteristic
23#14583836—24no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C23×D173—21no—
27C2×Dic346—21no—
28C2×Dic346C2×C411yescharacteristic
29Q8×D173—42no—
30C2×D692C2×C411yescharacteristic
31C2×C22⋊D173—21no—
32D4×D173—42no—
33C2×C4×D173—21no—
34C2×C4×D173C2×C411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39#14583803—21no—
40#14583803—21no—
41#14583803—21no—
42#14583803—21no—
43C4×Dic173D411yescharacteristic
44C4×Dic173D411yescharacteristic
45C4×Dic173C2311yescharacteristic
46C2×Q8×C173C2×C411yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C42×C173C2×C411yescharacteristic
51#13063443—24no—
52#13063443—42no—
53C22×D173—21no—
54C22×D173—42no—
55C22×D173—42no—
56C22×D173—42no—
57C22×D173C22⋊C411yescharacteristic
58Dic346—42no—
59Dic346—41no—
60D692—41no—
61C22⋊D173—42no—
62C2×Dic173—21no—
63C2×Dic173C22⋊C411yescharacteristic
64C2×Dic173C2×D411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C4×D173—22no—
67C4×D173—22no—
68C4×D173—41no—
69C4×D173—42no—
70C23×C173—21no—
71Q8×C173—41no—
72D4×C173—41no—
73C2×C4×C173—21no—
74C2×C4×C173C22⋊C411yescharacteristic
75C2×C4×C173C22⋊C411yescharacteristic
76C2×C4×C173C22×C411yescharacteristic
77C22×C173—41no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173C2×C22⋊C411yescharacteristic
81C2×D173—21no—
82C2×D173—21no—
83C2×D173—41no—
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—84no—
87C2×D173#10412yes—
88C4×C173—21no—
89C4×C173—21no—
90C4×C173—21no—
91C4×C173—41no—
92Dic173—21no—
93Dic173—21no—
94Dic173—41no—
95Dic173#10012yes—
96C2×C173—81no—
97C2×C173C2×#10011yescharacteristic
98C2×C173C2×#10411yescharacteristic
99C2×C173#42111yescharacteristic
100D173—22no—
101D173—81no—
102C173#152711yescharacteristic, complemented, Sylow
103#1527—1731nomaximal, Sylow
104C2×#124—1731no—
105#432—1731no—
106C4×M16—1731no—
107#329—1732no—
108#329—1732no—
109C22×Q8—1731no—
110C22×D4—1731no—
111C2×M16—3461no—
112#124—3461no—
113#124—3461no—
114#124—1731no—
115#124—1731no—
116#124—1731no—
117#124#1062192411yescharacteristic
118C2×C22⋊C4—3461no—
119C2×C22⋊C4—3461no—
120C2×C42—1731no—
121C4⋊C8—3461no—
122C22⋊C8—3461no—
123#97—1732no—
124#97—1732no—
125C4×C8—3461no—
126C24—3461no—
127C2×Q8—3461no—
128C2×Q8—1731no—
129C2×Q8—6922no—
130C2×Q8C2×#1062192411yescharacteristic
131C2×D4—3461no—
132C2×D4—1731no—
133C2×D4—6922no—
134C2×D4C2×#1062192411yescharacteristic
135C22×C4—3461no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4—1731no—
139M16—3464no—
140C2×C8—3462no—
141C2×C8—3462no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—3461no—
145C22⋊C4—3461no—
146C22⋊C4—3461no—
147C22⋊C4—3461no—
148C22⋊C4—3461no—
149C22⋊C4—3461no—
150C42—1731no—
151C42—1731no—
152C42—1731no—
153C42C2×#1062192411yescharacteristic
154C23—21no—
155C23—3461no—
156C23—1731no—
157C23—6922no—
158C23—6922no—
159C23—6922no—
160Q8—41no—
161Q8—6921no—
162Q8—6922no—
163D4—41no—
164D4—6921no—
165D4—6922no—
166C2×C4—21no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—3461no—
170C2×C4—1731no—
171C2×C4—1731no—
172C2×C4—1731no—
173C2×C4—6921no—
174C2×C4—6922no—
175C2×C4C22⋊#1062192411yescharacteristic
176C2×C4C22⋊#1062192411yescharacteristic
177C2×C4C22×#1062192411yescharacteristic
178C8—3464no—
179C8—6922no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—3461no—
184C22—3461no—
185C22—13844no—
186C22—1732no—
187C22—6921no—
188C22—6921no—
189C22—6921no—
190C22C2×C22⋊#1062192411yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—41no—
195C4—3461no—
196C4—3461no—
197C4—1732no—
198C4—6921no—
199C2—81no—
200C2—3462no—
201C2—13841no—
202C2D173⋊#10011yescharacteristic
203C2D173⋊#10411yescharacteristic
204C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C4
cosocleC4×Dic173C23
hypercenter#124#10621924
hypocenterC173#1527
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012514C2semidirect
2. C4×#14583836C2semidirect
3. #26012347C2semidirect
4. #26012346C2semidirect
9. #19149734C22semidirect
10. #19149734C22semidirect
102. C173#1527semidirect

Maximal subgroups

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

subgroupindex
#260125142
C4×#145838362
#260123472
#260123462
#124×D1732
#1527173