Finite Group Databasebeta

← #36354607

Subgroups — #36354607

Order 22144.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19149707C212yesmaximal, complemented
2#26012072C211yesmaximal, characteristic, complemented
3C2×#19149605C212yesmaximal, complemented
4#26011392C211yesmaximal, characteristic, complemented
5#343×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2212yes—
7C22×D692C2211yescharacteristic
8#19149707—24no—
9#19149707—24no—
10C2×#14583815—22no—
11C2×D1384—22no—
12#19149659—22no—
13#19149654—22no—
14#19149653—22no—
15C2×#14583803—22no—
16C2×C4×Dic173C2211yescharacteristic
17#19149605—24no—
18C2×M16×C173C2212yes—
19C2×C4⋊C4×C173C2211yescharacteristic
20C23×D173—21no—
21#14583826—22no—
22#14583826—22no—
23#14583826—22no—
24#14583826—22no—
25C2×Dic346D412yes—
26C2×D692—22no—
27C2×D692—22no—
28C2×D692—22no—
29C2×D692—21no—
30C2×D692D412yes—
31C2×D692D412yescomplemented
32C2×C22⋊D173—22no—
33C22×Dic173—21no—
34C2×C4×D173—22no—
35#14583815—28no—
36D1384—28no—
37C4⋊Dic173—22no—
38#14583803—24no—
39#14583803—44no—
40C4×Dic173—22no—
41C22×C4×C173D412yes—
42C22×C4×C173C2311yescharacteristic
43M16×C173—22no—
44M16×C173D414yescomplemented
45C2×C8×C173—22no—
46C4⋊C4×C173—22no—
47C22×D173—22no—
48C22×D173—22no—
49C22×D173—42no—
50C22×D173—42no—
51C22×D173—41no—
52C22×D173—41no—
53Dic346—22no—
54Dic346—22no—
55D692—22no—
56D692—24no—
57D692—22no—
58D692—42no—
59D692—82no—
60C22⋊D173—42no—
61C22⋊D173—42no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C4×D173—42no—
67C4×D173—42no—
68C23×C173C2×D411yescharacteristic
69C2×C4×C173—22no—
70C2×C4×C173—42no—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D412yes—
74C2×C4×C173#4111yescharacteristic
75C8×C173—24no—
76C22×C173—21no—
77C22×C173—21no—
78C22×C173#12011yescharacteristic
79C22×C173#12111yescharacteristic
80C22×C173#12711yescharacteristic
81C2×D173—44no—
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—42no—
86C2×D173—82no—
87C2×D173—81no—
88C2×D173—81no—
89C4×C173—42no—
90C4×C173#12012yes—
91C4×C173#12112yes—
92Dic173—42no—
93Dic173—42no—
94C2×C173—21no—
95C2×C173—21no—
96C2×C173#39211yescharacteristic
97C2×C173#45311yescharacteristic
98C2×C173#45411yescharacteristic
99D173—82no—
100D173—82no—
101C173#200611yescharacteristic, complemented, Sylow
102#2006—1731nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1732no—
104#537—1731no—
105C2×#104—1732no—
106#390—1731no—
107#343D17311yescharacteristic, complemented
108C2×#41—1732no—
109C22×D4—1731no—
110AGL(1,ℤ/8)—3464no—
111AGL(1,ℤ/8)—3464no—
112C2×SD16—3462no—
113C2×D8—3462no—
114C2×M16C2×D17312yes—
115C22⋊Q8—3462no—
116#121—3462no—
117#121—3462no—
118C2×C4⋊C4C2×D17311yescharacteristic
119C2×C22⋊C4—3462no—
120C2×C42—1731no—
121#104—3464no—
122C24—3461no—
123#41—3462no—
124#41—3462no—
125#41—3462no—
126#41—3462no—
127C2×Q8—1732no—
128C2×D4—3462no—
129C2×D4—3462no—
130C2×D4—3462no—
131C2×D4—3462no—
132C2×D4—3461no—
133C2×D4—1732no—
134C2×D4—1732no—
135C22×C4—3461no—
136C22×C4—3462no—
137C22×C4D69212yes—
138C22×C4C22×D17311yescharacteristic
139SD16—3468no—
140D8—3468no—
141M16—22no—
142M16D69214yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3462no—
146C22⋊C4—3464no—
147C22⋊C4—6924no—
148C42—3462no—
149C23—3462no—
150C23—3462no—
151C23—6922no—
152C23—6922no—
153C23—6921no—
154C23—6921no—
155C23C2×D69211yescharacteristic
156Q8—3462no—
157Q8—3462no—
158D4—3462no—
159D4—3464no—
160D4—3462no—
161D4—13842no—
162D4—6922no—
163D4—6922no—
164D4—6922no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—6921no—
170C2×C4—6921no—
171C2×C4—6922no—
172C2×C4—6922no—
173C2×C4D4×D17311yescharacteristic
174C2×C4D4×D17312yes—
175C2×C4C2×D69212yes—
176C2×C4#1458382711yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—13842no—
181C22—13841no—
182C22—13841no—
183C22—6924no—
184C22—6922no—
185C22—6922no—
186C22—6922no—
187C22—6922no—
188C22#1914964911yescharacteristic
189C22#1914965411yescharacteristic
190C22#1914967611yescharacteristic
191C4—42no—
192C4—6922no—
193C4—6922no—
194C4#1914964912yes—
195C4#1914965412yes—
196C2—21no—
197C2—21no—
198C2—13842no—
199C2—13842no—
200C2#2601140111yescharacteristic
201C2#2601166111yescharacteristic
202C2#2601166611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#343×C173C2
socleC22×C173#120
hypercenter#343D173
hypocenterC173#2006
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. #26012072C2semidirect
3. C2×#19149605C2semidirect
4. #26011392C2semidirect
5. #343×C173C2semidirect
31. C2×D692D4semidirect
44. M16×C173D4semidirect
101. C173#2006semidirect
107. #343D173semidirect
142. M16D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
#260120722
C2×#191496052
#260113922
#343×C1732
#2006173