Finite Group Databasebeta

← #36354612

Subgroups — #36354612

Order 22144.

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

normal subgroups only

This group has 370 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×#19149712C212yesmaximal, complemented
2#26012072C211yesmaximal, characteristic
3C2×#19149605C212yesmaximal, complemented
4#26011396C211yesmaximal, characteristic, complemented
5#343×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2212yes—
7C22×Dic346C2211yescharacteristic
8#19149712—24no—
9#19149712—24no—
10C2×Dic692—22no—
11C2×#14583815—22no—
12#19149659—22no—
13#19149654—22no—
14#19149653—22no—
15C2×C4×Dic173C2211yescharacteristic
16#19149605—24no—
17#19149586—22no—
18C2×M16×C173C2212yes—
19C2×C4⋊C4×C173C2211yescharacteristic
20#14583826—22no—
21#14583826—22no—
22#14583826—22no—
23#14583826—22no—
24C2×Dic346—22no—
25C2×Dic346—22no—
26C2×Dic346—21no—
27C2×Dic346D412yes—
28C2×Dic346D412yescomplemented
29C2×D692—22no—
30C2×D692D412yes—
31C2×C22⋊D173—22no—
32C22×Dic173—21no—
33C22×Dic173—21no—
34C2×C4×D173—22no—
35Dic692—28no—
36#14583815—28no—
37C4⋊Dic173—22no—
38#14583803—24no—
39C4×Dic173—22no—
40C22×C4×C173D412yes—
41C22×C4×C173C2311yescharacteristic
42M16×C173—22no—
43M16×C173D414yescomplemented
44C2×C8×C173—22no—
45C4⋊C4×C173—22no—
46C22×D173—22no—
47Dic346—24no—
48Dic346—22no—
49Dic346—22no—
50Dic346—42no—
51D692—22no—
52D692—22no—
53D692—82no—
54C22⋊D173—42no—
55C22⋊D173—42no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—41no—
60C2×Dic173—41no—
61C2×Dic173—41no—
62C2×Dic173—41no—
63C4×D173—42no—
64C4×D173—42no—
65C23×C173C2×D411yescharacteristic
66C2×C4×C173—22no—
67C2×C4×C173—42no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D412yes—
70C2×C4×C173C2×D412yes—
71C2×C4×C173#4111yescharacteristic
72C8×C173—24no—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173#12011yescharacteristic
76C22×C173#12111yescharacteristic
77C22×C173#12711yescharacteristic
78C2×D173—42no—
79C2×D173—42no—
80C2×D173—42no—
81C4×C173—42no—
82C4×C173#12012yes—
83C4×C173#12112yes—
84Dic173—42no—
85Dic173—42no—
86Dic173—42no—
87C2×C173—21no—
88C2×C173—21no—
89C2×C173#39211yescharacteristic
90C2×C173#45511yescharacteristic
91C2×C173#45611yescharacteristic
92D173—82no—
93C173#200711yescharacteristic, complemented, Sylow
94#2007—1731nomaximal, Sylow
95C2×#137—1732no—
96#537—1731no—
97C2×#104—1732no—
98#391—1731no—
99#343D17311yescharacteristic, complemented
100C2×#41—1732no—
101C22×Q8—1731no—
102#137—3464no—
103#137—3464no—
104C2×Q16—3462no—
105C2×SD16—3462no—
106C2×M16C2×D17312yes—
107C22⋊Q8—3462no—
108#121—3462no—
109#121—3462no—
110C2×C4⋊C4C2×D17311yescharacteristic
111C2×C42—1731no—
112#104—3464no—
113#95—3462no—
114#41—3462no—
115#41—3462no—
116#41—3462no—
117#41—3462no—
118C2×Q8—3461no—
119C2×Q8—3462no—
120C2×Q8—3462no—
121C2×Q8—1732no—
122C2×Q8—1732no—
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—1732no—
126C22×C4—3461no—
127C22×C4—3461no—
128C22×C4—3462no—
129C22×C4D69212yes—
130C22×C4C22×D17311yescharacteristic
131Q16—3468no—
132SD16—3468no—
133M16—22no—
134M16D69214yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—3462no—
138C22⋊C4—3464no—
139C42—3462no—
140C23—3462no—
141C23C2×D69211yescharacteristic
142Q8—3464no—
143Q8—3462no—
144Q8—3462no—
145Q8—6922no—
146D4—3462no—
147D4—3462no—
148D4—13842no—
149D4—6922no—
150D4—6922no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—3462no—
154C2×C4—3462no—
155C2×C4—3462no—
156C2×C4—6921no—
157C2×C4—6921no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4—6922no—
161C2×C4—6922no—
162C2×C4D4×D17311yescharacteristic
163C2×C4D4×D17312yes—
164C2×C4C2×D69212yes—
165C2×C4#1458382711yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—6922no—
170C22—6922no—
171C22—6922no—
172C22#1914964911yescharacteristic
173C22#1914965411yescharacteristic
174C22#1914967611yescharacteristic
175C4—42no—
176C4—6922no—
177C4—6922no—
178C4—6922no—
179C4#1914964912yes—
180C4#1914965412yes—
181C2—21no—
182C2—21no—
183C2—13842no—
184C2#2601140111yescharacteristic
185C2#2601167111yescharacteristic
186C2#2601167611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#343×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#343D173
hypocenterC173#2007
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×#19149712C2semidirect
3. C2×#19149605C2semidirect
4. #26011396C2semidirect
5. #343×C173C2semidirect
28. C2×Dic346D4semidirect
43. M16×C173D4semidirect
93. C173#2007semidirect
99. #343D173semidirect
134. M16D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
#260120722
C2×#191496052
#260113962
#343×C1732
#2007173