Finite Group Databasebeta

← #36354609

Subgroups — #36354609

Order 22144.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#19149710C212yesmaximal, complemented
2#26012072C211yesmaximal, characteristic
3C2×#19149607C212yesmaximal, complemented
4#26011260C211yesmaximal, characteristic, complemented
5#391×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2212yes—
7#19149710—24no—
8#19149710—24no—
9C2×#14583816—22no—
10C2×#14583813—22no—
11C2×#14583810C2212yes—
12#19149659—22no—
13#19149654—22no—
14#19149653—22no—
15#19149607—24no—
16C22×Q8×C173C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C2×C42×C173C2211yescharacteristic
19#95×C173—22no—
20#14583826—22no—
21#14583826—22no—
22#14583826—22no—
23#14583826—22no—
24C2×Dic346D412yes—
25C2×D692—22no—
26C2×D692D412yes—
27C2×C22⋊D173—22no—
28C2×C4×D173—22no—
29#14583816—28no—
30#14583813—28no—
31#14583810—22no—
32#14583810D414yescomplemented
33C2×#13063436—22no—
34C4⋊Dic173—22no—
35#14583803—24no—
36C2×Q8×C173—21no—
37C2×Q8×C173—22no—
38C2×Q8×C173—22no—
39C2×Q8×C173D412yescomplemented
40C22×C4×C173—21no—
41C22×C4×C173—21no—
42C22×C4×C173D412yes—
43C22×C4×C173C2311yescharacteristic
44C4⋊C4×C173—22no—
45C42×C173—22no—
46C22×D173—22no—
47Dic346—22no—
48Dic346—22no—
49D692—22no—
50D692—22no—
51D692—82no—
52C22⋊D173—42no—
53C22⋊D173—42no—
54C2×Dic173—22no—
55C4×D173—42no—
56C4×D173—42no—
57#13063436—24no—
58C23×C173C2×D411yescharacteristic
59Q8×C173—24no—
60Q8×C173—42no—
61C2×C4×C173—22no—
62C2×C4×C173—22no—
63C2×C4×C173—22no—
64C2×C4×C173—41no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173—41no—
68C2×C4×C173—42no—
69C2×C4×C173C2×D412yes—
70C2×C4×C173C2×D412yes—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173#4111yescharacteristic
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—42no—
83C4×C173—42no—
84C4×C173#12012yes—
85C4×C173#12112yes—
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#391D17311yescharacteristic, complemented
99#343—1731no—
100C2×#41—1732no—
101C22×Q8C2×D17311yescharacteristic
102#137—3464no—
103#137—3464no—
104C2×Q16—3462no—
105C2×SD16—3462no—
106C2×M16—1732no—
107C22⋊Q8—3462no—
108#121—3462no—
109#121—3462no—
110C2×C4⋊C4C2×D17311yescharacteristic
111C2×C42C2×D17311yescharacteristic
112#104—3464no—
113#95—22no—
114#41—3462no—
115#41—3462no—
116#41—3462no—
117#41—3462no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—1732no—
122C2×Q8D69212yescomplemented
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—1732no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—3462no—
129C22×C4C22⋊D17312yes—
130C22×C4C22×D17311yescharacteristic
131Q16—3468no—
132SD16—3468no—
133M16—3462no—
134M16—1734no—
135C2×C8—3462no—
136C4⋊C4—22no—
137C4⋊C4—3462no—
138C22⋊C4—3464no—
139C42—22no—
140C23—3462no—
141C23C2×C22⋊D17311yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—3462no—
145Q8—3462no—
146D4—3462no—
147D4—3462no—
148D4—13842no—
149D4—6922no—
150D4—6922no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—3462no—
160C2×C4—6922no—
161C2×C4—6922no—
162C2×C4D4×D17312yes—
163C2×C4D4×D17312yes—
164C2×C4C2×D69211yescharacteristic
165C2×C4#1458382611yescharacteristic
166C8—3464no—
167C22—21no—
168C22—21no—
169C22—6922no—
170C22—6922no—
171C22—6922no—
172C22#1914964711yescharacteristic
173C22#1914965311yescharacteristic
174C22#1914967511yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—6922no—
179C4#1914964912yes—
180C4#1914965212yes—
181C2—21no—
182C2—21no—
183C2—13842no—
184C2#2601140011yescharacteristic
185C2#2601166811yescharacteristic
186C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#391×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#391D173
hypocenterC173#2007
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#19149710C2semidirect
3. C2×#19149607C2semidirect
4. #26011260C2semidirect
5. #391×C173C2semidirect
32. #14583810D4semidirect
39. C2×Q8×C173D4semidirect
93. C173#2007semidirect
98. #391D173semidirect
122. C2×Q8D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
#260120722
C2×#191496072
#260112602
#391×C1732
#2007173