Finite Group Databasebeta

← #36354604

Subgroups — #36354604

Order 22144.

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

normal subgroups only

This group has 406 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×#19149705C212yesmaximal, complemented
2#26012072C211yesmaximal, characteristic, complemented
3C2×#19149607C212yesmaximal, complemented
4#26011260C211yesmaximal, characteristic, complemented
5#390×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2212yes—
7#19149705—24no—
8#19149705—24no—
9C2×#14583814—22no—
10C2×#14583811—22no—
11C2×#14583810C2212yes—
12#19149659—22no—
13#19149654—22no—
14#19149653—22no—
15#19149607—24no—
16C22×D4×C173C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C2×C22⋊C4×C173—22no—
19C2×C42×C173C2211yescharacteristic
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#14583814—28no—
30#14583811—28no—
31#14583810—22no—
32#14583810D414yescomplemented
33C2×#13063436—22no—
34C4⋊Dic173—22no—
35#14583803—24no—
36C24×C173—21no—
37C2×D4×C173—22no—
38C2×D4×C173—22no—
39C2×D4×C173—21no—
40C2×D4×C173D412yescomplemented
41C22×C4×C173—21no—
42C22×C4×C173D412yes—
43C22×C4×C173C2311yescharacteristic
44C4⋊C4×C173—22no—
45C22⋊C4×C173—44no—
46C42×C173—22no—
47C22×D173—22no—
48Dic346—22no—
49Dic346—22no—
50D692—22no—
51D692—22no—
52D692—82no—
53C22⋊D173—42no—
54C22⋊D173—42no—
55C2×Dic173—22no—
56C4×D173—42no—
57C4×D173—42no—
58#13063436—24no—
59C23×C173—22no—
60C23×C173—42no—
61C23×C173—42no—
62C23×C173—41no—
63C23×C173—41no—
64C23×C173C2×D411yescharacteristic
65D4×C173—24no—
66D4×C173—42no—
67C2×C4×C173—22no—
68C2×C4×C173—22no—
69C2×C4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173—42no—
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D412yes—
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173#4111yescharacteristic
76C22×C173—21no—
77C22×C173—21no—
78C22×C173—44no—
79C22×C173—42no—
80C22×C173—82no—
81C22×C173—81no—
82C22×C173—81no—
83C22×C173#12011yescharacteristic
84C22×C173#12111yescharacteristic
85C22×C173#12711yescharacteristic
86C2×D173—42no—
87C2×D173—42no—
88C2×D173—42no—
89C4×C173—42no—
90C4×C173—42no—
91C4×C173#12012yes—
92C4×C173#12112yes—
93Dic173—42no—
94C2×C173—21no—
95C2×C173—21no—
96C2×C173—82no—
97C2×C173#39211yescharacteristic
98C2×C173#45311yescharacteristic
99C2×C173#45411yescharacteristic
100D173—82no—
101C173#200611yescharacteristic, complemented, Sylow
102#2006—1731nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1732no—
104#537—1731no—
105C2×#104—1732no—
106#390D17311yescharacteristic, complemented
107#343—1731no—
108C2×#41—1732no—
109C22×D4C2×D17311yescharacteristic
110AGL(1,ℤ/8)—3464no—
111AGL(1,ℤ/8)—3464no—
112C2×SD16—3462no—
113C2×D8—3462no—
114C2×M16—1732no—
115C22⋊Q8—3462no—
116#121—3462no—
117#121—3462no—
118C2×C4⋊C4C2×D17311yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D17311yescharacteristic
121#104—3464no—
122C24—21no—
123#41—3462no—
124#41—3462no—
125#41—3462no—
126#41—3462no—
127C2×Q8—1732no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—3462no—
132C2×D4—3462no—
133C2×D4—1732no—
134C2×D4D69212yescomplemented
135C22×C4—21no—
136C22×C4—3462no—
137C22×C4C22⋊D17312yes—
138C22×C4C22×D17311yescharacteristic
139SD16—3468no—
140D8—3468no—
141M16—3462no—
142M16—1734no—
143C2×C8—3462no—
144C4⋊C4—22no—
145C4⋊C4—3462no—
146C22⋊C4—44no—
147C22⋊C4—3464no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—3462no—
155C23C2×C22⋊D17311yescharacteristic
156Q8—3462no—
157Q8—3462no—
158D4—24no—
159D4—42no—
160D4—3462no—
161D4—3462no—
162D4—13842no—
163D4—6922no—
164D4—6922no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—3462no—
171C2×C4—6922no—
172C2×C4—6922no—
173C2×C4D4×D17312yes—
174C2×C4D4×D17312yes—
175C2×C4C2×D69211yescharacteristic
176C2×C4#1458382611yescharacteristic
177C8—3464no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—6922no—
186C22—6922no—
187C22—6922no—
188C22#1914964711yescharacteristic
189C22#1914965311yescharacteristic
190C22#1914967511yescharacteristic
191C4—42no—
192C4—42no—
193C4—6922no—
194C4#1914964912yes—
195C4#1914965212yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—13842no—
200C2#2601140011yescharacteristic
201C2#2601165811yescharacteristic
202C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#390×C173C2
socleC22×C173#120
hypercenter#390D173
hypocenterC173#2006
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. #26012072C2semidirect
3. C2×#19149607C2semidirect
4. #26011260C2semidirect
5. #390×C173C2semidirect
32. #14583810D4semidirect
40. C2×D4×C173D4semidirect
101. C173#2006semidirect
106. #390D173semidirect
134. C2×D4D692semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
#260120722
C2×#191496072
#260112602
#390×C1732
#2006173