Finite Group Databasebeta

← #36369606

Subgroups — #36369606

Order 22144.

344 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012569C211yesmaximal, characteristic, complemented
2#26012483C212yesmaximal, complemented
3#26012471C211yesmaximal, characteristic, complemented
4#26012358C212yesmaximal, complemented
5#26012054C211yesmaximal, characteristic
6#19149766—24no—
7D173⋊M16C2212yes—
8C2×#14583835—22no—
9#19149758C2212yes—
10#19149741—24no—
11#19149737—22no—
12C2×#14583825C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14#19149675C411yescharacteristic, complemented
15#19149667C411yescharacteristic, complemented
16#19149666C411yescharacteristic, complemented
17#19149651—22no—
18#19149647—22no—
19#19149636C2211yescharacteristic
20#127×C173C411yescharacteristic, complemented
21#14583836—22no—
22#14583835—24no—
23#14583835—22no—
24D173⋊C8D414yescomplemented
25C2×#13063443—24no—
26C4⋊#10621924—22no—
27C22⋊#10621924—22no—
28C4×#10621924—22no—
29C23×D173—21no—
30#14583825—22no—
31#14583825—22no—
32C2×Dic346C2×C411yescharacteristic
33C2×D692C2×C411yescharacteristic
34C2×C22⋊D173—21no—
35C2×C22⋊D173—22no—
36C2×C22⋊D173D412yes—
37D4×D173—22no—
38D4×D173—22no—
39C22×Dic173D412yes—
40C2×C4×D173C2311yescharacteristic
41#14583805—21no—
42C22⋊Dic173—22no—
43#14583803—21no—
44#14583803—22no—
45C4×Dic173C2×C411yescharacteristic
46C2×D4×C173—22no—
47C2×D4×C173C2×C411yescharacteristic
48C2×D4×C173C2×C411yescharacteristic
49C42×C173C2×C411yescharacteristic
50C2×#10621924—42no—
51#13063443—24no—
52C22×D173—21no—
53C22×D173—42no—
54C22×D173—82no—
55C22×D173C2×D411yescharacteristic
56Dic346—22no—
57D692—22no—
58C22⋊D173—24no—
59C22⋊D173—41no—
60C22⋊D173—41no—
61C22⋊D173—82no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—21no—
65C2×Dic173C22⋊C412yes—
66C2×Dic173C2×D411yescharacteristic
67C4×D173C2×D412yes—
68C4×D173#4112yes—
69C23×C173—21no—
70C23×C173C22⋊C412yes—
71D4×C173—22no—
72D4×C173—22no—
73D4×C173—82no—
74C2×C4×C173—21no—
75C2×C4×C173C22×C411yescharacteristic
76C22×C173—22no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173—41no—
80C22×C173C2×C22⋊C411yescharacteristic
81C2×D173—41no—
82C2×D173—42no—
83C2×D173—41no—
84C2×D173—84no—
85C2×D173#12112yes—
86#10621924—82no—
87C4×C173—41no—
88C4×C173C4×D412yes—
89Dic173—22no—
90Dic173—41no—
91Dic173#12011yescharacteristic
92Dic173#12311yescharacteristic
93C2×C173—21no—
94C2×C173—42no—
95C2×C173—81no—
96C2×C173#38611yescharacteristic
97D173—22no—
98D173—81no—
99C173#191011yescharacteristic, complemented, Sylow
100#1910—1731nomaximal, Sylow
101#568—1731no—
102#535—1731no—
103#418—1732no—
104#410—1731no—
105H(3,ℤ/4)—1732no—
106C2×#41—1731no—
107C22×D4—1731no—
108#131—3464no—
109C2×M16—3462no—
110C2×M16—1732no—
111#127—1731no—
112#127#1062192411yescharacteristic, complemented
113#124—1731no—
114#124—1731no—
115#121—3462no—
116#120—3462no—
117#117—1732no—
118#117—1731no—
119#102—3464no—
120#99—3462no—
121C24—3461no—
122#41—3462no—
123#41—3462no—
124C2×Q8—1731no—
125C2×D4—22no—
126C2×D4—3462no—
127C2×D4—3462no—
128C2×D4—3462no—
129C2×D4—3461no—
130C2×D4—1732no—
131C2×D4—1731no—
132C2×D4C2×#1062192411yescharacteristic
133C2×D4C2×#1062192411yescharacteristic
134C22×C4—1732no—
135C22×C4—1731no—
136M16—3464no—
137M16—3462no—
138M16—3462no—
139C2×C8—3464no—
140C2×C8—1734no—
141C4⋊C4—3461no—
142C4⋊C4—3462no—
143C22⋊C4—3461no—
144C22⋊C4—3462no—
145C22⋊C4—3462no—
146C22⋊C4—3462no—
147C42—3462no—
148C42—1731no—
149C42C2×#1062192411yescharacteristic
150C23—21no—
151C23—3461no—
152C23—13842no—
153C23—1731no—
154C23—6922no—
155C23C22⋊#1062192412yes—
156Q8—3462no—
157D4—22no—
158D4—22no—
159D4—82no—
160D4—3464no—
161D4—3462no—
162D4—13842no—
163D4—6921no—
164D4—6921no—
165C2×C4—21no—
166C2×C4—3462no—
167C2×C4—3462no—
168C2×C4—3461no—
169C2×C4—1732no—
170C2×C4—1731no—
171C2×C4—1732no—
172C2×C4—1732no—
173C2×C4—6922no—
174C2×C4C22×#1062192411yescharacteristic
175C8—3464no—
176C22—22no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—13844no—
181C22—1732no—
182C22—6921no—
183C22—6922no—
184C22—6921no—
185C22C2×C22⋊#1062192411yescharacteristic
186C4—41no—
187C4—3462no—
188C4—13842no—
189C4—1731no—
190C4—1731no—
191C4—6921no—
192C4D4×#1062192412yes—
193C2—21no—
194C2—42no—
195C2—81no—
196C2—3462no—
197C2—13841no—
198C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC2×C173#386
prime radical#26012054C2
cosocleC2×C4×D173C23
hypercenter#127#10621924
hypocenterC173#1910
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26012569C2semidirect
2. #26012483C2semidirect
3. #26012471C2semidirect
4. #26012358C2semidirect
14. #19149675C4semidirect
15. #19149667C4semidirect
16. #19149666C4semidirect
20. #127×C173C4semidirect
24. D173⋊C8D4semidirect
99. C173#1910semidirect
112. #127#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125692
#260124832
#260124712
#260123582
#260120542
#1910173