Finite Group Databasebeta

← #36369567

Subgroups — #36369567

Order 22144.

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

normal subgroups only

This group has 332 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#26012571C211yesmaximal, characteristic, complemented
2#26012489C212yesmaximal, complemented
3#26012472C211yesmaximal, characteristic, complemented
4#26012362C212yesmaximal, complemented
5#26012040C211yesmaximal, characteristic
6#19149767—24no—
7D173⋊M16—22no—
8D173⋊M16C2212yes—
9#19149758C2212yes—
10#19149743—24no—
11#19149738—22no—
12C2×#14583827C2211yescharacteristic
13C2×#14583825C2211yescharacteristic
14#19149677C411yescharacteristic, complemented
15#19149675C411yescharacteristic, complemented
16#19149667C411yescharacteristic, complemented
17#19149652—22no—
18#19149639—22no—
19C42×D173C2211yescharacteristic
20#124×C173C411yescharacteristic, complemented
21#14583836—24no—
22#14583836D414yescomplemented
23#14583835—22no—
24D173⋊C8—24no—
25D173⋊C8—22no—
26C4⋊#10621924—22no—
27C22⋊#10621924—22no—
28C4×#10621924—22no—
29#14583827—22no—
30#14583827—22no—
31#14583825—22no—
32#14583825—22no—
33C2×Dic346C2×C411yescharacteristic
34C2×D692C2×C411yescharacteristic
35C2×D692D412yes—
36C2×C22⋊D173—22no—
37C2×C22⋊D173—21no—
38C22×Dic173—21no—
39C2×C4×D173—21no—
40C2×C4×D173D412yes—
41C2×C4×D173C2311yescharacteristic
42#14583805—22no—
43#14583803—22no—
44C4×Dic173—21no—
45C4×Dic173C2×C411yescharacteristic
46C2×Q8×C173C2×C411yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173—22no—
49C42×C173C2×C411yescharacteristic
50C2×#10621924—42no—
51#13063443—24no—
52C22×D173C22⋊C412yes—
53C22×D173C2×D411yescharacteristic
54Dic346—22no—
55D692—24no—
56D692—22no—
57C22⋊D173—44no—
58C22⋊D173—42no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—41no—
62C2×Dic173—41no—
63C2×Dic173C2×D411yescharacteristic
64C4×D173—22no—
65C4×D173—22no—
66C4×D173—42no—
67C4×D173C2×D412yes—
68C4×D173#4112yes—
69C23×C173—21no—
70Q8×C173—22no—
71D4×C173—22no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C412yes—
74C2×C4×C173C22×C411yescharacteristic
75C22×C173—42no—
76C22×C173—41no—
77C22×C173C2×C22⋊C411yescharacteristic
78C2×D173—22no—
79C2×D173—41no—
80C2×D173—42no—
81C2×D173#12112yes—
82#10621924—82no—
83C4×C173—22no—
84C4×C173—41no—
85C4×C173C4×D412yes—
86Dic173—22no—
87Dic173—41no—
88Dic173#12011yescharacteristic
89Dic173#12311yescharacteristic
90C2×C173—21no—
91C2×C173—81no—
92C2×C173#38611yescharacteristic
93D173—22no—
94D173—42no—
95C173#188411yescharacteristic, complemented, Sylow
96#1884—1731nomaximal, Sylow
97#568—1731no—
98#532—1731no—
99#421—1732no—
100#410—1731no—
101#339—1732no—
102C2×#41—1731no—
103C2×#41—1731no—
104#131—3464no—
105C2×M16—3462no—
106C2×M16—1732no—
107C4⋊Q8—1731no—
108#127—1731no—
109#124—1731no—
110#124#1062192411yescharacteristic, complemented
111#121—3462no—
112C4×D4—3462no—
113#117—1732no—
114C2×C42—1731no—
115#104—3464no—
116#99—3462no—
117#41—3462no—
118#41—3462no—
119#41—3462no—
120#41—3462no—
121C2×Q8—1731no—
122C2×Q8C2×#1062192411yescharacteristic
123C2×D4—3462no—
124C2×D4—3461no—
125C2×D4—1732no—
126C2×D4—1731no—
127C2×D4C2×#1062192411yescharacteristic
128C22×C4—3461no—
129C22×C4—3461no—
130C22×C4—1732no—
131C22×C4—1731no—
132M16—3464no—
133M16—3462no—
134M16—1734no—
135C2×C8—3464no—
136C2×C8—3462no—
137C4⋊C4—3462no—
138C4⋊C4—3462no—
139C22⋊C4—22no—
140C22⋊C4—3462no—
141C22⋊C4—3462no—
142C42—3461no—
143C42—3462no—
144C42—1731no—
145C42C2×#1062192411yescharacteristic
146C23—21no—
147C23—1732no—
148C23—1731no—
149Q8—22no—
150Q8—3462no—
151D4—22no—
152D4—3464no—
153D4—3462no—
154D4—6924no—
155D4—6922no—
156C2×C4—21no—
157C2×C4—3462no—
158C2×C4—3462no—
159C2×C4—3461no—
160C2×C4—3461no—
161C2×C4—1731no—
162C2×C4—1732no—
163C2×C4—1732no—
164C2×C4—6922no—
165C2×C4—6922no—
166C2×C4—6921no—
167C2×C4—6921no—
168C2×C4C22⋊#1062192412yes—
169C2×C4C22×#1062192411yescharacteristic
170C8—3464no—
171C22—42no—
172C22—41no—
173C22—3462no—
174C22—1732no—
175C22—6921no—
176C22—6922no—
177C22C2×C22⋊#1062192411yescharacteristic
178C4—22no—
179C4—41no—
180C4—3462no—
181C4—13842no—
182C4—1731no—
183C4—1731no—
184C4—6921no—
185C4D4×#1062192412yes—
186C2—21no—
187C2—81no—
188C2—3462no—
189C2—6922no—
190C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173#386
prime radical#26012040C2
cosocleC2×C4×D173C23
hypercenter#124#10621924
hypocenterC173#1884
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012571C2semidirect
2. #26012489C2semidirect
3. #26012472C2semidirect
4. #26012362C2semidirect
14. #19149677C4semidirect
15. #19149675C4semidirect
16. #19149667C4semidirect
20. #124×C173C4semidirect
22. #14583836D4semidirect
95. C173#1884semidirect
110. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125712
#260124892
#260124722
#260123622
#260120402
#1884173