Finite Group Databasebeta

← #36369604

Subgroups — #36369604

Order 22144.

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

normal subgroups only

This group has 312 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
2#26012484C212yesmaximal, complemented
3#26012471C211yesmaximal, characteristic
4#26012358C212yesmaximal, complemented
5#26012061C211yesmaximal, characteristic
6#19149766—24no—
7D173⋊M16C2212yes—
8C2×#14583835—22no—
9#19149758C2212yes—
10#19149742—24no—
11#19149737—22no—
12C2×#14583825C2211yescharacteristic
13C2×Q8×D173C2211yescharacteristic
14#19149679C411yescharacteristic, complemented
15#19149677C411yescharacteristic, complemented
16#19149666C411yescharacteristic, complemented
17#19149661—22no—
18#19149656—22no—
19#19149636C2211yescharacteristic
20#124×C173C411yescharacteristic, complemented
21#14583836—22no—
22#14583835—24no—
23#14583835—22no—
24D173⋊C8D414yes—
25C2×#13063443—24no—
26C4⋊#10621924—22no—
27C22⋊#10621924—22no—
28C4×#10621924—22no—
29#14583825—22no—
30#14583825—22no—
31C2×Dic346—21no—
32C2×Dic346C2×C411yescharacteristic
33C2×Dic346C2×C411yescharacteristic
34Q8×D173—22no—
35Q8×D173—22no—
36C2×C22⋊D173D412yes—
37C22×Dic173D412yes—
38C2×C4×D173—21no—
39C2×C4×D173C2311yescharacteristic
40C4⋊Dic173—22no—
41#14583805—22no—
42#14583805—21no—
43C22⋊Dic173—22no—
44#14583803—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×D173C2×D411yescharacteristic
53Dic346—22no—
54Dic346—22no—
55Dic346—41no—
56Dic346—41no—
57C22⋊D173—24no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173C22⋊C412yes—
63C2×Dic173C2×D411yescharacteristic
64C4×D173—42no—
65C4×D173C2×D412yes—
66C4×D173#4112yes—
67C23×C173C22⋊C412yes—
68Q8×C173—22no—
69D4×C173—22no—
70C2×C4×C173—21no—
71C2×C4×C173—21no—
72C2×C4×C173C22×C411yescharacteristic
73C22×C173—22no—
74C22×C173—42no—
75C22×C173C2×C22⋊C411yescharacteristic
76C2×D173—41no—
77C2×D173#12112yes—
78#10621924—82no—
79C4×C173—41no—
80C4×C173—41no—
81C4×C173C4×D412yes—
82Dic173—22no—
83Dic173—41no—
84Dic173—41no—
85Dic173#12011yescharacteristic
86Dic173#12311yescharacteristic
87C2×C173—21no—
88C2×C173—42no—
89C2×C173#38611yescharacteristic
90D173—22no—
91C173#190911yescharacteristic, complemented, Sylow
92#1909—1731nomaximal, Sylow
93#568—1731no—
94#536—1731no—
95#419—1732no—
96#410—1731no—
97H(3,ℤ/4)—1732no—
98C2×#41—1731no—
99C22×Q8—1731no—
100#131—3464no—
101C2×M16—3462no—
102C2×M16—1732no—
103C4⋊Q8—1731no—
104C4⋊Q8—1731no—
105#124—1731no—
106#124#1062192411yescharacteristic, complemented
107#123—3462no—
108C22⋊Q8—3462no—
109#117—1732no—
110#117—1731no—
111#103—3464no—
112#99—3462no—
113#41—3462no—
114#41—3462no—
115C2×Q8—3461no—
116C2×Q8—3462no—
117C2×Q8—3462no—
118C2×Q8—1731no—
119C2×Q8—1731no—
120C2×Q8C2×#1062192411yescharacteristic
121C2×D4—1732no—
122C2×D4C2×#1062192411yescharacteristic
123C22×C4—3461no—
124C22×C4—1732no—
125C22×C4—1731no—
126M16—3464no—
127M16—3462no—
128M16—3462no—
129C2×C8—3464no—
130C2×C8—1734no—
131C4⋊C4—3461no—
132C4⋊C4—3462no—
133C4⋊C4—3462no—
134C4⋊C4—3462no—
135C22⋊C4—22no—
136C22⋊C4—3461no—
137C22⋊C4—3462no—
138C22⋊C4—3462no—
139C42—3462no—
140C42—1731no—
141C42C2×#1062192411yescharacteristic
142C23—1731no—
143C23C22⋊#1062192412yes—
144Q8—22no—
145Q8—3462no—
146Q8—3462no—
147Q8—6921no—
148Q8—6921no—
149D4—22no—
150D4—3464no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3462no—
154C2×C4—3462no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—1732no—
158C2×C4—1731no—
159C2×C4—1732no—
160C2×C4—1732no—
161C2×C4—6922no—
162C2×C4—6922no—
163C2×C4C22×#1062192411yescharacteristic
164C8—3464no—
165C22—22no—
166C22—42no—
167C22—1732no—
168C22—6921no—
169C22C2×C22⋊#1062192411yescharacteristic
170C4—41no—
171C4—41no—
172C4—3462no—
173C4—13842no—
174C4—1731no—
175C4—1731no—
176C4—6921no—
177C4—6921no—
178C4D4×#1062192412yes—
179C2—21no—
180C2—42no—
181C2—3462no—
182C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173#386
prime radicalC2×#14583825C22
cosocleC2×C4×D173C23
hypercenter#124#10621924
hypocenterC173#1909
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
2. #26012484C2semidirect
4. #26012358C2semidirect
14. #19149679C4semidirect
15. #19149677C4semidirect
16. #19149666C4semidirect
20. #124×C173C4semidirect
91. C173#1909semidirect
106. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125692
#260124842
#260124712
#260123582
#260120612
#1909173