Finite Group Databasebeta

← #36369611

Subgroups — #36369611

Order 22144.

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

normal subgroups only

This group has 314 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#26012559C211yesmaximal, characteristic, complemented
2#26012489C212yesmaximal
3D173⋊#101C211yesmaximal, characteristic, complemented
4#26012358C212yesmaximal, complemented
5#26012061C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7Q8×#10621924—24no—
8#19149758—22no—
9#19149758C2212yes—
10#19149743—24no—
11#19149740—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—22no—
23D173⋊C8—22no—
24C4⋊#10621924—24no—
25C4⋊#10621924—22no—
26C22⋊#10621924—22no—
27C4×#10621924—24no—
28C4×#10621924D414yes—
29#14583825—22no—
30#14583825—22no—
31C2×Dic346C2×C411yescharacteristic
32C2×Dic346C2×C411yescharacteristic
33C2×Dic346D412yes—
34Q8×D173—22no—
35Q8×D173—22no—
36C2×C22⋊D173—21no—
37C22×Dic173—21no—
38C2×C4×D173D412yes—
39C2×C4×D173C2311yescharacteristic
40C4⋊Dic173—22no—
41#14583805—21no—
42#14583805—22no—
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—24no—
51#13063443—42no—
52C22×D173C2×D411yescharacteristic
53Dic346—22no—
54Dic346—24no—
55Dic346—22no—
56C22⋊D173—41no—
57C22⋊D173—41no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—42no—
61C2×Dic173C22⋊C412yes—
62C2×Dic173C2×D411yescharacteristic
63C4×D173—22no—
64C4×D173—22no—
65C4×D173C2×D412yes—
66C4×D173#4112yes—
67C23×C173—21no—
68Q8×C173—22no—
69D4×C173—22no—
70C2×C4×C173—21no—
71C2×C4×C173C22⋊C412yes—
72C2×C4×C173C22×C411yescharacteristic
73C22×C173—42no—
74C22×C173—41no—
75C22×C173C2×C22⋊C411yescharacteristic
76C2×D173—41no—
77C2×D173#12011yescharacteristic
78C2×D173#12311yescharacteristic
79#10621924—44no—
80C4×C173—22no—
81C4×C173—41no—
82C4×C173C4×D412yes—
83Dic173—22no—
84Dic173—41no—
85Dic173—41no—
86Dic173#12112yes—
87C2×C173—21no—
88C2×C173—81no—
89C2×C173#38611yescharacteristic
90D173—21no—
91D173—21no—
92C173#191211yescharacteristic, complemented, Sylow
93#1912—1731nomaximal, Sylow
94#536—1731no—
95#519—1731no—
96#421—1732no—
97C2×#101—1731no—
98H(3,ℤ/4)—1732no—
99C2×#41—1731no—
100C22×Q8—1731no—
101C2×M16—1732no—
102C4⋊Q8—1731no—
103C4⋊Q8—1731no—
104#124—1731no—
105#124#1062192411yescharacteristic, complemented
106#123—3462no—
107C22⋊Q8—3462no—
108C4×Q8—3464no—
109#117—3462no—
110#117—1732no—
111#117—1731no—
112#104—3464no—
113#101—3462no—
114#41—3462no—
115#41—3462no—
116C2×Q8—3462no—
117C2×Q8—3462no—
118C2×Q8—1731no—
119C2×Q8—1732no—
120C2×Q8—1731no—
121C2×Q8C2×#1062192411yescharacteristic
122C2×D4—3461no—
123C2×D4C2×#1062192411yescharacteristic
124C22×C4—3461no—
125C22×C4—1732no—
126C22×C4—1731no—
127M16—3462no—
128M16—3462no—
129C2×C8—3462no—
130C4⋊C4—3461no—
131C4⋊C4—3462no—
132C4⋊C4—3464no—
133C4⋊C4—3462no—
134C4⋊C4—3462no—
135C22⋊C4—22no—
136C22⋊C4—3461no—
137C22⋊C4—3462no—
138C22⋊C4—3462no—
139C42—3464no—
140C42—1734no—
141C42—1731no—
142C42C2×#1062192411yescharacteristic
143C23—21no—
144C23—1731no—
145Q8—22no—
146Q8—3464no—
147Q8—3462no—
148Q8—3462no—
149D4—22no—
150D4—6921no—
151D4—6921no—
152C2×C4—21no—
153C2×C4—3462no—
154C2×C4—3462no—
155C2×C4—3464no—
156C2×C4—3461no—
157C2×C4—3461no—
158C2×C4—1732no—
159C2×C4—1731no—
160C2×C4—1732no—
161C2×C4—1732no—
162C2×C4—6922no—
163C2×C4C22⋊#1062192412yes—
164C2×C4C22×#1062192411yescharacteristic
165C8—6922no—
166C22—42no—
167C22—41no—
168C22—1731no—
169C22—1731no—
170C22—6921no—
171C22C2×C22⋊#1062192411yescharacteristic
172C4—22no—
173C4—41no—
174C4—3462no—
175C4—1732no—
176C4—6921no—
177C4—6924no—
178C4—6921no—
179C4D4×#1062192412yes—
180C2—21no—
181C2—81no—
182C2—3461no—
183C2—3461no—
184C2#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#1912
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. #26012559C2semidirect
3. D173⋊#101C2semidirect
4. #26012358C2semidirect
14. #19149679C4semidirect
15. #19149677C4semidirect
16. #19149666C4semidirect
20. #124×C173C4semidirect
92. C173#1912semidirect
105. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125592
#260124892
D173⋊#1012
#260123582
#260120612
#1912173