Finite Group Databasebeta

← #36369866

Subgroups — #36369866

Order 22144.

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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#26012478C212yesmaximal
2#26012402C212yesmaximal, complemented
3#26012401C212yesmaximal, complemented
4#26012061C211yesmaximal, characteristic
5C2×#14583835—22no—
6#19149740C2214yes—
7#19149739—22no—
8C2×#14583825C2211yescharacteristic
9C2×Q8×D173C411yescharacteristic
10#19149679C2211yescharacteristic
11#19149677C2211yescharacteristic
12#19149666C411yescharacteristic
13#19149661—22no—
14#19149656—22no—
15#19149636C411yescharacteristic
16#124×C173C411yescharacteristic
17#14583835—24no—
18#14583835—42no—
19C2×#13063443—42no—
20#14583825—42no—
21C2×Dic346—21no—
22C2×Dic346C2×C411yescharacteristic
23C2×Dic346C2311yescharacteristic
24Q8×D173—42no—
25C2×C22⋊D173D412yes—
26C22×Dic173D412yes—
27C2×C4×D173—21no—
28C2×C4×D173C2×C411yescharacteristic
29C4⋊Dic173—22no—
30#14583805—21no—
31#14583805—22no—
32C22⋊Dic173—22no—
33#14583803—21no—
34C4×Dic173C2×C411yescharacteristic
35C2×Q8×C173C2×C411yescharacteristic
36C2×D4×C173C2×C411yescharacteristic
37C22⋊C4×C173—22no—
38C42×C173C2×C411yescharacteristic
39#13063443—44no—
40C22×D173C22⋊C411yescharacteristic
41Dic346—41no—
42Dic346—42no—
43Dic346—41no—
44C22⋊D173—42no—
45C2×Dic173—22no—
46C2×Dic173—22no—
47C2×Dic173—21no—
48C2×Dic173—21no—
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C2×D412yes—
51C4×D173—22no—
52C4×D173—42no—
53C23×C173C22⋊C412yes—
54Q8×C173—41no—
55D4×C173—41no—
56C2×C4×C173—21no—
57C2×C4×C173—21no—
58C2×C4×C173C22×C411yescharacteristic
59C22×C173—22no—
60C22×C173—42no—
61C22×C173C2×C22⋊C411yescharacteristic
62C2×D173—21no—
63C2×D173—41no—
64C4×C173—21no—
65C4×C173—41no—
66C4×C173—41no—
67Dic173—22no—
68Dic173—41no—
69Dic173—41no—
70Dic173#9912yes—
71C2×C173—21no—
72C2×C173—42no—
73C2×C173C2×#9911yescharacteristic
74D173—41no—
75C173#213511yescharacteristic, complemented, Sylow
76#2135—1731nomaximal, Sylow
77#536—1731no—
78#413—1732no—
79#356—1732no—
80#356—1732no—
81C2×#41—1731no—
82C22×Q8—1731no—
83C2×M16—3462no—
84C4⋊Q8—1731no—
85C4⋊Q8—1731no—
86#124—1731no—
87#124#1062192411yescharacteristic
88#123—3462no—
89C22⋊Q8—3462no—
90#117—1731no—
91#101—1734no—
92#100—3462no—
93#41—6922no—
94C2×Q8—3461no—
95C2×Q8—1731no—
96C2×Q8—1731no—
97C2×Q8—6922no—
98C2×Q8C2×#1062192411yescharacteristic
99C2×D4—1732no—
100C2×D4C2×#1062192411yescharacteristic
101C22×C4—3461no—
102C22×C4—1732no—
103C22×C4—1731no—
104M16—3464no—
105M16—6922no—
106C2×C8—6922no—
107C4⋊C4—3461no—
108C4⋊C4—3462no—
109C4⋊C4—3462no—
110C22⋊C4—22no—
111C22⋊C4—3461no—
112C22⋊C4—3462no—
113C42—1731no—
114C42C2×#1062192411yescharacteristic
115C23—1731no—
116C23C22⋊#1062192412yes—
117Q8—41no—
118Q8—6921no—
119Q8—6921no—
120Q8—6922no—
121D4—41no—
122D4—6922no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—3462no—
126C2×C4—3462no—
127C2×C4—3462no—
128C2×C4—3461no—
129C2×C4—3461no—
130C2×C4—1732no—
131C2×C4—1731no—
132C2×C4—6922no—
133C2×C4C22×#1062192411yescharacteristic
134C8—6924no—
135C22—22no—
136C22—42no—
137C22—3461no—
138C22—6921no—
139C22C2×C22⋊#1062192411yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—3462no—
144C4—1732no—
145C4—6921no—
146C4—6921no—
147C2—21no—
148C2—42no—
149C2—6921no—
150C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173C2×#99
prime radicalC2×#14583825C22
cosocleC2×Dic346C23
hypercenter#124#10621924
hypocenterC173#2135
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #26012402C2semidirect
3. #26012401C2semidirect
75. C173#2135semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124022
#260124012
#260120612
#2135173