Finite Group Databasebeta

← #36369855

Subgroups — #36369855

Order 22144.

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

normal subgroups only

This group has 250 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, complemented
2#26012402C212yesmaximal, complemented
3#26012399C212yesmaximal, complemented
4#26012039C211yesmaximal, characteristic
5C2×#14583835—22no—
6#19149740C2214yes—
7#19149739—22no—
8C2×#14583825C411yescharacteristic
9C2×#14583825C2211yescharacteristic
10#19149679C2211yescharacteristic
11#19149666C411yescharacteristic
12#19149666C2211yescharacteristic
13#19149650—22no—
14D4×Dic173—22no—
15C42×D173C411yescharacteristic
16#127×C173C411yescharacteristic
17#14583835—24no—
18#14583835—42no—
19C2×#13063443—42no—
20#14583825—41no—
21#14583825—41no—
22#14583825—41no—
23#14583825—41no—
24C2×Dic346C2×C411yescharacteristic
25C2×Dic346C2311yescharacteristic
26C2×C22⋊D173—21no—
27C2×C22⋊D173D412yes—
28C22×Dic173—21no—
29C22×Dic173D412yes—
30C2×C4×D173—21no—
31C2×C4×D173C2×C411yescharacteristic
32C4⋊Dic173—22no—
33C22⋊Dic173—22no—
34C22⋊Dic173—22no—
35C4×Dic173—21no—
36C4×Dic173C2×C411yescharacteristic
37C2×D4×C173—22no—
38C2×D4×C173C2×C411yescharacteristic
39C2×D4×C173C2×C411yescharacteristic
40C42×C173C2×C411yescharacteristic
41#13063443—44no—
42C22×D173C22⋊C411yescharacteristic
43Dic346—41no—
44Dic346—41no—
45C22⋊D173—42no—
46C22⋊D173—42no—
47C2×Dic173—22no—
48C2×Dic173—22no—
49C2×Dic173—21no—
50C2×Dic173—21no—
51C2×Dic173—41no—
52C2×Dic173—41no—
53C2×Dic173C22⋊C411yescharacteristic
54C2×Dic173C2×D412yes—
55C4×D173—21no—
56C4×D173—21no—
57C4×D173—42no—
58C23×C173—21no—
59C23×C173C22⋊C412yes—
60D4×C173—41no—
61D4×C173—41no—
62D4×C173—44no—
63C2×C4×C173—21no—
64C2×C4×C173C22×C411yescharacteristic
65C22×C173—22no—
66C22×C173—42no—
67C22×C173—42no—
68C22×C173—41no—
69C22×C173C2×C22⋊C411yescharacteristic
70C2×D173—21no—
71C2×D173—41no—
72C4×C173—22no—
73C4×C173—21no—
74Dic173—22no—
75Dic173—41no—
76Dic173—41no—
77Dic173#9911yescharacteristic
78Dic173#9911yescharacteristic
79C2×C173—21no—
80C2×C173—42no—
81C2×C173—81no—
82C2×C173C2×#9911yescharacteristic
83D173—41no—
84C173#213211yescharacteristic, complemented, Sylow
85#2132—1731nomaximal, Sylow
86#532—1731no—
87#413—1732no—
88#356—1732no—
89#355—1732no—
90C2×#41—1731no—
91C2×#41—1731no—
92C2×M16—3462no—
93C4⋊Q8—1731no—
94#127#1062192411yescharacteristic
95#124—1731no—
96#124—1731no—
97#121—3462no—
98C4×D4—3462no—
99C2×C42—1731no—
100#101—1734no—
101#100—3462no—
102#41—6921no—
103#41—6921no—
104#41—6921no—
105#41—6921no—
106C2×Q8—1731no—
107C2×Q8—1731no—
108C2×D4—22no—
109C2×D4—3461no—
110C2×D4—1732no—
111C2×D4C2×#1062192411yescharacteristic
112C2×D4C2×#1062192411yescharacteristic
113C22×C4—3461no—
114C22×C4—3461no—
115C22×C4—1732no—
116C22×C4—1731no—
117M16—3464no—
118M16—6922no—
119C2×C8—6922no—
120C4⋊C4—3462no—
121C22⋊C4—3462no—
122C22⋊C4—3462no—
123C42—3461no—
124C42—1731no—
125C42C2×#1062192411yescharacteristic
126C23—21no—
127C23—1731no—
128C23C22⋊#1062192412yes—
129Q8—6921no—
130Q8—6921no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—6922no—
135D4—6922no—
136C2×C4—21no—
137C2×C4—3461no—
138C2×C4—3461no—
139C2×C4—3462no—
140C2×C4—3462no—
141C2×C4—3461no—
142C2×C4—3461no—
143C2×C4—1732no—
144C2×C4—1731no—
145C2×C4—6922no—
146C2×C4—6921no—
147C2×C4—6921no—
148C2×C4C22×#1062192411yescharacteristic
149C8—6924no—
150C22—22no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22—3461no—
155C22—6921no—
156C22C2×C22⋊#1062192411yescharacteristic
157C4—22no—
158C4—21no—
159C4—3462no—
160C4—1731no—
161C4—1731no—
162C4—6921no—
163C4—6921no—
164C2—21no—
165C2—42no—
166C2—81no—
167C2—6921no—
168C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC2×C173C2×#99
prime radical#26012039C2
cosocleC2×Dic346C23
hypercenter#127#10621924
hypocenterC173#2132
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26012478C2semidirect
2. #26012402C2semidirect
3. #26012399C2semidirect
84. C173#2132semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124022
#260123992
#260120392
#2132173