Finite Group Databasebeta

← #36369863

Subgroups — #36369863

Order 22144.

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

normal subgroups only

This group has 262 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#26012400C212yesmaximal, complemented
3#26012399C212yesmaximal, complemented
4#26012054C211yesmaximal, characteristic
5C2×#14583835—22no—
6#19149740C2214yescomplemented
7#19149739—22no—
8C2×#14583825C2211yescharacteristic
9C2×D4×D173C411yescharacteristic
10#19149675C411yescharacteristic
11#19149667C2211yescharacteristic
12#19149666C2211yescharacteristic
13#19149651—22no—
14#19149647—22no—
15#19149636C411yescharacteristic
16#127×C173C411yescharacteristic
17#14583835—24no—
18#14583835—42no—
19C2×#13063443—42no—
20C23×D173—21no—
21#14583825—42no—
22C2×Dic346C2311yescharacteristic
23C2×D692C2×C411yescharacteristic
24C2×C22⋊D173—21no—
25C2×C22⋊D173—22no—
26C2×C22⋊D173D412yes—
27D4×D173—42no—
28C22×Dic173D412yes—
29C2×C4×D173C2×C411yescharacteristic
30#14583805—21no—
31C22⋊Dic173—22no—
32#14583803—21no—
33#14583803—22no—
34C4×Dic173C2×C411yescharacteristic
35C2×D4×C173—22no—
36C2×D4×C173C2×C411yescharacteristic
37C2×D4×C173C2×C411yescharacteristic
38C42×C173C2×C411yescharacteristic
39#13063443—44no—
40C22×D173—21no—
41C22×D173—42no—
42C22×D173—82no—
43C22×D173C22⋊C411yescharacteristic
44Dic346—41no—
45D692—41no—
46C22⋊D173—42no—
47C22⋊D173—42no—
48C22⋊D173—82no—
49C2×Dic173—22no—
50C2×Dic173—22no—
51C2×Dic173—21no—
52C2×Dic173C22⋊C411yescharacteristic
53C2×Dic173C2×D412yes—
54C4×D173—22no—
55C23×C173—21no—
56C23×C173C22⋊C412yes—
57D4×C173—41no—
58D4×C173—41no—
59D4×C173—82no—
60C2×C4×C173—21no—
61C2×C4×C173C22×C411yescharacteristic
62C22×C173—22no—
63C22×C173—42no—
64C22×C173—42no—
65C22×C173—41no—
66C22×C173C2×C22⋊C411yescharacteristic
67C2×D173—21no—
68C2×D173—41no—
69C2×D173—42no—
70C2×D173—41no—
71C2×D173—84no—
72C4×C173—21no—
73C4×C173—41no—
74Dic173—22no—
75Dic173—41no—
76Dic173#9912yes—
77C2×C173—21no—
78C2×C173—42no—
79C2×C173—81no—
80C2×C173C2×#9911yescharacteristic
81D173—41no—
82D173—81no—
83C173#213411yescharacteristic, complemented, Sylow
84#2134—1731nomaximal, Sylow
85#535—1731no—
86#413—1732no—
87#355—1732no—
88#355—1732no—
89C2×#41—1731no—
90C22×D4—1731no—
91C2×M16—3462no—
92#127—1731no—
93#127#1062192411yescharacteristic
94#124—1731no—
95#124—1731no—
96#121—3462no—
97#120—3462no—
98#117—1731no—
99#101—1734no—
100#100—3462no—
101C24—3461no—
102#41—6922no—
103C2×Q8—1731no—
104C2×D4—22no—
105C2×D4—3462no—
106C2×D4—3461no—
107C2×D4—1732no—
108C2×D4—1731no—
109C2×D4—6922no—
110C2×D4C2×#1062192411yescharacteristic
111C2×D4C2×#1062192411yescharacteristic
112C22×C4—1732no—
113C22×C4—1731no—
114M16—3464no—
115M16—6922no—
116C2×C8—6922no—
117C4⋊C4—3461no—
118C22⋊C4—3461no—
119C22⋊C4—3462no—
120C22⋊C4—3462no—
121C42—1731no—
122C42C2×#1062192411yescharacteristic
123C23—21no—
124C23—3461no—
125C23—13842no—
126C23—1731no—
127C23—6922no—
128C23C22⋊#1062192412yes—
129Q8—6921no—
130D4—41no—
131D4—41no—
132D4—82no—
133D4—13842no—
134D4—6921no—
135D4—6922no—
136D4—6922no—
137C2×C4—21no—
138C2×C4—3462no—
139C2×C4—3462no—
140C2×C4—3462no—
141C2×C4—3461no—
142C2×C4—1732no—
143C2×C4—1731no—
144C2×C4C22×#1062192411yescharacteristic
145C8—6924no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22—41no—
150C22—3461no—
151C22—13844no—
152C22—6922no—
153C22—6921no—
154C22—6921no—
155C22C2×C22⋊#1062192411yescharacteristic
156C4—21no—
157C4—41no—
158C4—3462no—
159C4—1732no—
160C4—6921no—
161C2—21no—
162C2—42no—
163C2—81no—
164C2—13841no—
165C2—6921no—
166C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC2×C173C2×#99
prime radical#26012054C2
cosocleC2×Dic346C23
hypercenter#127#10621924
hypocenterC173#2134
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. #26012400C2semidirect
3. #26012399C2semidirect
6. #19149740C22semidirect
83. C173#2134semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124002
#260123992
#260120542
#2134173