Finite Group Databasebeta

← #36369835

Subgroups — #36369835

Order 22144.

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

normal subgroups only

This group has 258 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#26012472C212yesmaximal, complemented
2#26012398C212yesmaximal, complemented
3#26012395C212yesmaximal, complemented
4#26012040C211yesmaximal, characteristic
5#19149758—22no—
6#19149738—22no—
7#19149738C2214yes—
8C2×#14583827C2211yescharacteristic
9C2×#14583825C411yescharacteristic, complemented
10#19149677C411yescharacteristic, complemented
11#19149675C2211yescharacteristic
12#19149667C2211yescharacteristic
13#19149652—22no—
14#19149639—22no—
15C42×D173C411yescharacteristic, complemented
16#124×C173C411yescharacteristic, complemented
17C4⋊#10621924—42no—
18C22⋊#10621924—24no—
19C4×#10621924—42no—
20#14583827—41no—
21#14583827—41no—
22#14583825—41no—
23#14583825—41no—
24C2×Dic346C2×C411yescharacteristic
25C2×D692D412yes—
26C2×D692C2311yescharacteristic
27C2×C22⋊D173—21no—
28C2×C22⋊D173—22no—
29C22×Dic173—21no—
30C2×C4×D173—21no—
31C2×C4×D173C2×C411yescharacteristic
32C2×C4×D173D412yes—
33#14583805—22no—
34#14583803—22no—
35C4×Dic173—21no—
36C4×Dic173C2×C411yescharacteristic
37C2×Q8×C173C2×C411yescharacteristic
38C2×D4×C173C2×C411yescharacteristic
39C22⋊C4×C173—22no—
40C42×C173C2×C411yescharacteristic
41C2×#10621924—44no—
42C22×D173C22⋊C411yescharacteristic
43C22×D173C2×D412yes—
44Dic346—41no—
45D692—42no—
46D692—41no—
47C22⋊D173—44no—
48C22⋊D173—42no—
49C2×Dic173—21no—
50C2×Dic173—21no—
51C2×Dic173—41no—
52C2×Dic173—41no—
53C2×Dic173C22⋊C411yescharacteristic
54C4×D173—21no—
55C4×D173—21no—
56C4×D173—22no—
57C4×D173—22no—
58C4×D173—42no—
59C23×C173—21no—
60Q8×C173—41no—
61D4×C173—41no—
62C2×C4×C173—21no—
63C2×C4×C173C22⋊C412yes—
64C2×C4×C173C22×C411yescharacteristic
65C22×C173—42no—
66C22×C173—41no—
67C22×C173C2×C22⋊C411yescharacteristic
68C2×D173—21no—
69C2×D173—22no—
70C2×D173—41no—
71C2×D173—42no—
72#10621924—84no—
73C4×C173—22no—
74C4×C173—21no—
75C4×C173—41no—
76Dic173—22no—
77Dic173—41no—
78Dic173#9911yescharacteristic
79Dic173#9911yescharacteristic
80C2×C173—21no—
81C2×C173—81no—
82C2×C173C2×#9911yescharacteristic
83D173—42no—
84D173—41no—
85C173#212611yescharacteristic, complemented, Sylow
86#2126—1731nomaximal, Sylow
87#532—1731no—
88#410—1732no—
89#354—1732no—
90#353—1732no—
91C2×#41—1731no—
92C2×#41—1731no—
93C4⋊Q8—1731no—
94#127—1731no—
95#124—1731no—
96#124#1062192411yescharacteristic, complemented
97#121—3462no—
98C4×D4—3462no—
99#117—3462no—
100C2×C42—1731no—
101#99—3462no—
102#99—1734no—
103#41—6921no—
104#41—6921no—
105#41—6921no—
106#41—6921no—
107C2×Q8—1731no—
108C2×Q8C2×#1062192411yescharacteristic
109C2×D4—3462no—
110C2×D4—3461no—
111C2×D4—1732no—
112C2×D4—1731no—
113C2×D4C2×#1062192411yescharacteristic
114C22×C4—3461no—
115C22×C4—3461no—
116C22×C4—1732no—
117C22×C4—1731no—
118C4⋊C4—3462no—
119C4⋊C4—6922no—
120C22⋊C4—22no—
121C22⋊C4—3462no—
122C22⋊C4—3464no—
123C42—3461no—
124C42—1731no—
125C42—6922no—
126C42C2×#1062192411yescharacteristic
127C23—21no—
128C23—1731no—
129C23—1732no—
130Q8—41no—
131Q8—6921no—
132D4—41no—
133D4—6922no—
134D4—6921no—
135D4—6924no—
136D4—6922no—
137C2×C4—21no—
138C2×C4—3462no—
139C2×C4—3462no—
140C2×C4—3461no—
141C2×C4—3461no—
142C2×C4—3461no—
143C2×C4—3461no—
144C2×C4—1731no—
145C2×C4—6922no—
146C2×C4—6924no—
147C2×C4—6921no—
148C2×C4—6921no—
149C2×C4C22⋊#1062192412yes—
150C2×C4C22×#1062192411yescharacteristic
151C22—42no—
152C22—41no—
153C22—3462no—
154C22—3461no—
155C22—6921no—
156C22—6922no—
157C22C2×C22⋊#1062192411yescharacteristic
158C4—22no—
159C4—21no—
160C4—41no—
161C4—3462no—
162C4—13844no—
163C4—1731no—
164C4—1731no—
165C4—6921no—
166C2—21no—
167C2—81no—
168C2—6922no—
169C2—6921no—
170C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173C2×#99
prime radical#26012040C2
cosocleC2×D692C23
hypercenter#124#10621924
hypocenterC173#2126
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012472C2semidirect
2. #26012398C2semidirect
3. #26012395C2semidirect
9. C2×#14583825C4semidirect
10. #19149677C4semidirect
15. C42×D173C4semidirect
16. #124×C173C4semidirect
85. C173#2126semidirect
96. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124722
#260123982
#260123952
#260120402
#2126173