Finite Group Databasebeta

← #36369841

Subgroups — #36369841

Order 22144.

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

normal subgroups only

This group has 286 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
1D173⋊#99C212yesmaximal, complemented
2#26012397C212yesmaximal, complemented
3#26012396C212yesmaximal, complemented
4#26012059C211yesmaximal, characteristic
5C2×C22⋊#10621924—22no—
6#19149738C2214yes—
7#19149737—22no—
8C2×#14583827C411yescharacteristic, complemented
9C2×D4×D173C2211yescharacteristic
10#19149676C2211yescharacteristic
11#19149675C411yescharacteristic, complemented
12#19149668C2211yescharacteristic
13#19149652—22no—
14#19149649—22no—
15#19149636C411yescharacteristic, complemented
16#124×C173C411yescharacteristic, complemented
17C22×#10621924—42no—
18C22⋊#10621924—24no—
19C22⋊#10621924—42no—
20C23×D173D412yes—
21#14583827—42no—
22C2×D692—21no—
23C2×D692—22no—
24C2×D692C2×C411yescharacteristic, complemented
25C2×D692C2311yescharacteristic
26C2×C22⋊D173—22no—
27C2×C22⋊D173D412yes—
28D4×D173—42no—
29C2×C4×D173—21no—
30C2×C4×D173C2×C411yescharacteristic
31#14583805—21no—
32#14583803—21no—
33#14583803—22no—
34C4×Dic173C2×C411yescharacteristic, complemented
35C2×Q8×C173C2×C411yescharacteristic, complemented
36C2×D4×C173C2×C411yescharacteristic
37C22⋊C4×C173—22no—
38C42×C173C2×C411yescharacteristic, complemented
39C2×#10621924—44no—
40C2×#10621924—84no—
41C22×D173—22no—
42C22×D173—22no—
43C22×D173—21no—
44C22×D173—82no—
45C22×D173C22⋊C411yescharacteristic
46C22×D173C2×D412yes—
47D692—41no—
48D692—41no—
49D692—42no—
50D692—82no—
51C22⋊D173—42no—
52C22⋊D173—82no—
53C2×Dic173—21no—
54C2×Dic173C22⋊C411yescharacteristic
55C4×D173—22no—
56C4×D173—42no—
57C23×C173C22⋊C412yes—
58Q8×C173—41no—
59D4×C173—41no—
60C2×C4×C173—21no—
61C2×C4×C173—21no—
62C2×C4×C173C22×C411yescharacteristic
63C22×C173—22no—
64C22×C173—42no—
65C22×C173C2×C22⋊C411yescharacteristic
66C2×D173—22no—
67C2×D173—41no—
68C2×D173—44no—
69C2×D173—42no—
70C2×D173—42no—
71C2×D173—41no—
72C2×D173—82no—
73C2×D173#9912yes—
74#10621924—84no—
75C4×C173—21no—
76C4×C173—41no—
77C4×C173—41no—
78Dic173—21no—
79Dic173—41no—
80C2×C173—21no—
81C2×C173—42no—
82C2×C173C2×#9911yescharacteristic
83D173—22no—
84D173—42no—
85D173—81no—
86C173#212711yescharacteristic, complemented, Sylow
87#2127—1731nomaximal, Sylow
88#535—1731no—
89C2×#99—1732no—
90#354—1732no—
91#353—1732no—
92C2×#41—1731no—
93C22×D4—1731no—
94#127—1731no—
95#127—1731no—
96#124—1731no—
97#124#1062192411yescharacteristic, complemented
98#121—3462no—
99#120—3462no—
100#117—1731no—
101C2×C22⋊C4—3462no—
102#99—3462no—
103#99—1734no—
104C24—1732no—
105#41—6922no—
106C2×Q8C2×#1062192411yescharacteristic, complemented
107C2×D4—3462no—
108C2×D4—3461no—
109C2×D4—3462no—
110C2×D4—1732no—
111C2×D4—1731no—
112C2×D4—1731no—
113C2×D4—6922no—
114C2×D4C2×#1062192411yescharacteristic
115C22×C4—3461no—
116C22×C4—1731no—
117C22×C4—6922no—
118C4⋊C4—3461no—
119C22⋊C4—22no—
120C22⋊C4—3461no—
121C22⋊C4—3462no—
122C22⋊C4—3464no—
123C22⋊C4—6922no—
124C42—1731no—
125C42C2×#1062192411yescharacteristic, complemented
126C23—3462no—
127C23—3462no—
128C23—3461no—
129C23—13842no—
130C23—1732no—
131C23—1731no—
132C23C22⋊#1062192412yes—
133Q8—41no—
134D4—41no—
135D4—13842no—
136D4—13842no—
137D4—6921no—
138D4—6921no—
139D4—6922no—
140D4—6922no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—3462no—
144C2×C4—3461no—
145C2×C4—13844no—
146C2×C4—1731no—
147C2×C4—6922no—
148C2×C4—6924no—
149C2×C4C22×#1062192411yescharacteristic
150C22—22no—
151C22—42no—
152C22—3462no—
153C22—13842no—
154C22—1732no—
155C22—6921no—
156C22—6922no—
157C22—6922no—
158C22—6924no—
159C22—6921no—
160C22C2×C22⋊#1062192411yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—3461no—
165C4—13844no—
166C4—6921no—
167C2—21no—
168C2—42no—
169C2—3462no—
170C2—13841no—
171C2—6922no—
172C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173C2×#99
prime radical#26012059C2
cosocleC2×D692C23
hypercenter#124#10621924
hypocenterC173#2127
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
1. D173⋊#99C2semidirect
2. #26012397C2semidirect
3. #26012396C2semidirect
8. C2×#14583827C4semidirect
11. #19149675C4semidirect
15. #19149636C4semidirect
16. #124×C173C4semidirect
24. C2×D692C2×C4semidirect
34. C4×Dic173C2×C4semidirect
35. C2×Q8×C173C2×C4semidirect
38. C42×C173C2×C4semidirect
86. C173#2127semidirect
97. #124#10621924semidirect
106. C2×Q8C2×#10621924semidirect
125. C42C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊#992
#260123972
#260123962
#260120592
#2127173