Finite Group Databasebeta

← #36368992

Subgroups — #36368992

Order 22144.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012509C211yesmaximal, characteristic, complemented
2#26012435C211yesmaximal, characteristic, complemented
3#26012342C214yesmaximal, complemented
4#26012039C211yesmaximal, characteristic
5C2×C4⋊#10621924—21no—
6#19149746—21no—
7#19149736C2212yes—
8C4×#13063443—21no—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4C2212yes—
11C2×#14583825—21no—
12#19149679C411yescharacteristic, complemented
13#19149666—21no—
14#19149650—21no—
15#19149650C2212yes—
16D4×Dic173—21no—
17D4×Dic173C412yescomplemented
18C42×D173C2211yescharacteristic
19#127×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21C2×#13063443—22no—
22C4⋊#10621924—42no—
23#14583825—42no—
24#14583825—42no—
25C2×Dic346—21no—
26C2×C22⋊D173—22no—
27C22×Dic173—22no—
28C2×C4×D173D411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C2×C4×D173C2311yescharacteristic
31C4⋊Dic173—21no—
32C4⋊Dic173C2×C412yes—
33C22⋊Dic173—22no—
34C22⋊Dic173—22no—
35C4×Dic173C2×C411yescharacteristic
36C4×Dic173D411yescharacteristic
37C4×Dic173D411yescharacteristic
38C2×D4×C173—21no—
39C2×D4×C173—21no—
40C2×D4×C173C2×C412yes—
41C42×C173C2×C411yescharacteristic
42C2×#10621924—42no—
43C2×#10621924—42no—
44C2×#10621924—42no—
45#13063443—42no—
46C22×D173C2×D411yescharacteristic
47Dic346—42no—
48C22⋊D173—42no—
49C22⋊D173—42no—
50C2×Dic173—22no—
51C2×Dic173—44no—
52C2×Dic173C22⋊C411yescharacteristic
53C2×Dic173C22⋊C411yescharacteristic
54C2×Dic173C2×D411yescharacteristic
55C4×D173—41no—
56C4×D173—41no—
57C4×D173D812yes—
58C4×D173SD1612yes—
59C23×C173—22no—
60D4×C173—42no—
61D4×C173—42no—
62D4×C173—42no—
63C2×C4×C173C22⋊C411yescharacteristic
64C2×C4×C173C22⋊C411yescharacteristic
65C2×C4×C173C22×C411yescharacteristic
66C22×C173—42no—
67C22×C173—42no—
68C22×C173—42no—
69C22×C173C2×C22⋊C411yescharacteristic
70C2×D173—21no—
71C2×D173—21no—
72C2×D173C2×D811yescharacteristic
73C2×D173C2×SD1611yescharacteristic
74#10621924—82no—
75C4×C173—21no—
76C4×C173—21no—
77C4×C173#10212yes—
78Dic173—21no—
79Dic173—21no—
80Dic173—42no—
81Dic173#10212yes—
82C2×C173—82no—
83C2×C173#41011yescharacteristic
84C2×C173C2×#10211yescharacteristic
85C2×C173#42111yescharacteristic
86D173—21no—
87D173—21no—
88C173#151711yescharacteristic, complemented, Sylow
89#1517—1731nomaximal, Sylow
90#532—1731no—
91#431—1731no—
92#382—1731no—
93#327—1734no—
94C2×#41—3461no—
95C4⋊Q8—1731no—
96#127#1062192411yescharacteristic, complemented
97#124—3461no—
98#121—3461no—
99#121—1732no—
100C4×D4—3461no—
101C4×D4—1732no—
102C2×C4⋊C4—3461no—
103C2×C42—1731no—
104C4⋊C8—3461no—
105C22⋊C8—1732no—
106C4×C8—3461no—
107#95—3461no—
108#95—1732no—
109#41—6922no—
110#41—6922no—
111C2×Q8—3461no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—3462no—
115C2×D4C2×#1062192412yes—
116C22×C4—3462no—
117C22×C4—3462no—
118C22×C4—1731no—
119C22×C4—1731no—
120C22×C4—1731no—
121C2×C8—3462no—
122C4⋊C4—3461no—
123C4⋊C4—1732no—
124C4⋊C4—6922no—
125C22⋊C4—3462no—
126C22⋊C4—3462no—
127C42—1731no—
128C42—1731no—
129C42—1731no—
130C42C2×#1062192411yescharacteristic
131C23—22no—
132C23—1731no—
133Q8—6922no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—6922no—
138D4—6922no—
139C2×C4—3462no—
140C2×C4—1731no—
141C2×C4—1731no—
142C2×C4—1731no—
143C2×C4—1732no—
144C2×C4—1732no—
145C2×C4—6921no—
146C2×C4—6921no—
147C2×C4—6922no—
148C2×C4—6922no—
149C2×C4—6922no—
150C2×C4—6924no—
151C2×C4C22⋊#1062192411yescharacteristic
152C2×C4C22⋊#1062192411yescharacteristic
153C2×C4C22×#1062192411yescharacteristic
154C8—6922no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22—3461no—
159C22—3461no—
160C22—1731no—
161C22—1731no—
162C22C2×C22⋊#1062192411yescharacteristic
163C4—21no—
164C4—21no—
165C4—3461no—
166C4—3461no—
167C4—13842no—
168C4—1732no—
169C4—6922no—
170C4#1914974112yes—
171C2—82no—
172C2—3461no—
173C2—3461no—
174C2#2601247111yescharacteristic
175C2C2×#1914974111yescharacteristic
176C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012039C2
cosocleC2×C4×D173C23
hypercenter#127#10621924
hypocenterC173#1517
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26012509C2semidirect
2. #26012435C2semidirect
3. #26012342C2semidirect
12. #19149679C4semidirect
17. D4×Dic173C4semidirect
19. #127×C173C4semidirect
88. C173#1517semidirect
96. #127#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125092
#260124352
#260123422
#260120392
#1517173