Finite Group Databasebeta

← #36368935

Subgroups — #36368935

Order 22144.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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#26012427C211yesmaximal, characteristic, complemented
3#26012345C212yesmaximal, complemented
4#26012342C212yesmaximal
5#26012020C211yesmaximal, characteristic
6C2×C4×#10621924—21no—
7#19149746—21no—
8#19149736C2212yes—
9C4×#13063443—21no—
10(C2×#10621924)⋊C4—21no—
11(C2×#10621924)⋊C4C2212yes—
12#19149672—21no—
13#19149671C411yescharacteristic, complemented
14#19149661—21no—
15#19149655C2211yescharacteristic
16#19149650C2211yescharacteristic
17Q8×Dic173C411yescharacteristic, complemented
18D4×Dic173C411yescharacteristic, complemented
19#19149639—21no—
20#19149635—21no—
21C42×D173C2211yescharacteristic
22#124×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24C2×#13063443—22no—
25C4×#10621924—24no—
26C2×C22⋊D173—21no—
27C22×Dic173—21no—
28C2×C4×D173D411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C2×C4×D173C2311yescharacteristic
31C4⋊Dic173—21no—
32C4⋊Dic173C2×C411yescharacteristic
33C4⋊Dic173C2×C411yescharacteristic
34#14583805—21no—
35#14583805—21no—
36#14583805—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39#14583803—21no—
40C4×Dic173—21no—
41C4×Dic173C2×C411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C2×Q8×C173C2×C411yescharacteristic
45C2×D4×C173C2×C411yescharacteristic
46C22⋊C4×C173—21no—
47C22⋊C4×C173—21no—
48C42×C173C2×C411yescharacteristic
49C2×#10621924—24no—
50C2×#10621924—42no—
51C2×#10621924—42no—
52#13063443—42no—
53C22×D173C2×D411yescharacteristic
54C22⋊D173—42no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—42no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C22⋊C411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C4×D173—22no—
62C4×D173—22no—
63C4×D173—41no—
64C23×C173—21no—
65Q8×C173—41no—
66D4×C173—41no—
67C2×C4×C173—21no—
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22×C411yescharacteristic
71C22×C173—41no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173C2×C22⋊C411yescharacteristic
75C2×D173—21no—
76C2×D173—21no—
77C2×D173#13512yes—
78#10621924—44no—
79C4×C173—21no—
80C4×C173—21no—
81C4×C173—21no—
82C4×C173—41no—
83Dic173—21no—
84Dic173—21no—
85Dic173—41no—
86Dic173—41no—
87Dic173#10412yes—
88C2×C173—81no—
89C2×C173#41011yescharacteristic
90C2×C173#41611yescharacteristic
91C2×C173C2×#10411yescharacteristic
92D173—22no—
93C173#150011yescharacteristic, complemented, Sylow
94#1500—1731nomaximal, Sylow
95#529—1731no—
96#431—1731no—
97#376—1731no—
98#328—1732no—
99#327—1732no—
100#126—3461no—
101#125—1731no—
102#124#1062192411yescharacteristic, complemented
103#123—3461no—
104C22⋊Q8—1731no—
105#121—1731no—
106C4×Q8—1731no—
107C4×D4—3461no—
108C4×D4—1731no—
109#117—3461no—
110C2×C42—3461no—
111C2×C42—1731no—
112C4⋊C8—3461no—
113C22⋊C8—1732no—
114C4×C8—3461no—
115#95—3461no—
116#95—1732no—
117C2×Q8C2×#1062192411yescharacteristic
118C2×D4—3461no—
119C2×D4C2×#1062192411yescharacteristic
120C22×C4—3462no—
121C22×C4—3461no—
122C22×C4—1731no—
123C22×C4—1731no—
124C22×C4—1731no—
125C2×C8—3462no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—1731no—
131C4⋊C4—1731no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—3461no—
135C22⋊C4—3461no—
136C22⋊C4—3461no—
137C42—3461no—
138C42—3464no—
139C42—1731no—
140C42—1731no—
141C42—1731no—
142C42C2×#1062192411yescharacteristic
143C23—21no—
144C23—1731no—
145Q8—41no—
146D4—41no—
147D4—6922no—
148C2×C4—21no—
149C2×C4—3462no—
150C2×C4—3462no—
151C2×C4—3464no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—1731no—
155C2×C4—1731no—
156C2×C4—1731no—
157C2×C4—6921no—
158C2×C4—6922no—
159C2×C4—6922no—
160C2×C4—6922no—
161C2×C4C22⋊#1062192411yescharacteristic
162C2×C4C22⋊#1062192411yescharacteristic
163C2×C4C22×#1062192411yescharacteristic
164C8—6922no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—3461no—
169C22—3461no—
170C22—1732no—
171C22C2×C22⋊#1062192411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—3461no—
177C4—3461no—
178C4—1732no—
179C4—6924no—
180C4—6921no—
181C4—6921no—
182C2—81no—
183C2—3462no—
184C2#2601247111yescharacteristic
185C2#2601248111yescharacteristic
186C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radical#19149650C22
cosocleC2×C4×D173C23
hypercenter#124#10621924
hypocenterC173#1500
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012509C2semidirect
2. #26012427C2semidirect
3. #26012345C2semidirect
13. #19149671C4semidirect
17. Q8×Dic173C4semidirect
18. D4×Dic173C4semidirect
22. #124×C173C4semidirect
93. C173#1500semidirect
102. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125092
#260124272
#260123452
#260123422
#260120202
#1500173