Finite Group Databasebeta

← #36368875

Subgroups — #36368875

Order 22144.

212 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 60 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#26012510C211yesmaximal, characteristic, complemented
2#26012510C211yesmaximal, characteristic, complemented
3Q8⋊#13063443C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5#26012020C211yesmaximal, characteristic
6C4⋊#13063443—21no—
7C4⋊#13063443C2212yes—
8#19149736—21no—
9#19149736—21no—
10C4×#13063443—21no—
11C4×#13063443C2212yes—
12#19149672—21no—
13#19149671C411yescharacteristic
14#19149661—21no—
15#19149655C411yescharacteristic
16#19149650C411yescharacteristic
17Q8×Dic173C2211yescharacteristic
18D4×Dic173C2211yescharacteristic
19#19149639—21no—
20#19149635—21no—
21C42×D173C2211yescharacteristic
22#124×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25C2×C22⋊D173—21no—
26C22×Dic173—21no—
27C2×C4×D173C2×C411yescharacteristic
28C2×C4×D173D411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C411yescharacteristic
32C4⋊Dic173C2×C411yescharacteristic
33#14583805—21no—
34#14583805—21no—
35#14583805—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38#14583803—21no—
39C4×Dic173—21no—
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173C2311yescharacteristic
43C2×Q8×C173C2×C411yescharacteristic
44C2×D4×C173C2×C411yescharacteristic
45C22⋊C4×C173—21no—
46C22⋊C4×C173—21no—
47C42×C173C2×C411yescharacteristic
48#13063443—42no—
49#13063443—42no—
50C22×D173C22⋊C411yescharacteristic
51C22⋊D173—42no—
52C2×Dic173—21no—
53C2×Dic173—21no—
54C2×Dic173—42no—
55C2×Dic173C22⋊C411yescharacteristic
56C2×Dic173C2×D411yescharacteristic
57C2×Dic173C2×D411yescharacteristic
58C4×D173—22no—
59C4×D173—22no—
60C4×D173—41no—
61C23×C173—21no—
62Q8×C173—22no—
63D4×C173—22no—
64C2×C4×C173—21no—
65C2×C4×C173C22⋊C411yescharacteristic
66C2×C4×C173C22⋊C411yescharacteristic
67C2×C4×C173C22×C411yescharacteristic
68C22×C173—41no—
69C22×C173—41no—
70C22×C173—41no—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—21no—
73C2×D173—21no—
74C2×D173—21no—
75C4×C173—21no—
76C4×C173—21no—
77C4×C173—41no—
78C4×C173#13112yes—
79Dic173—21no—
80Dic173—21no—
81Dic173—41no—
82Dic173—41no—
83Dic173#13512yes—
84C2×C173—81no—
85C2×C173#40811yescharacteristic
86C2×C173#41611yescharacteristic
87C2×C173#42111yescharacteristic
88D173—41no—
89C173#148211yescharacteristic, complemented, Sylow
90#1482—1731nomaximal, Sylow
91#529—1731no—
92#431—1731no—
93#431—1731no—
94Q8⋊C8—1732no—
95D4⋊C8—1732no—
96#126—3461no—
97#125—1731no—
98#124#1062192411yescharacteristic
99#123—3461no—
100C22⋊Q8—1731no—
101#121—1731no—
102C4×Q8—1731no—
103C4×D4—3461no—
104C4×D4—1731no—
105#117—3461no—
106C2×C42—1731no—
107C4⋊C8—3461no—
108C4⋊C8—1732no—
109C22⋊C8—3461no—
110C22⋊C8—3461no—
111C4×C8—3461no—
112C4×C8—1732no—
113C2×Q8C2×#1062192411yescharacteristic
114C2×D4—3461no—
115C2×D4C2×#1062192411yescharacteristic
116C22×C4—3461no—
117C22×C4—1731no—
118C22×C4—1731no—
119C22×C4—1731no—
120C2×C8—3462no—
121C2×C8—3462no—
122C4⋊C4—3461no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—3461no—
126C4⋊C4—1731no—
127C4⋊C4—1731no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—3461no—
131C22⋊C4—3461no—
132C22⋊C4—3461no—
133C42—3461no—
134C42—1731no—
135C42—1731no—
136C42—1731no—
137C42C2×#1062192411yescharacteristic
138C23—21no—
139C23—1731no—
140Q8—22no—
141D4—22no—
142D4—6922no—
143C2×C4—21no—
144C2×C4—3462no—
145C2×C4—3462no—
146C2×C4—3461no—
147C2×C4—3461no—
148C2×C4—1731no—
149C2×C4—1731no—
150C2×C4—1731no—
151C2×C4—6921no—
152C2×C4—6922no—
153C2×C4C22⋊#1062192411yescharacteristic
154C2×C4C22⋊#1062192411yescharacteristic
155C2×C4C22×#1062192411yescharacteristic
156C8—6922no—
157C8—6922no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—3461no—
162C22—3461no—
163C22—3461no—
164C22C2×C22⋊#1062192411yescharacteristic
165C4—21no—
166C4—21no—
167C4—41no—
168C4—3461no—
169C4—3461no—
170C4—1732no—
171C4—6921no—
172C4—6921no—
173C4#1914976612yes—
174C2—81no—
175C2—6921no—
176C2#2601246511yescharacteristic
177C2#2601248111yescharacteristic
178C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radical#19149650C4
cosocleC4×Dic173C23
hypercenter#124#10621924
hypocenterC173#1482
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012510C2semidirect
2. #26012510C2semidirect
3. Q8⋊#13063443C2semidirect
4. D4⋊#13063443C2semidirect
89. C173#1482semidirect

Maximal subgroups

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

subgroupindex
#260125102
Q8⋊#130634432
D4⋊#130634432
#260120202
#1482173