Finite Group Databasebeta

← #36369014

Subgroups — #36369014

Order 22144.

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

normal subgroups only

This group has 234 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
1C4×#14583835C211yesmaximal, characteristic, complemented
2C2×#19149734C211yesmaximal, characteristic, complemented
3#26012348C212yesmaximal, complemented
4#26012346C212yesmaximal, complemented
5#26012022C211yesmaximal, characteristic
6C2×#14583835—21no—
7C22×#13063443—21no—
8#19149734—21no—
9#19149734C2212yes—
10#19149734C2212yes—
11C4×#13063443—21no—
12#19149672—21no—
13#19149670C2211yescharacteristic
14#19149666C2211yescharacteristic
15#19149660—21no—
16#19149658C411yescharacteristic
17#19149644C411yescharacteristic
18D4×Dic173—21no—
19D4×Dic173C411yescharacteristic
20#19149635—21no—
21C2×C4×Dic173C2211yescharacteristic
22#121×C173C411yescharacteristic
23#14583835—24no—
24C2×#13063443—24no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27C2×Dic346C2×C411yescharacteristic
28C22×Dic173—21no—
29C22×Dic173D411yescharacteristic
30C22×Dic173D411yescharacteristic
31C4⋊Dic173—21no—
32C4⋊Dic173C2×C411yescharacteristic
33#14583805—21no—
34#14583805—21no—
35#14583805—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39C22⋊Dic173—21no—
40C4×Dic173—21no—
41C4×Dic173C2×C411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C4×Dic173C2311yescharacteristic
45C2×D4×C173—21no—
46C2×D4×C173C2×C411yescharacteristic
47C22×C4×C173C2×C411yescharacteristic
48C4⋊C4×C173C2×C411yescharacteristic
49C22⋊C4×C173—21no—
50#13063443—24no—
51#13063443—24no—
52Dic346—41no—
53C2×Dic173—22no—
54C2×Dic173—22no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—42no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C22⋊C411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C23×C173—21no—
63C23×C173C22⋊C411yescharacteristic
64D4×C173—41no—
65D4×C173—42no—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22×C411yescharacteristic
70C22×C173—21no—
71C22×C173—21no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173#10412yes—
76C22×C173C2×C22⋊C411yescharacteristic
77C4×C173—21no—
78C4×C173—21no—
79C4×C173—41no—
80Dic173—21no—
81Dic173—21no—
82Dic173—21no—
83Dic173—41no—
84Dic173—41no—
85Dic173#10412yes—
86C2×C173—22no—
87C2×C173—81no—
88C2×C173#41311yescharacteristic
89C2×C173C2×#10411yescharacteristic
90C2×C173C2×#10411yescharacteristic
91C173#152411yescharacteristic, complemented, Sylow
92#1524—1731nomaximal, Sylow
93#529—1731no—
94C4×M16—1731no—
95C2×#97—1731no—
96#330—1732no—
97#329—1732no—
98C2×M16—3461no—
99C22×C8—3461no—
100#126—3461no—
101#125—1731no—
102#124—1731no—
103#123—3461no—
104C22⋊Q8—1731no—
105#121#1062192411yescharacteristic
106C4×Q8—1731no—
107C4×D4—3461no—
108C4×D4—1731no—
109#117—3461no—
110C2×C42—1731no—
111#97—3461no—
112#97—1732no—
113#97—1732no—
114C4×C8—3461no—
115C2×Q8—1731no—
116C2×D4—21no—
117C2×D4C2×#1062192411yescharacteristic
118C22×C4—3461no—
119C22×C4—1731no—
120C22×C4—1731no—
121C22×C4C2×#1062192411yescharacteristic
122M16—3464no—
123C2×C8—3464no—
124C2×C8—3462no—
125C2×C8—3462no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—1731no—
131C4⋊C4C2×#1062192411yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3461no—
134C22⋊C4—3461no—
135C22⋊C4—3461no—
136C22⋊C4—3461no—
137C42—3461no—
138C42—1731no—
139C42—1731no—
140C42—1731no—
141C42—1731no—
142C23—21no—
143C23C22⋊#1062192411yescharacteristic
144Q8—6921no—
145D4—41no—
146D4—42no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3462no—
150C2×C4—3462no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—1731no—
154C2×C4—1731no—
155C2×C4—1731no—
156C2×C4—1731no—
157C2×C4—6922no—
158C2×C4C22⋊#1062192411yescharacteristic
159C2×C4C22×#1062192411yescharacteristic
160C8—3464no—
161C8—3464no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22#1914974312yes—
168C22C2×C22⋊#1062192411yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—3461no—
173C4—3461no—
174C4—3461no—
175C4—1732no—
176C4—6921no—
177C4—6921no—
178C2—22no—
179C2—81no—
180C2#2601247811yescharacteristic
181C2C2×#1914974311yescharacteristic
182C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting, prime radical#121×C173C4
socleC22×C173C2×C22⋊C4
cosocleC4×Dic173C23
hypercenter#121#10621924
hypocenterC173#1524
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C4×#14583835C2semidirect
2. C2×#19149734C2semidirect
3. #26012348C2semidirect
4. #26012346C2semidirect
91. C173#1524semidirect

Maximal subgroups

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

subgroupindex
C4×#145838352
C2×#191497342
#260123482
#260123462
#260120222
#1524173