Finite Group Databasebeta

← #36361829

Subgroups — #36361829

Order 22144.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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#26011432C212yesmaximal
2#26011430C211yesmaximal, characteristic
3#26011425C212yesmaximal
4#26011424C212yesmaximal
5#26011422C212yesmaximal
6#26011410C212yesmaximal
7#26011360C211yesmaximal, characteristic
8#26011358C212yesmaximal
9#395×C173C211yesmaximal, characteristic
10C2×C4⋊Dic173C2211yescharacteristic
11C2×C4⋊Dic173C2211yescharacteristic
12C2×#14583805C2212yes—
13C2×#14583805C2212yes—
14C2×#14583805C2212yes—
15C2×#14583805C2212yes—
16C2×#14583805C2212yes—
17C2×C4×Dic173C2211yescharacteristic
18#19149586C2212yes—
19#19149586C2212yes—
20#19149586C2212yes—
21#19149586C2211yescharacteristic
22#19149586C2212yes—
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2212yes—
26C22×Q8×C173C2211yescharacteristic
27C2×C4⋊C4×C173C2212yes—
28C2×C4⋊C4×C173C2211yescharacteristic
29#95×C173C2212yes—
30#95×C173C2211yescharacteristic
31C22×Dic173C2312yes—
32C22×Dic173C2312yes—
33C22×Dic173C2312yes—
34C22×Dic173C2312yes—
35C4⋊Dic173—41no—
36C4⋊Dic173—41no—
37#14583805—42no—
38#14583805—42no—
39#14583805—42no—
40#14583805—42no—
41#14583805—42no—
42C4×Dic173Q814yescomplemented
43C2×Q8×C173—41no—
44C2×Q8×C173—42no—
45C22×C4×C173C2312yes—
46C22×C4×C173C2311yescharacteristic
47C22×C4×C173C2312yes—
48C22×C4×C173C2311yescharacteristic
49C22×C4×C173C2311yescharacteristic
50C4⋊C4×C173—41no—
51C4⋊C4×C173—42no—
52C2×Dic173—22no—
53C2×Dic173—22no—
54C2×Dic173—22no—
55C2×Dic173—22no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—22no—
61C2×Dic173—24no—
62C2×Dic173C2×Q814yes—
63C23×C173C2411yescharacteristic
64Q8×C173—44no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—22no—
69C2×C4×C173—22no—
70C2×C4×C173—22no—
71C2×C4×C173—21no—
72C2×C4×C173—22no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—22no—
76C2×C4×C173—22no—
77C2×C4×C173C2×Q812yes—
78C22×C173C22×Q811yescharacteristic
79C22×C173H(5,𝔽2)12yes—
80C22×C173H(5,𝔽2)11yescharacteristic
81C22×C1732−1+411yescharacteristic
82C22×C1732−1+412yes—
83C4×C173—22no—
84C4×C173—41no—
85C4×C173—41no—
86C4×C173—42no—
87C4×C173—42no—
88Dic173—24no—
89Dic173—42no—
90Dic173—42no—
91Dic173—42no—
92C2×C173#54412yes—
93C2×C173#54411yescharacteristic
94C2×C173#56011yescharacteristic
95C2×C173#56311yescharacteristic
96C2×C173#56312yes—
97C173#284311yescharacteristic, complemented, Sylow
98#2843—1731nomaximal, Sylow
99#400—1731no—
100#400—1732no—
101#398—1732no—
102#398—1732no—
103#398—1732no—
104#395—1732no—
105#395D17311yescharacteristic
106#384—1731no—
107#383—1732no—
108C22×Q8C2×D17311yescharacteristic
109C2×C4⋊C4—1732no—
110C2×C4⋊C4—1732no—
111C2×C4⋊C4—1732no—
112C2×C4⋊C4—1732no—
113C2×C4⋊C4—1732no—
114C2×C4⋊C4—1731no—
115C2×C4⋊C4—1731no—
116C2×C4⋊C4C2×D17311yescharacteristic
117C2×C4⋊C4C2×D17312yes—
118C2×C42—1731no—
119#95—1732no—
120#95—1732no—
121#95—1732no—
122#95—1732no—
123#95—1732no—
124#95—1732no—
125#95—1731no—
126#95—1732no—
127#95C2×D17312yes—
128#95C2×D17311yescharacteristic
129C2×Q8—41no—
130C2×Q8—42no—
131C22×C4—1732no—
132C22×C4—1732no—
133C22×C4—1732no—
134C22×C4—1732no—
135C22×C4C22×D17312yes—
136C22×C4C22×D17312yes—
137C22×C4C22×D17311yescharacteristic
138C22×C4C22×D17311yescharacteristic
139C22×C4C22×D17311yescharacteristic
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—6922no—
143C4⋊C4—6922no—
144C4⋊C4—6922no—
145C4⋊C4—6922no—
146C4⋊C4—6922no—
147C4⋊C4—6921no—
148C4⋊C4—6921no—
149C42—1734no—
150C23C23×D17311yescharacteristic
151Q8—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—3462no—
165C2×C4—3462no—
166C2×C4—3462no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3462no—
171C2×C4—3462no—
172C2×C4—3464no—
173C2×C4—3462no—
174C2×C4—1734no—
175C2×C4Q8×D17312yes—
176C22C2×Q8×D17311yescharacteristic
177C22#1914972411yescharacteristic
178C22#1914972412yes—
179C22#1914972611yescharacteristic
180C22#1914972712yes—
181C4—22no—
182C4—41no—
183C4—42no—
184C4—41no—
185C4—42no—
186C4—3464no—
187C4—6922no—
188C4—6922no—
189C4—6922no—
190C2#2601211812yes—
191C2#2601212011yescharacteristic
192C2#2601223011yescharacteristic
193C2#2601224011yescharacteristic
194C2#2601224212yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, prime radical, cosocleC23×C173C24
Fitting#395×C173C2
hypercenter#395D173
hypocenterC173#2843
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
42. C4×Dic173Q8semidirect
97. C173#2843semidirect

Maximal subgroups

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

subgroupindex
#260114322
#260114302
#260114252
#260114242
#260114222
#260114102
#260113602
#260113582
#395×C1732
#2843173