Finite Group Databasebeta

← #36369071

Subgroups — #36369071

Order 22144.

344 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 70 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#26012496C211yesmaximal, characteristic, complemented
3#26012349C214yesmaximal, complemented
4#127×D173C211yesmaximal, characteristic
5C2×#14583836—21no—
6#19149746—21no—
7#19149746C2212yescomplemented
8#19149746C2212yescomplemented
9#19149736—21no—
10C4×#13063443—21no—
11C2×D4×D173—21no—
12C2×D4×D173—21no—
13C2×D4×D173C412yes—
14#19149676C411yescharacteristic
15#19149675—21no—
16#19149675—21no—
17#19149675C2212yes—
18C42×D173C2211yescharacteristic
19#127×C173C411yescharacteristic
20#14583836—42no—
21C2×#13063443—22no—
22C2×#13063443—22no—
23C23×D173—22no—
24C2×D692—21no—
25C2×D692—21no—
26C2×D692C2×C412yes—
27C2×C22⋊D173—22no—
28C2×C22⋊D173—22no—
29C2×C22⋊D173—22no—
30D4×D173—44no—
31D4×D173—44no—
32D4×D173—42no—
33D4×D173—42no—
34C2×C4×D173C2×C411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C4×Dic173D411yescharacteristic
38C4×Dic173D411yescharacteristic
39C4×Dic173C2311yescharacteristic
40C2×D4×C173—21no—
41C2×D4×C173—21no—
42C2×D4×C173C2×C412yes—
43C42×C173C2×C411yescharacteristic
44#13063443—42no—
45#13063443—42no—
46C22×D173—22no—
47C22×D173—44no—
48C22×D173—44no—
49C22×D173—44no—
50C22×D173C22⋊C411yescharacteristic
51D692—42no—
52D692—42no—
53D692—42no—
54C22⋊D173—44no—
55C22⋊D173—42no—
56C22⋊D173—42no—
57C22⋊D173—44no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C4×D173—22no—
62C4×D173—22no—
63C4×D173D812yes—
64C4×D173SD1612yes—
65C23×C173—22no—
66D4×C173—42no—
67D4×C173—42no—
68D4×C173—42no—
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22×C411yescharacteristic
72C22×C173—42no—
73C22×C173—42no—
74C22×C173—42no—
75C22×C173C2×C22⋊C411yescharacteristic
76C2×D173—21no—
77C2×D173—21no—
78C2×D173—42no—
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—88no—
82C2×D173#10212yes—
83C4×C173—21no—
84C4×C173—21no—
85C4×C173#10212yes—
86Dic173#10012yes—
87Dic173C2×D811yescharacteristic
88Dic173C2×SD1611yescharacteristic
89Dic173AGL(1,ℤ/8)12yes—
90C2×C173—82no—
91C2×C173C2×#10011yescharacteristic
92C2×C173C2×#10211yescharacteristic
93C2×C173#41811yescharacteristic
94D173—22no—
95D173—82no—
96C173#154511yescharacteristic, complemented, Sylow
97#1545—1731nomaximal, Sylow
98C2×#127—1731no—
99#431—1731no—
100#423—1731no—
101#331—1734no—
102C22×D4—3461no—
103C22×D4—3461no—
104C22×D4—1732no—
105C2×M16—3461no—
106#127—3461no—
107#127—3461no—
108#127—1732no—
109#127—1731no—
110#127#1062192411yescharacteristic
111C2×C42—1731no—
112C4⋊C8—3461no—
113C4⋊C8—1732no—
114C4⋊C8—1732no—
115C22⋊C8—3461no—
116C4×C8—3461no—
117C24—3462no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—3462no—
121C2×D4—3462no—
122C2×D4—3462no—
123C2×D4—3461no—
124C2×D4—3461no—
125C2×D4—1732no—
126C2×D4—6924no—
127C2×D4—6924no—
128C2×D4—6922no—
129C2×D4—6922no—
130C2×D4C2×#1062192412yes—
131C22×C4—1731no—
132C22×C4—1731no—
133C22×C4—1731no—
134M16—6922no—
135C2×C8—3462no—
136C2×C8—3462no—
137C42—1731no—
138C42—1731no—
139C42—1731no—
140C42C2×#1062192411yescharacteristic
141C23—22no—
142C23—3462no—
143C23—1731no—
144C23—6924no—
145C23—6924no—
146C23—6924no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—6922no—
151D4—6922no—
152D4—6924no—
153D4—6922no—
154D4—6922no—
155D4—6924no—
156D4—6922no—
157C2×C4—3462no—
158C2×C4—3462no—
159C2×C4—1731no—
160C2×C4—1731no—
161C2×C4—1731no—
162C2×C4—1732no—
163C2×C4—1732no—
164C2×C4C22⋊#1062192411yescharacteristic
165C2×C4C22⋊#1062192411yescharacteristic
166C2×C4C22×#1062192411yescharacteristic
167C8—6922no—
168C8—6922no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—3461no—
173C22—3461no—
174C22—13848no—
175C22—1732no—
176C22—6922no—
177C22—6922no—
178C22—6922no—
179C22C2×C22⋊#1062192411yescharacteristic
180C4—21no—
181C4—21no—
182C4—1732no—
183C4—1731no—
184C4—1731no—
185C4—1732no—
186C4#1914974112yes—
187C2—82no—
188C2—3462no—
189C2—13842no—
190C2D173⋊#10011yescharacteristic
191C2C2×#1914974111yescharacteristic
192C2#2601248311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC22×C173C2×C22⋊C4
prime radical#127×D173C2
cosocleC4×Dic173C23
hypercenter#127#10621924
hypocenterC173#1545
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. #26012496C2semidirect
3. #26012349C2semidirect
7. #19149746C22semidirect
8. #19149746C22semidirect
96. C173#1545semidirect

Maximal subgroups

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

subgroupindex
#260125092
#260124962
#260123492
#127×D1732
#1545173