Finite Group Databasebeta

← #36369083

Subgroups — #36369083

Order 22144.

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

normal subgroups only

This group has 252 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#26012514C211yesmaximal, characteristic, complemented
2D173⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
3#26012351C212yesmaximal, complemented
4#26012349C212yesmaximal, complemented
5#26012040C211yesmaximal, characteristic
6C2×D173⋊C8—21no—
7#19149746C2212yes—
8#19149746C2212yes—
9C4⋊#13063443—21no—
10#19149736—21no—
11#19149734—21no—
12C2×#14583827C411yescharacteristic
13C2×#14583825C411yescharacteristic
14#19149677C2211yescharacteristic
15#19149675C2211yescharacteristic
16#19149667C411yescharacteristic
17#19149652—21no—
18#19149652—21no—
19#19149639—21no—
20#19149639—21no—
21C42×D173C2211yescharacteristic
22#124×C173C411yescharacteristic
23D173⋊C8—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26#14583827—42no—
27#14583825—42no—
28C2×Dic346C2×C411yescharacteristic
29C2×D692—21no—
30C2×D692C2×C411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C22×Dic173—21no—
35C2×C4×D173—21no—
36C2×C4×D173C2×C411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39#14583805—21no—
40#14583805—21no—
41#14583803—21no—
42#14583803—21no—
43C4×Dic173D411yescharacteristic
44C4×Dic173D411yescharacteristic
45C4×Dic173C2311yescharacteristic
46C2×Q8×C173C2×C411yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C42×C173C2×C411yescharacteristic
51#13063443—42no—
52#13063443—42no—
53C22×D173—21no—
54C22×D173C22⋊C411yescharacteristic
55Dic346—41no—
56D692—41no—
57D692—42no—
58C22⋊D173—42no—
59C22⋊D173—42no—
60C22⋊D173—41no—
61C22⋊D173—41no—
62C2×Dic173—21no—
63C2×Dic173—42no—
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C2×Dic173C2×D411yescharacteristic
67C4×D173—22no—
68C4×D173—22no—
69C4×D173—21no—
70C4×D173—21no—
71C4×D173—42no—
72C23×C173—21no—
73Q8×C173—41no—
74D4×C173—41no—
75C2×C4×C173—21no—
76C2×C4×C173C22⋊C411yescharacteristic
77C2×C4×C173C22⋊C411yescharacteristic
78C2×C4×C173C22×C411yescharacteristic
79C22×C173—41no—
80C22×C173—41no—
81C22×C173—41no—
82C22×C173C2×C22⋊C411yescharacteristic
83C2×D173—21no—
84C2×D173—21no—
85C2×D173—41no—
86C2×D173—41no—
87C2×D173—41no—
88C2×D173#10012yes—
89C4×C173—21no—
90C4×C173—21no—
91C4×C173—21no—
92C4×C173—41no—
93Dic173—21no—
94Dic173—41no—
95Dic173#13512yes—
96Dic173AGL(1,ℤ/8)11yescharacteristic
97Dic173#13711yescharacteristic
98C2×C173—81no—
99C2×C173C2×#10011yescharacteristic
100C2×C173#41611yescharacteristic
101C2×C173#41711yescharacteristic
102D173—22no—
103D173—81no—
104C173#155111yescharacteristic, complemented, Sylow
105#1551—1731nomaximal, Sylow
106#532—1731no—
107#432—1731no—
108C2×C4⋊C8—1731no—
109#333—1732no—
110#331—1732no—
111C2×#41—1731no—
112C2×#41—1731no—
113C22×C8—3461no—
114C4⋊Q8—1731no—
115#127—1731no—
116#124—1731no—
117#124#1062192411yescharacteristic
118#121—3461no—
119#121—3461no—
120C4×D4—3461no—
121C4×D4—3461no—
122C2×C42—1731no—
123C4⋊C8—3461no—
124C4⋊C8—1732no—
125C4⋊C8—1732no—
126C22⋊C8—3461no—
127#97—3461no—
128#41—6922no—
129#41—6922no—
130C2×Q8—1731no—
131C2×Q8C2×#1062192411yescharacteristic
132C2×D4—3461no—
133C2×D4—3461no—
134C2×D4—3461no—
135C2×D4—3461no—
136C2×D4—1731no—
137C2×D4C2×#1062192411yescharacteristic
138C22×C4—3461no—
139C22×C4—3461no—
140C22×C4—1731no—
141C22×C4—1731no—
142C22×C4—1731no—
143C2×C8—3462no—
144C2×C8—3462no—
145C2×C8—6922no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C22⋊C4—3461no—
151C22⋊C4—3461no—
152C42—1731no—
153C42—1731no—
154C42—1731no—
155C42C2×#1062192411yescharacteristic
156C23—21no—
157C23—3461no—
158C23—1731no—
159Q8—41no—
160Q8—6921no—
161D4—41no—
162D4—6921no—
163D4—6922no—
164D4—6922no—
165D4—6921no—
166D4—6921no—
167D4—6922no—
168C2×C4—21no—
169C2×C4—3462no—
170C2×C4—3461no—
171C2×C4—3462no—
172C2×C4—3461no—
173C2×C4—3461no—
174C2×C4—1731no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4—6922no—
178C2×C4—6922no—
179C2×C4C22⋊#1062192411yescharacteristic
180C2×C4C22⋊#1062192411yescharacteristic
181C2×C4C22×#1062192411yescharacteristic
182C8—6922no—
183C8—6922no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—3461no—
188C22—3461no—
189C22—1732no—
190C22—6921no—
191C22—6921no—
192C22—6921no—
193C22C2×C22⋊#1062192411yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—3461no—
199C4—1732no—
200C4—1731no—
201C4—1731no—
202C4—6921no—
203C2—81no—
204C2—3462no—
205C2—13841no—
206C2D173⋊#10011yescharacteristic
207C2#2601248111yescharacteristic
208C2#2601248211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012040C2
cosocleC4×Dic173C23
hypercenter#124#10621924
hypocenterC173#1551
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. #26012514C2semidirect
2. D173⋊(C4⋊C8)C2semidirect
3. #26012351C2semidirect
4. #26012349C2semidirect
104. C173#1551semidirect

Maximal subgroups

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

subgroupindex
#260125142
D173⋊(C4⋊C8)2
#260123512
#260123492
#260120402
#1551173