Finite Group Databasebeta

← #36355234

Subgroups — #36355234

Order 22144.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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#26012055C211yesmaximal, characteristic
2#26011465C211yesmaximal, characteristic, complemented
3#26011283C214yesmaximal, complemented
4#410×C173C211yesmaximal, characteristic, complemented
5C22×D692C411yescharacteristic, complemented
6#19149675C2212yes—
7#19149668C412yescomplemented
8#19149653—21no—
9#19149653—21no—
10#19149647—21no—
11#19149647—21no—
12#19149638—21no—
13#19149638C411yescharacteristic, complemented
14#19149594—21no—
15#19149594C2212yescomplemented
16C2×#41×C173C2211yescharacteristic
17#117×C173—21no—
18#99×C173—21no—
19#99×C173C2212yescomplemented
20C23×D173—21no—
21C2×D692—21no—
22C2×D692—41no—
23C2×D692—41no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—22no—
26C2×C22⋊D173—22no—
27C4⋊Dic173—21no—
28C4⋊Dic173—41no—
29C22⋊Dic173—21no—
30C22⋊Dic173—22no—
31#14583803—22no—
32#14583803—22no—
33C4×Dic173—41no—
34C4×Dic173C2×C412yes—
35#41×C173—41no—
36#41×C173—41no—
37C2×Q8×C173C2×C411yescharacteristic
38C2×D4×C173D411yescharacteristic
39C2×D4×C173D411yescharacteristic
40C2×D4×C173C2311yescharacteristic
41C22×C4×C173C2×C411yescharacteristic
42C22×C4×C173D411yescharacteristic
43C22×C4×C173D411yescharacteristic
44C4⋊C4×C173—41no—
45C22⋊C4×C173—22no—
46C42×C173—41no—
47C22×D173—22no—
48C22×D173—42no—
49C22×D173—82no—
50D692—42no—
51D692—41no—
52D692—41no—
53C22⋊D173—82no—
54C22⋊D173—82no—
55C2×Dic173—22no—
56C2×Dic173—42no—
57C23×C173C22⋊C411yescharacteristic
58C23×C173C2×D411yescharacteristic
59C23×C173C2×D411yescharacteristic
60Q8×C173—41no—
61D4×C173—41no—
62D4×C173—41no—
63D4×C173—41no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—42no—
71C2×C4×C173C22⋊C411yescharacteristic
72C2×C4×C173C22⋊C411yescharacteristic
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22×C411yescharacteristic
75C22×C173—21no—
76C22×C173—21no—
77C22×C173—21no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173C2×C22⋊C411yescharacteristic
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—84no—
86C4×C173—21no—
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—82no—
90C4×C173#9911yescharacteristic
91C4×C173#9911yescharacteristic
92Dic173—42no—
93Dic173—82no—
94C2×C173—21no—
95C2×C173—41no—
96C2×C173—41no—
97C2×C173—41no—
98C2×C173C2×#9911yescharacteristic
99D173—82no—
100C173#212811yescharacteristic, complemented, Sylow
101#2128—1731nomaximal, Sylow
102#535—1731no—
103#410—1731no—
104#410D17311yescharacteristic, complemented
105#353—1734no—
106C2×#41C2×D17311yescharacteristic
107C22×D4—1731no—
108#127—1732no—
109#124—1732no—
110#121—3461no—
111#121—3461no—
112#120—3461no—
113#120—3461no—
114#117—21no—
115#117—3461no—
116#117—1731no—
117#99—21no—
118#99—3461no—
119#99—1732no—
120#99C2×D17312yescomplemented
121C24—3461no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D17311yescharacteristic
125C2×D4—3462no—
126C2×D4—3462no—
127C2×D4—3461no—
128C2×D4—1732no—
129C2×D4—6921no—
130C2×D4—6921no—
131C2×D4C22⋊D17311yescharacteristic
132C2×D4D69211yescharacteristic
133C2×D4C22×D17311yescharacteristic
134C22×C4C4×D17311yescharacteristic
135C22×C4C22⋊D17311yescharacteristic
136C22×C4D69211yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—3461no—
139C4⋊C4—6921no—
140C22⋊C4—22no—
141C22⋊C4—3461no—
142C22⋊C4—3462no—
143C22⋊C4—3462no—
144C22⋊C4—3462no—
145C42—41no—
146C42—1732no—
147C42—6921no—
148C23—3462no—
149C23—13842no—
150C23—6922no—
151C23#1458380311yescharacteristic
152C23C2×C22⋊D17311yescharacteristic
153C23C2×D69211yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—13842no—
159D4—13842no—
160D4—6922no—
161D4—6921no—
162D4—6921no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—3462no—
171C2×C4—6922no—
172C2×C4#1458380311yescharacteristic
173C2×C4#1458380311yescharacteristic
174C2×C4#1458380311yescharacteristic
175C2×C4C2×C4×D17311yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—13844no—
183C22—6922no—
184C22—6922no—
185C22—6922no—
186C22C2×#1458380311yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—13842no—
192C4—6922no—
193C4#1914959311yescharacteristic
194C4#1914959311yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—13842no—
200C2C2×#1914959311yescharacteristic
subgroupG/H
centerC2C2×#19149593
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#410×C173C2
socleC2×C173C2×#99
prime radical#26012055C2
cosocleC2×D4×C173C23
hypercenter#410D173
hypocenterC173#2128
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26011465C2semidirect
3. #26011283C2semidirect
4. #410×C173C2semidirect
5. C22×D692C4semidirect
7. #19149668C4semidirect
13. #19149638C4semidirect
15. #19149594C22semidirect
19. #99×C173C22semidirect
100. C173#2128semidirect
104. #410D173semidirect
120. #99C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120552
#260114652
#260112832
#410×C1732
#2128173