Finite Group Databasebeta

← #36355229

Subgroups — #36355229

Order 22144.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012054C211yesmaximal, characteristic
2#99×D173C212yesmaximal, complemented
3#26011285C212yesmaximal, complemented
4#26011284C211yesmaximal, characteristic, complemented
5#353×C173C211yesmaximal, characteristic, complemented
6C2×#14583825C411yescharacteristic, complemented
7C2×D4×D173C2211yescharacteristic
8#19149675C411yescharacteristic, complemented
9#19149667C411yescharacteristic, complemented
10#19149666C2211yescharacteristic
11#19149651—22no—
12#19149647—22no—
13#19149636C411yescharacteristic, complemented
14C22⋊C4×D173—22no—
15#19149594C2212yes—
16#19149593—22no—
17#127×C173C2211yescharacteristic
18#99×C173C2212yes—
19C23×D173D412yes—
20#14583825—42no—
21C2×Dic346C2×C411yescharacteristic, complemented
22C2×D692C2×C411yescharacteristic
23C2×C22⋊D173—21no—
24C2×C22⋊D173—22no—
25C2×C22⋊D173D412yes—
26D4×D173—42no—
27C22×Dic173—21no—
28C2×C4×D173—42no—
29C2×C4×D173C2×C411yescharacteristic
30#14583805—21no—
31C22⋊Dic173—22no—
32C22⋊Dic173—22no—
33#14583803—21no—
34#14583803—22no—
35#14583803—42no—
36C4×Dic173C2×C411yescharacteristic, complemented
37C2×D4×C173—22no—
38C2×D4×C173C2×C411yescharacteristic, complemented
39C2×D4×C173C2311yescharacteristic
40C22⋊C4×C173—22no—
41C42×C173C2×C411yescharacteristic, complemented
42C22×D173—22no—
43C22×D173—22no—
44C22×D173—82no—
45C22×D173C22⋊C412yes—
46C22×D173C22⋊C411yescharacteristic
47Dic346—41no—
48D692—41no—
49C22⋊D173—42no—
50C22⋊D173—42no—
51C22⋊D173—82no—
52C2×Dic173—21no—
53C2×Dic173—21no—
54C2×Dic173—42no—
55C2×Dic173—42no—
56C2×Dic173C22⋊C411yescharacteristic
57C4×D173—22no—
58C4×D173—84no—
59C23×C173—21no—
60C23×C173C2×D412yes—
61D4×C173—41no—
62D4×C173—41no—
63D4×C173—82no—
64C2×C4×C173—21no—
65C2×C4×C173—42no—
66C2×C4×C173C22×C411yescharacteristic
67C22×C173—22no—
68C22×C173—42no—
69C22×C173—42no—
70C22×C173—41no—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—22no—
73C2×D173—41no—
74C2×D173—44no—
75C2×D173—42no—
76C2×D173—82no—
77C2×D173#9912yes—
78C4×C173—21no—
79C4×C173—41no—
80C4×C173—82no—
81Dic173—21no—
82Dic173—41no—
83Dic173—41no—
84Dic173—82no—
85C2×C173—21no—
86C2×C173—42no—
87C2×C173—81no—
88C2×C173C2×#9911yescharacteristic
89D173—22no—
90D173—42no—
91C173#212711yescharacteristic, complemented, Sylow
92#2127—1731nomaximal, Sylow
93#535—1731no—
94C2×#99—1732no—
95#354—1732no—
96#353—1731no—
97#353D17311yescharacteristic, complemented
98C2×#41—1731no—
99C22×D4—1731no—
100#127—1731no—
101#127C2×D17311yescharacteristic
102#124—1731no—
103#124—1731no—
104#121—3462no—
105#120—3462no—
106#117—1731no—
107C2×C22⋊C4—3462no—
108#99—3462no—
109#99—1732no—
110#99C2×D17312yes—
111C24—1732no—
112#41—6922no—
113C2×Q8—1731no—
114C2×D4—22no—
115C2×D4—3462no—
116C2×D4—3461no—
117C2×D4—1732no—
118C2×D4—1731no—
119C2×D4—6922no—
120C2×D4C4×D17311yescharacteristic, complemented
121C2×D4C22×D17311yescharacteristic
122C22×C4—3461no—
123C22×C4—1731no—
124C22×C4—6922no—
125C4⋊C4—3461no—
126C22⋊C4—22no—
127C22⋊C4—3461no—
128C22⋊C4—3462no—
129C22⋊C4—3462no—
130C22⋊C4—3462no—
131C22⋊C4—6922no—
132C42—1731no—
133C42C4×D17311yescharacteristic, complemented
134C23—21no—
135C23—3462no—
136C23—3462no—
137C23—13842no—
138C23—1732no—
139C23—1731no—
140C23D4×D17312yes—
141Q8—6921no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—13842no—
146D4—6921no—
147D4—6922no—
148D4—6922no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—3462no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—13844no—
155C2×C4—1731no—
156C2×C4—6922no—
157C2×C4—6922no—
158C2×C4C2×C4×D17311yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—3462no—
164C22—13842no—
165C22—1732no—
166C22—6921no—
167C22—6922no—
168C22—6924no—
169C22C22⋊C4×D17311yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—3461no—
174C4—13842no—
175C4—6921no—
176C4—6921no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—3462no—
181C2—6922no—
182C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#353×C173C2
socleC2×C173C2×#99
prime radical#26012054C2
cosocleC2×D4×C173C23
hypercenter#353D173
hypocenterC173#2127
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D173C2semidirect
3. #26011285C2semidirect
4. #26011284C2semidirect
5. #353×C173C2semidirect
6. C2×#14583825C4semidirect
8. #19149675C4semidirect
9. #19149667C4semidirect
13. #19149636C4semidirect
21. C2×Dic346C2×C4semidirect
36. C4×Dic173C2×C4semidirect
38. C2×D4×C173C2×C4semidirect
41. C42×C173C2×C4semidirect
91. C173#2127semidirect
97. #353D173semidirect
120. C2×D4C4×D173semidirect
133. C42C4×D173semidirect

Maximal subgroups

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

subgroupindex
#260120542
#99×D1732
#260112852
#260112842
#353×C1732
#2127173