Finite Group Databasebeta

← #36355233

Subgroups — #36355233

Order 22144.

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

normal subgroups only

This group has 270 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#26011463C212yesmaximal, complemented
3#26011284C211yesmaximal, characteristic, complemented
4#26011283C212yesmaximal, complemented
5#353×C173C211yesmaximal, characteristic, complemented
6C2×#14583825C2211yescharacteristic
7C2×D4×D173C411yescharacteristic, complemented
8#19149675C2211yescharacteristic
9#19149667C411yescharacteristic, complemented
10#19149666C411yescharacteristic, complemented
11#19149651—22no—
12#19149647—22no—
13#19149636C411yescharacteristic, complemented
14#19149635—22no—
15#19149594C2212yescomplemented
16#19149593—22no—
17#127×C173C2211yescharacteristic
18#99×C173C2212yescomplemented
19C23×D173—21no—
20#14583825—42no—
21C2×Dic346C2×C411yescharacteristic
22C2×D692C2×C411yescharacteristic
23C2×C22⋊D173—21no—
24C2×C22⋊D173—22no—
25C2×C22⋊D173D412yes—
26D4×D173—42no—
27C22×Dic173D412yes—
28C2×C4×D173C2×C411yescharacteristic
29#14583805—21no—
30#14583805—42no—
31C22⋊Dic173—22no—
32C22⋊Dic173—22no—
33#14583803—21no—
34#14583803—22no—
35C4×Dic173—42no—
36C4×Dic173C2×C411yescharacteristic
37C2×D4×C173—22no—
38C2×D4×C173C2×C411yescharacteristic
39C2×D4×C173C2311yescharacteristic
40C22⋊C4×C173—22no—
41C42×C173C2×C411yescharacteristic
42C22×D173—21no—
43C22×D173—42no—
44C22×D173—82no—
45C22×D173C22⋊C411yescharacteristic
46Dic346—41no—
47D692—41no—
48C22⋊D173—42no—
49C22⋊D173—42no—
50C22⋊D173—82no—
51C2×Dic173—22no—
52C2×Dic173—22no—
53C2×Dic173—21no—
54C2×Dic173—42no—
55C2×Dic173C22⋊C412yes—
56C2×Dic173C22⋊C411yescharacteristic
57C4×D173—22no—
58C23×C173—21no—
59C23×C173C2×D412yes—
60D4×C173—41no—
61D4×C173—41no—
62D4×C173—82no—
63C2×C4×C173—21no—
64C2×C4×C173—42no—
65C2×C4×C173C22×C411yescharacteristic
66C22×C173—22no—
67C22×C173—42no—
68C22×C173—42no—
69C22×C173—41no—
70C22×C173C2×C22⋊C411yescharacteristic
71C2×D173—21no—
72C2×D173—41no—
73C2×D173—42no—
74C2×D173—41no—
75C2×D173—84no—
76C4×C173—21no—
77C4×C173—41no—
78C4×C173—82no—
79Dic173—22no—
80Dic173—41no—
81Dic173—82no—
82Dic173#9912yes—
83C2×C173—21no—
84C2×C173—42no—
85C2×C173—81no—
86C2×C173C2×#9911yescharacteristic
87D173—41no—
88D173—81no—
89C173#212811yescharacteristic, complemented, Sylow
90#2128—1731nomaximal, Sylow
91#535—1731no—
92#410—1732no—
93#353—1731no—
94#353—1732no—
95#353D17311yescharacteristic, complemented
96C2×#41—1731no—
97C22×D4—1731no—
98#127—1731no—
99#127C2×D17311yescharacteristic
100#124—1731no—
101#124—1731no—
102#121—3462no—
103#120—3462no—
104#117—3462no—
105#117—1731no—
106#99—3462no—
107#99—1732no—
108#99C2×D17312yescomplemented
109C24—3461no—
110#41—6922no—
111C2×Q8—1731no—
112C2×D4—22no—
113C2×D4—3462no—
114C2×D4—3461no—
115C2×D4—1732no—
116C2×D4—1731no—
117C2×D4—6922no—
118C2×D4C4×D17311yescharacteristic
119C2×D4C22×D17311yescharacteristic
120C22×C4—1732no—
121C22×C4—1731no—
122C4⋊C4—3461no—
123C4⋊C4—6922no—
124C22⋊C4—22no—
125C22⋊C4—3461no—
126C22⋊C4—3462no—
127C22⋊C4—3462no—
128C22⋊C4—3462no—
129C42—1731no—
130C42—6922no—
131C42C4×D17311yescharacteristic
132C23—21no—
133C23—3461no—
134C23—13842no—
135C23—1731no—
136C23—6922no—
137C23D4×D17312yes—
138Q8—6921no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—13842no—
143D4—6921no—
144D4—6922no—
145D4—6922no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—3462no—
149C2×C4—3462no—
150C2×C4—3462no—
151C2×C4—3461no—
152C2×C4—1732no—
153C2×C4—1731no—
154C2×C4—6922no—
155C2×C4C2×C4×D17311yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—3461no—
161C22—13844no—
162C22—6922no—
163C22—6921no—
164C22—6921no—
165C22C22⋊C4×D17311yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—3462no—
170C4—13842no—
171C4—1732no—
172C4—6921no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—13841no—
177C2—6921no—
178C2#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#2128
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. #26011463C2semidirect
3. #26011284C2semidirect
4. #26011283C2semidirect
5. #353×C173C2semidirect
7. C2×D4×D173C4semidirect
9. #19149667C4semidirect
10. #19149666C4semidirect
13. #19149636C4semidirect
15. #19149594C22semidirect
18. #99×C173C22semidirect
89. C173#2128semidirect
95. #353D173semidirect
108. #99C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120542
#260114632
#260112842
#260112832
#353×C1732
#2128173