Finite Group Databasebeta

← #36355227

Subgroups — #36355227

Order 22144.

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

normal subgroups only

This group has 258 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#26012040C211yesmaximal, characteristic
2#26011463C212yesmaximal, complemented
3#26011286C211yesmaximal, characteristic, complemented
4#26011283C212yesmaximal, complemented
5#354×C173C211yesmaximal, characteristic, complemented
6C2×#14583827C411yescharacteristic, complemented
7C2×#14583825C2211yescharacteristic
8#19149677C411yescharacteristic, complemented
9#19149675C2211yescharacteristic
10#19149667C411yescharacteristic, complemented
11#19149652—22no—
12#19149639—22no—
13#19149635—22no—
14C42×D173C411yescharacteristic, complemented
15#19149594C2212yes—
16#19149593—22no—
17#124×C173C2211yescharacteristic
18#99×C173C2212yes—
19#14583827—41no—
20#14583827—41no—
21#14583825—41no—
22#14583825—41no—
23C2×Dic346C2×C411yescharacteristic
24C2×D692—21no—
25C2×D692C2×C411yescharacteristic
26C2×C22⋊D173—22no—
27C2×C22⋊D173D412yes—
28C22×Dic173D412yes—
29C2×C4×D173—21no—
30C2×C4×D173—21no—
31C2×C4×D173C2×C411yescharacteristic
32#14583805—22no—
33#14583805—42no—
34C22⋊Dic173—22no—
35#14583803—22no—
36C4×Dic173—21no—
37C4×Dic173—42no—
38C4×Dic173C2×C411yescharacteristic
39C2×Q8×C173C2×C411yescharacteristic
40C2×D4×C173C2311yescharacteristic
41C22⋊C4×C173—22no—
42C22⋊C4×C173—22no—
43C42×C173C2×C411yescharacteristic
44C22×D173—21no—
45C22×D173C22⋊C411yescharacteristic
46Dic346—41no—
47D692—41no—
48D692—42no—
49C22⋊D173—42no—
50C22⋊D173—44no—
51C2×Dic173—21no—
52C2×Dic173—22no—
53C2×Dic173—22no—
54C2×Dic173—42no—
55C2×Dic173C22⋊C412yes—
56C2×Dic173C22⋊C411yescharacteristic
57C4×D173—21no—
58C4×D173—21no—
59C4×D173—42no—
60C4×D173—41no—
61C4×D173—41no—
62C23×C173C2×D412yes—
63Q8×C173—41no—
64D4×C173—41no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—42no—
68C2×C4×C173C22×C411yescharacteristic
69C22×C173—22no—
70C22×C173—42no—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—21no—
73C2×D173—41no—
74C2×D173—42no—
75C2×D173—41no—
76C4×C173—21no—
77C4×C173—41no—
78C4×C173—41no—
79C4×C173—82no—
80Dic173—22no—
81Dic173—22no—
82Dic173—82no—
83Dic173#9911yescharacteristic
84Dic173#9911yescharacteristic
85C2×C173—21no—
86C2×C173—42no—
87C2×C173C2×#9911yescharacteristic
88D173—41no—
89D173—81no—
90C173#212611yescharacteristic, complemented, Sylow
91#2126—1731nomaximal, Sylow
92#532—1731no—
93#410—1732no—
94#354—1731no—
95#354D17311yescharacteristic, complemented
96#353—1732no—
97C2×#41—1731no—
98C2×#41—1731no—
99C4⋊Q8—1731no—
100#127—1731no—
101#124—1731no—
102#124C2×D17311yescharacteristic
103#121—3462no—
104C4×D4—3462no—
105#117—3462no—
106C2×C42—1731no—
107#99—3462no—
108#99—1732no—
109#99C2×D17312yes—
110#41—6921no—
111#41—6921no—
112#41—6921no—
113#41—6921no—
114C2×Q8—1731no—
115C2×Q8C4×D17311yescharacteristic
116C2×D4—3461no—
117C2×D4—3462no—
118C2×D4—1732no—
119C2×D4—1731no—
120C2×D4C22×D17311yescharacteristic
121C22×C4—3461no—
122C22×C4—3461no—
123C22×C4—1732no—
124C22×C4—1731no—
125C4⋊C4—3462no—
126C4⋊C4—6922no—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C22⋊C4—3462no—
130C22⋊C4—3462no—
131C42—3461no—
132C42—1731no—
133C42—6922no—
134C42C4×D17311yescharacteristic
135C23—3461no—
136C23—1731no—
137C23D4×D17312yes—
138Q8—41no—
139Q8—6921no—
140D4—41no—
141D4—6921no—
142D4—6922no—
143D4—6922no—
144D4—6924no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—3461no—
149C2×C4—3461no—
150C2×C4—3461no—
151C2×C4—3462no—
152C2×C4—3462no—
153C2×C4—1732no—
154C2×C4—1731no—
155C2×C4—6922no—
156C2×C4—6921no—
157C2×C4—6921no—
158C2×C4—6922no—
159C2×C4C2×C4×D17311yescharacteristic
160C22—22no—
161C22—42no—
162C22—3461no—
163C22—6922no—
164C22—6921no—
165C22—6921no—
166C22C22⋊C4×D17311yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—82no—
171C4—3462no—
172C4—3462no—
173C4—13842no—
174C4—1731no—
175C4—1731no—
176C2—21no—
177C2—42no—
178C2—13841no—
179C2—6921no—
180C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#354×C173C2
socleC2×C173C2×#99
prime radical#26012040C2
cosocleC2×D4×C173C23
hypercenter#354D173
hypocenterC173#2126
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #26011463C2semidirect
3. #26011286C2semidirect
4. #26011283C2semidirect
5. #354×C173C2semidirect
6. C2×#14583827C4semidirect
8. #19149677C4semidirect
10. #19149667C4semidirect
14. C42×D173C4semidirect
90. C173#2126semidirect
95. #354D173semidirect

Maximal subgroups

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

subgroupindex
#260120402
#260114632
#260112862
#260112832
#354×C1732
#2126173