Finite Group Databasebeta

← #36353776

Subgroups — #36353776

Order 22144.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#26011968C211yesmaximal, characteristic
2#26011531C211yesmaximal, characteristic
3C2×#19149597C211yesmaximal, characteristic
4#26011396C211yesmaximal, characteristic
5#26011252C211yesmaximal, characteristic
6#26011250C211yesmaximal, characteristic
7#428×C173C211yesmaximal, characteristic
8C22×Dic346C2211yescharacteristic
9C2×#14583810C2211yescharacteristic
10C4×Dic346—21no—
11C4×Dic346—21no—
12#19149644—21no—
13#19149644—21no—
14#19149638C2211yescharacteristic
15#19149635—22no—
16C2×C4×Dic173C2211yescharacteristic
17#19149612—21no—
18#19149610—21no—
19#19149597C414yescomplemented
20#19149586—22no—
21C2×M16×C173C2211yescharacteristic
22#117×C173C2211yescharacteristic
23C2×C4⋊C4×C173C2211yescharacteristic
24#107×C173—21no—
25#106×C173—21no—
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346C2×C412yes—
31C2×Dic346Q812yes—
32C22×Dic173—21no—
33C22×Dic173D412yes—
34#14583810—21no—
35#14583810C2×C412yes—
36C2×#13063436—21no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—22no—
40#14583805—22no—
41C22⋊Dic173—21no—
42C4×Dic173—22no—
43C4×Dic173—22no—
44C4×Dic173—21no—
45C4×Dic173—21no—
46C4×Dic173—21no—
47C22×C4×C173—21no—
48C22×C4×C173C2311yescharacteristic
49M16×C173—21no—
50M16×C173C2×C412yes—
51C2×C8×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173D411yescharacteristic
55C4⋊C4×C173D411yescharacteristic
56C22⋊C4×C173—21no—
57C42×C173—21no—
58Dic346—22no—
59Dic346—22no—
60Dic346—22no—
61Dic346—22no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—22no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173C4⋊C414yes—
70#13063436—22no—
71C23×C173C2×D411yescharacteristic
72C2×C4×C173—21no—
73C2×C4×C173—21no—
74C2×C4×C173—42no—
75C2×C4×C173—41no—
76C2×C4×C173C22×C411yescharacteristic
77C2×C4×C173C2×D411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×Q811yescharacteristic
80C2×C4×C173#4111yescharacteristic
81C2×C4×C173#4111yescharacteristic
82C8×C173—22no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173C2×C4⋊C411yescharacteristic
86C22×C173#12011yescharacteristic
87C22×C173#12311yescharacteristic
88C4×C173—41no—
89C4×C173—41no—
90C4×C173—81no—
91C4×C173C4×D411yescharacteristic
92C4×C173C4×D411yescharacteristic
93C4×C173C22⋊Q811yescharacteristic
94C4×C173C22⋊Q811yescharacteristic
95Dic173—24no—
96Dic173—42no—
97Dic173—41no—
98Dic173—41no—
99C2×C173—21no—
100C2×C173—21no—
101C2×C173#38511yescharacteristic
102C2×C173#45411yescharacteristic
103C2×C173#45611yescharacteristic
104C173#185711yescharacteristic, complemented, Sylow
105#1857—1731nomaximal, Sylow
106#519—1731no—
107#428—1731no—
108#428D17311yescharacteristic
109C2×#101—1731no—
110#391—1731no—
111#339—1731no—
112#339—1731no—
113C22×Q8—1731no—
114C2×M16—1731no—
115C2×M16C2×D17311yescharacteristic
116C4×Q8—3461no—
117C4×Q8—3461no—
118C4×Q8—3461no—
119C4×Q8—3461no—
120#117—3462no—
121#117—1731no—
122#117C2×D17311yescharacteristic
123C2×C4⋊C4C2×D17311yescharacteristic
124C2×C42—1731no—
125#107—21no—
126#107—3461no—
127#106—21no—
128#106—3461no—
129#101—1734no—
130#95—3462no—
131C2×Q8—3461no—
132C2×Q8—3461no—
133C2×Q8—3461no—
134C2×Q8—3461no—
135C2×Q8—1732no—
136C2×Q8—1732no—
137C22×C4—21no—
138C22×C4—3461no—
139C22×C4—1732no—
140C22×C4C22×D17311yescharacteristic
141M16—21no—
142M16—3461no—
143M16—1732no—
144M16C4×D17312yes—
145C2×C8—21no—
146C2×C8—3461no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—3462no—
150C4⋊C4—3462no—
151C4⋊C4—1731no—
152C4⋊C4—1731no—
153C4⋊C4C22⋊D17311yescharacteristic
154C4⋊C4C22⋊D17311yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3461no—
157C42—21no—
158C42—3462no—
159C42—3462no—
160C42—3461no—
161C42—3461no—
162C42—3461no—
163C23D4×D17311yescharacteristic
164Q8—3462no—
165Q8—3462no—
166Q8—3462no—
167Q8—3462no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—3462no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3461no—
176C2×C4—3461no—
177C2×C4—1734no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4C2×C4×D17311yescharacteristic
181C2×C4D4×D17311yescharacteristic
182C2×C4C2×C22⋊D17311yescharacteristic
183C2×C4Q8×D17311yescharacteristic
184C2×C4#1458382611yescharacteristic
185C2×C4#1458382711yescharacteristic
186C8—22no—
187C8—3462no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D17311yescharacteristic
191C22#1914964711yescharacteristic
192C22#1914966411yescharacteristic
193C4—41no—
194C4—41no—
195C4—81no—
196C4—3464no—
197C4—6922no—
198C4—6921no—
199C4—6921no—
200C4C4×C22⋊D17311yescharacteristic
201C4#1914964111yescharacteristic
202C4#1914965511yescharacteristic
203C4#1914965711yescharacteristic
204C2—21no—
205C2—21no—
206C2#2601136211yescharacteristic
207C2#2601166211yescharacteristic
208C2#2601167211yescharacteristic
subgroupG/H
centerC22#19149647
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#428×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
cosocleC22×C4×C173C23
hypercenter#428D173
hypocenterC173#1857
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
19. #19149597C4semidirect
104. C173#1857semidirect

Maximal subgroups

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

subgroupindex
#260119682
#260115312
C2×#191495972
#260113962
#260112522
#260112502
#428×C1732
#1857173