Finite Group Databasebeta

← #36352060

Subgroups — #36352060

Order 22144.

240 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 71 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#26012043C211yesmaximal, characteristic
2#26011541C211yesmaximal, characteristic
3#26011342C211yesmaximal, characteristic
4#26011229C211yesmaximal, characteristic
5#26011228C211yesmaximal, characteristic
6#26011227C211yesmaximal, characteristic
7#328×C173C211yesmaximal, characteristic
8#19149678C411yescharacteristic, complemented
9#19149677C411yescharacteristic, complemented
10#19149669—21no—
11#19149658C2211yescharacteristic
12#19149656—21no—
13Q8×Dic173C411yescharacteristic, complemented
14#19149644—21no—
15#19149644C411yescharacteristic, complemented
16#19149635—21no—
17C2×#14583805—21no—
18C2×C4×Dic173—21no—
19C2×C4×Dic173C2211yescharacteristic
20#19149608—21no—
21C22⋊#13063436C2211yescharacteristic
22C8×Dic173—21no—
23#19149586C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346C2×C411yescharacteristic
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173C2×C411yescharacteristic
35#14583805—24no—
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
42C4×Dic173—21no—
43C4×Dic173—21no—
44C4×Dic173—42no—
45C4×Dic173C2×C411yescharacteristic
46C4×Dic173C2×C411yescharacteristic
47C4×Dic173D411yescharacteristic
48C4×Dic173D411yescharacteristic
49C2×Q8×C173C2×C411yescharacteristic
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C2×C8×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173C2×C411yescharacteristic
55C22⋊C4×C173—21no—
56Dic346—42no—
57Dic346—41no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—41no—
63C2×Dic173—41no—
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C22⋊C411yescharacteristic
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173SD1612yes—
69C2×Dic173Q1612yes—
70#13063436—41no—
71C23×C173C2×D411yescharacteristic
72Q8×C173—41no—
73C2×C4×C173—22no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173C22×C411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C8×C173—41no—
82C22×C173—21no—
83C22×C173—21no—
84C22×C173C2×C22⋊C411yescharacteristic
85C22×C173C2×SD1611yescharacteristic
86C22×C173C2×Q1611yescharacteristic
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—42no—
90C4×C173—41no—
91C4×C173—41no—
92Dic173—21no—
93Dic173—42no—
94Dic173—41no—
95Dic173—41no—
96Dic173#10312yes—
97Dic173#10312yes—
98Dic173#10412yes—
99C2×C173—21no—
100C2×C173—21no—
101C2×C173#41011yescharacteristic
102C2×C173C2×#10311yescharacteristic
103C2×C173C2×#10411yescharacteristic
104C173#150311yescharacteristic, complemented, Sylow
105#1503—1731nomaximal, Sylow
106#533—1731no—
107#431—1731no—
108C4×C4⋊C4—1731no—
109#328—1731no—
110#328—1731no—
111#328—1731no—
112#328D17311yescharacteristic
113C4⋊Q8—1731no—
114C4⋊Q8—1731no—
115#125—3461no—
116C22⋊Q8—3461no—
117C22⋊Q8—1731no—
118C22⋊Q8C2×D17311yescharacteristic
119C4×Q8—3461no—
120C4×Q8—1731no—
121C4×Q8—1731no—
122#117—3461no—
123C2×C4⋊C4—3461no—
124C2×C42—3461no—
125C2×C42—1731no—
126C4⋊C8—3461no—
127C22⋊C8—1731no—
128C22⋊C8C2×D17311yescharacteristic
129C4×C8—3461no—
130#95—1731no—
131#95C2×D17311yescharacteristic
132C2×Q8—3461no—
133C2×Q8—1731no—
134C2×Q8C4×D17311yescharacteristic
135C22×C4—21no—
136C22×C4—3461no—
137C22×C4—1731no—
138C22×C4—1731no—
139C22×C4C22×D17311yescharacteristic
140C2×C8—21no—
141C2×C8—3461no—
142C4⋊C4—21no—
143C4⋊C4—3464no—
144C4⋊C4—3461no—
145C4⋊C4—3461no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C4⋊C4—3461no—
149C4⋊C4—3461no—
150C4⋊C4—1731no—
151C4⋊C4C4×D17311yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3461no—
154C42—3461no—
155C42—3461no—
156C42—1731no—
157C42—1731no—
158C42—1731no—
159C42—1731no—
160C42—6922no—
161C23D4×D17311yescharacteristic
162Q8—41no—
163Q8—6921no—
164Q8—6922no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—3462no—
172C2×C4—3462no—
173C2×C4—3461no—
174C2×C4—3461no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—1731no—
179C2×C4—1732no—
180C2×C4—1732no—
181C2×C4—6921no—
182C2×C4—6921no—
183C2×C4C2×C4×D17311yescharacteristic
184C2×C4D4×D17311yescharacteristic
185C8—41no—
186C8—6921no—
187C22—21no—
188C22—21no—
189C22C22⋊C4×D17311yescharacteristic
190C22SD16×D17311yescharacteristic
191C22Q16×D17311yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—3461no—
198C4—1732no—
199C4—1732no—
200C4—1732no—
201C4—6922no—
202C4—6921no—
203C4—6921no—
204C2—21no—
205C2—21no—
206C2#2601146311yescharacteristic
207C2#103×D17311yescharacteristic
208C2#104×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC23×C173C2×D4
cosocleC22×C4×C173C23
hypercenter#328D173
hypocenterC173#1503
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
8. #19149678C4semidirect
9. #19149677C4semidirect
13. Q8×Dic173C4semidirect
15. #19149644C4semidirect
104. C173#1503semidirect

Maximal subgroups

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

subgroupindex
#260120432
#260115412
#260113422
#260112292
#260112282
#260112272
#328×C1732
#1503173