Finite Group Databasebeta

← #36352125

Subgroups — #36352125

Order 22144.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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#26011355C211yesmaximal, 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×Dic173C2211yescharacteristic
19#19149608—21no—
20C22⋊#13063436C2211yescharacteristic
21C8×Dic173—21no—
22#19149586—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—21no—
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—42no—
41C22⋊Dic173—21no—
42C4×Dic173—21no—
43C4×Dic173—21no—
44C4×Dic173C2×C411yescharacteristic
45C4×Dic173C2×C411yescharacteristic
46C4×Dic173D411yescharacteristic
47C4×Dic173D411yescharacteristic
48C2×Q8×C173C2×C411yescharacteristic
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173C2×C411yescharacteristic
54C22⋊C4×C173—21no—
55Dic346—42no—
56Dic346—41no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—41no—
60C2×Dic173—41no—
61C2×Dic173—41no—
62C2×Dic173—41no—
63C2×Dic173C22⋊C411yescharacteristic
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C22⋊C411yescharacteristic
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173SD1612yes—
68C2×Dic173Q1612yes—
69#13063436—41no—
70C23×C173C2×D411yescharacteristic
71Q8×C173—41no—
72C2×C4×C173—21no—
73C2×C4×C173—21no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173C22×C411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C8×C173—41no—
81C22×C173—21no—
82C22×C173—21no—
83C22×C173C2×C22⋊C411yescharacteristic
84C22×C173C2×SD1611yescharacteristic
85C22×C173C2×Q1611yescharacteristic
86C4×C173—21no—
87C4×C173—21no—
88C4×C173—41no—
89C4×C173—41no—
90C4×C173—81no—
91Dic173—21no—
92Dic173—21no—
93Dic173—41no—
94Dic173—41no—
95Dic173—81no—
96Dic173#10312yes—
97Dic173#10312yes—
98C2×C173—21no—
99C2×C173—21no—
100C2×C173#41011yescharacteristic
101C2×C173C2×#10311yescharacteristic
102C2×C173#42111yescharacteristic
103C173#151811yescharacteristic, complemented, Sylow
104#1518—1731nomaximal, Sylow
105#533—1731no—
106#431—1731no—
107#382—1731no—
108#328—1731no—
109#328—1731no—
110#328—1731no—
111#328D17311yescharacteristic
112C4⋊Q8—1731no—
113C4⋊Q8—1731no—
114#125—3461no—
115C22⋊Q8—3461no—
116C22⋊Q8—1731no—
117C22⋊Q8C2×D17311yescharacteristic
118C4×Q8—3461no—
119C4×Q8—1731no—
120C4×Q8—1731no—
121#117—3461no—
122C2×C4⋊C4—3461no—
123C2×C42—1731no—
124C4⋊C8—3461no—
125C22⋊C8—1731no—
126C22⋊C8C2×D17311yescharacteristic
127C4×C8—3461no—
128#95—3461no—
129#95—1731no—
130#95C2×D17311yescharacteristic
131C2×Q8—3461no—
132C2×Q8—1731no—
133C2×Q8C4×D17311yescharacteristic
134C22×C4—21no—
135C22×C4—3461no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4C22×D17311yescharacteristic
139C2×C8—21no—
140C2×C8—3461no—
141C4⋊C4—21no—
142C4⋊C4—3461no—
143C4⋊C4—3461no—
144C4⋊C4—3461no—
145C4⋊C4—3461no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C4⋊C4—1731no—
149C4⋊C4—6922no—
150C4⋊C4C4×D17311yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—3461no—
153C42—3461no—
154C42—3461no—
155C42—1731no—
156C42—1731no—
157C42—1731no—
158C42—1731no—
159C23D4×D17311yescharacteristic
160Q8—41no—
161Q8—6921no—
162Q8—6922no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3461no—
170C2×C4—3461no—
171C2×C4—1731no—
172C2×C4—1731no—
173C2×C4—1731no—
174C2×C4—1731no—
175C2×C4—1732no—
176C2×C4—1732no—
177C2×C4—6921no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6921no—
181C2×C4C2×C4×D17311yescharacteristic
182C2×C4D4×D17311yescharacteristic
183C8—41no—
184C8—6921no—
185C22—21no—
186C22—21no—
187C22C22⋊C4×D17311yescharacteristic
188C22SD16×D17311yescharacteristic
189C22Q16×D17311yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—81no—
195C4—3461no—
196C4—3461no—
197C4—13841no—
198C4—1732no—
199C4—1732no—
200C4—6921no—
201C4—6921no—
202C2—21no—
203C2—21no—
204C2#2601146311yescharacteristic
205C2#103×D17311yescharacteristic
206C2#2601150611yescharacteristic
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#1518
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
103. C173#1518semidirect

Maximal subgroups

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

subgroupindex
#260120432
#260115412
#260113552
#260112292
#260112282
#260112272
#328×C1732
#1518173