Finite Group Databasebeta

← #36353522

Subgroups — #36353522

Order 22144.

552 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#19149606C212yesmaximal, complemented
2#26011492C212yesmaximal, complemented
3C2×#19149598C211yesmaximal, characteristic, complemented
4#26011398C211yesmaximal, characteristic, complemented
5C2×2−1+4×C173C211yesmaximal, characteristic
6C2×#14583810C2212yes—
7C2×C4⋊Dic173C2211yescharacteristic
8C2×C4×Dic173C2211yescharacteristic
9#19149606—24no—
10#19149603—22no—
11#19149600—22no—
12#19149598—22no—
13#19149586—22no—
142−1+4×C173—24no—
152−1+4×C173—22no—
162−1+4×C173C414yescomplemented
17C2×#41×C173—22no—
18C2×#41×C173—22no—
19C2×#41×C173C2212yes—
20C22×Q8×C173—22no—
21C22×Q8×C173C2211yescharacteristic
22C22×Dic173—21no—
23C22×Dic173—21no—
24#14583810—22no—
25#14583810—22no—
26C2×#13063436—22no—
27C4⋊Dic173—22no—
28C4×Dic173—22no—
29#41×C173—28no—
30#41×C173—28no—
31#41×C173—28no—
32#41×C173—28no—
33#41×C173—22no—
34#41×C173—22no—
35#41×C173C2×C414yes—
36#41×C173D414yes—
37C2×Q8×C173—22no—
38C2×Q8×C173—28no—
39C2×Q8×C173—28no—
40C2×Q8×C173—22no—
41C2×Q8×C173—22no—
42C2×Q8×C173C2×C412yes—
43C2×Q8×C173D412yes—
44C2×Q8×C173D412yes—
45C2×D4×C173—22no—
46C2×D4×C173—22no—
47C2×D4×C173D412yes—
48C22×C4×C173—24no—
49C22×C4×C173—22no—
50C22×C4×C173D412yes—
51C22×C4×C173C2311yescharacteristic
52C2×Dic173—22no—
53C2×Dic173—41no—
54C2×Dic173—41no—
55C2×Dic173—42no—
56C2×Dic173—41no—
57#13063436—42no—
58C23×C173—22no—
59C23×C173C2×D411yescharacteristic
60Q8×C173—28no—
61Q8×C173—24no—
62Q8×C173—22no—
63Q8×C173—24no—
64Q8×C173C22⋊C414yes—
65D4×C173—28no—
66D4×C173—24no—
67D4×C173—24no—
68D4×C173—22no—
69D4×C173C22⋊C414yes—
70C2×C4×C173—216no—
71C2×C4×C173—22no—
72C2×C4×C173—22no—
73C2×C4×C173—22no—
74C2×C4×C173—24no—
75C2×C4×C173—24no—
76C2×C4×C173C22⋊C414yes—
77C2×C4×C173C22×C411yescharacteristic
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173C2×D411yescharacteristic
81C22×C173—21no—
82C22×C173—21no—
83C22×C173—24no—
84C22×C173—42no—
85C22×C173—42no—
86C22×C173C2×C22⋊C411yescharacteristic
87C22×C173#12011yescharacteristic
88C22×C173#12011yescharacteristic
89C4×C173—24no—
90C4×C173—24no—
91C4×C173C2×C22⋊C412yes—
92C4×C173#12012yes—
93Dic173—42no—
94Dic173—81no—
95C2×C173—21no—
96C2×C173—21no—
97C2×C173—44no—
98C2×C173#37911yescharacteristic
99C2×C173#45411yescharacteristic
100C2×C173#45611yescharacteristic
101C173#179311yescharacteristic, complemented, Sylow
102#1793—1731nomaximal, Sylow
103C2×2−1+4D17311yescharacteristic
104C2×#104—1732no—
105#417—1732no—
106C2×#101—1731no—
107#391—1731no—
1082−1+4—24no—
1092−1+4—22no—
1102−1+4Dic17314yescomplemented
111C2×#41—22no—
112C2×#41—22no—
113C2×#41C2×D17312yes—
114C22×Q8—22no—
115C22×Q8C2×D17311yescharacteristic
116C2×M16—1732no—
117C2×C4⋊C4—1731no—
118C2×C42—1731no—
119#104—3464no—
120#103—3462no—
121#102—3462no—
122#101—3462no—
123#95—3462no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—22no—
128#41—22no—
129#41—28no—
130#41C2×Dic17314yes—
131#41C22⋊D17314yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×Dic17312yes—
138C2×Q8C22⋊D17312yes—
139C2×Q8C22⋊D17312yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22⋊D17312yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—3461no—
146C22×C4—3461no—
147C22×C4C22⋊D17312yes—
148C22×C4C22×D17311yescharacteristic
149M16—3462no—
150M16—3462no—
151C2×C8—3462no—
152C4⋊C4—3462no—
153C42—3462no—
154C23—22no—
155C23C2×C22⋊D17311yescharacteristic
156Q8—28no—
157Q8—24no—
158Q8—22no—
159Q8—24no—
160Q8C22⋊Dic17314yes—
161D4—28no—
162D4—24no—
163D4—24no—
164D4—22no—
165D4C22⋊Dic17314yes—
166C2×C4—216no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—3462no—
173C2×C4—6922no—
174C2×C4—6921no—
175C2×C4—6921no—
176C2×C4—6921no—
177C2×C4C22⋊Dic17314yes—
178C2×C4C22×Dic17311yescharacteristic
179C2×C4C2×C22⋊D17312yes—
180C2×C4C2×C22⋊D17312yes—
181C2×C4C2×C22⋊D17311yescharacteristic
182C8—6922no—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—42no—
187C22—42no—
188C22C2×C22⋊Dic17311yescharacteristic
189C22#1914964811yescharacteristic
190C22#1914964811yescharacteristic
191C4—24no—
192C4—24no—
193C4—13841no—
194C4—6922no—
195C4C2×C22⋊Dic17312yes—
196C4#1914964812yes—
197C2—21no—
198C2—21no—
199C2—44no—
200C2#2601134711yescharacteristic
201C2#2601166411yescharacteristic
202C2#2601167411yescharacteristic
subgroupG/H
centerC22#19149648
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radicalC2×2−1+4×C173C2
socleC22×C173#120
cosocleC22×C4×C173C23
hypercenterC2×2−1+4D173
hypocenterC173#1793
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#19149606C2semidirect
2. #26011492C2semidirect
3. C2×#19149598C2semidirect
4. #26011398C2semidirect
16. 2−1+4×C173C4semidirect
101. C173#1793semidirect
110. 2−1+4Dic173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496062
#260114922
C2×#191495982
#260113982
C2×2−1+4×C1732
#1793173