Finite Group Databasebeta

← #36354097

Subgroups — #36354097

Order 22144.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#26012062C211yesmaximal, characteristic
2#26011968C211yesmaximal, characteristic, complemented
3#26011508C211yesmaximal, characteristic
4C2×#19149597C211yesmaximal, characteristic, complemented
5#26011248C211yesmaximal, characteristic, complemented
6#26011247C211yesmaximal, characteristic, complemented
7#421×C173C211yesmaximal, characteristic
8C22×Dic346C2211yescharacteristic
9C2×#14583810C2211yescharacteristic
10#19149677C412yescomplemented
11#19149666C412yescomplemented
12#19149660—22no—
13#19149658—22no—
14C4×Dic346—22no—
15#19149644—22no—
16#19149638C2211yescharacteristic
17#19149638C2211yescharacteristic
18#19149635—22no—
19#19149606—22no—
20#19149597—22no—
21C2×#41×C173C2211yescharacteristic
22C2×M16×C173C2211yescharacteristic
23#117×C173C2211yescharacteristic
24#104×C173—22no—
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346C2×C412yes—
30C2×Dic346D412yes—
31C22×Dic173D412yes—
32#14583810—21no—
33#14583810—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37#14583805—24no—
38#14583805—24no—
39C22⋊Dic173—24no—
40C22⋊Dic173—21no—
41C22⋊Dic173—21no—
42C4×Dic173—24no—
43C4×Dic173C2×C412yes—
44C4×Dic173D412yes—
45#41×C173—21no—
46#41×C173—21no—
47#41×C173—21no—
48#41×C173—21no—
49C2×Q8×C173C2×C411yescharacteristic
50C2×D4×C173—21no—
51C2×D4×C173C2×C411yescharacteristic
52C22×C4×C173—21no—
53C22×C4×C173C2311yescharacteristic
54M16×C173—21no—
55M16×C173—21no—
56C2×C8×C173—21no—
57C4⋊C4×C173—21no—
58C22⋊C4×C173—21no—
59C42×C173D412yes—
60Dic346—22no—
61Dic346—22no—
62Dic346—22no—
63Dic346—22no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67C2×Dic173—22no—
68C2×Dic173C22⋊C414yes—
69#13063436—41no—
70C23×C173—21no—
71C23×C173C2×D411yescharacteristic
72Q8×C173—21no—
73Q8×C173—21no—
74D4×C173—21no—
75D4×C173—21no—
76D4×C173—41no—
77D4×C173—41no—
78C2×C4×C173—21no—
79C2×C4×C173—22no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173C22×C411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173#4111yescharacteristic
87C2×C4×C173#4111yescharacteristic
88C8×C173—41no—
89C22×C173—41no—
90C22×C173—42no—
91C22×C173—41no—
92C22×C173C2×C22⋊C411yescharacteristic
93C22×C173#12011yescharacteristic
94C22×C173#12311yescharacteristic
95C4×C173—42no—
96C4×C173—41no—
97C4×C173C4×D411yescharacteristic
98C4×C173C4×D411yescharacteristic
99C4×C173#12111yescharacteristic
100C4×C173#12111yescharacteristic
101Dic173—24no—
102Dic173—42no—
103Dic173—42no—
104C2×C173—21no—
105C2×C173—21no—
106C2×C173—21no—
107C2×C173—81no—
108C2×C173#38611yescharacteristic
109C173#191211yescharacteristic, complemented, Sylow
110#1912—1731nomaximal, Sylow
111#536—1731no—
112#519—1731no—
113#421—1731no—
114#421D17311yescharacteristic
115C2×#101—1731no—
116H(3,ℤ/4)—1731no—
117H(3,ℤ/4)—1731no—
118C2×#41C2×D17311yescharacteristic
119C22×Q8—1731no—
120C2×M16—1731no—
121C2×M16C2×D17311yescharacteristic
122C4⋊Q8—1732no—
123#124—1732no—
124#123—3462no—
125C22⋊Q8—3462no—
126C4×Q8—3462no—
127C4×Q8—3462no—
128#117—3462no—
129#117—1731no—
130#117—1731no—
131#117C2×D17311yescharacteristic
132#104—22no—
133#104—3462no—
134#101—3462no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—3461no—
140C2×Q8—3461no—
141C2×Q8—3461no—
142C2×Q8—3461no—
143C2×Q8—1732no—
144C2×Q8—1732no—
145C2×Q8C4×D17311yescharacteristic
146C2×D4—21no—
147C2×D4C4×D17311yescharacteristic
148C22×C4—21no—
149C22×C4—1732no—
150C22×C4C22×D17311yescharacteristic
151M16—21no—
152M16—21no—
153M16—3461no—
154M16—3461no—
155C2×C8—21no—
156C2×C8—3461no—
157C4⋊C4—21no—
158C4⋊C4—3464no—
159C4⋊C4—3464no—
160C4⋊C4—3461no—
161C4⋊C4—3461no—
162C22⋊C4—21no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3464no—
166C42—3464no—
167C42—1732no—
168C42—1732no—
169C42C22⋊D17312yes—
170C23—21no—
171C23D4×D17311yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—3462no—
175Q8—3462no—
176Q8—3462no—
177Q8—3462no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3462no—
187C2×C4—3462no—
188C2×C4—3462no—
189C2×C4—3462no—
190C2×C4—1734no—
191C2×C4C2×C4×D17311yescharacteristic
192C2×C4D4×D17311yescharacteristic
193C2×C4D4×D17311yescharacteristic
194C2×C4C2×C22⋊D17311yescharacteristic
195C2×C4#1458382511yescharacteristic
196C2×C4#1458382611yescharacteristic
197C8—41no—
198C8—6921no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D17311yescharacteristic
203C22#1914964711yescharacteristic
204C22#1914966111yescharacteristic
205C4—42no—
206C4—41no—
207C4—3464no—
208C4—6922no—
209C4—6922no—
210C4#1914963911yescharacteristic
211C4C4×C22⋊D17311yescharacteristic
212C4#1914965011yescharacteristic
213C4#1914965211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2601136711yescharacteristic
subgroupG/H
centerC2#26011367
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#421×C173C2
socleC2×C173#386
prime radicalC2×#41×C173C22
cosocleC22×C4×C173C23
hypercenter#421D173
hypocenterC173#1912
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #26011968C2semidirect
4. C2×#19149597C2semidirect
5. #26011248C2semidirect
6. #26011247C2semidirect
10. #19149677C4semidirect
11. #19149666C4semidirect
109. C173#1912semidirect

Maximal subgroups

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

subgroupindex
#260120622
#260119682
#260115082
C2×#191495972
#260112482
#260112472
#421×C1732
#1912173