Finite Group Databasebeta

← #36369570

Subgroups — #36369570

Order 22144.

444 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012558C211yesmaximal, characteristic, complemented
2#26012483C212yesmaximal, complemented
3D173⋊#100C211yesmaximal, characteristic, complemented
4#26012362C212yesmaximal, complemented
5#127×D173C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7D4×#10621924—24no—
8#19149758C2212yes—
9C2×C22⋊#10621924—22no—
10#19149741—24no—
11#19149739—22no—
12C2×D4×D173—22no—
13C2×D4×D173C2211yescharacteristic
14C2×D4×D173C2211yescharacteristic
15#19149676C411yescharacteristic, complemented
16#19149675—22no—
17#19149675C411yescharacteristic, complemented
18#19149675C411yescharacteristic, complemented
19C42×D173C2211yescharacteristic
20#127×C173C411yescharacteristic, complemented
21C22×#10621924—24no—
22#14583836—22no—
23#14583835—22no—
24D173⋊C8—22no—
25C4⋊#10621924D414yescomplemented
26C22⋊#10621924—24no—
27C22⋊#10621924—22no—
28C4×#10621924—22no—
29C23×D173—21no—
30C23×D173D412yes—
31C2×D692—22no—
32C2×D692C2×C411yescharacteristic
33C2×D692C2×C411yescharacteristic, complemented
34C2×C22⋊D173—22no—
35C2×C22⋊D173—21no—
36C2×C22⋊D173—22no—
37C2×C22⋊D173D412yes—
38D4×D173—22no—
39D4×D173—22no—
40D4×D173—22no—
41D4×D173—22no—
42D4×D173—48no—
43C2×C4×D173—21no—
44C2×C4×D173C2311yescharacteristic
45C4×Dic173—21no—
46C4×Dic173C2×C411yescharacteristic, complemented
47C2×D4×C173—22no—
48C2×D4×C173C2×C411yescharacteristic
49C2×D4×C173C2×C411yescharacteristic, complemented
50C42×C173C2×C411yescharacteristic, complemented
51C2×#10621924—24no—
52C2×#10621924—84no—
53#13063443—42no—
54C22×D173—22no—
55C22×D173—22no—
56C22×D173—21no—
57C22×D173—44no—
58C22×D173—44no—
59C22×D173—41no—
60C22×D173—41no—
61C22×D173C22⋊C412yes—
62C22×D173C2×D411yescharacteristic
63D692—22no—
64D692—22no—
65D692—44no—
66C22⋊D173—24no—
67C22⋊D173—44no—
68C22⋊D173—44no—
69C22⋊D173—42no—
70C2×Dic173—21no—
71C2×Dic173C2×D411yescharacteristic
72C4×D173—24no—
73C4×D173C2×D412yes—
74C4×D173#4112yes—
75C23×C173—21no—
76C23×C173C22⋊C412yes—
77D4×C173—22no—
78D4×C173—22no—
79D4×C173—44no—
80C2×C4×C173—21no—
81C2×C4×C173C22×C411yescharacteristic
82C22×C173—22no—
83C22×C173—42no—
84C22×C173—42no—
85C22×C173—41no—
86C22×C173C2×C22⋊C411yescharacteristic
87C2×D173—22no—
88C2×D173—22no—
89C2×D173—44no—
90C2×D173—42no—
91C2×D173—42no—
92C2×D173—41no—
93C2×D173—82no—
94C2×D173—82no—
95C2×D173—82no—
96C2×D173#12011yescharacteristic
97C2×D173#12311yescharacteristic
98#10621924—44no—
99C4×C173—22no—
100C4×C173C4×D412yes—
101Dic173—22no—
102Dic173#12112yes—
103C2×C173—21no—
104C2×C173—42no—
105C2×C173—81no—
106C2×C173#38611yescharacteristic
107D173—21no—
108D173—42no—
109D173—81no—
110D173#45312yes—
111C173#188711yescharacteristic, complemented, Sylow
112#1887—1731nomaximal, Sylow
113C2×#127—1731no—
114#518—1731no—
115#418—1732no—
116C2×#100—1731no—
117#339—1732no—
118C22×D4—3462no—
119C22×D4—1731no—
120C22×D4—1731no—
121C2×M16—1732no—
122#127—3462no—
123#127—1731no—
124#127—1731no—
125#127—1731no—
126#127#1062192411yescharacteristic, complemented
127C4×D4—3464no—
128#117—1732no—
129C2×C22⋊C4—3462no—
130C2×C42—1731no—
131#102—3464no—
132#100—3462no—
133C24—3461no—
134C24—1732no—
135C2×D4—22no—
136C2×D4—3462no—
137C2×D4—3462no—
138C2×D4—3462no—
139C2×D4—3462no—
140C2×D4—3462no—
141C2×D4—3462no—
142C2×D4—3462no—
143C2×D4—3461no—
144C2×D4—1732no—
145C2×D4—1731no—
146C2×D4—1731no—
147C2×D4—6928no—
148C2×D4C2×#1062192411yescharacteristic
149C2×D4C2×#1062192411yescharacteristic, complemented
150C22×C4—3461no—
151C22×C4—3464no—
152C22×C4—1731no—
153M16—3462no—
154M16—3462no—
155C2×C8—3462no—
156C4⋊C4—1734no—
157C22⋊C4—3464no—
158C22⋊C4—3462no—
159C42—3461no—
160C42—3462no—
161C42—1731no—
162C42C2×#1062192411yescharacteristic, complemented
163C23—21no—
164C23—3462no—
165C23—3462no—
166C23—3461no—
167C23—1731no—
168C23—1732no—
169C23—6924no—
170C23—6924no—
171C23—6921no—
172C23—6921no—
173C23C22⋊#1062192412yes—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—3464no—
178D4—3462no—
179D4—3462no—
180D4—6924no—
181D4—6924no—
182D4—6924no—
183D4—6922no—
184C2×C4—21no—
185C2×C4—3464no—
186C2×C4—3461no—
187C2×C4—3464no—
188C2×C4—13844no—
189C2×C4—1731no—
190C2×C4—1732no—
191C2×C4—1732no—
192C2×C4C22×#1062192411yescharacteristic
193C8—6922no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—3462no—
199C22—3462no—
200C22—13842no—
201C22—13842no—
202C22—13842no—
203C22—1731no—
204C22—1731no—
205C22—6922no—
206C22—6922no—
207C22—6924no—
208C22—6921no—
209C22C2×C22⋊#1062192411yescharacteristic
210C4—22no—
211C4—3462no—
212C4—1732no—
213C4—6924no—
214C4D4×#1062192412yes—
215C2—21no—
216C2—42no—
217C2—81no—
218C2—3461no—
219C2—13841no—
220C2—1732no—
221C2—6922no—
222C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC2×C173#386
prime radical#127×D173C2
cosocleC2×C4×D173C23
hypercenter#127#10621924
hypocenterC173#1887
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26012558C2semidirect
2. #26012483C2semidirect
3. D173⋊#100C2semidirect
4. #26012362C2semidirect
15. #19149676C4semidirect
17. #19149675C4semidirect
18. #19149675C4semidirect
20. #127×C173C4semidirect
25. C4⋊#10621924D4semidirect
33. C2×D692C2×C4semidirect
46. C4×Dic173C2×C4semidirect
49. C2×D4×C173C2×C4semidirect
50. C42×C173C2×C4semidirect
111. C173#1887semidirect
126. #127#10621924semidirect
149. C2×D4C2×#10621924semidirect
162. C42C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125582
#260124832
D173⋊#1002
#260123622
#127×D1732
#1887173