Finite Group Databasebeta

← #36369615

Subgroups — #36369615

Order 22144.

308 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012569C211yesmaximal, characteristic
2#26012478C211yesmaximal, characteristic
3C2×#19149740C212yesmaximal, complemented
4#26012372C212yesmaximal, complemented
5#26012063C211yesmaximal, characteristic
6#19149766—24no—
7D173⋊M16—21no—
8C2×#14583835—21no—
9C2×#14583835C2212yes—
10C2×#14583835C2212yes—
11#19149740—24no—
12#19149740—21no—
13#19149739—21no—
14C2×#14583825C2211yescharacteristic
15C22×Dic346C2211yescharacteristic
16#19149678—21no—
17#19149665—21no—
18#19149663C411yescharacteristic
19#19149662C411yescharacteristic
20#19149660C411yescharacteristic
21#19149657—21no—
22#19149656—21no—
23#19149635C2211yescharacteristic
24#123×C173C411yescharacteristic
25#14583836—22no—
26#14583835—22no—
27#14583835—22no—
28#14583835—22no—
29#14583835—22no—
30#14583835D414yes—
31D173⋊C8—22no—
32C2×#13063443—22no—
33C2×#13063443—22no—
34C2×#13063443—22no—
35#14583825—22no—
36#14583825—22no—
37C2×Dic346—21no—
38C2×Dic346—21no—
39C2×Dic346—22no—
40C2×Dic346—22no—
41C2×Dic346D411yescharacteristic
42C2×C22⋊D173C2×C411yescharacteristic
43C2×C22⋊D173D411yescharacteristic
44C22×Dic173C2×C411yescharacteristic
45C22×Dic173D411yescharacteristic
46C22×Dic173C2311yescharacteristic
47C2×C4×D173D411yescharacteristic
48C4⋊Dic173—21no—
49#14583805—21no—
50#14583805—21no—
51#14583805—21no—
52C22⋊Dic173—21no—
53C22⋊Dic173C2×C411yescharacteristic
54#14583803—21no—
55#14583803—21no—
56C4×Dic173—21no—
57C2×D4×C173C2×C411yescharacteristic
58C22×C4×C173C2×C411yescharacteristic
59C4⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173C2×C411yescharacteristic
62#13063443—24no—
63#13063443—42no—
64C22×D173C22⋊C411yescharacteristic
65Dic346—22no—
66Dic346—41no—
67Dic346—42no—
68Dic346—41no—
69C22⋊D173—22no—
70C22⋊D173—22no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—22no—
76C2×Dic173C22⋊C411yescharacteristic
77C2×Dic173C2×D412yes—
78C2×Dic173C2×D411yescharacteristic
79C2×Dic173C2×D411yescharacteristic
80C2×Dic173#4112yes—
81C4×D173—21no—
82C4×D173—21no—
83C23×C173C22⋊C411yescharacteristic
84C23×C173C22×C411yescharacteristic
85D4×C173—22no—
86C2×C4×C173—21no—
87C2×C4×C173—21no—
88C2×C4×C173—22no—
89C2×C4×C173C22⋊C411yescharacteristic
90C22×C173—21no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173C2×C22⋊C411yescharacteristic
94C22×C173C4×D412yes—
95C2×D173—21no—
96C2×D173—41no—
97C4×C173—21no—
98C4×C173—41no—
99C4×C173—41no—
100Dic173—21no—
101Dic173—41no—
102Dic173—41no—
103Dic173#12011yescharacteristic
104Dic173#12112yes—
105Dic173#12311yescharacteristic
106C2×C173—21no—
107C2×C173—22no—
108C2×C173—41no—
109C2×C173#38611yescharacteristic
110D173—41no—
111C173#191511yescharacteristic, complemented, Sylow
112#1915—1731nomaximal, Sylow
113#568—1731no—
114#536—1731no—
115#413—1731no—
116C2×#101—1732no—
117#344—1732no—
118C2×#41—1731no—
119C22×Q8—1731no—
120#131—3464no—
121C2×M16—3461no—
122C2×M16—3461no—
123C2×M16—1732no—
124C2×M16—1732no—
125C4⋊Q8—3461no—
126#124—3461no—
127#123—1731no—
128#123—1731no—
129#123—1731no—
130#123#1062192411yescharacteristic
131C22⋊Q8—3461no—
132C22⋊Q8—3461no—
133#117—1731no—
134#101—3461no—
135#101—3464no—
136#100—3461no—
137#41—3462no—
138#41—3462no—
139C2×Q8—3462no—
140C2×Q8—3461no—
141C2×Q8—3461no—
142C2×Q8—3462no—
143C2×Q8—1731no—
144C2×D4—1731no—
145C2×D4—1731no—
146C2×D4C2×#1062192411yescharacteristic
147C22×C4—1731no—
148C22×C4—1731no—
149C22×C4—1731no—
150C22×C4—1731no—
151C22×C4C2×#1062192411yescharacteristic
152M16—3462no—
153M16—3462no—
154M16—3462no—
155M16—3462no—
156M16—3462no—
157M16—1734no—
158C2×C8—3462no—
159C2×C8—3462no—
160C2×C8—3462no—
161C2×C8—3462no—
162C4⋊C4—21no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C22⋊C4—21no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C22⋊C4—1731no—
172C22⋊C4C2×#1062192411yescharacteristic
173C42—3461no—
174C23—1731no—
175C23C22⋊#1062192411yescharacteristic
176C23C22×#1062192411yescharacteristic
177Q8—3462no—
178Q8—6921no—
179Q8—6922no—
180Q8—6921no—
181D4—22no—
182D4—3462no—
183D4—3462no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—3461no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—3461no—
193C2×C4—3462no—
194C2×C4—1731no—
195C2×C4—1731no—
196C2×C4—1731no—
197C2×C4—1732no—
198C2×C4—1732no—
199C2×C4C22⋊#1062192411yescharacteristic
200C8—3464no—
201C8—6922no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—3461no—
206C22—6921no—
207C22C2×C22⋊#1062192411yescharacteristic
208C22D4×#1062192412yes—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—3461no—
213C4—1731no—
214C4—1732no—
215C4—1731no—
216C4—6921no—
217C4—6921no—
218C2—21no—
219C2—22no—
220C2—41no—
221C2—6921no—
222C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#123×C173C4
socleC2×C173#386
prime radicalC2×#14583825C22
cosocleC22×Dic173C23
hypercenter#123#10621924
hypocenterC173#1915
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
3. C2×#19149740C2semidirect
4. #26012372C2semidirect
111. C173#1915semidirect

Maximal subgroups

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

subgroupindex
#260125692
#260124782
C2×#191497402
#260123722
#260120632
#1915173