Finite Group Databasebeta

← #36355201

Subgroups — #36355201

Order 22144.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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#26011997C211yesmaximal, characteristic
2#26011472C211yesmaximal, characteristic, complemented
3#26011463C211yesmaximal, characteristic, complemented
4#26011282C211yesmaximal, characteristic, complemented
5#26011280C211yesmaximal, characteristic, complemented
6#26011278C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C173C211yesmaximal, characteristic, complemented
8C2×#14583825C2211yescharacteristic
9C23×Dic173C411yescharacteristic, complemented
10M16×D173—21no—
11#19149663C411yescharacteristic, complemented
12#19149660C2211yescharacteristic
13#19149656—21no—
14#19149651—21no—
15#19149648C411yescharacteristic, complemented
16D4×Dic173—21no—
17#19149639—21no—
18#19149635—21no—
19#19149635C411yescharacteristic, complemented
20#19149597—21no—
21#19149596C2211yescharacteristic
22#19149594C2211yescharacteristic
23#19149593—21no—
24#120×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27#14583825—41no—
28#14583825—41no—
29C2×Dic346D411yescharacteristic
30C2×C22⋊D173—21no—
31C2×C22⋊D173C2×C411yescharacteristic
32C2×C22⋊D173D411yescharacteristic
33C22×Dic173—21no—
34C22×Dic173—21no—
35C22×Dic173—41no—
36C22×Dic173—41no—
37C22×Dic173C2×C411yescharacteristic
38C22×Dic173C2×C411yescharacteristic
39C22×Dic173D411yescharacteristic
40C2×C4×D173D411yescharacteristic
41#14583810—21no—
42#14583809—41no—
43C8×D173—41no—
44C4⋊Dic173—21no—
45#14583805—21no—
46#14583805—21no—
47#14583805—41no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—21no—
51C22⋊Dic173C2×C411yescharacteristic
52#14583803—21no—
53C4×Dic173—21no—
54C4×Dic173—41no—
55C24×C173C2×C411yescharacteristic
56C2×D4×C173—21no—
57C2×D4×C173C2311yescharacteristic
58M16×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173C2×C411yescharacteristic
62C22×D173C22⋊C411yescharacteristic
63Dic346—41no—
64C22⋊D173—41no—
65C22⋊D173—41no—
66C22⋊D173—42no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—42no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76C2×Dic173—41no—
77C2×Dic173—41no—
78C2×Dic173C22⋊C411yescharacteristic
79C2×Dic173C22⋊C411yescharacteristic
80C2×Dic173C22⋊C411yescharacteristic
81C4×D173—21no—
82C4×D173—21no—
83#13063436—41no—
84C23×C173—21no—
85C23×C173—41no—
86C23×C173—42no—
87C23×C173C22×C411yescharacteristic
88C23×C173C2×D411yescharacteristic
89D4×C173—41no—
90D4×C173—42no—
91C2×C4×C173—21no—
92C2×C4×C173—41no—
93C2×C4×C173C2×D411yescharacteristic
94C8×C173—41no—
95C22×C173—22no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—42no—
102C22×C173—41no—
103C22×C173—42no—
104C22×C173C2×C22⋊C411yescharacteristic
105C2×D173—21no—
106C2×D173—41no—
107C4×C173—21no—
108C4×C173—41no—
109C4×C173—81no—
110Dic173—21no—
111Dic173—21no—
112Dic173—41no—
113Dic173—41no—
114Dic173—81no—
115Dic173#9911yescharacteristic
116Dic173#9911yescharacteristic
117C2×C173—21no—
118C2×C173—41no—
119C2×C173—42no—
120C2×C173—41no—
121C2×C173C2×#9911yescharacteristic
122D173—41no—
123C173#212011yescharacteristic, complemented, Sylow
124#2120—1731nomaximal, Sylow
125#525—1731no—
126#413—1731no—
127#410—1731no—
128#352—1731no—
129#352—1731no—
130AΣL(1,𝔽16)—1731no—
131AΣL(1,𝔽16)D17311yescharacteristic, complemented
132C2×#41—1731no—
133C23×C4—1731no—
134C2×M16—3461no—
135#123—1731no—
136#123—1731no—
137C22⋊Q8—3461no—
138#121—3461no—
139#120—1731no—
140#120C2×D17311yescharacteristic
141C4×D4—3461no—
142C4×D4—3461no—
143#117—3461no—
144#117—1731no—
145#101—3461no—
146#100—1731no—
147#100C2×D17311yescharacteristic
148#99—3461no—
149#99—1731no—
150#99C2×D17311yescharacteristic
151C24C4×D17311yescharacteristic
152#41—6921no—
153#41—6921no—
154C2×Q8—1731no—
155C2×D4—21no—
156C2×D4—3461no—
157C2×D4—1731no—
158C2×D4—1731no—
159C2×D4C22×D17311yescharacteristic
160C22×C4—3461no—
161C22×C4—3461no—
162C22×C4—1731no—
163C22×C4—1731no—
164C22×C4—1731no—
165C22×C4—1731no—
166C22×C4—6921no—
167C22×C4—6921no—
168M16—21no—
169M16—3461no—
170M16—6921no—
171C2×C8—6921no—
172C4⋊C4—3461no—
173C4⋊C4—3461no—
174C4⋊C4—3461no—
175C4⋊C4—6921no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—3461no—
179C22⋊C4—3461no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—1731no—
183C22⋊C4C4×D17311yescharacteristic
184C42—3461no—
185C42—6921no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—1731no—
190C23C2×C4×D17311yescharacteristic
191C23D4×D17311yescharacteristic
192Q8—6921no—
193D4—41no—
194D4—42no—
195D4—6921no—
196D4—6921no—
197D4—6922no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3461no—
208C2×C4—1731no—
209C2×C4—1731no—
210C2×C4—1731no—
211C2×C4—6922no—
212C2×C4—6921no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4D4×D17311yescharacteristic
217C8—41no—
218C8—6921no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—3461no—
229C22—6921no—
230C22C22⋊C4×D17311yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—3461no—
235C4—3461no—
236C4—13841no—
237C4—1731no—
238C4—1731no—
239C4—6921no—
240C4—6921no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—6921no—
246C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
FittingAΣL(1,𝔽16)×C173C2
socleC2×C173C2×#99
prime radical#26011997C2
cosocleC2×D4×C173C23
hypercenterAΣL(1,𝔽16)D173
hypocenterC173#2120
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #26011472C2semidirect
3. #26011463C2semidirect
4. #26011282C2semidirect
5. #26011280C2semidirect
6. #26011278C2semidirect
7. AΣL(1,𝔽16)×C173C2semidirect
9. C23×Dic173C4semidirect
11. #19149663C4semidirect
15. #19149648C4semidirect
19. #19149635C4semidirect
123. C173#2120semidirect
131. AΣL(1,𝔽16)D173semidirect

Maximal subgroups

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

subgroupindex
#260119972
#260114722
#260114632
#260112822
#260112802
#260112782
AΣL(1,𝔽16)×C1732
#2120173