Finite Group Databasebeta

← #36355211

Subgroups — #36355211

Order 22144.

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

normal subgroups only

This group has 266 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#26012056C211yesmaximal, characteristic
2#26011472C211yesmaximal, characteristic, complemented
3#26011463C211yesmaximal, characteristic, complemented
4#26011279C211yesmaximal, characteristic, complemented
5#26011278C211yesmaximal, characteristic, complemented
6#26011277C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C173C211yesmaximal, characteristic, complemented
8C2×#14583825C2211yescharacteristic
9C22×C22⋊D173C411yescharacteristic, complemented
10M16×D173—21no—
11#19149675—21no—
12#19149665—21no—
13#19149652—21no—
14#19149651—21no—
15#19149649C411yescharacteristic, complemented
16#19149648C411yescharacteristic, complemented
17#19149647C2211yescharacteristic
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
27C23×D173C2×C411yescharacteristic
28#14583825—41no—
29#14583825—41no—
30C2×Dic346D411yescharacteristic
31C2×D692—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36C2×C22⋊D173—41no—
37C2×C22⋊D173—41no—
38C2×C22⋊D173C2×C411yescharacteristic
39C2×C22⋊D173D411yescharacteristic
40C22×Dic173C2×C411yescharacteristic
41C22×Dic173D411yescharacteristic
42C2×C4×D173D411yescharacteristic
43#14583810—21no—
44#14583809—41no—
45C8×D173—41no—
46#14583805—21no—
47#14583805—41no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173C2×C411yescharacteristic
51#14583803—21no—
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×D173—21no—
63C22×D173—41no—
64C22×D173—81no—
65C22×D173C22⋊C411yescharacteristic
66Dic346—41no—
67D692—81no—
68C22⋊D173—41no—
69C22⋊D173—41no—
70C22⋊D173—42no—
71C22⋊D173—41no—
72C22⋊D173—41no—
73C22⋊D173—81no—
74C22⋊D173—81no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—41no—
81C2×Dic173C22⋊C411yescharacteristic
82C2×Dic173C22⋊C411yescharacteristic
83C2×Dic173C22⋊C411yescharacteristic
84C4×D173—21no—
85C4×D173—21no—
86#13063436—41no—
87C23×C173—21no—
88C23×C173—41no—
89C23×C173—81no—
90C23×C173C22×C411yescharacteristic
91C23×C173C2×D411yescharacteristic
92D4×C173—41no—
93D4×C173—81no—
94C2×C4×C173—21no—
95C2×C4×C173—41no—
96C2×C4×C173C2×D411yescharacteristic
97C8×C173—41no—
98C22×C173—21no—
99C22×C173—21no—
100C22×C173—41no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173—82no—
106C22×C173C2×C22⋊C411yescharacteristic
107C2×D173—21no—
108C2×D173—41no—
109C2×D173—41no—
110C2×D173—41no—
111C2×D173—41no—
112C2×D173—82no—
113C4×C173—21no—
114C4×C173—41no—
115C4×C173—81no—
116Dic173—21no—
117Dic173—21no—
118Dic173—41no—
119Dic173—81no—
120Dic173#9911yescharacteristic
121Dic173#9911yescharacteristic
122C2×C173—21no—
123C2×C173—41no—
124C2×C173—41no—
125C2×C173—81no—
126C2×C173C2×#9911yescharacteristic
127D173—41no—
128D173—81no—
129C173#212211yescharacteristic, complemented, Sylow
130#2122—1731nomaximal, Sylow
131#535—1731no—
132#413—1731no—
133#410—1731no—
134AΣL(1,𝔽16)—1731no—
135AΣL(1,𝔽16)—1731no—
136AΣL(1,𝔽16)—1731no—
137AΣL(1,𝔽16)D17311yescharacteristic, complemented
138C2×#41—1731no—
139C22×D4—1731no—
140C2×M16—3461no—
141#127—3461no—
142#124—3461no—
143#121—3461no—
144#121—3461no—
145#120—1731no—
146#120—1731no—
147#120—1731no—
148#120C2×D17311yescharacteristic
149#117—3461no—
150#117—1731no—
151#101—3461no—
152#100—1731no—
153#100C2×D17311yescharacteristic
154#99—3461no—
155#99—1731no—
156#99C2×D17311yescharacteristic
157C24—1731no—
158C24C4×D17311yescharacteristic
159#41—6921no—
160#41—6921no—
161C2×Q8—1731no—
162C2×D4—21no—
163C2×D4—3461no—
164C2×D4—3461no—
165C2×D4—3461no—
166C2×D4—3461no—
167C2×D4—3461no—
168C2×D4—1731no—
169C2×D4—1731no—
170C2×D4—6921no—
171C2×D4—6921no—
172C2×D4C22×D17311yescharacteristic
173C22×C4—1731no—
174C22×C4—1731no—
175C22×C4—1731no—
176M16—21no—
177M16—3461no—
178M16—6921no—
179C2×C8—6921no—
180C4⋊C4—3461no—
181C4⋊C4—6921no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—3461no—
185C22⋊C4—3461no—
186C22⋊C4—3461no—
187C22⋊C4—3461no—
188C22⋊C4—1731no—
189C22⋊C4C4×D17311yescharacteristic
190C42—3461no—
191C42—6921no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—3461no—
196C23—13841no—
197C23—1731no—
198C23—6921no—
199C23C2×C4×D17311yescharacteristic
200C23D4×D17311yescharacteristic
201Q8—6921no—
202D4—41no—
203D4—81no—
204D4—13841no—
205D4—13841no—
206D4—13841no—
207D4—6921no—
208D4—6921no—
209D4—6922no—
210D4—6921no—
211D4—6921no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—3461no—
215C2×C4—3461no—
216C2×C4—3461no—
217C2×C4—3461no—
218C2×C4—3461no—
219C2×C4—3461no—
220C2×C4—3461no—
221C2×C4—1731no—
222C2×C4—1731no—
223C2×C4—1731no—
224C2×C4—6921no—
225C2×C4D4×D17311yescharacteristic
226C8—41no—
227C8—6921no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—3461no—
237C22—13842no—
238C22—6921no—
239C22—6921no—
240C22—6921no—
241C22—6921no—
242C22C22⋊C4×D17311yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—3461no—
247C4—3461no—
248C4—13841no—
249C4—1731no—
250C4—1731no—
251C4—6921no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—13841no—
257C2—6921no—
258C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
FittingAΣL(1,𝔽16)×C173C2
socleC2×C173C2×#99
prime radical#26012056C2
cosocleC2×D4×C173C23
hypercenterAΣL(1,𝔽16)D173
hypocenterC173#2122
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. #26011279C2semidirect
5. #26011278C2semidirect
6. #26011277C2semidirect
7. AΣL(1,𝔽16)×C173C2semidirect
9. C22×C22⋊D173C4semidirect
15. #19149649C4semidirect
16. #19149648C4semidirect
19. #19149635C4semidirect
129. C173#2122semidirect
137. AΣL(1,𝔽16)D173semidirect

Maximal subgroups

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

subgroupindex
#260120562
#260114722
#260114632
#260112792
#260112782
#260112772
AΣL(1,𝔽16)×C1732
#2122173