Finite Group Databasebeta

← #36355569

Subgroups — #36355569

Order 22144.

260 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012228C211yesmaximal, characteristic, complemented
2#26011681C211yesmaximal, characteristic, complemented
3#26011678C211yesmaximal, characteristic, complemented
4#26011284C211yesmaximal, characteristic, complemented
5#26011282C211yesmaximal, characteristic, complemented
6#26011279C211yesmaximal, characteristic, complemented
7#453×C173C211yesmaximal, characteristic, complemented
8#19149673—21no—
9#19149660—21no—
10#19149660—21no—
11#19149660C2211yescharacteristic
12#19149651—21no—
13#19149651—21no—
14#19149651—21no—
15#19149647—21no—
16#19149647—21no—
17#19149647C2211yescharacteristic
18#19149594—21no—
19#19149594—21no—
20#19149594C2211yescharacteristic
21#19149594C2211yescharacteristic
22H(5,𝔽2)×C173C2211yescharacteristic
23AGL(1,ℤ/8)×C173—21no—
24#127×C173C2211yescharacteristic
25#104×C173—21no—
26#100×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×C22⋊D173—21no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—41no—
32C22×Dic173—21no—
33C22×Dic173D411yescharacteristic, complemented
34#14583805—21no—
35#14583805—21no—
36#14583805—21no—
37#14583805—41no—
38C22⋊Dic173—21no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173—41no—
43C22⋊Dic173D411yescharacteristic
44C22⋊Dic173D411yescharacteristic, complemented
45#14583803—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—41no—
49#41×C173—21no—
50#41×C173—41no—
51C2×D4×C173—21no—
52C2×D4×C173—21no—
53C2×D4×C173—21no—
54C2×D4×C173—21no—
55C2×D4×C173—41no—
56C2×D4×C173D411yescharacteristic, complemented
57C2×D4×C173C2311yescharacteristic
58SD16×C173—41no—
59D8×C173—41no—
60M16×C173—21no—
61C42×C173D411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—41no—
64C22×D173—81no—
65C22⋊D173—81no—
66C22⋊D173—81no—
67C22⋊D173—81no—
68C22⋊D173—82no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—41no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C2×Dic173—81no—
76C23×C173—21no—
77C23×C173—21no—
78C23×C173—21no—
79C23×C173C2×D411yescharacteristic
80C23×C173C2×D411yescharacteristic
81Q8×C173—21no—
82D4×C173—21no—
83D4×C173—41no—
84D4×C173—41no—
85D4×C173—41no—
86D4×C173—41no—
87D4×C173—41no—
88D4×C173—81no—
89D4×C173—81no—
90C2×C4×C173—21no—
91C2×C4×C173—21no—
92C2×C4×C173—21no—
93C2×C4×C173—41no—
94C2×C4×C173C2×D411yescharacteristic
95C8×C173—41no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173—41no—
104C22×C173—81no—
105C22×C173—81no—
106C22×C173#12011yescharacteristic
107C2×D173—41no—
108C2×D173—41no—
109C2×D173—42no—
110C2×D173—41no—
111C2×D173—81no—
112C4×C173—21no—
113C4×C173—41no—
114C4×C173—41no—
115Dic173—41no—
116Dic173—41no—
117Dic173—41no—
118Dic173—81no—
119Dic173—81no—
120C2×C173—21no—
121C2×C173—41no—
122C2×C173—41no—
123C2×C173—81no—
124C2×C173—81no—
125C2×C173B(4,𝔽2)11yescharacteristic
126D173—81no—
127C173#220011yescharacteristic, complemented, Sylow
128#2200—1731nomaximal, Sylow
129#560—1731no—
130#458—1731no—
131B(4,𝔽2)—1731no—
132#453D17311yescharacteristic, complemented
133#353—1731no—
134#352—1731no—
135AΣL(1,𝔽16)—1731no—
136H(5,𝔽2)C2×D17311yescharacteristic
137AGL(1,ℤ/8)—21no—
138#127C2×D17311yescharacteristic
139#126—3461no—
140#123—3461no—
141#123—3461no—
142#123—1731no—
143#121—3461no—
144#121—3461no—
145#121—3461no—
146#120—3461no—
147#120—3461no—
148#120—1731no—
149#104—21no—
150#100C2×D17311yescharacteristic
151#99—3461no—
152#99—3461no—
153#99—1731no—
154#99—1731no—
155C24—1731no—
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—3461no—
164C2×D4—3461no—
165C2×D4—3461no—
166C2×D4—6921no—
167C2×D4C22⋊D17311yescharacteristic, complemented
168C2×D4C22×D17311yescharacteristic
169C22×C4—3461no—
170C22×C4—1731no—
171SD16—41no—
172D8—41no—
173M16—21no—
174C4⋊C4—3461no—
175C4⋊C4—3461no—
176C4⋊C4—3461no—
177C4⋊C4—6921no—
178C22⋊C4—3461no—
179C22⋊C4—3461no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—3461no—
185C22⋊C4—1731no—
186C22⋊C4—1731no—
187C22⋊C4—6921no—
188C22⋊C4—6921no—
189C42C22⋊D17311yescharacteristic, complemented
190C23—21no—
191C23—21no—
192C23—21no—
193C23—3461no—
194C23—13841no—
195C23—6921no—
196C23D4×D17311yescharacteristic
197C23D4×D17311yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—81no—
206D4—81no—
207D4—13842no—
208D4—13841no—
209D4—13841no—
210D4—13841no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—3461no—
216C2×C4—3461no—
217C2×C4—3461no—
218C2×C4—13841no—
219C2×C4—6921no—
220C2×C4—6921no—
221C2×C4—6921no—
222C2×C4C2×C22⋊D17311yescharacteristic
223C8—41no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—81no—
233C22—81no—
234C22—13841no—
235C22—6921no—
236C22—6921no—
237C22—6922no—
238C22—6921no—
239C22#1914964711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—13841no—
244C4—13841no—
245C4—6921no—
246C4—6921no—
247C4—6921no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—13841no—
254C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#453×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#453D173
hypocenterC173#2200
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26012228C2semidirect
2. #26011681C2semidirect
3. #26011678C2semidirect
4. #26011284C2semidirect
5. #26011282C2semidirect
6. #26011279C2semidirect
7. #453×C173C2semidirect
33. C22×Dic173D4semidirect
44. C22⋊Dic173D4semidirect
56. C2×D4×C173D4semidirect
61. C42×C173D4semidirect
127. C173#2200semidirect
132. #453D173semidirect
167. C2×D4C22⋊D173semidirect
189. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122282
#260116812
#260116782
#260112842
#260112822
#260112792
#453×C1732
#2200173