Finite Group Databasebeta

← #36355567

Subgroups — #36355567

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#26012231C211yesmaximal, characteristic, complemented
2#26011678C211yesmaximal, characteristic, complemented
3#26011659C211yesmaximal, characteristic, complemented
4#26011283C211yesmaximal, characteristic, complemented
5#26011281C211yesmaximal, characteristic, complemented
6#26011277C211yesmaximal, characteristic, complemented
7#458×C173C211yesmaximal, characteristic, complemented
8#19149704—21no—
9#19149675C2211yescharacteristic
10#19149674—21no—
11#19149661—21no—
12#19149654—21no—
13#19149653—21no—
14#19149652—21no—
15#19149649—21no—
16#19149647—21no—
17#19149647C2211yescharacteristic
18#19149606—21no—
19#19149596C2211yescharacteristic
20#19149594—21no—
21#19149594C2211yescharacteristic
22H(5,𝔽2)×C173C2211yescharacteristic
23#123×C173—21no—
24#123×C173C2211yescharacteristic
25#99×C173—21no—
26#99×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692D411yescharacteristic, complemented
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—41no—
35C2×C4×D173—21no—
36#14583813—41no—
37#14583811—41no—
38#14583810—21no—
39C4⋊Dic173—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173D411yescharacteristic
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—41no—
48C4×Dic173D411yescharacteristic, complemented
49#41×C173—21no—
50#41×C173—41no—
51C2×D4×C173—21no—
52C2×D4×C173—21no—
53C2×D4×C173—41no—
54C2×D4×C173C2311yescharacteristic
55C22×C4×C173D411yescharacteristic, complemented
56C4⋊C4×C173—21no—
57C4⋊C4×C173—41no—
58C22⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—41no—
61C22⋊C4×C173D411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—41no—
65C22×D173—81no—
66D692—41no—
67D692—81no—
68D692—81no—
69C22⋊D173—81no—
70C22⋊D173—81no—
71C22⋊D173—81no—
72C22⋊D173—82no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—41no—
76C4×D173—81no—
77#13063436—41no—
78C23×C173—21no—
79C23×C173—21no—
80C23×C173C2×D411yescharacteristic
81C23×C173C2×D411yescharacteristic
82Q8×C173—21no—
83D4×C173—21no—
84D4×C173—41no—
85D4×C173—41no—
86D4×C173—41no—
87D4×C173—41no—
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—21no—
91C2×C4×C173—21no—
92C2×C4×C173—41no—
93C2×C4×C173—41no—
94C2×C4×C173—41no—
95C2×C4×C173C2×D411yescharacteristic
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—81no—
102C22×C173—81no—
103C22×C173#12011yescharacteristic
104C2×D173—41no—
105C2×D173—41no—
106C2×D173—41no—
107C2×D173—41no—
108C2×D173—42no—
109C2×D173—41no—
110C2×D173—41no—
111C2×D173—81no—
112C4×C173—21no—
113C4×C173—41no—
114C4×C173—41no—
115C4×C173—41no—
116C4×C173—81no—
117Dic173—41no—
118Dic173—41no—
119Dic173—81no—
120C2×C173—21no—
121C2×C173—41no—
122C2×C173—41no—
123C2×C173—81no—
124C2×C173B(4,𝔽2)11yescharacteristic
125D173—81no—
126D173—81no—
127C173#220011yescharacteristic, complemented, Sylow
128#2200—1731nomaximal, Sylow
129#560—1731no—
130#458D17311yescharacteristic, complemented
131B(4,𝔽2)—1731no—
132#453—1731no—
133#353—1731no—
134#352—1731no—
135AΣL(1,𝔽16)—1731no—
136H(5,𝔽2)C2×D17311yescharacteristic
137AGL(1,ℤ/8)—3461no—
138#127—1731no—
139#126—3461no—
140#123—21no—
141#123—3461no—
142#123C2×D17311yescharacteristic
143#121—3461no—
144#121—3461no—
145#121—3461no—
146#120—3461no—
147#120—3461no—
148#120—1731no—
149#104—3461no—
150#100—1731no—
151#99—21no—
152#99—3461no—
153#99—1731no—
154#99C2×D17311yescharacteristic
155C24—1731no—
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—3461no—
162C2×D4—3461no—
163C2×D4—3461no—
164C2×D4—3461no—
165C2×D4—3461no—
166C2×D4—1731no—
167C2×D4—6921no—
168C2×D4C22×D17311yescharacteristic
169C22×C4—3461no—
170C22×C4C22⋊D17311yescharacteristic, complemented
171SD16—6921no—
172D8—6921no—
173M16—3461no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—3461no—
177C4⋊C4—3461no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—41no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—3461no—
185C22⋊C4—3461no—
186C22⋊C4—1731no—
187C22⋊C4—6921no—
188C22⋊C4C22⋊D17311yescharacteristic, complemented
189C42—1731no—
190C23—21no—
191C23—21no—
192C23—3461no—
193C23—3461no—
194C23—13841no—
195C23—6921no—
196C23D4×D17311yescharacteristic
197C23C2×C22⋊D17311yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—13841no—
205D4—13841no—
206D4—13842no—
207D4—13841no—
208D4—13841no—
209D4—13841no—
210D4—6921no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—3461no—
219C2×C4—3461no—
220C2×C4—13841no—
221C2×C4—6921no—
222C2×C4D4×D17311yescharacteristic
223C8—6921no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—81no—
230C22—81no—
231C22—13841no—
232C22—6921no—
233C22—6921no—
234C22—6921no—
235C22—6921no—
236C22—6922no—
237C22—6921no—
238C22—6921no—
239C22#1914964711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—13841no—
246C4—6921no—
247C4—6921no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—13841no—
253C2—13841no—
254C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#458×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#458D173
hypocenterC173#2200
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #26012231C2semidirect
2. #26011678C2semidirect
3. #26011659C2semidirect
4. #26011283C2semidirect
5. #26011281C2semidirect
6. #26011277C2semidirect
7. #458×C173C2semidirect
30. C2×D692D4semidirect
48. C4×Dic173D4semidirect
55. C22×C4×C173D4semidirect
61. C22⋊C4×C173D4semidirect
127. C173#2200semidirect
130. #458D173semidirect
170. C22×C4C22⋊D173semidirect
188. C22⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122312
#260116782
#260116592
#260112832
#260112812
#260112772
#458×C1732
#2200173