Finite Group Databasebeta

← #36355568

Subgroups — #36355568

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#26012230C211yesmaximal, characteristic, complemented
2#26011681C211yesmaximal, characteristic, complemented
3#26011659C211yesmaximal, characteristic, complemented
4#26011283C211yesmaximal, characteristic, complemented
5#26011280C211yesmaximal, characteristic, complemented
6#26011278C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C173C211yesmaximal, characteristic, complemented
8#19149704—21no—
9#19149675C2211yescharacteristic
10#19149672—21no—
11#19149661—21no—
12#19149660—21no—
13#19149660C2211yescharacteristic
14#19149652—21no—
15#19149651—21no—
16#19149650—21no—
17#19149648—21no—
18#19149606—21no—
19#19149596C2211yescharacteristic
20#19149594—21no—
21#19149594C2211yescharacteristic
22H(5,𝔽2)×C173C2211yescharacteristic
23#120×C173—21no—
24#120×C173C2211yescharacteristic
25#99×C173—21no—
26#99×C173C2211yescharacteristic
27C2×D692D411yescharacteristic, complemented
28C2×C22⋊D173—21no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C22×Dic173D411yescharacteristic, complemented
33C2×C4×D173—21no—
34#14583813—41no—
35#14583811—41no—
36#14583810—21no—
37C4⋊Dic173—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—41no—
41C22⋊Dic173—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—41no—
46C22⋊Dic173D411yescharacteristic, complemented
47#14583803—21no—
48C4×Dic173D411yescharacteristic, complemented
49C24×C173D411yescharacteristic
50#41×C173—21no—
51#41×C173—41no—
52C2×D4×C173—21no—
53C2×D4×C173—21no—
54C2×D4×C173—21no—
55C2×D4×C173—41no—
56C2×D4×C173—41no—
57C2×D4×C173C2311yescharacteristic
58C22⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—41no—
61C22⋊C4×C173D411yescharacteristic
62C22×D173—21no—
63D692—41no—
64C22⋊D173—81no—
65C22⋊D173—81no—
66C22⋊D173—81no—
67C22⋊D173—81no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C4×D173—81no—
74#13063436—41no—
75C23×C173—21no—
76C23×C173—21no—
77C23×C173—21no—
78C23×C173—41no—
79C23×C173—81no—
80C23×C173C2×D411yescharacteristic
81C23×C173C2×D411yescharacteristic
82Q8×C173—21no—
83D4×C173—21no—
84D4×C173—41no—
85D4×C173—41no—
86D4×C173—41no—
87D4×C173—41no—
88D4×C173—81no—
89D4×C173—82no—
90C2×C4×C173—21no—
91C2×C4×C173—21no—
92C2×C4×C173—21no—
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—41no—
102C22×C173—42no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173—81no—
106C22×C173—81no—
107C22×C173—81no—
108C22×C173#12011yescharacteristic
109C2×D173—41no—
110C2×D173—41no—
111C2×D173—41no—
112C4×C173—21no—
113C4×C173—41no—
114C4×C173—41no—
115C4×C173—81no—
116Dic173—41no—
117Dic173—41no—
118Dic173—41no—
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)D17311yescharacteristic, complemented
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—3461no—
141#123—3461no—
142#123—1731no—
143#121—3461no—
144#121—3461no—
145#121—3461no—
146#120—21no—
147#120—3461no—
148#120C2×D17311yescharacteristic
149#104—3461no—
150#100—1731no—
151#99—21no—
152#99—3461no—
153#99—1731no—
154#99C2×D17311yescharacteristic
155C24C22⋊D17311yescharacteristic
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—3461no—
164C2×D4—3461no—
165C2×D4—3461no—
166C2×D4—3461no—
167C2×D4—1731no—
168C2×D4C22×D17311yescharacteristic
169C22×C4—3461no—
170C22×C4—1731no—
171SD16—6921no—
172D8—6921no—
173M16—3461no—
174C4⋊C4—3461no—
175C4⋊C4—3461no—
176C4⋊C4—3461no—
177C4⋊C4—6921no—
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
189C42—1731no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—3461no—
196C23D4×D17311yescharacteristic
197C23C2×C22⋊D17311yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—82no—
206D4—13841no—
207D4—13841no—
208D4—13841no—
209D4—13841no—
210D4—6921no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—3461no—
217C2×C4—3461no—
218C2×C4—3461no—
219C2×C4—13841no—
220C2×C4—6921no—
221C2×C4—6921no—
222C2×C4D4×D17311yescharacteristic
223C8—6921no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—6921no—
237C22—6921no—
238C22—6921no—
239C22#1914964711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—81no—
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
FittingB(4,𝔽2)×C173C2
socleC2×C173B(4,𝔽2)
hypercenterB(4,𝔽2)D173
hypocenterC173#2200
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26012230C2semidirect
2. #26011681C2semidirect
3. #26011659C2semidirect
4. #26011283C2semidirect
5. #26011280C2semidirect
6. #26011278C2semidirect
7. B(4,𝔽2)×C173C2semidirect
27. C2×D692D4semidirect
32. C22×Dic173D4semidirect
46. C22⋊Dic173D4semidirect
48. C4×Dic173D4semidirect
127. C173#2200semidirect
131. B(4,𝔽2)D173semidirect

Maximal subgroups

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

subgroupindex
#260122302
#260116812
#260116592
#260112832
#260112802
#260112782
B(4,𝔽2)×C1732
#2200173