Finite Group Databasebeta

← #36355562

Subgroups — #36355562

Order 22144.

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

normal subgroups only

This group has 270 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#26012234C211yesmaximal, characteristic, complemented
2#26011678C211yesmaximal, characteristic, complemented
3#26011669C211yesmaximal, characteristic, complemented
4#26011285C211yesmaximal, characteristic, complemented
5#26011278C211yesmaximal, characteristic, complemented
6#26011277C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C173C211yesmaximal, characteristic, complemented
8#19149709—21no—
9#19149666—21no—
10#19149666C2211yescharacteristic
11#19149665—21no—
12#19149665—21no—
13#19149665—21no—
14#19149648—21no—
15#19149647—21no—
16#19149647—21no—
17#19149647C2211yescharacteristic
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
27C23×D173D411yescharacteristic, complemented
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic, complemented
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—41no—
34#14583816—41no—
35#14583814—41no—
36#14583810—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173D411yescharacteristic, complemented
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—41no—
47C4×Dic173—21no—
48C4×Dic173D411yescharacteristic, complemented
49C24×C173D411yescharacteristic, complemented
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, complemented
62C22×D173—21no—
63C22×D173—41no—
64C22×D173—81no—
65Dic346—41no—
66Dic346—81no—
67C22⋊D173—81no—
68C22⋊D173—81no—
69C22⋊D173—81no—
70C22⋊D173—82no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—41no—
75#13063436—41no—
76C23×C173—21no—
77C23×C173—21no—
78C23×C173—21no—
79C23×C173—41no—
80C23×C173—81no—
81C23×C173C2×D411yescharacteristic
82C23×C173C2×D411yescharacteristic
83Q8×C173—21no—
84D4×C173—21no—
85D4×C173—41no—
86D4×C173—41no—
87D4×C173—41no—
88D4×C173—41no—
89D4×C173—81no—
90D4×C173—82no—
91C2×C4×C173—21no—
92C2×C4×C173—21no—
93C2×C4×C173—21no—
94C2×C4×C173—41no—
95C2×C4×C173—41no—
96C2×C4×C173C2×D411yescharacteristic
97C22×C173—21no—
98C22×C173—21no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173—42no—
104C22×C173—41no—
105C22×C173—41no—
106C22×C173—81no—
107C22×C173—81no—
108C22×C173—81no—
109C22×C173#12011yescharacteristic
110C2×D173—41no—
111C2×D173—41no—
112C2×D173—42no—
113C2×D173—41no—
114C2×D173—81no—
115C4×C173—21no—
116C4×C173—41no—
117C4×C173—41no—
118C4×C173—81no—
119Dic173—41no—
120Dic173—41no—
121Dic173—41no—
122Dic173—81no—
123C2×C173—21no—
124C2×C173—41no—
125C2×C173—41no—
126C2×C173—81no—
127C2×C173—81no—
128C2×C173B(4,𝔽2)11yescharacteristic
129D173—81no—
130C173#219911yescharacteristic, complemented, Sylow
131#2199—1731nomaximal, Sylow
132H(3,𝔽4)—1731no—
133B(4,𝔽2)—1731no—
134B(4,𝔽2)D17311yescharacteristic, complemented
135#455—1731no—
136#354—1731no—
137AΣL(1,𝔽16)—1731no—
138AΣL(1,𝔽16)—1731no—
139H(5,𝔽2)C2×D17311yescharacteristic
140#137—3461no—
141#124—3461no—
142#124—3461no—
143#124—3461no—
144#124—3461no—
145#124—1731no—
146#120—21no—
147#120—3461no—
148#120—3461no—
149#120—3461no—
150#120—1731no—
151#120C2×D17311yescharacteristic
152#104—3461no—
153#100—1731no—
154#99—21no—
155#99—3461no—
156#99—1731no—
157#99C2×D17311yescharacteristic
158C24—1731no—
159C24C22⋊D17311yescharacteristic, complemented
160#41—21no—
161#41—41no—
162C2×Q8—3461no—
163C2×Q8—1731no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—3461no—
172C2×D4—6921no—
173C2×D4C22×D17311yescharacteristic
174Q16—6921no—
175SD16—6921no—
176M16—3461no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—41no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—3461no—
185C22⋊C4—3461no—
186C22⋊C4—3461no—
187C22⋊C4—3461no—
188C22⋊C4—1731no—
189C22⋊C4—6921no—
190C22⋊C4C22⋊D17311yescharacteristic, complemented
191C42—3461no—
192C42—1731no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23—3461no—
199C23—13841no—
200C23—6921no—
201C23D4×D17311yescharacteristic
202C23C2×C22⋊D17311yescharacteristic
203Q8—21no—
204Q8—13841no—
205Q8—6921no—
206D4—21no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—81no—
212D4—82no—
213D4—13842no—
214D4—13841no—
215D4—13841no—
216D4—13841no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3461no—
223C2×C4—3461no—
224C2×C4—3461no—
225C2×C4—6921no—
226C2×C4D4×D17311yescharacteristic
227C8—6921no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—13841no—
241C22—6921no—
242C22—6921no—
243C22—6922no—
244C22—6921no—
245C22#1914964711yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—13841no—
251C4—6921no—
252C4—6921no—
253C4—6921no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—81no—
259C2—13841no—
260C2#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#2199
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. #26012234C2semidirect
2. #26011678C2semidirect
3. #26011669C2semidirect
4. #26011285C2semidirect
5. #26011278C2semidirect
6. #26011277C2semidirect
7. B(4,𝔽2)×C173C2semidirect
27. C23×D173D4semidirect
29. C2×Dic346D4semidirect
42. C22⋊Dic173D4semidirect
48. C4×Dic173D4semidirect
49. C24×C173D4semidirect
61. C22⋊C4×C173D4semidirect
130. C173#2199semidirect
134. B(4,𝔽2)D173semidirect
159. C24C22⋊D173semidirect
190. C22⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122342
#260116782
#260116692
#260112852
#260112782
#260112772
B(4,𝔽2)×C1732
#2199173