Finite Group Databasebeta

← #36355552

Subgroups — #36355552

Order 22144.

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

normal subgroups only

This group has 258 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#26012171C211yesmaximal, characteristic, complemented
2#26011681C211yesmaximal, characteristic, complemented
3#26011678C211yesmaximal, characteristic
4#26011286C211yesmaximal, characteristic, complemented
5#26011282C211yesmaximal, characteristic, complemented
6#26011279C211yesmaximal, characteristic, complemented
7#455×C173C211yesmaximal, characteristic, complemented
8#19149672—21no—
9#19149661—21no—
10#19149660—21no—
11#19149660C2211yescharacteristic
12#19149655—21no—
13#19149647—21no—
14#19149647C2211yescharacteristic
15#19149639—21no—
16#19149636—21no—
17C22⋊C4×D173—21no—
18#19149594—21no—
19#19149594—21no—
20#19149594C2211yescharacteristic
21#19149594C2211yescharacteristic
22H(5,𝔽2)×C173C2211yescharacteristic
23#137×C173—21no—
24#124×C173C2211yescharacteristic
25#104×C173—21no—
26#100×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×C22⋊D173—21no—
29C2×C22⋊D173—41no—
30C22×Dic173D411yescharacteristic, complemented
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33C4⋊Dic173—21no—
34#14583805—21no—
35#14583805—21no—
36#14583805—21no—
37#14583805—41no—
38C22⋊Dic173—21no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—41no—
42C22⋊Dic173D411yescharacteristic
43C22⋊Dic173D411yescharacteristic, complemented
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—41no—
48C4×Dic173—21no—
49#41×C173—21no—
50#41×C173—41no—
51C2×Q8×C173D411yescharacteristic, complemented
52C2×D4×C173—21no—
53C2×D4×C173—21no—
54C2×D4×C173—41no—
55C2×D4×C173C2311yescharacteristic
56Q16×C173—41no—
57SD16×C173—41no—
58M16×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C42×C173D411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—42no—
64C22×D173—41no—
65C22⋊D173—42no—
66C22⋊D173—82no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C4×D173—42no—
74C4×D173—81no—
75C23×C173—21no—
76C23×C173—21no—
77C23×C173C2×D411yescharacteristic
78C23×C173C2×D411yescharacteristic
79Q8×C173—21no—
80Q8×C173—41no—
81D4×C173—21no—
82D4×C173—41no—
83D4×C173—41no—
84D4×C173—41no—
85D4×C173—41no—
86C2×C4×C173—21no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—41no—
91C2×C4×C173C2×D411yescharacteristic
92C8×C173—41no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—81no—
99C22×C173—81no—
100C22×C173#12011yescharacteristic
101C2×D173—22no—
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—42no—
105C2×D173—81no—
106C4×C173—21no—
107C4×C173—41no—
108C4×C173—41no—
109C4×C173—41no—
110Dic173—41no—
111Dic173—41no—
112Dic173—41no—
113Dic173—81no—
114Dic173—81no—
115C2×C173—21no—
116C2×C173—41no—
117C2×C173—41no—
118C2×C173—81no—
119C2×C173B(4,𝔽2)11yescharacteristic
120D173—42no—
121C173#219711yescharacteristic, complemented, Sylow
122#2197—1731nomaximal, Sylow
123#551—1731no—
124#458—1731no—
125B(4,𝔽2)—1731no—
126#455D17311yescharacteristic, complemented
127#354—1731no—
128#352—1731no—
129AΣL(1,𝔽16)—1731no—
130H(5,𝔽2)C2×D17311yescharacteristic
131#137—21no—
132#126—3461no—
133#124C2×D17311yescharacteristic
134#123—3461no—
135#123—3461no—
136#123—1731no—
137C22⋊Q8—3461no—
138#120—3461no—
139#120—1731no—
140C4×D4—3461no—
141#117—3461no—
142C2×C22⋊C4—3461no—
143#104—21no—
144#100C2×D17311yescharacteristic
145#99—3461no—
146#99—3461no—
147#99—1731no—
148#99—1731no—
149C24—1731no—
150#41—21no—
151#41—41no—
152C2×Q8C22⋊D17311yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—3461no—
157C2×D4—6921no—
158C2×D4C22×D17311yescharacteristic
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4—1731no—
162Q16—41no—
163SD16—41no—
164M16—21no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—6921no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—1731no—
179C22⋊C4—1731no—
180C22⋊C4—6921no—
181C22⋊C4—6921no—
182C42—3461no—
183C42C22⋊D17311yescharacteristic, complemented
184C23—21no—
185C23—21no—
186C23—3461no—
187C23—6921no—
188C23—6922no—
189C23D4×D17311yescharacteristic
190C23D4×D17311yescharacteristic
191Q8—21no—
192Q8—41no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—13842no—
199D4—6922no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3461no—
208C2×C4—13841no—
209C2×C4—6922no—
210C2×C4—6921no—
211C2×C4—6921no—
212C2×C4—6921no—
213C2×C4C2×C22⋊D17311yescharacteristic
214C8—41no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—3462no—
223C22—13841no—
224C22—6921no—
225C22—6922no—
226C22—6921no—
227C22#1914964711yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—13841no—
233C4—13841no—
234C4—6921no—
235C4—6921no—
236C4—6921no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—81no—
241C2—6922no—
242C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#455×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011678C2
hypercenter#455D173
hypocenterC173#2197
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #26012171C2semidirect
2. #26011681C2semidirect
4. #26011286C2semidirect
5. #26011282C2semidirect
6. #26011279C2semidirect
7. #455×C173C2semidirect
30. C22×Dic173D4semidirect
43. C22⋊Dic173D4semidirect
51. C2×Q8×C173D4semidirect
61. C42×C173D4semidirect
121. C173#2197semidirect
126. #455D173semidirect
152. C2×Q8C22⋊D173semidirect
183. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260121712
#260116812
#260116782
#260112862
#260112822
#260112792
#455×C1732
#2197173