Finite Group Databasebeta

← #36355554

Subgroups — #36355554

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#26012169C211yesmaximal, characteristic, complemented
2#26011680C211yesmaximal, characteristic, complemented
3#26011679C211yesmaximal, characteristic
4#26011671C211yesmaximal, characteristic, complemented
5#26011285C211yesmaximal, characteristic, complemented
6#26011279C211yesmaximal, characteristic, complemented
7#352×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic
9#19149712—21no—
10#19149672—21no—
11#19149666C2211yescharacteristic
12#19149662—21no—
13#19149661—21no—
14#19149657—21no—
15#19149649—21no—
16#19149647C2211yescharacteristic
17C4×C22⋊D173—21no—
18#19149637—21no—
19C22⋊C4×D173—21no—
20#19149605—21no—
21#19149594C2211yescharacteristic
22#19149593—21no—
23#19149593—21no—
24#123×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28#14583826—21no—
29#14583825—41no—
30C2×Dic346D411yescharacteristic, complemented
31C2×D692—41no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35D4×D173—41no—
36C2×C4×D173—21no—
37C2×C4×D173—21no—
38Dic692—41no—
39#14583815—41no—
40C4⋊Dic173—21no—
41#14583805—21no—
42#14583805—21no—
43#14583805—41no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173D411yescharacteristic
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51#14583803—41no—
52#14583803—41no—
53C4×Dic173—21no—
54C4×Dic173D411yescharacteristic, complemented
55C2×D4×C173C2311yescharacteristic
56C22×C4×C173D411yescharacteristic, complemented
57M16×C173—21no—
58C4⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173D411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—42no—
66C22×D173—41no—
67Dic346—21no—
68Dic346—41no—
69D692—21no—
70D692—82no—
71C22⋊D173—42no—
72C22⋊D173—41no—
73C22⋊D173—41no—
74C22⋊D173—41no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—41no—
81C4×D173—41no—
82C4×D173—42no—
83C4×D173—81no—
84C23×C173C2×D411yescharacteristic
85C23×C173C2×D411yescharacteristic
86D4×C173—41no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—41no—
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#12011yescharacteristic
98C2×D173—22no—
99C2×D173—41no—
100C2×D173—41no—
101C2×D173—42no—
102C2×D173—41no—
103C2×D173—81no—
104C2×D173—81no—
105C2×D173—81no—
106C4×C173—21no—
107C4×C173—41no—
108C4×C173—41no—
109C4×C173—81no—
110Dic173—41no—
111Dic173—41no—
112Dic173—41no—
113Dic173—41no—
114Dic173—81no—
115C2×C173—21no—
116C2×C173—41no—
117C2×C173—41no—
118C2×C173B(4,𝔽2)11yescharacteristic
119D173—42no—
120D173—81no—
121C173#219711yescharacteristic, complemented, Sylow
122#2197—1731nomaximal, Sylow
123#551—1731no—
124#458—1731no—
125B(4,𝔽2)—1731no—
126#455—1731no—
127#354—1731no—
128#352D17311yescharacteristic, complemented
129AΣL(1,𝔽16)—1731no—
130H(5,𝔽2)—1731no—
131#137—3461no—
132#126—3461no—
133#124—1731no—
134#123—3461no—
135#123—3461no—
136#123C2×D17311yescharacteristic
137C22⋊Q8—3461no—
138#120—3461no—
139#120—1731no—
140C4×D4—3461no—
141#117—3461no—
142C2×C22⋊C4—3461no—
143#104—3461no—
144#100C2×D17311yescharacteristic
145#99—3461no—
146#99—3461no—
147#99—1731no—
148#99C2×D17311yescharacteristic
149C24—1731no—
150#41—3461no—
151#41—6921no—
152C2×Q8—1731no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—6921no—
157C2×D4—6921no—
158C2×D4C22×D17311yescharacteristic
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4D69211yescharacteristic, complemented
162Q16—6921no—
163SD16—6921no—
164M16—21no—
165C4⋊C4—21no—
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—6921no—
180C22⋊C4—6921no—
181C22⋊C4D69211yescharacteristic, complemented
182C42—3461no—
183C42—1731no—
184C23—3461no—
185C23—3461no—
186C23—3461no—
187C23—6921no—
188C23—6922no—
189C23D4×D17311yescharacteristic
190C23C2×D69211yescharacteristic
191Q8—3461no—
192Q8—6921no—
193D4—41no—
194D4—3461no—
195D4—13842no—
196D4—6921no—
197D4—6921no—
198D4—6922no—
199D4—6921no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3461no—
208C2×C4—3461no—
209C2×C4—13841no—
210C2×C4—6921no—
211C2×C4—6922no—
212C2×C4—6921no—
213C2×C4D4×D17311yescharacteristic
214C8—41no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—3462no—
220C22—13841no—
221C22—13841no—
222C22—13841no—
223C22—6921no—
224C22—6922no—
225C22—6921no—
226C22—6921no—
227C22#1914964911yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—13841no—
233C4—6921no—
234C4—6921no—
235C4—6921no—
236C4—6921no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—13841no—
241C2—6922no—
242C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#352×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011679C2
hypercenter#352D173
hypocenterC173#2197
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #26012169C2semidirect
2. #26011680C2semidirect
4. #26011671C2semidirect
5. #26011285C2semidirect
6. #26011279C2semidirect
7. #352×C173C2semidirect
30. C2×Dic346D4semidirect
54. C4×Dic173D4semidirect
56. C22×C4×C173D4semidirect
61. C22⋊C4×C173D4semidirect
121. C173#2197semidirect
128. #352D173semidirect
161. C22×C4D692semidirect
181. C22⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260121692
#260116802
#260116792
#260116712
#260112852
#260112792
#352×C1732
#2197173