Finite Group Databasebeta

← #36355551

Subgroups — #36355551

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#26012172C211yesmaximal, characteristic, complemented
2#26011681C211yesmaximal, characteristic, complemented
3#26011669C211yesmaximal, characteristic, complemented
4#26011285C211yesmaximal, characteristic, complemented
5#26011280C211yesmaximal, characteristic, complemented
6#26011278C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C173C211yesmaximal, characteristic
8#19149709—21no—
9#19149672—21no—
10#19149666C2211yescharacteristic
11#19149660—21no—
12#19149660—21no—
13#19149660C2211yescharacteristic
14#19149656—21no—
15D4×Dic173—21no—
16#19149635—21no—
17C2×C22⋊Dic173—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×Dic346D411yescharacteristic, complemented
28C22×Dic173—21no—
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic, complemented
31#14583816—41no—
32#14583814—41no—
33#14583810—21no—
34C4⋊Dic173—21no—
35#14583805—21no—
36#14583805—21no—
37#14583805—21no—
38#14583805—41no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—41no—
46C22⋊Dic173D411yescharacteristic, complemented
47C4×Dic173—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
62Dic346—41no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—42no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—81no—
71#13063436—41no—
72C23×C173—21no—
73C23×C173—21no—
74C23×C173—21no—
75C23×C173—41no—
76C23×C173—42no—
77C23×C173C2×D411yescharacteristic
78C23×C173C2×D411yescharacteristic
79Q8×C173—21no—
80D4×C173—21no—
81D4×C173—41no—
82D4×C173—42no—
83D4×C173—41no—
84D4×C173—41no—
85D4×C173—41no—
86D4×C173—82no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—41no—
91C2×C4×C173—41no—
92C2×C4×C173C2×D411yescharacteristic
93C22×C173—22no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—42no—
101C22×C173—41no—
102C22×C173—81no—
103C22×C173—81no—
104C22×C173—81no—
105C22×C173#12011yescharacteristic
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×C173—42no—
119C2×C173—81no—
120C2×C173B(4,𝔽2)11yescharacteristic
121C173#219711yescharacteristic, complemented, Sylow
122#2197—1731nomaximal, Sylow
123#551—1731no—
124#458—1731no—
125B(4,𝔽2)D17311yescharacteristic
126#455—1731no—
127#354—1731no—
128#352—1731no—
129AΣL(1,𝔽16)—1731no—
130H(5,𝔽2)C2×D17311yescharacteristic
131#137—3461no—
132#126—3461no—
133#124—1731no—
134#123—3461no—
135#123—3461no—
136#123—1731no—
137C22⋊Q8—3461no—
138#120—21no—
139#120C2×D17311yescharacteristic
140C4×D4—3461no—
141#117—3461no—
142C2×C22⋊C4—3461no—
143#104—3461no—
144#100—1731no—
145#99—21no—
146#99—3461no—
147#99—1731no—
148#99C2×D17311yescharacteristic
149C24C22⋊D17311yescharacteristic
150#41—21no—
151#41—41no—
152C2×Q8—1731no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C22×D17311yescharacteristic
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4—1731no—
162Q16—6921no—
163SD16—6921no—
164M16—3461no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—6921no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—41no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—1731no—
180C22⋊C4—6921no—
181C22⋊C4C22⋊D17311yescharacteristic
182C42—3461no—
183C42—1731no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23D4×D17311yescharacteristic
190C23C2×C22⋊D17311yescharacteristic
191Q8—21no—
192Q8—6921no—
193D4—21no—
194D4—41no—
195D4—42no—
196D4—41no—
197D4—41no—
198D4—41no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
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—6921no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#1914964711yescharacteristic
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—42no—
241C2—81no—
242C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting, prime radicalB(4,𝔽2)×C173C2
socleC2×C173B(4,𝔽2)
hypercenterB(4,𝔽2)D173
hypocenterC173#2197
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. #26012172C2semidirect
2. #26011681C2semidirect
3. #26011669C2semidirect
4. #26011285C2semidirect
5. #26011280C2semidirect
6. #26011278C2semidirect
27. C2×Dic346D4semidirect
30. C22×Dic173D4semidirect
46. C22⋊Dic173D4semidirect
48. C4×Dic173D4semidirect
121. C173#2197semidirect

Maximal subgroups

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

subgroupindex
#260121722
#260116812
#260116692
#260112852
#260112802
#260112782
B(4,𝔽2)×C1732
#2197173