Finite Group Databasebeta

← #36352541

Subgroups — #36352541

Order 22144.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012079C211yesmaximal, characteristic
2#26011647C211yesmaximal, characteristic, complemented
3#26011630C211yesmaximal, characteristic, complemented
4#26011227C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011215C211yesmaximal, characteristic, complemented
7#393×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—21no—
9C2×#14583813—21no—
10#19149672—21no—
11#19149665—21no—
12#19149661—21no—
13#19149658C2211yescharacteristic
14#19149653C2211yescharacteristic
15#19149652—21no—
16#19149648—21no—
17C4×C22⋊D173—21no—
18#19149603—21no—
19#19149603—21no—
20C22⋊#13063436C2211yescharacteristic
21C22⋊#13063436C2211yescharacteristic
22C22×Q8×C173C2211yescharacteristic
23C2×C22⋊C4×C173—21no—
24C2×C22⋊C4×C173C2211yescharacteristic
25#95×C173—21no—
26#95×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C4×D173—21no—
33#14583816—42no—
34#14583813—42no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C24×C173D411yescharacteristic
48C2×Q8×C173—21no—
49C2×Q8×C173—21no—
50C2×Q8×C173—42no—
51C22×C4×C173—21no—
52C22×C4×C173—21no—
53C22×C4×C173D411yescharacteristic
54C22×C4×C173C2311yescharacteristic
55C22⋊C4×C173—21no—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—42no—
58C22×D173—21no—
59Dic346—41no—
60D692—41no—
61C22⋊D173—41no—
62C22⋊D173—41no—
63C22⋊D173—81no—
64C22⋊D173—81no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C4×D173—81no—
69#13063436—41no—
70#13063436—41no—
71C23×C173—21no—
72C23×C173—21no—
73C23×C173—21no—
74C23×C173—41no—
75C23×C173—41no—
76C23×C173C2×D411yescharacteristic
77Q8×C173—22no—
78Q8×C173—22no—
79Q8×C173—41no—
80Q8×C173—41no—
81C2×C4×C173—22no—
82C2×C4×C173—21no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2×D411yescharacteristic
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—42no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—81no—
99C22×C173#12011yescharacteristic
100C22×C173#13511yescharacteristic
101C22×C173#13711yescharacteristic
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—41no—
105C4×C173—21no—
106C4×C173—21no—
107C4×C173—41no—
108C4×C173—42no—
109C4×C173—81no—
110Dic173—41no—
111Dic173—41no—
112Dic173—41no—
113C2×C173—21no—
114C2×C173—21no—
115C2×C173—81no—
116C2×C173#45211yescharacteristic
117C2×C173#45511yescharacteristic
118C2×C173B(4,𝔽2)11yescharacteristic
119D173—81no—
120C173#160611yescharacteristic, complemented, Sylow
121#1606—1731nomaximal, Sylow
122#538—1731no—
123#451—1731no—
124#448—1731no—
125#393D17311yescharacteristic, complemented
126#328—1731no—
127#327—1731no—
128#323—1731no—
129C22×Q8C2×D17311yescharacteristic
130C2×Q16—3461no—
131C2×SD16—3461no—
132#126—3461no—
133#124—3461no—
134#123—3461no—
135C22⋊Q8—1731no—
136#121—3461no—
137#121—1731no—
138#120—3461no—
139C4×D4—3461no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C2×D17311yescharacteristic
142#103—3461no—
143#103—3461no—
144C22⋊C8—1731no—
145C22⋊C8—1731no—
146#95—21no—
147#95C2×D17311yescharacteristic
148C24C22⋊D17311yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—1731no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—1731no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—3461no—
160C22×C4C22⋊D17311yescharacteristic
161C22×C4C22×D17311yescharacteristic
162Q16—6922no—
163SD16—6922no—
164C2×C8—3461no—
165C2×C8—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—1731no—
169C4⋊C4—1731no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—42no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C42—3461no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3461no—
185C23C2×C22⋊D17311yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—6921no—
191D4—13841no—
192D4—13841no—
193D4—6921no—
194D4—6921no—
195D4—6921no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3461no—
208C2×C4—13841no—
209C2×C4D4×D17311yescharacteristic
210C2×C4D4×D17311yescharacteristic
211C8—6921no—
212C8—6921no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—6921no—
221C22—6921no—
222C22—6921no—
223C22#1914964711yescharacteristic
224C22#1914969911yescharacteristic
225C22#1914971011yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—81no—
231C4—6921no—
232C4—6921no—
233C4—6921no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—13841no—
238C2#2601165111yescharacteristic
239C2#2601166811yescharacteristic
240C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#393×C173C2
socleC22×C173#120
prime radical#26012079C2
hypercenter#393D173
hypocenterC173#1606
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #26011647C2semidirect
3. #26011630C2semidirect
4. #26011227C2semidirect
5. #26011224C2semidirect
6. #26011215C2semidirect
7. #393×C173C2semidirect
120. C173#1606semidirect
125. #393D173semidirect

Maximal subgroups

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

subgroupindex
#260120792
#260116472
#260116302
#260112272
#260112242
#260112152
#393×C1732
#1606173