Finite Group Databasebeta

← #36352614

Subgroups — #36352614

Order 22144.

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

normal subgroups only

This group has 240 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#26012081C211yesmaximal, characteristic
2#26011824C211yesmaximal, characteristic, complemented
3#26011810C211yesmaximal, characteristic, complemented
4#26011428C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8#19149674—21no—
9#19149668—21no—
10#19149664—21no—
11#19149654—21no—
12#19149653C2211yescharacteristic
13#19149649—21no—
14#19149639—21no—
15C2×C4⋊Dic173—21no—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×#14583803—21no—
18C2×#14583803C2211yescharacteristic
19#19149612—21no—
20#19149611—21no—
21#19149603—21no—
22#19149601—21no—
23C22⋊#13063436C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C22×Dic173D411yescharacteristic
34C2×C4×D173—21no—
35C2×#13063436—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173—42no—
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—42no—
48C4×Dic173—21no—
49C2×Q8×C173D411yescharacteristic
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C2×C8×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173D411yescharacteristic
55C22⋊C4×C173—21no—
56C22×D173—21no—
57C22×D173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173—41no—
61C22×D173—41no—
62D692—41no—
63D692—81no—
64D692—81no—
65C22⋊D173—41no—
66C22⋊D173—41no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C4×D173—81no—
74#13063436—41no—
75C23×C173C2×D411yescharacteristic
76Q8×C173—41no—
77C2×C4×C173—22no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173—41no—
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C8×C173—41no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173#12011yescharacteristic
89C22×C173#13511yescharacteristic
90C22×C173#13711yescharacteristic
91C2×D173—42no—
92C2×D173—41no—
93C2×D173—41no—
94C2×D173—41no—
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—41no—
98C2×D173—81no—
99C4×C173—21no—
100C4×C173—21no—
101C4×C173—42no—
102C4×C173—41no—
103C4×C173—41no—
104Dic173—41no—
105Dic173—41no—
106Dic173—81no—
107C2×C173—21no—
108C2×C173—21no—
109C2×C173#45211yescharacteristic
110C2×C173#45311yescharacteristic
111C2×C173#45811yescharacteristic
112D173—81no—
113D173—81no—
114C173#161811yescharacteristic, complemented, Sylow
115#1618—1731nomaximal, Sylow
116#538—1731no—
117#484—1731no—
118#481—1731no—
119#399—1731no—
120#328D17311yescharacteristic, complemented
121#327—1731no—
122#323—1731no—
123#126—3461no—
124#124—3461no—
125#123—3461no—
126C22⋊Q8C2×D17311yescharacteristic
127#121—3461no—
128#121—1731no—
129#120—3461no—
130C4×D4—3461no—
131C2×C4⋊C4—3461no—
132C2×C4⋊C4—1731no—
133C2×C22⋊C4—3461no—
134C2×C22⋊C4—1731no—
135#107—3461no—
136#106—3461no—
137#103—3461no—
138#102—3461no—
139C22⋊C8—1731no—
140C22⋊C8C2×D17311yescharacteristic
141#95C2×D17311yescharacteristic
142C24—1731no—
143C2×Q8D69211yescharacteristic
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—3461no—
147C2×D4—1731no—
148C22×C4—21no—
149C22×C4—3461no—
150C22×C4—3461no—
151C22×C4—1731no—
152C22×C4C22×D17311yescharacteristic
153C2×C8—21no—
154C2×C8—3461no—
155C4⋊C4—21no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—3461no—
159C4⋊C4—1731no—
160C4⋊C4—6922no—
161C4⋊C4D69211yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—6922no—
170C42—3461no—
171C23—3461no—
172C23—3461no—
173C23—3461no—
174C23—3461no—
175C23—6921no—
176C23—6921no—
177C23D4×D17311yescharacteristic
178Q8—41no—
179D4—13841no—
180D4—13841no—
181D4—6921no—
182D4—6921no—
183D4—6921no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—13841no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4—6921no—
197C2×C4D4×D17311yescharacteristic
198C2×C4C2×D69211yescharacteristic
199C8—41no—
200C8—6921no—
201C22—21no—
202C22—21no—
203C22—13841no—
204C22—6922no—
205C22—6921no—
206C22—6921no—
207C22—6921no—
208C22—6921no—
209C22—6921no—
210C22—6921no—
211C22#1914964911yescharacteristic
212C22#1914970011yescharacteristic
213C22#1914971111yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—13841no—
220C4—6921no—
221C4—6921no—
222C2—21no—
223C2—21no—
224C2—13841no—
225C2—13841no—
226C2#2601165611yescharacteristic
227C2#2601166011yescharacteristic
228C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radical#26012081C2
hypercenter#328D173
hypocenterC173#1618
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26011824C2semidirect
3. #26011810C2semidirect
4. #26011428C2semidirect
5. #26011224C2semidirect
6. #26011214C2semidirect
7. #328×C173C2semidirect
114. C173#1618semidirect
120. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120812
#260118242
#260118102
#260114282
#260112242
#260112142
#328×C1732
#1618173