Finite Group Databasebeta

← #36352586

Subgroups — #36352586

Order 22144.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012116C211yesmaximal, characteristic
2#26011826C211yesmaximal, characteristic, complemented
3#26011426C211yesmaximal, characteristic
4#26011229C211yesmaximal, characteristic, complemented
5#26011228C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic
7#484×C173C211yesmaximal, characteristic, complemented
8#19149659—21no—
9#19149658C2211yescharacteristic
10#19149657—21no—
11#19149657—21no—
12#19149656—21no—
13#19149655—21no—
14C2×#14583805—21no—
15C2×#14583803—21no—
16C2×#14583803C2211yescharacteristic
17C22⋊C4×D173—21no—
18#19149612—21no—
19#19149602—21no—
20C22⋊#13063436C2211yescharacteristic
21#19149586C2211yescharacteristic
22C22⋊Q8×C173C2211yescharacteristic
23C2×C4⋊C4×C173C2211yescharacteristic
24#107×C173—21no—
25#103×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic
30C22×Dic173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D411yescharacteristic
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—42no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—42no—
47C2×Q8×C173D411yescharacteristic
48C22×C4×C173—21no—
49C22×C4×C173C2311yescharacteristic
50C2×C8×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173D411yescharacteristic
55C22⋊C4×C173—21no—
56C22×D173—21no—
57C22×D173—21no—
58C22×D173—41no—
59C22×D173—41no—
60C22×D173Q1612yes—
61Dic346—41no—
62Dic346—81no—
63C2×Dic173—22no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C4×D173—81no—
71C4×D173—81no—
72#13063436—41no—
73C23×C173C2×D411yescharacteristic
74Q8×C173—41no—
75C2×C4×C173—21no—
76C2×C4×C173—21no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C8×C173—41no—
84C22×C173—21no—
85C22×C173—21no—
86C22×C173#12011yescharacteristic
87C22×C173C2×Q1611yescharacteristic
88C22×C173#13711yescharacteristic
89C2×D173—22no—
90C2×D173—41no—
91C2×D173—41no—
92C2×D173—42no—
93C2×D173—81no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—41no—
97C4×C173—41no—
98C4×C173—81no—
99Dic173—42no—
100Dic173—41no—
101Dic173—41no—
102Dic173—41no—
103C2×C173—21no—
104C2×C173—21no—
105C2×C173#45111yescharacteristic
106C2×C173#45311yescharacteristic
107C2×C173#45811yescharacteristic
108D173—42no—
109C173#161411yescharacteristic, complemented, Sylow
110#1614—1731nomaximal, Sylow
111#543—1731no—
112#484—1731no—
113#484D17311yescharacteristic, complemented
114#399—1731no—
115#328—1731no—
116#328—1731no—
117#323—1731no—
118C22⋊Q8—3461no—
119C22⋊Q8—3461no—
120C22⋊Q8—3461no—
121C22⋊Q8—3461no—
122C22⋊Q8—3461no—
123C22⋊Q8—1731no—
124C22⋊Q8C2×D17311yescharacteristic
125C2×C4⋊C4—3461no—
126C2×C4⋊C4C2×D17311yescharacteristic
127C2×C22⋊C4—3461no—
128C2×C22⋊C4—3461no—
129C2×C22⋊C4—1731no—
130#107—21no—
131#107—3461no—
132#103—21no—
133#103—3461no—
134C22⋊C8—1731no—
135C22⋊C8C2×D17311yescharacteristic
136#95—1731no—
137C24—1731no—
138C2×Q8—3461no—
139C2×Q8—1731no—
140C2×Q8C22⋊D17311yescharacteristic
141C22×C4—21no—
142C22×C4—3461no—
143C22×C4—3461no—
144C22×C4—3461no—
145C22×C4—1731no—
146C22×C4C22×D17311yescharacteristic
147C2×C8—21no—
148C2×C8—3461no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—1731no—
157C4⋊C4—6922no—
158C4⋊C4C22⋊D17311yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—6922no—
166C23—3461no—
167C23—3461no—
168C23—1732no—
169C23—6921no—
170C23—6921no—
171C23D4×D17311yescharacteristic
172Q8—41no—
173Q8—13841no—
174Q8—6921no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—3462no—
182C2×C4—3461no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4—13841no—
186C2×C4—13841no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4D4×D17311yescharacteristic
191C2×C4C2×C22⋊D17311yescharacteristic
192C8—41no—
193C8—6921no—
194C22—21no—
195C22—21no—
196C22—3462no—
197C22—13841no—
198C22—6922no—
199C22—6921no—
200C22—6921no—
201C22#1914964711yescharacteristic
202C22Q16×D17311yescharacteristic
203C22#1914971111yescharacteristic
204C4—21no—
205C4—21no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—6922no—
210C4—6921no—
211C4—6921no—
212C4—6921no—
213C2—21no—
214C2—21no—
215C2—6922no—
216C2#2601164511yescharacteristic
217C2#2601165711yescharacteristic
218C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#484×C173C2
socleC22×C173#120
prime radicalC23×D173D4
hypercenter#484D173
hypocenterC173#1614
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #26011826C2semidirect
4. #26011229C2semidirect
5. #26011228C2semidirect
7. #484×C173C2semidirect
109. C173#1614semidirect
113. #484D173semidirect

Maximal subgroups

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

subgroupindex
#260121162
#260118262
#260114262
#260112292
#260112282
#260112142
#484×C1732
#1614173