Finite Group Databasebeta

← #36352854

Subgroups — #36352854

Order 22144.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012235C211yesmaximal, characteristic, complemented
2#26011878C211yesmaximal, characteristic, complemented
3#26011855C211yesmaximal, characteristic, complemented
4#26011232C211yesmaximal, characteristic, complemented
5#26011231C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic, complemented
7#496×C173C211yesmaximal, characteristic, complemented
8C2×#14583814—21no—
9C2×#14583811—21no—
10#19149671C2211yescharacteristic
11#19149667C2211yescharacteristic
12#19149663—21no—
13#19149661—21no—
14#19149656—21no—
15#19149655—21no—
16#19149652—21no—
17#19149651—21no—
18#19149600—21no—
19#19149600—21no—
20#19149590C2211yescharacteristic
21#19149590C2211yescharacteristic
22C2×SD16×C173—21no—
23C2×D8×C173—21no—
24#127×C173C2211yescharacteristic
25#124×C173C2211yescharacteristic
26#97×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C22×Dic173—21no—
32C2×C4×D173—21no—
33#14583814—42no—
34#14583811—42no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173D411yescharacteristic
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43C22⋊Dic173—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47C2×Q8×C173D411yescharacteristic
48C2×D4×C173—21no—
49C2×D4×C173—21no—
50C2×D4×C173—21no—
51C2×D4×C173D411yescharacteristic
52SD16×C173—42no—
53D8×C173—42no—
54C2×C8×C173—21no—
55C22⋊C4×C173—21no—
56C22⋊C4×C173—21no—
57C42×C173C2311yescharacteristic
58C22×D173—21no—
59Dic346—41no—
60D692—41no—
61C22⋊D173—81no—
62C22⋊D173—81no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—81no—
67C4×D173—81no—
68#13063436—22no—
69#13063436—41no—
70C23×C173—21no—
71C23×C173—41no—
72Q8×C173—41no—
73D4×C173—41no—
74D4×C173—41no—
75D4×C173—41no—
76D4×C173—41no—
77D4×C173—41no—
78D4×C173—42no—
79C2×C4×C173—21no—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C8×C173—22no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—41no—
87C22×C173—81no—
88C22×C173—81no—
89C22×C173—81no—
90C22×C173#12011yescharacteristic
91C2×D173—41no—
92C2×D173—41no—
93C2×D173—41no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—21no—
97C4×C173—41no—
98Dic173—41no—
99Dic173—41no—
100Dic173—41no—
101C2×C173—81no—
102C2×C173—161no—
103C2×C173#45311yescharacteristic
104C2×C173#45311yescharacteristic
105C2×C173#45411yescharacteristic
106D173—81no—
107C173#165611yescharacteristic, complemented, Sylow
108#1656—1731nomaximal, Sylow
109#562—1731no—
110#496—1731no—
111#496D17311yescharacteristic, complemented
112#490—1731no—
113#330—1731no—
114#329—1731no—
115#329—1731no—
116C2×SD16—21no—
117C2×SD16—3461no—
118C2×D8—21no—
119C2×D8—3461no—
120#127C2×D17311yescharacteristic
121#125—1731no—
122#124—1731no—
123#124C2×D17311yescharacteristic
124#123—3461no—
125#123—3461no—
126C22⋊Q8—3461no—
127C22⋊Q8—3461no—
128#121—3461no—
129#121—3461no—
130#102—3461no—
131#102—3461no—
132#97—1731no—
133#97—1731no—
134#97C2×D17311yescharacteristic
135C2×Q8—1731no—
136C2×Q8C22⋊D17311yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—1731no—
143C2×D4C22⋊D17311yescharacteristic
144C22×C4—3461no—
145C22×C4—3461no—
146SD16—42no—
147SD16—6922no—
148D8—42no—
149D8—6922no—
150C2×C8—21no—
151C2×C8—3461no—
152C2×C8—3461no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—1731no—
158C4⋊C4—1731no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C42C22×D17311yescharacteristic
166C23—21no—
167C23—41no—
168C23—3461no—
169Q8—41no—
170Q8—6921no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—42no—
177D4—13841no—
178D4—13841no—
179D4—6921no—
180C2×C4—21no—
181C2×C4—3461no—
182C2×C4—3461no—
183C2×C4—3461no—
184C2×C4—13841no—
185C2×C4—13841no—
186C2×C4D4×D17311yescharacteristic
187C2×C4D4×D17311yescharacteristic
188C2×C4C2×C22⋊D17311yescharacteristic
189C8—22no—
190C8—3462no—
191C8—6921no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—6921no—
199C22—6921no—
200C22—6921no—
201C22#1914964711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—6921no—
207C4—6921no—
208C4—6921no—
209C2—81no—
210C2—161no—
211C2—13841no—
212C2#2601165711yescharacteristic
213C2#2601165811yescharacteristic
214C2#2601166211yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#496×C173C2
socleC22×C173#120
hypercenter#496D173
hypocenterC173#1656
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26012235C2semidirect
2. #26011878C2semidirect
3. #26011855C2semidirect
4. #26011232C2semidirect
5. #26011231C2semidirect
6. #26011230C2semidirect
7. #496×C173C2semidirect
107. C173#1656semidirect
111. #496D173semidirect

Maximal subgroups

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

subgroupindex
#260122352
#260118782
#260118552
#260112322
#260112312
#260112302
#496×C1732
#1656173