Finite Group Databasebeta

← #36355587

Subgroups — #36355587

Order 22144.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 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#26012203C211yesmaximal, characteristic, complemented
2#26011674C211yesmaximal, characteristic, complemented
3#26011664C211yesmaximal, characteristic, complemented
4#26011289C211yesmaximal, characteristic, complemented
5#26011288C211yesmaximal, characteristic, complemented
6#26011287C211yesmaximal, characteristic, complemented
7#454×C173C211yesmaximal, characteristic, complemented
8#19149709—21no—
9#19149704—21no—
10#19149677C2211yescharacteristic
11#19149668C2211yescharacteristic
12#19149665—21no—
13#19149652—21no—
14#19149651—21no—
15#19149639—21no—
16#19149636—21no—
17#19149635—21no—
18#19149606—21no—
19#19149606—21no—
20#19149598C2211yescharacteristic
21#19149598C2211yescharacteristic
222−1+4×C173C2211yescharacteristic
23AGL(1,ℤ/8)×C173—21no—
24#124×C173C2211yescharacteristic
25#104×C173—21no—
26#101×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C22×Dic173—21no—
32C2×C4×D173—21no—
33#14583816—41no—
34#14583814—41no—
35#14583813—41no—
36#14583811—41no—
37#14583810—21no—
38#14583810—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42C22⋊Dic173—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C4×Dic173D411yescharacteristic
48C4×Dic173D411yescharacteristic
49#41×C173—21no—
50#41×C173—21no—
51#41×C173—21no—
52#41×C173—41no—
53C2×Q8×C173—41no—
54C2×Q8×C173C2311yescharacteristic
55C2×D4×C173D411yescharacteristic
56SD16×C173—41no—
57D8×C173—41no—
58M16×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C42×C173D411yescharacteristic
62C22×D173—21no—
63Dic346—41no—
64D692—41no—
65C22⋊D173—42no—
66C22⋊D173—81no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—42no—
71C4×D173—42no—
72#13063436—41no—
73#13063436—41no—
74C23×C173—21no—
75Q8×C173—21no—
76Q8×C173—21no—
77Q8×C173—21no—
78Q8×C173—41no—
79D4×C173—21no—
80D4×C173—21no—
81D4×C173—21no—
82D4×C173—41no—
83D4×C173—41no—
84C2×C4×C173—21no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173C2×D411yescharacteristic
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D411yescharacteristic
91C8×C173—41no—
92C22×C173—41no—
93C22×C173—41no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173#12011yescharacteristic
97C2×D173—41no—
98C2×D173—41no—
99C2×D173—41no—
100C4×C173—21no—
101C4×C173—21no—
102C4×C173—21no—
103C4×C173—41no—
104C4×C173—41no—
105Dic173—22no—
106Dic173—41no—
107Dic173—41no—
108C2×C173—21no—
109C2×C173—81no—
110C2×C173—81no—
111C2×C173B(4,𝔽2)11yescharacteristic
112D173—81no—
113C173#220411yescharacteristic, complemented, Sylow
114#2204—1731nomaximal, Sylow
115#555—1731no—
116#456—1731no—
117#454—1731no—
118#454D17311yescharacteristic, complemented
119#356—1731no—
120#355—1731no—
121#355—1731no—
1222−1+4C2×D17311yescharacteristic
123#137—3461no—
124AGL(1,ℤ/8)—21no—
125AGL(1,ℤ/8)—3461no—
126C4⋊Q8—1731no—
127#124—3461no—
128#124—1731no—
129#124C2×D17311yescharacteristic
130#121—3461no—
131#121—3461no—
132C4×D4—3461no—
133#117—3461no—
134#117—3461no—
135#104—21no—
136#104—3461no—
137#104—3461no—
138#101—1731no—
139#101—1731no—
140#101C2×D17311yescharacteristic
141#41—21no—
142#41—21no—
143#41—21no—
144#41—41no—
145C2×Q8—41no—
146C2×Q8—1731no—
147C2×Q8C22×D17311yescharacteristic
148C2×D4—3461no—
149C2×D4—3461no—
150C2×D4—1731no—
151C2×D4C22⋊D17311yescharacteristic
152C22×C4—3461no—
153C22×C4—3461no—
154Q16—6921no—
155SD16—41no—
156SD16—6921no—
157SD16—6921no—
158D8—41no—
159D8—6921no—
160M16—21no—
161M16—3461no—
162M16—3461no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C42—3461no—
173C42—1731no—
174C42—1731no—
175C42C22⋊D17311yescharacteristic
176C23—21no—
177C23—3461no—
178Q8—21no—
179Q8—21no—
180Q8—21no—
181Q8—41no—
182Q8—6921no—
183D4—21no—
184D4—21no—
185D4—21no—
186D4—41no—
187D4—41no—
188D4—13841no—
189D4—6921no—
190D4—6922no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—3461no—
198C2×C4—6922no—
199C2×C4—6922no—
200C2×C4D4×D17311yescharacteristic
201C2×C4D4×D17311yescharacteristic
202C2×C4C2×C22⋊D17311yescharacteristic
203C8—41no—
204C8—6921no—
205C8—6921no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—6921no—
211C22—6921no—
212C22—6921no—
213C22#1914964711yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—3462no—
220C4—6921no—
221C4—6921no—
222C2—21no—
223C2—81no—
224C2—81no—
225C2—13841no—
226C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#454×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#454D173
hypocenterC173#2204
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26012203C2semidirect
2. #26011674C2semidirect
3. #26011664C2semidirect
4. #26011289C2semidirect
5. #26011288C2semidirect
6. #26011287C2semidirect
7. #454×C173C2semidirect
113. C173#2204semidirect
118. #454D173semidirect

Maximal subgroups

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

subgroupindex
#260122032
#260116742
#260116642
#260112892
#260112882
#260112872
#454×C1732
#2204173