Finite Group Databasebeta

← #36355515

Subgroups — #36355515

Order 22144.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#26012288C211yesmaximal, characteristic
2#26011934C211yesmaximal, characteristic, complemented
3#26011931C211yesmaximal, characteristic, complemented
4#26011750C211yesmaximal, characteristic, complemented
5#26011300C211yesmaximal, characteristic, complemented
6#26011272C211yesmaximal, characteristic, complemented
7#360×C173C211yesmaximal, characteristic, complemented
8#41×D173—21no—
9#19149711—21no—
10#19149708—21no—
11#19149706—21no—
12#19149703—21no—
13#19149701C2211yescharacteristic
14#19149699—21no—
15C2×Dic692C2211yescharacteristic, complemented
16C2×#14583809C2211yescharacteristic
17#19149659—21no—
18Dic1384—21no—
19#19149625—21no—
20#19149624—21no—
21#19149623—21no—
22#19149619C2211yescharacteristic, complemented
23#19149602—21no—
24#135×C173C2211yescharacteristic
25M32×C173C2211yescharacteristic, complemented
26#106×C173C2211yescharacteristic, complemented
27#14583827—21no—
28#14583826—21no—
29#14583826—41no—
30#14583825—21no—
31#14583825—41no—
32C2×Dic346—21no—
33Q8×D173—21no—
34D4×D173—21no—
35D4×D173—41no—
36C2×C4×D173—41no—
37C2×C4×D173D411yescharacteristic
38Dic692—21no—
39Dic692—21no—
40Dic692D411yescharacteristic
41#14583816—21no—
42#14583815—21no—
43#14583814—21no—
44#14583813—21no—
45D1384D411yescharacteristic
46#14583811—21no—
47#14583809—21no—
48#14583809—21no—
49C2×#13063436D411yescharacteristic
50C4⋊Dic173—41no—
51#14583803—41no—
52#14583801—21no—
53#41×C173—21no—
54Q16×C173D411yescharacteristic
55SD16×C173—21no—
56D8×C173D411yescharacteristic
57C2×C8×C173C2311yescharacteristic
58C4⋊C4×C173—21no—
59C16×C173—21no—
60C22×D173—41no—
61C22×D173D811yescharacteristic
62Dic346—21no—
63Dic346—41no—
64Dic346—41no—
65Dic346—41no—
66D692—21no—
67D692—41no—
68C22⋊D173—41no—
69C22⋊D173—41no—
70C22⋊D173—41no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C2×Dic173D811yescharacteristic
74C4×D173—21no—
75C4×D173—21no—
76C4×D173—41no—
77C4×D173—41no—
78C4×D173—41no—
79#13063436—21no—
80Q8×C173—21no—
81D4×C173—21no—
82D4×C173—41no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173C2×D411yescharacteristic
86C8×C173C2×D411yescharacteristic
87C8×C173C2×D411yescharacteristic
88C22×C173—41no—
89C22×C173C2×D811yescharacteristic
90C2×D173—21no—
91C2×D173—41no—
92C2×D173—41no—
93C2×D173—82no—
94C4×C173—41no—
95C4×C173—81no—
96C4×C173#12011yescharacteristic
97C4×C173AGL(1,ℤ/8)11yescharacteristic
98Dic173—21no—
99Dic173—41no—
100Dic173—81no—
101C2×C173—21no—
102C2×C173—81no—
103C2×C173#44711yescharacteristic
104D173—41no—
105D173—81no—
106C173#219111yescharacteristic, complemented, Sylow
107#2191—1731nomaximal, Sylow
108#575—1731no—
109#510—1731no—
110#510—1731no—
111#470—1731no—
112#360—1731no—
113#360D17311yescharacteristic, complemented
114#349—1731no—
115C2×#41—3461no—
116#137—3461no—
117#137—3461no—
118AGL(1,ℤ/8)—3461no—
119AGL(1,ℤ/8)—3461no—
120#135—3461no—
121#135—1731no—
122#135C2×D17311yescharacteristic
123C2×Q16—1731no—
124C2×M16—1731no—
125C22⋊Q8—3461no—
126Q32—3461no—
127Q32—3461no—
128SD32—3461no—
129SD32—3461no—
130M32—1731no—
131M32C2×D17311yescharacteristic, complemented
132#106C2×D17311yescharacteristic, complemented
133#103—3461no—
134#41—21no—
135#41—3461no—
136#41—3461no—
137#41—3461no—
138#41—6921no—
139#41—6921no—
140C2×Q8—3461no—
141C2×Q8—3461no—
142C2×D4—3461no—
143C2×D4—6921no—
144C22×C4—1731no—
145C22×C4—6921no—
146Q16—3461no—
147Q16—3461no—
148Q16—3461no—
149Q16—1731no—
150Q16D69211yescharacteristic
151SD16—21no—
152SD16—3461no—
153SD16—3461no—
154SD16—3461no—
155D8—3461no—
156D8—1731no—
157D8D69211yescharacteristic
158M16—3461no—
159M16—3461no—
160C2×C8—1731no—
161C2×C8C22×D17311yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—6921no—
164C22⋊C4—6921no—
165C16—21no—
166C16—3461no—
167C23—1731no—
168C23—6921no—
169Q8—21no—
170Q8—3461no—
171Q8—6921no—
172Q8—6921no—
173Q8—6921no—
174D4—21no—
175D4—41no—
176D4—3461no—
177D4—6921no—
178D4—6921no—
179D4—6921no—
180D4—6921no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4—1731no—
186C2×C4—6921no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4—6921no—
191C2×C4D4×D17311yescharacteristic
192C8—3461no—
193C8D4×D17311yescharacteristic
194C8C2×D69211yescharacteristic
195C22—41no—
196C22—3461no—
197C22—13842no—
198C22—6921no—
199C22—6921no—
200C22D8×D17311yescharacteristic
201C4—41no—
202C4—81no—
203C4—3461no—
204C4—13841no—
205C4—6921no—
206C4#1914964911yescharacteristic
207C4#1914970611yescharacteristic
208C2—21no—
209C2—81no—
210C2—13841no—
211C2—6921no—
212C2#2601162611yescharacteristic
subgroupG/H
centerC2#26011626
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting#360×C173C2
socleC2×C173#447
prime radical#26012288C2
hypercenter#360D173
hypocenterC173#2191
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #26011934C2semidirect
3. #26011931C2semidirect
4. #26011750C2semidirect
5. #26011300C2semidirect
6. #26011272C2semidirect
7. #360×C173C2semidirect
15. C2×Dic692C22semidirect
22. #19149619C22semidirect
25. M32×C173C22semidirect
26. #106×C173C22semidirect
106. C173#2191semidirect
113. #360D173semidirect
131. M32C2×D173semidirect
132. #106C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122882
#260119342
#260119312
#260117502
#260113002
#260112722
#360×C1732
#2191173