Finite Group Databasebeta

← #30770081

Subgroups — #30770081

Order 17152.

224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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#22562465C211yesmaximal, characteristic, complemented
2#22562290C211yesmaximal, characteristic, complemented
3#22562285C211yesmaximal, characteristic
4#22558285C211yesmaximal, characteristic, complemented
5#22558247C211yesmaximal, characteristic, complemented
6#22558245C211yesmaximal, characteristic, complemented
7C67×#2170C211yesmaximal, characteristic, complemented
8#16863836—21no—
9#16863831—21no—
10C2×#14523988—21no—
11C2×#14523988—21no—
12#16863776C2211yescharacteristic
13#16863773C2211yescharacteristic
14#16863217C2211yescharacteristic
15#16863217C2211yescharacteristic
16#16863211—21no—
17#16863208—21no—
18#504×C67—21no—
19#443×C67C2211yescharacteristic
20#364×C67C2211yescharacteristic
21#350×C67—21no—
22C4×C16×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25C2×D536—21no—
26C2×D536—21no—
27#14524043—21no—
28#14524040—21no—
29#14523989—41no—
30#14523988—42no—
31#14523988—41no—
32#14523988—41no—
33#14523987—41no—
34#14523986—41no—
35#14523984—41no—
36#14523983—21no—
37#14523983—21no—
38#135×C67D411yescharacteristic
39#131×C67—21no—
40M32×C67—21no—
41M32×C67—41no—
42C2×C16×C67—41no—
43C2×C16×C67D411yescharacteristic
44C2×C16×C67D411yescharacteristic
45#108×C67D411yescharacteristic
46#104×C67—21no—
47C4×C8×C67C2311yescharacteristic
48C2×Dic134—41no—
49C2×D268—41no—
50C2×D268—81no—
51Dic268—41no—
52Dic268—41no—
53Dic268—41no—
54Dic268—41no—
55D536—41no—
56D536—41no—
57D536—41no—
58D536—41no—
59C4⋊Dic67—81no—
60C67⋊C16—41no—
61C67⋊C16—41no—
62#41×C67—21no—
63Q16×C67D811yescharacteristic
64SD16×C67—21no—
65D8×C67D811yescharacteristic
66M16×C67—21no—
67M16×C67—41no—
68C2×C8×C67—41no—
69C2×C8×C67C2×D411yescharacteristic
70C2×C8×C67C2×D411yescharacteristic
71C42×C67#4111yescharacteristic
72C16×C67—21no—
73C16×C67—41no—
74C16×C67D811yescharacteristic
75C16×C67D811yescharacteristic
76C22×D67—81no—
77Q8×C67—21no—
78D4×C67—21no—
79D4×C67—41no—
80C2×C4×C67—21no—
81C2×C4×C67—41no—
82C2×C4×C67#12111yescharacteristic
83C2×Dic67—81no—
84D268—81no—
85D268—81no—
86D268—162no—
87Dic134—81no—
88Dic134—81no—
89C8×C67—41no—
90C8×C67C2×D811yescharacteristic
91C8×C67C2×D811yescharacteristic
92C8×C67#13511yescharacteristic
93C8×C67AGL(1,ℤ/8)11yescharacteristic
94C22×C67—41no—
95C22×C67#47211yescharacteristic
96C2×D67—162no—
97C2×D67—161no—
98C4×C67—22no—
99C4×C67—41no—
100C4×C67#45911yescharacteristic
101C4×C67#46611yescharacteristic
102Dic67—161no—
103#12438—671nomaximal, Sylow
104C2×C67—21no—
105C2×C67—81no—
106C2×C67#166711yescharacteristic
107D67—321no—
108#2248—671no—
109#2214—671no—
110#2213—671no—
111#2170D6711yescharacteristic, complemented
112#1373—671no—
113#1358—671no—
114#1357—671no—
115C67#1243811yescharacteristic, complemented, Sylow
116#510—1341no—
117#509—1341no—
118C2×SD32—1341no—
119C2×SD32—1341no—
120#504—21no—
121#494—671no—
122#493—671no—
123#443C2×D6711yescharacteristic
124#364—671no—
125#364—671no—
126#364C2×D6711yescharacteristic
127#362—1341no—
128#361—1341no—
129#350—21no—
130C4×C16C2×D6711yescharacteristic
131#135D26811yescharacteristic
132C2×Q16—1341no—
133C2×Q16—1341no—
134C2×D8—1341no—
135C2×D8—1341no—
136#131—21no—
137C4⋊Q8—1341no—
138#127—1341no—
139Q32—2681no—
140SD32—2682no—
141SD32—2681no—
142SD32—2681no—
143SD32—2681no—
144SD32—2681no—
145D16—2681no—
146M32—21no—
147M32—41no—
148M32—1341no—
149M32—1341no—
150C2×C16—41no—
151C2×C16C22⋊D6711yescharacteristic
152C2×C16C22⋊D6711yescharacteristic
153#108D26811yescharacteristic
154#104—21no—
155C4×C8C22×D6711yescharacteristic
156#41—21no—
157C2×Q8—2681no—
158C2×D4—2681no—
159C2×D4—5361no—
160Q16—2681no—
161Q16—2681no—
162Q16—2681no—
163Q16—2681no—
164Q16D53611yescharacteristic
165SD16—21no—
166D8—2681no—
167D8—2681no—
168D8—2681no—
169D8—2681no—
170D8D53611yescharacteristic
171M16—21no—
172M16—41no—
173C2×C8—41no—
174C2×C8C2×C22⋊D6711yescharacteristic
175C2×C8C2×D26811yescharacteristic
176C4⋊C4—5361no—
177C42#1176162811yescharacteristic
178C16—21no—
179C16—41no—
180C16—2681no—
181C16—2681no—
182C16#1176161311yescharacteristic
183C16#1176161311yescharacteristic
184C23—5361no—
185Q8—21no—
186Q8—5361no—
187Q8—5361no—
188D4—21no—
189D4—41no—
190D4—5361no—
191D4—5361no—
192D4—10722no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—5361no—
196C2×C4#1452401711yescharacteristic
197C8—41no—
198C8C2×#1176161311yescharacteristic
199C8C2×D53611yescharacteristic
200C8#1452406511yescharacteristic
201C8#1452406811yescharacteristic
202C22—41no—
203C22—10722no—
204C22—10721no—
205C22#1686366611yescharacteristic
206C4—22no—
207C4—41no—
208C4—10721no—
209C4#1686359111yescharacteristic
210C4#1686363211yescharacteristic
211C2—21no—
212C2—81no—
213C2—21441no—
214C2#2255956411yescharacteristic
subgroupG/H
centerC2#22559564
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2170C2
socleC2×C67#1667
prime radical#22562285C2
hypercenter#2170D67
hypocenterC67#12438
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2170

Complements

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

NH ≅ G/Ntype
1. #22562465C2semidirect
2. #22562290C2semidirect
4. #22558285C2semidirect
5. #22558247C2semidirect
6. #22558245C2semidirect
7. C67×#2170C2semidirect
111. #2170D67semidirect
115. C67#12438semidirect

Maximal subgroups

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

subgroupindex
#225624652
#225622902
#225622852
#225582852
#225582472
#225582452
C67×#21702
#1243867