Finite Group Databasebeta

← #30770609

Subgroups — #30770609

Order 17152.

284 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 75 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#22562466C211yesmaximal, characteristic
2#22562436C211yesmaximal, characteristic, complemented
3#22562433C211yesmaximal, characteristic
4#22559816C211yesmaximal, characteristic
5#22559803C211yesmaximal, characteristic, complemented
6#22558128C211yesmaximal, characteristic, complemented
7C67×#2247C211yesmaximal, characteristic, complemented
8C2×#14523986—22no—
9#16863801C2211yescharacteristic
10#16863792—21no—
11#16863773C2211yescharacteristic
12#16863748—21no—
13#16863744—21no—
14#16863744—21no—
15#16863200—22no—
16#16863198—22no—
17C4×C67⋊C16C2211yescharacteristic
18C8×C67⋊C8C2211yescharacteristic
19C2×Q32×C67—22no—
20#494×C67C2211yescharacteristic
21#494×C67C2211yescharacteristic
22C4×C16×C67C2211yescharacteristic
23C2×D536—21no—
24C2×D536—21no—
25#14524043—21no—
26#14524040—21no—
27#14523986—44no—
28C2×C67⋊C16D412yes—
29#14523977—21no—
30#14523977—21no—
31#14523974—42no—
32#14523966—42no—
33#14523963—42no—
34#14523963—42no—
35C4×C67⋊C8D411yescharacteristic
36C4×C67⋊C8D411yescharacteristic
37C2×Q16×C67—21no—
38C2×Q16×C67—21no—
39C2×Q16×C67—21no—
40C2×Q16×C67—21no—
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43Q32×C67—44no—
44C2×C16×C67D412yes—
45C4×C8×C67C2311yescharacteristic
46C2×Dic134—41no—
47C2×D268—41no—
48C2×D268—41no—
49C2×D268—41no—
50D536—42no—
51D536—42no—
52C2×C67⋊C8D812yes—
53C2×C67⋊C8D812yes—
54C4⋊Dic67—41no—
55C4⋊Dic67—41no—
56C67⋊C16—22no—
57C2×Q8×C67—41no—
58C2×Q8×C67—41no—
59Q16×C67—42no—
60Q16×C67—42no—
61Q16×C67—42no—
62Q16×C67—42no—
63C2×C8×C67C2×D411yescharacteristic
64C2×C8×C67C2×D411yescharacteristic
65C4⋊C4×C67—41no—
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—41no—
69C42×C67C2×D411yescharacteristic
70C16×C67—22no—
71C22×D67—81no—
72Q8×C67—81no—
73Q8×C67—81no—
74Q8×C67—81no—
75Q8×C67—81no—
76C2×C4×C67—81no—
77C2×C4×C67—81no—
78C2×C4×C67#12711yescharacteristic
79C2×C4×C67C2×D811yescharacteristic
80C2×C4×C67C2×D811yescharacteristic
81C2×Dic67—81no—
82D268—81no—
83D268—81no—
84D268—82no—
85D268—82no—
86Dic134—82no—
87C8×C67AGL(1,ℤ/8)12yes—
88C8×C67AGL(1,ℤ/8)12yes—
89C67⋊C8—22no—
90C67⋊C8—22no—
91C22×C67#49311yescharacteristic
92C2×D67—161no—
93C2×D67—161no—
94C2×D67—161no—
95C4×C67—161no—
96C4×C67—161no—
97C4×C67#49611yescharacteristic
98C4×C67#49611yescharacteristic
99C4×C67#50812yes—
100C4×C67#51012yes—
101Dic67—161no—
102#12530—671nomaximal, Sylow
103C2×C67#172111yescharacteristic
104C2×C67#224911yescharacteristic
105C2×C67#225111yescharacteristic
106D67—321no—
107#2248—671no—
108#2247D6711yescharacteristic, complemented
109#2241—671no—
110#2240—671no—
111#1709—671no—
112#1707—671no—
113#1312—671no—
114C67#1253011yescharacteristic, complemented, Sylow
115C2×Q32—22no—
116C2×SD32—1342no—
117#500—671no—
118#498—1341no—
119#494C2×D6711yescharacteristic
120#494C2×D6711yescharacteristic
121#493—671no—
122#487—1341no—
123#486—1341no—
124#486—1341no—
125#358—1342no—
126#357—1342no—
127C4×C16—671no—
128C4×C16C2×D6711yescharacteristic
129C82—671no—
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—1341no—
135C2×D8—1341no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—1341no—
139#127—1341no—
140Q32—44no—
141SD32—2684no—
142C2×C16—672no—
143C2×C16C22⋊D6712yes—
144#107—1341no—
145#107—1341no—
146#106—2682no—
147#103—2682no—
148#102—2682no—
149#102—2682no—
150C4×C8—671no—
151C4×C8—671no—
152C4×C8C22×D6711yescharacteristic
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—2681no—
156C2×D4—2681no—
157C2×D4—2681no—
158C2×D4—2681no—
159Q16—42no—
160Q16—42no—
161Q16—42no—
162Q16—42no—
163D8—2682no—
164D8—2682no—
165C2×C8—672no—
166C2×C8—672no—
167C2×C8D4×D6711yescharacteristic
168C2×C8C2×C22⋊D6711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—2681no—
174C4⋊C4—2681no—
175C42D4×D6711yescharacteristic
176C16—22no—
177C16—1342no—
178C23—5361no—
179Q8—81no—
180Q8—81no—
181Q8—81no—
182Q8—81no—
183Q8—5362no—
184D4—5361no—
185D4—5361no—
186D4—5362no—
187D4—5362no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—5361no—
191C2×C4#1452403911yescharacteristic
192C2×C4D8×D6711yescharacteristic
193C2×C4D8×D6711yescharacteristic
194C8—1342no—
195C8—1342no—
196C8#1452406712yes—
197C8#1452406912yes—
198C22—10721no—
199C22—10721no—
200C22—10721no—
201C22#1686377111yescharacteristic
202C4—161no—
203C4—161no—
204C4—10721no—
205C4#1686378311yescharacteristic
206C4#1686378411yescharacteristic
207C4#1686382912yes—
208C4#1686383812yes—
209C2—21441no—
210C2#2255987411yescharacteristic
211C2#2256247111yescharacteristic
212C2#2256247711yescharacteristic
subgroupG/H
centerC22#16863771
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2247C2
socleC22×C67#493
prime radical#16863773C22
hypercenter#2247D67
hypocenterC67#12530
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2247

Complements

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

NH ≅ G/Ntype
2. #22562436C2semidirect
5. #22559803C2semidirect
6. #22558128C2semidirect
7. C67×#2247C2semidirect
108. #2247D67semidirect
114. C67#12530semidirect

Maximal subgroups

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

subgroupindex
#225624662
#225624362
#225624332
#225598162
#225598032
#225581282
C67×#22472
#1253067