Finite Group Databasebeta

← #30770608

Subgroups — #30770608

Order 17152.

284 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 79 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#22562461C211yesmaximal, characteristic, complemented
3#22562438C211yesmaximal, characteristic, complemented
4#22562434C211yesmaximal, characteristic
5#22559815C211yesmaximal, characteristic
6#22558127C211yesmaximal, characteristic, complemented
7C67×#1707C211yesmaximal, characteristic, complemented
8C2×#14523989—22no—
9C2×#14523986—22no—
10#16863776C2211yescharacteristic
11#16863773C2211yescharacteristic
12#16863743—21no—
13#16863743—21no—
14#16863199—22no—
15#16863196—22no—
16C4×C67⋊C16C2211yescharacteristic
17C4×C67⋊C16C2211yescharacteristic
18#500×C67C2211yescharacteristic
19#498×C67—21no—
20#494×C67C2211yescharacteristic
21#487×C67—21no—
22C82×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25C2×D536—21no—
26C2×D536—21no—
27#14524043—21no—
28#14524040—21no—
29#14523989—44no—
30#14523986—44no—
31C2×C67⋊C16D412yes—
32C2×C67⋊C16D412yes—
33#14523965—42no—
34#14523965—42no—
35C2×Q16×C67—21no—
36C2×Q16×C67—21no—
37C4⋊Q8×C67—21no—
38C4⋊Q8×C67—21no—
39#107×C67—21no—
40#107×C67—21no—
41#106×C67—42no—
42#103×C67—42no—
43C4×C8×C67D411yescharacteristic
44C4×C8×C67D411yescharacteristic
45C4×C8×C67C2311yescharacteristic
46C2×Dic134—41no—
47C2×D268—41no—
48C2×D268—41no—
49C2×D268—41no—
50Dic268—42no—
51Dic268—42no—
52D536—42no—
53D536—42no—
54C4⋊Dic67—41no—
55C4⋊Dic67—41no—
56C67⋊C16—22no—
57C67⋊C16—22no—
58C2×Q8×C67—41no—
59C2×Q8×C67—41no—
60Q16×C67—42no—
61Q16×C67—42no—
62C2×C8×C67D812yes—
63C2×C8×C67D812yes—
64C2×C8×C67C2×D411yescharacteristic
65C2×C8×C67C2×D411yescharacteristic
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—41no—
69C4⋊C4×C67—41no—
70C42×C67C2×D411yescharacteristic
71C22×D67—81no—
72Q8×C67—81no—
73Q8×C67—81no—
74Q8×C67—82no—
75C2×C4×C67—81no—
76C2×C4×C67—81no—
77C2×C4×C67#12711yescharacteristic
78C2×C4×C67C2×D811yescharacteristic
79C2×C4×C67C2×D811yescharacteristic
80C2×Dic67—81no—
81D268—81no—
82D268—81no—
83D268—82no—
84D268—82no—
85Dic134—81no—
86Dic134—81no—
87C8×C67—22no—
88C8×C67—22no—
89C8×C67AGL(1,ℤ/8)12yes—
90C8×C67AGL(1,ℤ/8)12yes—
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#2247—671no—
109#2241—671no—
110#2240—671no—
111#1709—671no—
112#1707D6711yescharacteristic, complemented
113#1312—671no—
114C67#1253011yescharacteristic, complemented, Sylow
115C2×Q32—1342no—
116C2×SD32—1342no—
117#500C2×D6711yescharacteristic
118#498—21no—
119#494—671no—
120#494C2×D6711yescharacteristic
121#493—671no—
122#487—21no—
123#486—1341no—
124#486—1341no—
125#358—1342no—
126#357—1342no—
127C4×C16—671no—
128C4×C16—671no—
129C82C2×D6711yescharacteristic
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—1341no—
133C2×Q16—1341no—
134C2×D8—1341no—
135C2×D8—1341no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—1341no—
139#127—1341no—
140Q32—2684no—
141SD32—2684no—
142C2×C16—672no—
143C2×C16—672no—
144#107—21no—
145#107—21no—
146#106—42no—
147#103—42no—
148#102—2682no—
149#102—2682no—
150C4×C8C22⋊D6711yescharacteristic
151C4×C8C22⋊D6711yescharacteristic
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—2682no—
162Q16—2682no—
163D8—2682no—
164D8—2682no—
165C2×C8#1176161312yes—
166C2×C8#1176161312yes—
167C2×C8D4×D6711yescharacteristic
168C2×C8D4×D6711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—2681no—
174C4⋊C4—2681no—
175C42C2×C22⋊D6711yescharacteristic
176C16—1342no—
177C16—1342no—
178C23—5361no—
179Q8—81no—
180Q8—81no—
181Q8—82no—
182Q8—5361no—
183Q8—5361no—
184D4—5361no—
185D4—5361no—
186D4—5362no—
187D4—5362no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—5361no—
191C2×C4#1452403911yescharacteristic
192C2×C4C2×#1176161311yescharacteristic
193C2×C4C2×#1176161311yescharacteristic
194C8—22no—
195C8—22no—
196C8#1452407012yes—
197C8#1452407012yes—
198C22—10721no—
199C22—10721no—
200C22—10721no—
201C22#1686377211yescharacteristic
202C4—161no—
203C4—161no—
204C4—10721no—
205C4#1686378211yescharacteristic
206C4#1686378211yescharacteristic
207C4#1686382612yes—
208C4#1686383712yes—
209C2—21441no—
210C2#2255987311yescharacteristic
211C2#2256246811yescharacteristic
212C2#2256247811yescharacteristic
subgroupG/H
centerC22#16863772
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1707C2
socleC22×C67#493
prime radical#16863773C22
hypercenter#1707D67
hypocenterC67#12530
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1707

Complements

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

NH ≅ G/Ntype
2. #22562461C2semidirect
3. #22562438C2semidirect
6. #22558127C2semidirect
7. C67×#1707C2semidirect
112. #1707D67semidirect
114. C67#12530semidirect

Maximal subgroups

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

subgroupindex
#225624662
#225624612
#225624382
#225624342
#225598152
#225581272
C67×#17072
#1253067