Finite Group Databasebeta

← #30770612

Subgroups — #30770612

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#22562464C211yesmaximal, characteristic
2#22562461C211yesmaximal, characteristic, complemented
3#22562436C211yesmaximal, characteristic, complemented
4#22562432C211yesmaximal, characteristic
5#22559806C211yesmaximal, characteristic, complemented
6#22558127C211yesmaximal, characteristic, complemented
7C67×#1709C211yesmaximal, characteristic
8C2×#14523989—22no—
9C2×#14523987—22no—
10#16863801C2211yescharacteristic
11#16863794—21no—
12#16863776C2211yescharacteristic
13#16863747—21no—
14#16863200—22no—
15#16863197—22no—
16C4×C67⋊C16C2211yescharacteristic
17C4×C67⋊C16C2211yescharacteristic
18#494×C67C2211yescharacteristic
19#493×C67C2211yescharacteristic
20#486×C67—21no—
21#486×C67—21no—
22C82×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25#14524043—21no—
26#14524043—21no—
27#14523989—44no—
28#14523987—44no—
29C2×C67⋊C16D412yes—
30C2×C67⋊C16D412yes—
31#14523977—21no—
32#14523977—21no—
33#14523975—42no—
34#14523968—42no—
35C2×Q16×C67—21no—
36C2×Q16×C67—21no—
37C2×D8×C67—21no—
38C2×D8×C67—21no—
39C4⋊Q8×C67—21no—
40#127×C67—21no—
41#102×C67—42no—
42#102×C67—42no—
43C4×C8×C67D411yescharacteristic
44C4×C8×C67D411yescharacteristic
45C4×C8×C67C2311yescharacteristic
46C2×Dic134—41no—
47C2×Dic134—41no—
48Dic268—42no—
49Dic268—42no—
50C4⋊Dic67—41no—
51C4⋊Dic67—41no—
52C4⋊Dic67—41no—
53C4⋊Dic67—41no—
54C67⋊C16—22no—
55C67⋊C16—22no—
56C2×Q8×C67—41no—
57C2×D4×C67—41no—
58C2×D4×C67—41no—
59C2×D4×C67—41no—
60Q16×C67—42no—
61Q16×C67—42no—
62D8×C67—42no—
63D8×C67—42no—
64C2×C8×C67D812yes—
65C2×C8×C67D812yes—
66C2×C8×C67C2×D411yescharacteristic
67C2×C8×C67C2×D411yescharacteristic
68C4⋊C4×C67—41no—
69C4⋊C4×C67—41no—
70C42×C67C2×D411yescharacteristic
71C23×C67—81no—
72Q8×C67—81no—
73Q8×C67—81no—
74D4×C67—81no—
75D4×C67—81no—
76D4×C67—82no—
77D4×C67—82no—
78C2×C4×C67—81no—
79C2×C4×C67#12711yescharacteristic
80C2×C4×C67C2×D811yescharacteristic
81C2×C4×C67C2×D811yescharacteristic
82C2×Dic67—81no—
83C2×Dic67—81no—
84Dic134—81no—
85Dic134—81no—
86Dic134—82no—
87C8×C67—22no—
88C8×C67—22no—
89C8×C67AGL(1,ℤ/8)12yes—
90C8×C67AGL(1,ℤ/8)12yes—
91C22×C67—161no—
92C22×C67—161no—
93C22×C67—161no—
94C22×C67#49311yescharacteristic
95C4×C67—161no—
96C4×C67#49611yescharacteristic
97C4×C67#49611yescharacteristic
98C4×C67#50812yes—
99C4×C67#51012yes—
100Dic67—161no—
101Dic67—161no—
102#12530—671nomaximal, Sylow
103C2×C67—321no—
104C2×C67#172111yescharacteristic
105C2×C67#224911yescharacteristic
106C2×C67#225111yescharacteristic
107#2248—671no—
108#2247—671no—
109#2241—671no—
110#2240—671no—
111#1709D6711yescharacteristic
112#1707—671no—
113#1312—671no—
114C67#1253011yescharacteristic, complemented, Sylow
115C2×Q32—1342no—
116C2×SD32—1342no—
117#500—671no—
118#498—1341no—
119#494—671no—
120#494C2×D6711yescharacteristic
121#493C2×D6711yescharacteristic
122#487—1341no—
123#486—21no—
124#486—21no—
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—21no—
135C2×D8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—1341no—
138C4⋊Q8—1341no—
139#127—21no—
140Q32—2684no—
141SD32—2684no—
142C2×C16—672no—
143C2×C16—672no—
144#107—1341no—
145#107—1341no—
146#106—2682no—
147#103—2682no—
148#102—42no—
149#102—42no—
150C4×C8C22⋊D6711yescharacteristic
151C4×C8C22⋊D6711yescharacteristic
152C4×C8C22×D6711yescharacteristic
153C2×Q8—41no—
154C2×Q8—2681no—
155C2×Q8—2681no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159Q16—42no—
160Q16—42no—
161Q16—2682no—
162Q16—2682no—
163D8—42no—
164D8—42no—
165C2×C8#1176161312yes—
166C2×C8#1176161312yes—
167C2×C8D4×D6711yescharacteristic
168C2×C8D4×D6711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—2681no—
172C4⋊C4—2681no—
173C4⋊C4—2681no—
174C4⋊C4—2681no—
175C42C2×C22⋊D6711yescharacteristic
176C16—1342no—
177C16—1342no—
178C23—81no—
179Q8—81no—
180Q8—81no—
181Q8—5361no—
182Q8—5361no—
183Q8—5362no—
184D4—81no—
185D4—81no—
186D4—82no—
187D4—82no—
188C2×C4—81no—
189C2×C4—5361no—
190C2×C4—5361no—
191C2×C4#1452403911yescharacteristic
192C2×C4C2×#1176161311yescharacteristic
193C2×C4C2×#1176161311yescharacteristic
194C8—22no—
195C8—22no—
196C8#1452406712yes—
197C8#1452406712yes—
198C22—161no—
199C22—161no—
200C22—161no—
201C22#1686377211yescharacteristic
202C4—161no—
203C4—10721no—
204C4—10721no—
205C4#1686378411yescharacteristic
206C4#1686378411yescharacteristic
207C4#1686382612yes—
208C4#1686383612yes—
209C2—321no—
210C2#2255987611yescharacteristic
211C2#2256246811yescharacteristic
212C2#2256248111yescharacteristic
subgroupG/H
centerC22#16863772
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1709C2
socleC22×C67#493
prime radical#493×C67C22
hypercenter#1709D67
hypocenterC67#12530
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1709

Complements

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

NH ≅ G/Ntype
2. #22562461C2semidirect
3. #22562436C2semidirect
5. #22559806C2semidirect
6. #22558127C2semidirect
114. C67#12530semidirect

Maximal subgroups

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

subgroupindex
#225624642
#225624612
#225624362
#225624322
#225598062
#225581272
C67×#17092
#1253067