Finite Group Databasebeta

← #30770611

Subgroups — #30770611

Order 17152.

284 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 83 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
2#22562462C211yesmaximal, characteristic, complemented
3#22562437C211yesmaximal, characteristic, complemented
4#22562433C211yesmaximal, characteristic
5#22559815C211yesmaximal, characteristic
6#22559806C211yesmaximal, characteristic, complemented
7C67×#1312C211yesmaximal, characteristic, complemented
8C2×Dic536—22no—
9C2×#14523988—22no—
10#16863801C2211yescharacteristic
11#16863794—21no—
12#16863776C2211yescharacteristic
13#16863776C2211yescharacteristic
14#16863773C2211yescharacteristic
15#16863747—21no—
16#16863743—21no—
17#16863743—21no—
18#16863201—22no—
19#16863198—22no—
20C4×C16×C67C2211yescharacteristic
21C4×C16×C67C2211yescharacteristic
22C82×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25C2×Dic268—21no—
26C2×Dic268—21no—
27C2×D536—21no—
28C2×D536—21no—
29#14524043—21no—
30#14524043—21no—
31#14524043—21no—
32#14524040—21no—
33Dic536—44no—
34#14523988—44no—
35#14523977—21no—
36#14523977—21no—
37#14523975—42no—
38#14523968—42no—
39#14523965—42no—
40#14523965—42no—
41C2×C16×C67D412yes—
42C2×C16×C67D412yes—
43C4×C8×C67D411yescharacteristic
44C4×C8×C67D411yescharacteristic
45C4×C8×C67C2311yescharacteristic
46C2×Dic134—41no—
47C2×Dic134—41no—
48C2×Dic134—41no—
49C2×D268—41no—
50C2×D268—41no—
51C2×D268—41no—
52Dic268—42no—
53Dic268—42no—
54Dic268—42no—
55Dic268—42no—
56D536—42no—
57D536—42no—
58C4⋊Dic67—41no—
59C4⋊Dic67—41no—
60C4⋊Dic67—41no—
61C4⋊Dic67—41no—
62C4⋊Dic67—41no—
63C4⋊Dic67—41no—
64C2×C8×C67D812yes—
65C2×C8×C67D812yes—
66C2×C8×C67C2×D411yescharacteristic
67C2×C8×C67C2×D411yescharacteristic
68C42×C67C2×D411yescharacteristic
69C16×C67—22no—
70C16×C67—22no—
71C22×D67—81no—
72C2×C4×C67#12711yescharacteristic
73C2×C4×C67C2×D811yescharacteristic
74C2×C4×C67C2×D811yescharacteristic
75C2×Dic67—81no—
76C2×Dic67—81no—
77C2×Dic67—81no—
78D268—81no—
79D268—81no—
80D268—82no—
81D268—82no—
82Dic134—81no—
83Dic134—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#49311yescharacteristic
92C2×D67—161no—
93C2×D67—161no—
94C2×D67—161no—
95C4×C67#49611yescharacteristic
96C4×C67#49611yescharacteristic
97C4×C67#50812yes—
98C4×C67#51012yes—
99Dic67—161no—
100Dic67—161no—
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#1707—671no—
113#1312D6711yescharacteristic, complemented
114C67#1253011yescharacteristic, complemented, Sylow
115C2×Q32—1342no—
116C2×SD32—1342no—
117#500—671no—
118#498—1341no—
119#494—671no—
120#494—671no—
121#493—671no—
122#487—1341no—
123#486—1341no—
124#486—1341no—
125#358—1342no—
126#357—1342no—
127C4×C16C2×D6711yescharacteristic
128C4×C16C2×D6711yescharacteristic
129C82C2×D6711yescharacteristic
130C2×Q16—1341no—
131C2×Q16—1341no—
132C2×Q16—1341no—
133C2×Q16—1341no—
134C2×D8—1341no—
135C2×D8—1341no—
136C4⋊Q8—1341no—
137C4⋊Q8—1341no—
138C4⋊Q8—1341no—
139#127—1341no—
140Q32—2684no—
141SD32—2684no—
142C2×C16D26812yes—
143C2×C16D26812yes—
144#107—1341no—
145#107—1341no—
146#106—2682no—
147#103—2682no—
148#102—2682no—
149#102—2682no—
150C4×C8D26811yescharacteristic
151C4×C8D26811yescharacteristic
152C4×C8C22×D6711yescharacteristic
153C2×Q8—2681no—
154C2×Q8—2681no—
155C2×Q8—2681no—
156C2×D4—2681no—
157C2×D4—2681no—
158C2×D4—2681no—
159Q16—2682no—
160Q16—2682no—
161Q16—2682no—
162Q16—2682no—
163D8—2682no—
164D8—2682no—
165C2×C8D53612yes—
166C2×C8D53612yes—
167C2×C8C2×D26811yescharacteristic
168C2×C8C2×D26811yescharacteristic
169C4⋊C4—2681no—
170C4⋊C4—2681no—
171C4⋊C4—2681no—
172C4⋊C4—2681no—
173C4⋊C4—2681no—
174C4⋊C4—2681no—
175C42C2×D26811yescharacteristic
176C16—22no—
177C16—22no—
178C23—5361no—
179Q8—5361no—
180Q8—5361no—
181Q8—5361no—
182Q8—5361no—
183Q8—5362no—
184D4—5361no—
185D4—5361no—
186D4—5362no—
187D4—5362no—
188C2×C4—5361no—
189C2×C4—5361no—
190C2×C4—5361no—
191C2×C4#1452404011yescharacteristic
192C2×C4C2×D53611yescharacteristic
193C2×C4C2×D53611yescharacteristic
194C8—22no—
195C8—22no—
196C8#1452407112yes—
197C8#1452407112yes—
198C22—10721no—
199C22—10721no—
200C22—10721no—
201C22#1686377311yescharacteristic
202C4—10721no—
203C4—10721no—
204C4—10721no—
205C4#1686378511yescharacteristic
206C4#1686378511yescharacteristic
207C4#1686382812yes—
208C4#1686383912yes—
209C2—21441no—
210C2#2255987511yescharacteristic
211C2#2256247011yescharacteristic
212C2#2256248011yescharacteristic
subgroupG/H
centerC22#16863773
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1312C2
socleC22×C67#493
prime radical#16863773C22
hypercenter#1312D67
hypocenterC67#12530
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1312

Complements

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

NH ≅ G/Ntype
2. #22562462C2semidirect
3. #22562437C2semidirect
6. #22559806C2semidirect
7. C67×#1312C2semidirect
113. #1312D67semidirect
114. C67#12530semidirect

Maximal subgroups

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

subgroupindex
#225624652
#225624622
#225624372
#225624332
#225598152
#225598062
C67×#13122
#1253067