Finite Group Databasebeta

← #30758936

Subgroups — #30758936

Order 17152.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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#22561037C211yesmaximal, characteristic
2#22558768C212yesmaximal
3#22558067C212yesmaximal, complemented
4#22558042C211yesmaximal, characteristic, complemented
5C67×#2864C211yesmaximal, characteristic, complemented
6#16863946C2211yescharacteristic
7#16863449C2212yes—
8#16863420C2211yescharacteristic
9C2×#14523971—22no—
10#16863129—24no—
11#389×C67—22no—
12#387×C67—24no—
13#382×C67C2212yes—
14C43×C67C2211yescharacteristic
15C2×#11761628—21no—
16C2×#11761612—21no—
17#14524043Q811yescharacteristic
18#14524040Q811yescharacteristic
19#14524031—21no—
20#14524017—22no—
21C4×D268—22no—
22C4⋊(C67⋊C8)—22no—
23C4⋊(C67⋊C8)—22no—
24#14523971—44no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—22no—
27C2×C4⋊C4×C67—24no—
28C2×C4⋊C4×C67—22no—
29C2×C42×C67—22no—
30C2×C42×C67D412yes—
31C2×C42×C67C2311yescharacteristic
32#95×C67—24no—
33#95×C67—22no—
34#95×C67D414yes—
35#11761628—42no—
36#11761628—42no—
37C2×Dic134—21no—
38C2×D268—21no—
39C2×D268—22no—
40C2×C22⋊D67—41no—
41C2×C4×D67—41no—
42#11761612—42no—
43C2×C67⋊C8—42no—
44C2×C67⋊C8—41no—
45C4⋊Dic67—22no—
46(C2×D67)⋊C4—42no—
47C22×C4×C67—22no—
48C22×C4×C67—24no—
49C22×C4×C67C2×D412yes—
50C22×C4×C67C2×D411yescharacteristic
51C4⋊C4×C67—48no—
52C4⋊C4×C67—42no—
53C4⋊C4×C67—42no—
54C42×C67—22no—
55C42×C67—22no—
56C42×C67—44no—
57C42×C67C2×Q811yescharacteristic
58C42×C67#4111yescharacteristic
59C42×C67#4112yes—
60C23×C67#12011yescharacteristic
61C22×D67—41no—
62C2×C4×C67—24no—
63C2×C4×C67—42no—
64C2×C4×C67—42no—
65C2×C4×C67—42no—
66C2×C4×C67—48no—
67C2×C4×C67—44no—
68C2×C4×C67C22⋊Q812yes—
69C2×C4×C67C22⋊Q811yescharacteristic
70C2×C4×C67#12412yes—
71C2×C4×C67#12411yescharacteristic
72C2×C4×C67#13512yes—
73C2×C4×C67AGL(1,ℤ/8)12yes—
74C22⋊D67—82no—
75C2×Dic67—41no—
76D268—42no—
77D268—82no—
78C4×D67—81no—
79C4×D67—81no—
80Dic134—42no—
81C67⋊C8—82no—
82C67⋊C8—81no—
83C22×C67—22no—
84C22×C67#39311yescharacteristic
85C22×C67#44912yes—
86C2×D67—81no—
87C2×D67—82no—
88C4×C67—22no—
89C4×C67—22no—
90C4×C67—44no—
91C4×C67—84no—
92C4×C67#47812yes—
93C4×C67#48811yescharacteristic
94C4×C67#49011yescharacteristic
95Dic67—81no—
96#10366—671nomaximal, Sylow
97C2×C67—22no—
98C2×C67#202511yescharacteristic
99C2×C67#203112yes—
100D67—161no—
101#2864D6711yescharacteristic, complemented
102#1962—671no—
103#1517—672no—
104#1286—672no—
105#1276—671no—
106C67#1036611yescharacteristic, complemented, Sylow
107#532—671no—
108#431—672no—
109#423—671no—
110C2×#104—1342no—
111#389—22no—
112#387—24no—
113#382C2×D6712yes—
114C43C2×D6711yescharacteristic
115#327—1344no—
116C2×#41—1341no—
117C2×M16—1341no—
118C4⋊Q8—671no—
119#127—671no—
120#124—1341no—
121#121—1342no—
122C4×D4—1342no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42C22⋊D6712yes—
127C2×C42C22×D6711yescharacteristic
128C4⋊C8—1342no—
129C4⋊C8—1342no—
130#104—2684no—
131C22⋊C8—1342no—
132C4×C8—1342no—
133#95—24no—
134#95—22no—
135#95C22⋊D6714yes—
136#41—2682no—
137#41—2682no—
138C2×Q8—1341no—
139C2×D4—2681no—
140C2×D4—1341no—
141C2×D4—1342no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—2681no—
145C22×C4C2×C22⋊D6712yes—
146C22×C4C2×C22⋊D6711yescharacteristic
147M16—2682no—
148C2×C8—2682no—
149C2×C8—2681no—
150C4⋊C4—48no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—1342no—
154C22⋊C4—2682no—
155C42—22no—
156C42—22no—
157C42—44no—
158C42Q8×D6711yescharacteristic
159C42#1176162911yescharacteristic
160C42#1176162912yes—
161C23—2681no—
162C23#1452401211yescharacteristic
163Q8—2682no—
164D4—2682no—
165D4—5362no—
166D4—5362no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—2681no—
174C2×C4—5361no—
175C2×C4—5361no—
176C2×C4#1452401912yes—
177C2×C4#1452401911yescharacteristic
178C2×C4#1452403212yes—
179C2×C4#1452403211yescharacteristic
180C2×C4#1452406312yes—
181C2×C4#1452406812yes—
182C8—5362no—
183C8—5361no—
184C22—22no—
185C22—5361no—
186C22—5362no—
187C22#1686330711yescharacteristic
188C22#1686354112yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—5361no—
194C4#1686369312yes—
195C4#1686375311yescharacteristic
196C4#1686376211yescharacteristic
197C2—22no—
198C2—10721no—
199C2#2256135211yescharacteristic
200C2#2256137812yes—
subgroupG/H
centerC22#16863307
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2864C2
socleC22×C67#393
prime radical#16863946C22
hypercenter#2864D67
hypocenterC67#10366
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2864

Complements

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

NH ≅ G/Ntype
3. #22558067C2semidirect
4. #22558042C2semidirect
5. C67×#2864C2semidirect
101. #2864D67semidirect
106. C67#10366semidirect

Maximal subgroups

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

subgroupindex
#225610372
#225587682
#225580672
#225580422
C67×#28642
#1036667