Finite Group Databasebeta

← #30759930

Subgroups — #30759930

Order 17152.

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

normal subgroups only

This group has 344 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#22571554C211yesmaximal, characteristic
2#22558907C212yesmaximal
3#22558720C212yesmaximal
4#22558079C211yesmaximal, characteristic
5C67×#1832C211yesmaximal, characteristic
6#16863953C2212yes—
7#16863692—22no—
8#16863637—22no—
9#16863607—22no—
10#16863453C2212yes—
11#16863143—22no—
12#16863141—22no—
13#16863132—24no—
14C2×C4⋊C8×C67C2211yescharacteristic
15C2×C4⋊C8×C67C2211yescharacteristic
16#376×C67C2211yescharacteristic
17#336×C67—22no—
18C2×Dic268—22no—
19#14524043D412yes—
20#14524043D412yes—
21#14524035—22no—
22#14524022—22no—
23#14524022—22no—
24C4×Dic134—22no—
25C4×Dic134—22no—
26#14524002—22no—
27#14523977—22no—
28C4⋊(C67⋊C8)—22no—
29#14523968—24no—
30C22⋊(C67⋊C8)—22no—
31#14523954—22no—
32C22×C8×C67—21no—
33C22×C8×C67D412yes—
34C2×C42×C67—21no—
35C2×C42×C67C2311yescharacteristic
36C4⋊C8×C67—22no—
37C4⋊C8×C67—22no—
38#95×C67—21no—
39#95×C67D412yes—
40C2×Dic134—22no—
41C2×Dic134—22no—
42Dic268—82no—
43C2×C67⋊C8—42no—
44C4⋊Dic67—22no—
45C4⋊Dic67—22no—
46C4⋊Dic67—22no—
47C4⋊Dic67—22no—
48Dic67⋊C4—42no—
49Dic67⋊C4—42no—
50C22⋊Dic67—42no—
51C4×Dic67—42no—
52C22×C4×C67—22no—
53C22×C4×C67C2×D411yescharacteristic
54C22×C4×C67C2×D411yescharacteristic
55C22×C4×C67#4111yescharacteristic
56C2×C8×C67—22no—
57C2×C8×C67—42no—
58C2×C8×C67—42no—
59C2×C8×C67—41no—
60C42×C67—41no—
61C42×C67—41no—
62C42×C67C2×D412yes—
63C42×C67#4112yes—
64C23×C67#12111yescharacteristic
65C2×C4×C67—22no—
66C2×C4×C67—21no—
67C2×C4×C67—21no—
68C2×C4×C67—41no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—42no—
72C2×C4×C67#12011yescharacteristic
73C2×C4×C67#12112yes—
74C2×C4×C67#12312yes—
75C2×C4×C67#12411yescharacteristic
76C2×Dic67—42no—
77C2×Dic67—42no—
78Dic134—42no—
79Dic134—42no—
80Dic134—82no—
81C8×C67—42no—
82C8×C67—81no—
83C67⋊C8—82no—
84C22×C67—21no—
85C22×C67—21no—
86C22×C67#39411yescharacteristic
87C22×C67#47312yes—
88C4×C67—22no—
89C4×C67—21no—
90C4×C67—22no—
91C4×C67—82no—
92C4×C67#45411yescharacteristic
93C4×C67#45511yescharacteristic
94Dic67—82no—
95Dic67—82no—
96#10517—671nomaximal, Sylow
97C2×C67—22no—
98C2×C67#204011yescharacteristic
99C2×C67#204711yescharacteristic
100C2×C67#205311yescharacteristic
101#3249—671no—
102#1832D6711yescharacteristic
103#1552—672no—
104#1506—672no—
105#1290—671no—
106C67#1051711yescharacteristic, complemented, Sylow
107#533—672no—
108#477—1342no—
109#467—1342no—
110#462—1342no—
111#432—672no—
112C2×C4⋊C8C2×D6711yescharacteristic
113C2×C4⋊C8C2×D6711yescharacteristic
114#376C2×D6711yescharacteristic
115#336—22no—
116#332—1342no—
117#332—1342no—
118#328—1344no—
119C2×Q16—1342no—
120C22×C8—21no—
121C22×C8C22⋊D6712yes—
122C4⋊Q8—672no—
123C4⋊Q8—672no—
124#125—1342no—
125C22⋊Q8—1342no—
126C22⋊Q8—1342no—
127C4×Q8—1342no—
128C4×Q8—1342no—
129#117—1342no—
130C2×C42—21no—
131C2×C42C22×D6711yescharacteristic
132#107—1342no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—1342no—
136#103—1344no—
137C22⋊C8—1342no—
138#97—1342no—
139#95—21no—
140#95D26812yes—
141C2×Q8—1342no—
142C2×Q8—1342no—
143C22×C4—22no—
144C22×C4C2×C22⋊D6711yescharacteristic
145C22×C4C2×D26811yescharacteristic
146C22×C4#1176162811yescharacteristic
147Q16—5362no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—41no—
152C2×C8—2682no—
153C4⋊C4—2682no—
154C4⋊C4—2682no—
155C4⋊C4—1342no—
156C4⋊C4—1342no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C22⋊C4—2682no—
160C42—41no—
161C42—41no—
162C42—2682no—
163C42D4×D6712yes—
164C42#1176162912yes—
165C23#1452401711yescharacteristic
166Q8—2682no—
167Q8—2682no—
168Q8—5362no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—2682no—
177C2×C4—2682no—
178C2×C4#1452401111yescharacteristic
179C2×C4#1452401812yes—
180C2×C4#1452402812yes—
181C2×C4#1452403211yescharacteristic
182C8—42no—
183C8—81no—
184C8—5362no—
185C22—21no—
186C22—21no—
187C22#1686331311yescharacteristic
188C22#1686367112yes—
189C4—21no—
190C4—22no—
191C4—22no—
192C4—82no—
193C4—5362no—
194C4—5362no—
195C4#1686356811yescharacteristic
196C4#1686357311yescharacteristic
197C2—22no—
198C2#2256143811yescharacteristic
199C2#2256147711yescharacteristic
200C2#2256151511yescharacteristic
subgroupG/H
centerC22#16863313
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1832C2
socleC22×C67#394
prime radicalC23×C67#121
hypercenter#1832D67
hypocenterC67#10517
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
106. C67#10517semidirect

Maximal subgroups

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

subgroupindex
#225715542
#225589072
#225587202
#225580792
C67×#18322
#1051767