Finite Group Databasebeta

← #30759925

Subgroups — #30759925

Order 17152.

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

normal subgroups only

This group has 364 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×#1843C211yesmaximal, 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×C67—22no—
15C2×C4⋊C8×C67C2211yescharacteristic
16C2×C4⋊C8×C67C2211yescharacteristic
17#376×C67C2211yescharacteristic
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—
38C4⋊C8×C67—44no—
39#95×C67—21no—
40#95×C67D412yes—
41C2×Dic134—22no—
42C2×Dic134—22no—
43Dic268—82no—
44C2×C67⋊C8—42no—
45C4⋊Dic67—22no—
46C4⋊Dic67—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67—22no—
49Dic67⋊C4—42no—
50Dic67⋊C4—42no—
51C22⋊Dic67—42no—
52C4×Dic67—42no—
53C22×C4×C67—22no—
54C22×C4×C67C2×D411yescharacteristic
55C22×C4×C67C2×D411yescharacteristic
56C22×C4×C67#4111yescharacteristic
57C2×C8×C67—22no—
58C2×C8×C67—42no—
59C2×C8×C67—42no—
60C2×C8×C67—41no—
61C42×C67—22no—
62C42×C67—22no—
63C42×C67C2×D412yes—
64C42×C67#4112yes—
65C23×C67#12111yescharacteristic
66C2×C4×C67—22no—
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—24no—
70C2×C4×C67—41no—
71C2×C4×C67—42no—
72C2×C4×C67—42no—
73C2×C4×C67#12011yescharacteristic
74C2×C4×C67#12112yes—
75C2×C4×C67#12312yes—
76C2×C4×C67#12411yescharacteristic
77C2×Dic67—42no—
78C2×Dic67—42no—
79Dic134—42no—
80Dic134—42no—
81Dic134—82no—
82C8×C67—42no—
83C8×C67—81no—
84C67⋊C8—82no—
85C22×C67—21no—
86C22×C67—21no—
87C22×C67#39411yescharacteristic
88C22×C67#47312yes—
89C4×C67—22no—
90C4×C67—21no—
91C4×C67—22no—
92C4×C67—44no—
93C4×C67#45311yescharacteristic
94C4×C67#45611yescharacteristic
95Dic67—82no—
96Dic67—82no—
97#10516—671nomaximal, Sylow
98C2×C67—22no—
99C2×C67#203911yescharacteristic
100C2×C67#204711yescharacteristic
101C2×C67#205311yescharacteristic
102#3249—671no—
103#1843D6711yescharacteristic
104#1552—672no—
105#1506—672no—
106#1290—671no—
107C67#1051611yescharacteristic, complemented, Sylow
108#533—672no—
109#477—1342no—
110#467—1342no—
111#462—1342no—
112#432—672no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8C2×D6711yescharacteristic
115C2×C4⋊C8C2×D6711yescharacteristic
116#376C2×D6711yescharacteristic
117#332—1342no—
118#332—1342no—
119#328—1344no—
120C2×Q16—1342no—
121C22×C8—21no—
122C22×C8C22⋊D6712yes—
123C4⋊Q8—672no—
124C4⋊Q8—672no—
125#125—1342no—
126C22⋊Q8—1342no—
127C22⋊Q8—1342no—
128C4×Q8—1342no—
129C4×Q8—1342no—
130#117—1342no—
131C2×C42—21no—
132C2×C42C22×D6711yescharacteristic
133#107—1342no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—44no—
137C4⋊C8—1342no—
138#103—1344no—
139C22⋊C8—1342no—
140#97—1342no—
141#95—21no—
142#95D26812yes—
143C2×Q8—1342no—
144C2×Q8—1342no—
145C22×C4—22no—
146C22×C4C2×C22⋊D6711yescharacteristic
147C22×C4C2×D26811yescharacteristic
148C22×C4#1176162811yescharacteristic
149Q16—5362no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—2682no—
155C4⋊C4—2682no—
156C4⋊C4—2682no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C4⋊C4—1342no—
160C4⋊C4—1342no—
161C22⋊C4—2682no—
162C42—22no—
163C42—22no—
164C42—2682no—
165C42D4×D6712yes—
166C42#1176162912yes—
167C23#1452401711yescharacteristic
168Q8—2682no—
169Q8—2682no—
170Q8—5362no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—2682no—
179C2×C4—2682no—
180C2×C4#1452401111yescharacteristic
181C2×C4#1452401812yes—
182C2×C4#1452402812yes—
183C2×C4#1452403211yescharacteristic
184C8—42no—
185C8—81no—
186C8—5362no—
187C22—21no—
188C22—21no—
189C22#1686331311yescharacteristic
190C22#1686367112yes—
191C4—21no—
192C4—22no—
193C4—22no—
194C4—44no—
195C4—5362no—
196C4—5362no—
197C4#1686356311yescharacteristic
198C4#1686357811yescharacteristic
199C2—22no—
200C2#2256143311yescharacteristic
201C2#2256147711yescharacteristic
202C2#2256151511yescharacteristic
subgroupG/H
centerC22#16863313
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1843C2
socleC22×C67#394
prime radicalC23×C67#121
hypercenter#1843D67
hypocenterC67#10516
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1843

Complements

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

NH ≅ G/Ntype
107. C67#10516semidirect

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×#18432
#1051667