Finite Group Databasebeta

← #30748856

Subgroups — #30748856

Order 17152.

404 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 101 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#22568920C211yesmaximal, characteristic
2#22558967C212yesmaximal, complemented
3#22558942C212yesmaximal
4C67⋊#1309C211yesmaximal, characteristic, complemented
5C67×#1309C211yesmaximal, characteristic, complemented
6C2×#14524051C412yes—
7#16863946C2211yescharacteristic
8#16863473—22no—
9#16863461—22no—
10#16863420C2212yes—
11#16863364C412yes—
12C2×C4×C67⋊C8C2211yescharacteristic
13#16863148—22no—
14#16863148—22no—
15#16863144—22no—
16#16863139—22no—
17#423×C67C2212yes—
18#423×C67C2211yescharacteristic
19C2×#11761628C2×C412yes—
20#14524051—44no—
21C2×#11761612—22no—
22C2×#11761611—22no—
23C22×C67⋊C8C2×C411yescharacteristic
24C22×C67⋊C8C2×C411yescharacteristic
25C2×C8×D67—22no—
26#14524043D411yescharacteristic
27#14524040D411yescharacteristic
28#14524031Q812yes—
29#14524017—22no—
30C4×D268—22no—
31C4⋊(C67⋊C8)—22no—
32C4⋊(C67⋊C8)—22no—
33(C2×D67)⋊C8—24no—
34C4×C67⋊C8D412yes—
35C4×C67⋊C8D412yes—
36C2×M16×C67—22no—
37C2×M16×C67C2×C412yes—
38C2×C42×C67C2311yescharacteristic
39C4⋊C8×C67—22no—
40C4⋊C8×C67—22no—
41C4⋊C8×C67—22no—
42#11761628—42no—
43#11761628—42no—
44C2×Dic134C4⋊C412yes—
45C2×D268—22no—
46C2×D268C4⋊C412yes—
47C2×C22⋊D67—22no—
48C2×C4×D67—22no—
49#11761612—82no—
50#11761611—44no—
51C2×C67⋊C8—22no—
52C2×C67⋊C8—41no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8C22⋊C412yes—
55C2×C67⋊C8C22⋊C411yescharacteristic
56C2×C67⋊C8C22⋊C411yescharacteristic
57C8×D67—44no—
58C4⋊Dic67—22no—
59(C2×D67)⋊C4—24no—
60C22×C4×C67C22×C411yescharacteristic
61C22×C4×C67#4112yes—
62M16×C67—42no—
63M16×C67—82no—
64C2×C8×C67—22no—
65C2×C8×C67—42no—
66C42×C67C2×D412yes—
67C42×C67C2×D411yescharacteristic
68C42×C67C2×Q811yescharacteristic
69C23×C67#11711yescharacteristic
70C22×D67—22no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67C2×C22⋊C411yescharacteristic
76C2×C4×C67C2×C4⋊C411yescharacteristic
77C2×C4×C67#12112yes—
78C2×C4×C67C22⋊Q812yes—
79C22⋊D67—44no—
80C2×Dic67—22no—
81D268—42no—
82D268—42no—
83D268—42no—
84C4×D67—42no—
85C4×D67—42no—
86Dic134—42no—
87C8×C67—42no—
88C8×C67—82no—
89C67⋊C8—21no—
90C67⋊C8—22no—
91C67⋊C8—82no—
92C67⋊C8#10012yes—
93C22×C67—21no—
94C22×C67—22no—
95C22×C67#38011yescharacteristic
96C2×D67—42no—
97C2×D67—44no—
98C4×C67—21no—
99C4×C67—22no—
100C4×C67C2×#10011yescharacteristic
101C4×C67#41311yescharacteristic
102C4×C67#47212yes—
103C4×C67#47312yes—
104Dic67—42no—
105#8638—671nomaximal, Sylow
106C2×C67—41no—
107C2×C67#180011yescharacteristic
108C2×C67#181612yes—
109D67—82no—
110#2931—671no—
111#1567—672no—
112#1560—672no—
113#1309—671no—
114#1309D6711yescharacteristic, complemented
115C67#863811yescharacteristic, complemented, Sylow
116C2×#131—672no—
117#532—671no—
118#436—1342no—
119C8×D4—1342no—
120#423—672no—
121#423C2×D6712yes—
122#423C2×D6711yescharacteristic
123#408—672no—
124C2×C4×C8—671no—
125#335—1342no—
126#335—1342no—
127#333—1342no—
128#331—1342no—
129C2×#41—672no—
130#131—2684no—
131C2×M16—22no—
132C2×M16—1342no—
133C2×M16—1342no—
134C2×M16C4×D6712yes—
135C22×C8—671no—
136C22×C8—671no—
137C22×C8—1342no—
138C4⋊Q8—671no—
139#127—671no—
140#124—672no—
141#121—1342no—
142C4×D4—1342no—
143C2×C42C22×D6711yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—1342no—
148C4⋊C8—1342no—
149C22⋊C8—1344no—
150C4×C8—672no—
151C4×C8—672no—
152#41—2682no—
153#41—2682no—
154C2×Q8—672no—
155C2×D4—672no—
156C2×D4—1342no—
157C2×D4—1342no—
158C22×C4—1342no—
159C22×C4C2×C4×D6711yescharacteristic
160C22×C4#1176162812yes—
161M16—42no—
162M16—82no—
163M16—2684no—
164M16—5362no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—2684no—
168C2×C8—2682no—
169C2×C8—2681no—
170C2×C8—671no—
171C2×C8—672no—
172C2×C8—671no—
173C2×C8—1342no—
174C4⋊C4—1342no—
175C22⋊C4—1344no—
176C42D4×D6712yes—
177C42D4×D6711yescharacteristic
178C42Q8×D6711yescharacteristic
179C23—1342no—
180C23#1452400011yescharacteristic
181Q8—2682no—
182D4—2682no—
183D4—2684no—
184D4—2682no—
185D4—2682no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—2682no—
191C2×C4—2682no—
192C2×C4—1342no—
193C2×C4C22⋊C4×D6711yescharacteristic
194C2×C4C4⋊C4×D6711yescharacteristic
195C2×C4#1452401612yes—
196C2×C4#1452402112yes—
197C8—42no—
198C8—82no—
199C8—5362no—
200C8—672no—
201C8—1341no—
202C8—1342no—
203C22—22no—
204C22—21no—
205C22—2684no—
206C22—2682no—
207C22#1686325511yescharacteristic
208C4—22no—
209C4—21no—
210C4—2682no—
211C4#100×D6711yescharacteristic
212C4#1686337911yescharacteristic
213C4#1686366512yes—
214C4#1686367012yes—
215C2—41no—
216C2—5362no—
217C2#2256020011yescharacteristic
218C2#2256027212yes—
subgroupG/H
centerC22#16863255
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1309C2
socleC22×C67#380
prime radical#16863946C22
cosocleC2×C42×C67C23
hypercenter#1309D67
hypocenterC67#8638
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #22558967C2semidirect
4. C67⋊#1309C2semidirect
5. C67×#1309C2semidirect
114. #1309D67semidirect
115. C67#8638semidirect

Maximal subgroups

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

subgroupindex
#225689202
#225589672
#225589422
C67⋊#13092
C67×#13092
#863867