Finite Group Databasebeta

← #30760287

Subgroups — #30760287

Order 17152.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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#22560967C212yesmaximal
2#22560365C211yesmaximal, characteristic
3#22558092C211yesmaximal, characteristic
4#22558050C212yesmaximal
5C67×#2571C211yesmaximal, characteristic
6C2×#14523976—22no—
7C2×#14523974—22no—
8C2×C4⋊(C67⋊C8)C2212yes—
9C2×C4×C67⋊C8C2211yescharacteristic
10#16863266C2211yescharacteristic
11#16863158—22no—
12#400×C67—22no—
13#389×C67—22no—
14#384×C67C2211yescharacteristic
15#382×C67—22no—
16C4×C4⋊C4×C67C2212yes—
17C22×C67⋊C8—21no—
18C22×C67⋊C8—22no—
19C2×C4⋊Dic67—22no—
20C2×C4⋊Dic67—21no—
21#14523976—44no—
22#14523974—44no—
23C4⋊(C67⋊C8)—22no—
24C4⋊(C67⋊C8)Q814yes—
25C4×C67⋊C8—21no—
26C4×C67⋊C8Q812yes—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67—22no—
29C2×C4⋊C4×C67D412yes—
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67D412yes—
32C2×C42×C67—22no—
33C2×C42×C67C2311yescharacteristic
34#95×C67—22no—
35#95×C67—24no—
36#95×C67—22no—
37C22×Dic67—41no—
38C2×C67⋊C8—22no—
39C2×C67⋊C8—22no—
40C2×C67⋊C8—22no—
41C2×C67⋊C8—24no—
42C2×C67⋊C8—42no—
43C2×C67⋊C8—42no—
44C4⋊Dic67—42no—
45C4⋊Dic67—42no—
46C4⋊Dic67—42no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67—22no—
50C22×C4×C67C2×D411yescharacteristic
51C22×C4×C67C2×D412yes—
52C4⋊C4×C67—22no—
53C4⋊C4×C67—22no—
54C4⋊C4×C67—22no—
55C4⋊C4×C67—22no—
56C4⋊C4×C67—42no—
57C4⋊C4×C67—42no—
58C4⋊C4×C67—42no—
59C4⋊C4×C67—82no—
60C42×C67—42no—
61C42×C67—42no—
62C42×C67C2×Q811yescharacteristic
63C42×C67C2×Q812yes—
64C42×C67#4111yescharacteristic
65C23×C67#12011yescharacteristic
66C2×C4×C67—21no—
67C2×C4×C67—21no—
68C2×C4×C67—42no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—42no—
72C2×C4×C67—44no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67—42no—
76C2×C4×C67C22⋊Q811yescharacteristic
77C2×C4×C67C22⋊Q812yes—
78C2×C4×C67C4⋊Q811yescharacteristic
79C2×C4×C67C4⋊Q812yes—
80C2×C4×C67#13512yes—
81C2×C4×C67#13512yes—
82C2×C4×C67AGL(1,ℤ/8)12yes—
83C2×C4×C67#13712yes—
84C2×Dic67—82no—
85C2×Dic67—81no—
86C67⋊C8—24no—
87C67⋊C8—44no—
88C22×C67#39511yescharacteristic
89C22×C67#44912yes—
90C22×C67#45212yes—
91C22×C67#45311yescharacteristic
92C22×C67#45611yescharacteristic
93C4×C67—21no—
94C4×C67—21no—
95C4×C67—82no—
96C4×C67—82no—
97C4×C67—82no—
98C4×C67#47812yes—
99C4×C67#47912yes—
100C4×C67#49912yes—
101C4×C67#50112yes—
102Dic67—161no—
103#10577—671nomaximal, Sylow
104C2×C67#162411yescharacteristic
105C2×C67#163011yescharacteristic
106C2×C67#165212yes—
107C2×C67#205912yes—
108C2×C67#206011yescharacteristic
109#2571D6711yescharacteristic
110#1947—672no—
111#1844—671no—
112#1296—671no—
113#1279—672no—
114C67#1057711yescharacteristic, complemented, Sylow
115C2×#107—1342no—
116C2×#106—1342no—
117C2×C4⋊C8—672no—
118C2×C4×C8—671no—
119#400—22no—
120#389—22no—
121#384—671no—
122#384C2×D6711yescharacteristic
123#382—22no—
124C4×C4⋊C4C2×D6712yes—
125#340—1342no—
126C22×C8—1342no—
127C22×C8—1341no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—1342no—
131C2×C4⋊C4—1341no—
132C2×C4⋊C4C22⋊D6712yes—
133C2×C4⋊C4C22⋊D6712yes—
134C2×C4⋊C4C22⋊D6712yes—
135C2×C42—22no—
136C2×C42C22×D6711yescharacteristic
137#107—2684no—
138#106—2684no—
139C4⋊C8—674no—
140C4⋊C8—1342no—
141C4×C8—672no—
142C4×C8—1341no—
143#95—22no—
144#95—24no—
145#95—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—2681no—
150C22×C4C2×C22⋊D6711yescharacteristic
151C22×C4C2×C22⋊D6712yes—
152C2×C8—2682no—
153C2×C8—2682no—
154C2×C8—1344no—
155C2×C8—1342no—
156C2×C8—1342no—
157C2×C8—1342no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—2682no—
167C4⋊C4—2682no—
168C4⋊C4—2682no—
169C42—42no—
170C42—42no—
171C42Q8×D6711yescharacteristic
172C42Q8×D6712yes—
173C42#1176162911yescharacteristic
174C23#1452401211yescharacteristic
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—5362no—
186C2×C4—5361no—
187C2×C4#1452401911yescharacteristic
188C2×C4#1452401912yes—
189C2×C4#1452404111yescharacteristic
190C2×C4#1452404112yes—
191C2×C4#1452406312yes—
192C2×C4#1452406312yes—
193C2×C4#1452406812yes—
194C2×C4#1452407312yes—
195C8—2684no—
196C8—1344no—
197C22#1686331511yescharacteristic
198C22#1686354112yes—
199C22#1686355812yes—
200C22#1686356411yescharacteristic
201C22#1686357911yescharacteristic
202C4—21no—
203C4—21no—
204C4—82no—
205C4—82no—
206C4—82no—
207C4—10721no—
208C4#1686369312yes—
209C4#1686370012yes—
210C4#1686379512yes—
211C4#1686380612yes—
212C2#2255929211yescharacteristic
213C2#2255933011yescharacteristic
214C2#2255947912yes—
215C2#2256154412yes—
216C2#2256155111yescharacteristic
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2571C2
socle, prime radicalC23×C67#120
hypercenter#2571D67
hypocenterC67#10577
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
114. C67#10577semidirect

Maximal subgroups

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

subgroupindex
#225609672
#225603652
#225580922
#225580502
C67×#25712
#1057767