Finite Group Databasebeta

← #30760335

Subgroups — #30760335

Order 17152.

382 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 97 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#22560378C212yesmaximal
2#22560342C211yesmaximal, characteristic
3#22558090C212yesmaximal
4#22558090C211yesmaximal, characteristic
5C67×#2845C211yesmaximal, characteristic
6C2×#14523976—22no—
7C2×#14523974—22no—
8C2×#14523954C2211yescharacteristic
9C2×#14523954C2212yes—
10#16863266C2211yescharacteristic
11#16863158—22no—
12#398×C67—22no—
13#398×C67—22no—
14#398×C67—22no—
15#382×C67C2211yescharacteristic
16#382×C67C2212yes—
17C22×C67⋊C8—22no—
18C22×C67⋊C8—21no—
19C2×C4⋊Dic67—21no—
20C2×C4⋊Dic67—22no—
21#14523976—44no—
22#14523974—44no—
23#14523954—22no—
24#14523954—21no—
25#14523954Q814yes—
26#14523954Q812yes—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67—22no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67D412yes—
32C2×C4⋊C4×C67D412yes—
33C2×C42×C67C2311yescharacteristic
34#95×C67—22no—
35#95×C67—22no—
36#95×C67—22no—
37#95×C67—22no—
38C22×Dic67—41no—
39C2×C67⋊C8—24no—
40C2×C67⋊C8—24no—
41C2×C67⋊C8—24no—
42C2×C67⋊C8—22no—
43C2×C67⋊C8—41no—
44C2×C67⋊C8—41no—
45C4⋊Dic67—42no—
46C4⋊Dic67—42no—
47C4⋊Dic67—42no—
48C22×C4×C67—22no—
49C22×C4×C67—22no—
50C22×C4×C67—22no—
51C22×C4×C67C2×D412yes—
52C22×C4×C67C2×D411yescharacteristic
53C4⋊C4×C67—42no—
54C4⋊C4×C67—42no—
55C4⋊C4×C67—42no—
56C4⋊C4×C67—82no—
57C4⋊C4×C67—82no—
58C4⋊C4×C67—82no—
59C42×C67C2×Q812yes—
60C42×C67C2×Q811yescharacteristic
61C42×C67#4111yescharacteristic
62C23×C67#12011yescharacteristic
63C2×C4×C67—22no—
64C2×C4×C67—21no—
65C2×C4×C67—21no—
66C2×C4×C67—22no—
67C2×C4×C67—42no—
68C2×C4×C67—42no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—42no—
72C2×C4×C67—42no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67—42no—
76C2×C4×C67C22⋊Q812yes—
77C2×C4×C67C22⋊Q811yescharacteristic
78C2×C4×C67C4⋊Q812yes—
79C2×C4×C67C4⋊Q811yescharacteristic
80C2×Dic67—82no—
81C2×Dic67—81no—
82C67⋊C8—28no—
83C67⋊C8—42no—
84C22×C67#39511yescharacteristic
85C22×C67#45312yes—
86C22×C67#45411yescharacteristic
87C22×C67#45511yescharacteristic
88C22×C67#45612yes—
89C4×C67—22no—
90C4×C67—22no—
91C4×C67—21no—
92C4×C67—21no—
93C4×C67—82no—
94C4×C67—82no—
95C4×C67—82no—
96Dic67—161no—
97#10585—671nomaximal, Sylow
98C2×C67#165611yescharacteristic
99C2×C67#166112yes—
100C2×C67#166311yescharacteristic
101C2×C67#206012yes—
102C2×C67#206111yescharacteristic
103#2845D6711yescharacteristic
104#1847—672no—
105#1836—671no—
106#1295—671no—
107#1295—672no—
108C67#1058511yescharacteristic, complemented, Sylow
109C2×#107—1342no—
110C2×#106—1342no—
111C2×#97—671no—
112C2×#97—672no—
113#398—22no—
114#398—22no—
115#398—22no—
116#384—671no—
117#382C2×D6711yescharacteristic
118#382C2×D6712yes—
119#340—1342no—
120C22×C8—1341no—
121C22×C8—1342no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—1341no—
126C2×C4⋊C4—1342no—
127C2×C4⋊C4C22⋊D6712yes—
128C2×C4⋊C4C22⋊D6712yes—
129C2×C4⋊C4C22⋊D6712yes—
130C2×C42C22×D6711yescharacteristic
131#107—2684no—
132#106—2684no—
133#97—672no—
134#97—674no—
135#97—1341no—
136#97—1342no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—2681no—
145C22×C4C2×C22⋊D6712yes—
146C22×C4C2×C22⋊D6711yescharacteristic
147C2×C8—2681no—
148C2×C8—2681no—
149C2×C8—1342no—
150C2×C8—1344no—
151C2×C8—1344no—
152C2×C8—1344no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—2682no—
160C4⋊C4—2682no—
161C4⋊C4—2682no—
162C42Q8×D6712yes—
163C42Q8×D6711yescharacteristic
164C42#1176162911yescharacteristic
165C23#1452401211yescharacteristic
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—5362no—
180C2×C4—5361no—
181C2×C4#1452401912yes—
182C2×C4#1452401911yescharacteristic
183C2×C4#1452404112yes—
184C2×C4#1452404111yescharacteristic
185C8—2682no—
186C8—1348no—
187C22#1686331511yescharacteristic
188C22#1686356412yes—
189C22#1686356911yescharacteristic
190C22#1686357411yescharacteristic
191C22#1686357912yes—
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—82no—
197C4—82no—
198C4—82no—
199C4—10721no—
200C2#2255950211yescharacteristic
201C2#2255952612yes—
202C2#2255953711yescharacteristic
203C2#2256155112yes—
204C2#2256155611yescharacteristic
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2845C2
socle, prime radicalC23×C67#120
hypercenter#2845D67
hypocenterC67#10585
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2845

Complements

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

NH ≅ G/Ntype
108. C67#10585semidirect

Maximal subgroups

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

subgroupindex
#225603782
#225603422
#225580902
C67×#28452
#1058567