Finite Group Databasebeta

← #30760330

Subgroups — #30760330

Order 17152.

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

normal subgroups only

This group has 374 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#22560378C211yesmaximal, characteristic
2#22560342C212yesmaximal
3#22558090C211yesmaximal, characteristic
4#22558090C212yesmaximal
5C67×#2844C211yesmaximal, characteristic
6C2×#14523976—21no—
7C2×#14523974—21no—
8C2×#14523954C2212yes—
9C2×#14523954C2211yescharacteristic
10#16863266C2211yescharacteristic
11#16863158—22no—
12#16863158—22no—
13#400×C67—22no—
14#400×C67—21no—
15#395×C67—22no—
16#395×C67—21no—
17#382×C67C2212yes—
18#382×C67C2211yescharacteristic
19C22×C67⋊C8—21no—
20C22×C67⋊C8—22no—
21C2×C4⋊Dic67—22no—
22C2×C4⋊Dic67—21no—
23#14523976—42no—
24#14523974—42no—
25#14523954—21no—
26#14523954—22no—
27#14523954Q812yescomplemented
28#14523954Q814yescomplemented
29C22×Q8×C67—21no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67D412yes—
33C2×C4⋊C4×C67D412yes—
34C2×C4⋊C4×C67D411yescharacteristic
35C2×C4⋊C4×C67D411yescharacteristic
36C2×C42×C67C2311yescharacteristic
37#95×C67—21no—
38#95×C67—22no—
39#95×C67—21no—
40#95×C67—22no—
41#95×C67—22no—
42#95×C67—22no—
43C22×Dic67—41no—
44C2×C67⋊C8—22no—
45C2×C67⋊C8—22no—
46C2×C67⋊C8—22no—
47C2×C67⋊C8—24no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—42no—
50C4⋊Dic67—41no—
51C4⋊Dic67—41no—
52C4⋊Dic67—44no—
53C2×Q8×C67—81no—
54C2×Q8×C67—82no—
55C22×C4×C67—21no—
56C22×C4×C67—22no—
57C22×C4×C67—21no—
58C22×C4×C67—22no—
59C22×C4×C67C2×D411yescharacteristic
60C22×C4×C67C2×D412yes—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—42no—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—41no—
65C4⋊C4×C67—81no—
66C4⋊C4×C67—82no—
67C42×C67C2×Q811yescharacteristic
68C42×C67C2×Q812yes—
69C42×C67#4111yescharacteristic
70C23×C67#12011yescharacteristic
71Q8×C67—84no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—21no—
76C2×C4×C67—41no—
77C2×C4×C67—41no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67—42no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67C22⋊Q811yescharacteristic
87C2×C4×C67C22⋊Q812yes—
88C2×C4×C67C4⋊Q811yescharacteristic
89C2×C4×C67C4⋊Q812yes—
90C2×Dic67—81no—
91C2×Dic67—82no—
92C67⋊C8—24no—
93C67⋊C8—44no—
94C22×C67#39511yescharacteristic
95C22×C67#45311yescharacteristic
96C22×C67#45412yes—
97C22×C67#45512yes—
98C22×C67#45611yescharacteristic
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—22no—
102C4×C67—22no—
103C4×C67—81no—
104C4×C67—81no—
105C4×C67—82no—
106C4×C67—82no—
107Dic67—161no—
108#10584—671nomaximal, Sylow
109C2×C67#165711yescharacteristic
110C2×C67#165911yescharacteristic
111C2×C67#166412yes—
112C2×C67#206011yescharacteristic
113C2×C67#206112yes—
114#2844D6711yescharacteristic
115#1847—671no—
116#1836—672no—
117#1295—672no—
118#1295—671no—
119C67#1058411yescharacteristic, complemented, Sylow
120C2×#107—1341no—
121C2×#106—1341no—
122C2×#97—672no—
123C2×#97—671no—
124#400—22no—
125#400—21no—
126#395—22no—
127#395—21no—
128#384—671no—
129#382C2×D6712yes—
130#382C2×D6711yescharacteristic
131#340—1342no—
132#340—1342no—
133C22×Q8—21no—
134C22×C8—1342no—
135C22×C8—1341no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—1342no—
139C2×C4⋊C4—1341no—
140C2×C4⋊C4C22⋊D6712yes—
141C2×C4⋊C4C22⋊D6712yes—
142C2×C4⋊C4C22⋊D6711yescharacteristic
143C2×C4⋊C4C22⋊D6711yescharacteristic
144C2×C42C22×D6711yescharacteristic
145#107—2682no—
146#106—2682no—
147#97—674no—
148#97—672no—
149#97—1342no—
150#97—1341no—
151#95—22no—
152#95—21no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—2681no—
164C22×C4C2×C22⋊D6711yescharacteristic
165C22×C4C2×C22⋊D6712yes—
166C2×C8—2682no—
167C2×C8—2682no—
168C2×C8—1344no—
169C2×C8—1342no—
170C2×C8—1342no—
171C2×C8—1342no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—2681no—
179C4⋊C4—2681no—
180C4⋊C4—2684no—
181C42Q8×D6711yescharacteristic
182C42Q8×D6712yes—
183C42#1176162911yescharacteristic
184C23#1452401211yescharacteristic
185Q8—84no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—5361no—
201C2×C4—5362no—
202C2×C4#1452401911yescharacteristic
203C2×C4#1452401912yes—
204C2×C4#1452404111yescharacteristic
205C2×C4#1452404112yes—
206C8—2684no—
207C8—1344no—
208C22#1686331511yescharacteristic
209C22#1686356411yescharacteristic
210C22#1686356912yes—
211C22#1686357412yes—
212C22#1686357911yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—10721no—
222C2#2255950711yescharacteristic
223C2#2255951511yescharacteristic
224C2#2255954312yes—
225C2#2256155111yescharacteristic
226C2#2256155612yes—
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2844C2
socle, prime radicalC23×C67#120
hypercenter#2844D67
hypocenterC67#10584
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
27. #14523954Q8semidirect
28. #14523954Q8semidirect
119. C67#10584semidirect

Maximal subgroups

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

subgroupindex
#225603782
#225603422
#225580902
C67×#28442
#1058467