Finite Group Databasebeta

← #30747269

Subgroups — #30747269

Order 17152.

684 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 147 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#22558604C214yesmaximal, complemented
2#22558523C212yesmaximal
3C2×#541×C67C211yesmaximal, characteristic
4#16863449C2214yes—
5Q8⋊(C67⋊C8)—28no—
6#16863122—22no—
7#16863120—22no—
8#541×C67—28no—
9C2×C4⋊Q8×C67C411yescharacteristic
10C2×#126×C67—22no—
11C2×#125×C67—22no—
12C2×#125×C67C411yescharacteristic
13C2×C22⋊Q8×C67C412yes—
14C2×C4×Q8×C67C2212yes—
15C2×#117×C67C2211yescharacteristic
16C4⋊(C67⋊C8)—24no—
17C22⋊(C67⋊C8)—24no—
18C4×C67⋊C8—24no—
19C22×Q8×C67C2×C412yes—
20C23×C4×C67C2×C411yescharacteristic
21C4⋊Q8×C67—24no—
22#126×C67—216no—
23#125×C67—216no—
24#125×C67—24no—
25C22⋊Q8×C67—24no—
26C22⋊Q8×C67—24no—
27C4×Q8×C67—24no—
28C4×Q8×C67D418yes—
29#117×C67—24no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67C2×C412yes—
35C2×C4⋊C4×C67D412yes—
36C2×C22⋊C4×C67—22no—
37C2×C22⋊C4×C67D412yes—
38C2×C42×C67—22no—
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67C2311yescharacteristic
41C2×C67⋊C8—44no—
42C24×C67C22⋊C411yescharacteristic
43C2×Q8×C67—22no—
44C2×Q8×C67—28no—
45C2×Q8×C67C22⋊C414yes—
46C22×C4×C67—22no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67—24no—
50C22×C4×C67C22⋊C411yescharacteristic
51C22×C4×C67C22⋊C412yes—
52C22×C4×C67C22×C411yescharacteristic
53C22×C4×C67C2×D412yes—
54C4⋊C4×C67—28no—
55C4⋊C4×C67—28no—
56C4⋊C4×C67—28no—
57C4⋊C4×C67—22no—
58C4⋊C4×C67—28no—
59C4⋊C4×C67—24no—
60C4⋊C4×C67C22⋊C414yes—
61C22⋊C4×C67—28no—
62C22⋊C4×C67—24no—
63C42×C67—22no—
64C42×C67—28no—
65C42×C67C2×D414yes—
66C23×C67—21no—
67C23×C67—22no—
68C23×C67—42no—
69C23×C67C2×C22⋊C411yescharacteristic
70Q8×C67—28no—
71Q8×C67—44no—
72C2×C4×C67—22no—
73C2×C4×C67—24no—
74C2×C4×C67—24no—
75C2×C4×C67—21no—
76C2×C4×C67—22no—
77C2×C4×C67—42no—
78C2×C4×C67—44no—
79C2×C4×C67—44no—
80C2×C4×C67—44no—
81C2×C4×C67—42no—
82C2×C4×C67C2×C22⋊C412yes—
83C2×C4×C67#12014yes—
84C2×C4×C67#13114yes—
85C67⋊C8—84no—
86C22×C67—22no—
87C22×C67—42no—
88C22×C67—41no—
89C22×C67—82no—
90C22×C67#37911yescharacteristic
91C22×C67#40812yes—
92C4×C67—24no—
93C4×C67—42no—
94C4×C67—44no—
95C4×C67—41no—
96C4×C67—44no—
97C4×C67#40814yes—
98#8327—671nomaximal, Sylow
99C2×C67—22no—
100C2×C67—81no—
101C2×C67#178011yescharacteristic
102C2×C67#179612yes—
103C2×#541D6711yescharacteristic
104#1479—674no—
105#1458—672no—
106C67#832711yescharacteristic, complemented, Sylow
107#541—28no—
108C2×C4⋊Q8Dic6711yescharacteristic
109C2×#126—22no—
110C2×#125—22no—
111C2×#125Dic6711yescharacteristic
112C2×C22⋊Q8Dic6712yes—
113C2×C4×Q8C2×D6712yes—
114C2×#117C2×D6711yescharacteristic
115#431—674no—
116Q8⋊C8—1348no—
117#324—1342no—
118#323—1342no—
119C22×Q8C2×Dic6712yes—
120C23×C4C2×Dic6711yescharacteristic
121C4⋊Q8—24no—
122#126—216no—
123#125—216no—
124#125—24no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C4×Q8—24no—
128C4×Q8C22⋊D6718yes—
129#117—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2×Dic6712yes—
135C2×C4⋊C4C22⋊D6712yes—
136C2×C22⋊C4—22no—
137C2×C22⋊C4C22⋊D6712yes—
138C2×C42—22no—
139C2×C42C2×Dic6711yescharacteristic
140C2×C42C22×D6711yescharacteristic
141C4⋊C8—1344no—
142C22⋊C8—1344no—
143C4×C8—1344no—
144C24C22⋊Dic6711yescharacteristic
145C2×Q8—22no—
146C2×Q8—28no—
147C2×Q8C22⋊Dic6714yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4C22⋊Dic6711yescharacteristic
153C22×C4C22⋊Dic6712yes—
154C22×C4C22×Dic6711yescharacteristic
155C22×C4C2×C22⋊D6712yes—
156C2×C8—2684no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4C22⋊Dic6714yes—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C42—22no—
167C42—28no—
168C42C2×C22⋊D6714yes—
169C23—21no—
170C23—22no—
171C23—42no—
172C23C2×C22⋊Dic6711yescharacteristic
173Q8—28no—
174Q8—44no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4C2×C22⋊Dic6712yes—
186C2×C4#1452401214yes—
187C2×C4#1452405014yes—
188C8—5364no—
189C22—22no—
190C22—41no—
191C22—42no—
192C22—82no—
193C22#1686325211yescharacteristic
194C22#1686336312yes—
195C4—24no—
196C4—41no—
197C4—44no—
198C4—44no—
199C4—42no—
200C4#1686336314yes—
201C2—22no—
202C2—81no—
203C2#2256012111yescharacteristic
204C2#2256018412yes—
subgroupG/H
centerC22#16863252
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC2×#541×C67C2
socleC22×C67#379
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenterC2×#541D67
hypocenterC67#8327
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. #22558604C2semidirect
106. C67#8327semidirect

Maximal subgroups

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

subgroupindex
#225586042
#225585232
C2×#541×C672
#832767