Finite Group Databasebeta

← #30747266

Subgroups — #30747266

Order 17152.

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

normal subgroups only

This group has 812 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#22558600C214yesmaximal, complemented
2#22558523C212yesmaximal, complemented
3C2×#540×C67C211yesmaximal, characteristic
4#16863449C2214yes—
5D4⋊(C67⋊C8)—28no—
6#16863122—22no—
7#16863120—22no—
8#540×C67—28no—
9C2×#127×C67C411yescharacteristic
10C2×#125×C67C411yescharacteristic
11C2×#123×C67—22no—
12C2×#121×C67—22no—
13C2×#121×C67C412yes—
14C2×C4×D4×C67C2212yes—
15C2×#117×C67C2211yescharacteristic
16C4⋊(C67⋊C8)—24no—
17C22⋊(C67⋊C8)—24no—
18C4×C67⋊C8—24no—
19C22×D4×C67—22no—
20C22×D4×C67—22no—
21C22×D4×C67C2×C412yes—
22C23×C4×C67—22no—
23C23×C4×C67C2×C411yescharacteristic
24#127×C67—24no—
25#125×C67—24no—
26#123×C67—216no—
27#121×C67—216no—
28#121×C67—24no—
29#121×C67—24no—
30C4×D4×C67—24no—
31C4×D4×C67D418yes—
32#117×C67—24no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67C2×C412yes—
35C2×C4⋊C4×C67D412yes—
36C2×C22⋊C4×C67—22no—
37C2×C22⋊C4×C67—22no—
38C2×C22⋊C4×C67D412yes—
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67C2311yescharacteristic
41C2×C67⋊C8—44no—
42C24×C67—22no—
43C24×C67C22⋊C411yescharacteristic
44C2×D4×C67—28no—
45C2×D4×C67—28no—
46C2×D4×C67—22no—
47C2×D4×C67—28no—
48C2×D4×C67—84no—
49C2×D4×C67—84no—
50C2×D4×C67C22⋊C414yes—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—24no—
54C22×C4×C67—28no—
55C22×C4×C67—84no—
56C22×C4×C67C22⋊C411yescharacteristic
57C22×C4×C67C22⋊C412yes—
58C22×C4×C67C22×C411yescharacteristic
59C22×C4×C67C2×D412yes—
60C4⋊C4×C67—28no—
61C4⋊C4×C67—22no—
62C4⋊C4×C67—24no—
63C4⋊C4×C67C22⋊C414yes—
64C22⋊C4×C67—28no—
65C22⋊C4×C67—28no—
66C22⋊C4×C67—24no—
67C42×C67—22no—
68C42×C67C2×D414yes—
69C23×C67—21no—
70C23×C67—24no—
71C23×C67—22no—
72C23×C67—44no—
73C23×C67—44no—
74C23×C67—42no—
75C23×C67—44no—
76C23×C67C2×C22⋊C411yescharacteristic
77D4×C67—28no—
78D4×C67—44no—
79D4×C67—88no—
80D4×C67—88no—
81C2×C4×C67—22no—
82C2×C4×C67—21no—
83C2×C4×C67—24no—
84C2×C4×C67—22no—
85C2×C4×C67—42no—
86C2×C4×C67—44no—
87C2×C4×C67—44no—
88C2×C4×C67—42no—
89C2×C4×C67—88no—
90C2×C4×C67C2×C22⋊C412yes—
91C2×C4×C67#12014yes—
92C2×C4×C67#13114yes—
93C67⋊C8—84no—
94C22×C67—22no—
95C22×C67—44no—
96C22×C67—42no—
97C22×C67—44no—
98C22×C67—41no—
99C22×C67—44no—
100C22×C67—82no—
101C22×C67—88no—
102C22×C67#37911yescharacteristic
103C22×C67#40812yes—
104C4×C67—24no—
105C4×C67—44no—
106C4×C67—42no—
107C4×C67—41no—
108C4×C67#40814yes—
109#8326—671nomaximal, Sylow
110C2×C67—22no—
111C2×C67—81no—
112C2×C67—84no—
113C2×C67#178011yescharacteristic
114C2×C67#179512yes—
115C2×#540D6711yescharacteristic
116#1478—674no—
117#1458—672no—
118C67#832611yescharacteristic, complemented, Sylow
119#540—28no—
120C2×#127Dic6711yescharacteristic
121C2×#125Dic6711yescharacteristic
122C2×#123—22no—
123C2×#121—22no—
124C2×#121Dic6712yes—
125C2×C4×D4C2×D6712yes—
126C2×#117C2×D6711yescharacteristic
127#431—674no—
128D4⋊C8—1348no—
129#324—1342no—
130#323—1342no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C2×Dic6712yes—
134C23×C4—22no—
135C23×C4C2×Dic6711yescharacteristic
136#127—24no—
137#125—24no—
138#123—216no—
139#121—216no—
140#121—24no—
141#121—24no—
142C4×D4—24no—
143C4×D4C22⋊D6718yes—
144#117—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C2×Dic6712yes—
147C2×C4⋊C4C22⋊D6712yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4C22⋊D6712yes—
151C2×C42C2×Dic6711yescharacteristic
152C2×C42C22×D6711yescharacteristic
153C4⋊C8—1344no—
154C22⋊C8—1344no—
155C4×C8—1344no—
156C24—22no—
157C24C22⋊Dic6711yescharacteristic
158C2×D4—28no—
159C2×D4—28no—
160C2×D4—22no—
161C2×D4—28no—
162C2×D4—84no—
163C2×D4—84no—
164C2×D4C22⋊Dic6714yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—84no—
170C22×C4C22⋊Dic6711yescharacteristic
171C22×C4C22⋊Dic6712yes—
172C22×C4C22×Dic6711yescharacteristic
173C22×C4C2×C22⋊D6712yes—
174C2×C8—2684no—
175C4⋊C4—28no—
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4C22⋊Dic6714yes—
179C22⋊C4—28no—
180C22⋊C4—28no—
181C22⋊C4—24no—
182C42—22no—
183C42C2×C22⋊D6714yes—
184C23—21no—
185C23—24no—
186C23—22no—
187C23—44no—
188C23—44no—
189C23—42no—
190C23—44no—
191C23C2×C22⋊Dic6711yescharacteristic
192D4—28no—
193D4—44no—
194D4—88no—
195D4—88no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—88no—
205C2×C4C2×C22⋊Dic6712yes—
206C2×C4#1452401214yes—
207C2×C4#1452405014yes—
208C8—5364no—
209C22—22no—
210C22—41no—
211C22—44no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—82no—
216C22—88no—
217C22#1686325211yescharacteristic
218C22#1686336312yes—
219C4—24no—
220C4—41no—
221C4—44no—
222C4—42no—
223C4#1686336314yes—
224C2—22no—
225C2—84no—
226C2—81no—
227C2#2256012111yescharacteristic
228C2#2256017912yes—
subgroupG/H
centerC22#16863252
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
Fitting, prime radicalC2×#540×C67C2
socleC22×C67#379
cosocleC2×C42×C67C23
hypercenterC2×#540D67
hypocenterC67#8326
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. #22558600C2semidirect
2. #22558523C2semidirect
118. C67#8326semidirect

Maximal subgroups

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

subgroupindex
#225586002
#225585232
C2×#540×C672
#832667