Finite Group Databasebeta

← #30749172

Subgroups — #30749172

Order 17152.

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

normal subgroups only

This group has 830 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#22560119C212yesmaximal
2#22558052C214yesmaximal, complemented
3C2×#380×C67C211yesmaximal, characteristic
4C2×C22⋊(C67⋊C8)—22no—
5C2×C22⋊(C67⋊C8)—22no—
6C2×C22⋊(C67⋊C8)C2214yes—
7C24×C4×C67C2211yescharacteristic
8C22×C4⋊C4×C67C412yes—
9C22×C22⋊C4×C67C412yes—
10#380×C67—22no—
11#380×C67—22no—
12C2×#95×C67C2212yes—
13C22×C67⋊C8—24no—
14C22⋊(C67⋊C8)—28no—
15C22⋊(C67⋊C8)—44no—
16C22⋊(C67⋊C8)—44no—
17C25×C67C2×C411yescharacteristic
18C23×C4×C67—24no—
19C23×C4×C67—44no—
20C23×C4×C67C2×C411yescharacteristic
21C23×C4×C67C2×C414yes—
22C23×C4×C67D412yes—
23C23×C4×C67D412yes—
24C23×C4×C67C2311yescharacteristic
25C2×C4⋊C4×C67—22no—
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67—44no—
28C2×C22⋊C4×C67—22no—
29C2×C22⋊C4×C67—22no—
30C2×C22⋊C4×C67—44no—
31#95×C67—24no—
32C2×C67⋊C8—44no—
33C2×C67⋊C8—44no—
34C2×C67⋊C8—44no—
35C24×C67—22no—
36C24×C67—22no—
37C24×C67—22no—
38C24×C67—24no—
39C24×C67—42no—
40C24×C67C22⋊C411yescharacteristic
41C24×C67C22⋊C411yescharacteristic
42C24×C67C22×C411yescharacteristic
43C22×C4×C67—22no—
44C22×C4×C67—24no—
45C22×C4×C67—24no—
46C22×C4×C67—24no—
47C22×C4×C67—24no—
48C22×C4×C67—24no—
49C22×C4×C67—24no—
50C22×C4×C67—28no—
51C22×C4×C67—42no—
52C22×C4×C67—42no—
53C22×C4×C67—42no—
54C22×C4×C67—416no—
55C22×C4×C67—44no—
56C22×C4×C67—44no—
57C22×C4×C67C22⋊C412yes—
58C22×C4×C67C2×D412yes—
59C22×C4×C67#4114yes—
60C4⋊C4×C67—44no—
61C4⋊C4×C67—44no—
62C22⋊C4×C67—44no—
63C22⋊C4×C67—44no—
64C23×C67—22no—
65C23×C67—21no—
66C23×C67—21no—
67C23×C67—28no—
68C23×C67—21no—
69C23×C67—21no—
70C23×C67—22no—
71C23×C67—22no—
72C23×C67—22no—
73C23×C67—22no—
74C23×C67—21no—
75C23×C67—21no—
76C23×C67—48no—
77C23×C67—48no—
78C23×C67—44no—
79C23×C67—44no—
80C23×C67#10012yes—
81C23×C67#10112yes—
82C23×C67C2×C22⋊C411yescharacteristic
83C23×C67#11712yes—
84C23×C67#13114yes—
85C2×C4×C67—28no—
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—44no—
91C2×C4×C67—48no—
92C2×C4×C67—48no—
93C2×C4×C67—44no—
94C2×C4×C67—44no—
95C2×C4×C67—44no—
96C2×C4×C67—88no—
97C2×C4×C67#9914yes—
98C2×C4×C67#12314yes—
99C67⋊C8—84no—
100C22×C67—22no—
101C22×C67—22no—
102C22×C67—24no—
103C22×C67—24no—
104C22×C67—22no—
105C22×C67—22no—
106C22×C67—24no—
107C22×C67—24no—
108C22×C67—416no—
109C22×C67—41no—
110C22×C67—41no—
111C22×C67—41no—
112C22×C67—41no—
113C22×C67—42no—
114C22×C67—42no—
115C22×C67#38111yescharacteristic
116C22×C67#40812yes—
117C22×C67C2×#9912yes—
118C22×C67C2×#10011yescharacteristic
119C22×C67C2×#10111yescharacteristic
120C22×C67#43314yes—
121C4×C67—24no—
122C4×C67—24no—
123C4×C67—84no—
124#8704—671nomaximal, Sylow
125C2×C67—24no—
126C2×C67—42no—
127C2×C67—42no—
128C2×C67#147112yes—
129C2×C67#147212yes—
130C2×C67#181811yescharacteristic
131C2×C67#181911yescharacteristic
132C2×C67#182111yescharacteristic
133C2×#380D6711yescharacteristic
134#1779—672no—
135#1280—674no—
136C67#870411yescharacteristic, complemented, Sylow
137C24×C4C2×D6711yescharacteristic
138C22×C4⋊C4Dic6712yes—
139C22×C22⋊C4Dic6712yes—
140C2×C22⋊C8—674no—
141C2×C22⋊C8—1342no—
142C2×C22⋊C8—1342no—
143#380—22no—
144#380—22no—
145C2×#95C2×D6712yes—
146C25C2×Dic6711yescharacteristic
147C23×C4—24no—
148C23×C4—44no—
149C23×C4C2×Dic6711yescharacteristic
150C23×C4C2×Dic6714yes—
151C23×C4C22⋊D6712yes—
152C23×C4C22⋊D6712yes—
153C23×C4C22×D6711yescharacteristic
154C22×C8—1344no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—44no—
161C22⋊C8—2684no—
162C22⋊C8—2684no—
163C22⋊C8—1348no—
164#95—24no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—24no—
169C24—42no—
170C24C22⋊Dic6711yescharacteristic
171C24C22⋊Dic6711yescharacteristic
172C24C22×Dic6711yescharacteristic
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—416no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4C22⋊Dic6712yes—
188C22×C4C2×C22⋊D6712yes—
189C22×C4#1176162814yes—
190C2×C8—2684no—
191C2×C8—2684no—
192C2×C8—2684no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—28no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—48no—
210C23—48no—
211C23—44no—
212C23—44no—
213C23#1452396012yes—
214C23#1452396212yes—
215C23C2×C22⋊Dic6711yescharacteristic
216C23#1452400212yes—
217C23#1452405014yes—
218C2×C4—28no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—88no—
230C2×C4#1452395814yes—
231C2×C4#1452402614yes—
232C8—5364no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22—22no—
238C22—22no—
239C22—24no—
240C22—24no—
241C22—416no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22#1686325911yescharacteristic
249C22#1686336312yes—
250C22C2×#1452395812yes—
251C22C2×#1452396011yescharacteristic
252C22C2×#1452396211yescharacteristic
253C22#1686345714yes—
254C4—24no—
255C4—24no—
256C4—84no—
257C2—24no—
258C2—42no—
259C2—42no—
260C2#2255856912yes—
261C2#2255857312yes—
262C2#2256028011yescharacteristic
263C2#2256028311yescharacteristic
264C2#2256028911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC2×#380×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC25×C67C2×C4
cosocleC23×C4×C67C23
hypercenterC2×#380D67
hypocenterC67#8704
LCSG ▷ C24×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #22558052C2semidirect
136. C67#8704semidirect

Maximal subgroups

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

subgroupindex
#225601192
#225580522
C2×#380×C672
#870467