Finite Group Databasebeta

← #30749293

Subgroups — #30749293

Order 17152.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×#16863122C212yesmaximal
2#22558057C214yesmaximal
3C67×#2274C211yesmaximal, characteristic
4C2×C22⋊(C67⋊C8)C2214yes—
5#16863122—22no—
6#16863122—22no—
7C2×#117×C67C411yescharacteristic
8C2×#117×C67C411yescharacteristic
9C22×C4⋊C4×C67C2211yescharacteristic
10C4×C4⋊C4×C67—22no—
11C4×C4⋊C4×C67—22no—
12C4×C22⋊C4×C67C412yes—
13C4×C22⋊C4×C67C2212yes—
14C22×C67⋊C8—24no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—24no—
17C23×C4×C67C2×C411yescharacteristic
18C23×C4×C67C2×C411yescharacteristic
19C23×C4×C67C2311yescharacteristic
20#117×C67—22no—
21#117×C67—24no—
22#117×C67C814yescomplemented
23C2×C4⋊C4×C67—21no—
24C2×C4⋊C4×C67—21no—
25C2×C4⋊C4×C67—24no—
26C2×C4⋊C4×C67D412yes—
27C2×C4⋊C4×C67D412yes—
28C2×C22⋊C4×C67C2×C412yes—
29C2×C22⋊C4×C67C2×C412yes—
30C2×C42×C67—21no—
31C2×C42×C67—21no—
32C2×C42×C67—22no—
33C2×C42×C67—22no—
34#95×C67—22no—
35#95×C67—22no—
36C2×C67⋊C8—44no—
37C2×C67⋊C8—84no—
38C24×C67C22×C411yescharacteristic
39C22×C4×C67—22no—
40C22×C4×C67—22no—
41C22×C4×C67—22no—
42C22×C4×C67—22no—
43C22×C4×C67—22no—
44C22×C4×C67—81no—
45C22×C4×C67—81no—
46C22×C4×C67C22⋊C411yescharacteristic
47C22×C4×C67C22⋊C411yescharacteristic
48C22×C4×C67C22⋊C412yes—
49C22×C4×C67C2×C812yes—
50C22×C4×C67M1612yes—
51C22×C4×C67C2×D412yes—
52C22×C4×C67#4114yes—
53C4⋊C4×C67—24no—
54C4⋊C4×C67—24no—
55C4⋊C4×C67—24no—
56C4⋊C4×C67—24no—
57C22⋊C4×C67—22no—
58C22⋊C4×C67—22no—
59C22⋊C4×C67C2×C814yes—
60C22⋊C4×C67M1614yes—
61C42×C67—24no—
62C42×C67—24no—
63C42×C67—82no—
64C42×C67—82no—
65C23×C67—41no—
66C23×C67—41no—
67C23×C67C2×C22⋊C411yescharacteristic
68C23×C67#11712yes—
69C23×C67C22×C811yescharacteristic
70C23×C67C2×M1611yescharacteristic
71C23×C67C2×M1612yes—
72C2×C4×C67—21no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—24no—
77C2×C4×C67—24no—
78C2×C4×C67—22no—
79C2×C4×C67—24no—
80C2×C4×C67—24no—
81C2×C4×C67—42no—
82C2×C4×C67—44no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—82no—
89C2×C4×C67C22⋊C812yes—
90C2×C4×C67C22⋊C812yes—
91C2×C4×C67#12314yes—
92C67⋊C8—84no—
93C22×C67—21no—
94C22×C67—22no—
95C22×C67—21no—
96C22×C67—22no—
97C22×C67—41no—
98C22×C67—41no—
99C22×C67—42no—
100C22×C67#38111yescharacteristic
101C22×C67C2×C22⋊C811yescharacteristic
102C22×C67#40711yescharacteristic
103C22×C67#43112yes—
104C22×C67#43212yes—
105C4×C67—22no—
106C4×C67—22no—
107C4×C67—24no—
108C4×C67—44no—
109C4×C67—44no—
110C4×C67—42no—
111#8735—671nomaximal, Sylow
112C2×C67—21no—
113C2×C67—21no—
114C2×C67—22no—
115C2×C67—22no—
116C2×C67#181711yescharacteristic
117C2×C67#182811yescharacteristic
118C2×C67#183011yescharacteristic
119#2274D6711yescharacteristic
120C2×#324—672no—
121#1282—674no—
122C67#873511yescharacteristic, complemented, Sylow
123C2×#117Dic6711yescharacteristic
124C2×#117Dic6711yescharacteristic
125C22×C4⋊C4C2×D6711yescharacteristic
126C2×C22⋊C8—674no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C22⋊C4Dic6712yes—
130C4×C22⋊C4C2×D6712yes—
131#324—1342no—
132#324—1342no—
133C23×C4C2×Dic6711yescharacteristic
134C23×C4C2×Dic6711yescharacteristic
135C23×C4C22×D6711yescharacteristic
136C22×C8—1344no—
137#117—22no—
138#117—24no—
139#117C67⋊C814yescomplemented
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4C22⋊D6712yes—
144C2×C4⋊C4C22⋊D6712yes—
145C2×C22⋊C4C2×Dic6712yes—
146C2×C22⋊C4C2×Dic6712yes—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C22⋊C8—1344no—
152C22⋊C8—1344no—
153#95—22no—
154#95—22no—
155C24C22×Dic6711yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4C22⋊Dic6711yescharacteristic
164C22×C4C22⋊Dic6711yescharacteristic
165C22×C4C22⋊Dic6712yes—
166C22×C4C2×C67⋊C812yes—
167C22×C4#1176161212yes—
168C22×C4C2×C22⋊D6712yes—
169C22×C4#1176162814yes—
170C2×C8—2684no—
171C2×C8—5364no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4C2×C67⋊C814yes—
179C22⋊C4#1176161214yes—
180C42—24no—
181C42—24no—
182C42—82no—
183C42—82no—
184C23—41no—
185C23—41no—
186C23C2×C22⋊Dic6711yescharacteristic
187C23#1452400212yes—
188C23C22×C67⋊C811yescharacteristic
189C23C2×#1176161211yescharacteristic
190C23C2×#1176161212yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4C22⋊(C67⋊C8)12yes—
209C2×C4C22⋊(C67⋊C8)12yes—
210C2×C4#1452402614yes—
211C8—5364no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22#1686325911yescharacteristic
220C22C2×C22⋊(C67⋊C8)11yescharacteristic
221C22#1686336011yescharacteristic
222C22#1686344912yes—
223C22#1686345312yes—
224C4—22no—
225C4—22no—
226C4—24no—
227C4—44no—
228C4—44no—
229C4—42no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2#2256027711yescharacteristic
235C2#2256031711yescharacteristic
236C2#2256032311yescharacteristic
subgroupG/H
centerC22#16863259
commutatorC23×C67C22×C8
FrattiniC23×C4C22×D67
FittingC67×#2274C2
socleC22×C67#381
prime radicalC24×C67C22×C4
cosocleC23×C4×C67C23
hypercenter#2274D67
hypocenterC67#8735
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
22. #117×C67C8semidirect
122. C67#8735semidirect
139. #117C67⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#168631222
#225580572
C67×#22742
#873567