Finite Group Databasebeta

← #30753536

Subgroups — #30753536

Order 17152.

432 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 111 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#22569294C211yesmaximal, characteristic
2#22558926C212yesmaximal
3#22558553C211yesmaximal, characteristic
4#22558065C212yesmaximal
5C67×#2568C211yesmaximal, characteristic
6#16863527—24no—
7#16863453—21no—
8#16863453C2212yes—
9#16863453C2212yes—
10#16863449—21no—
11#16863142—24no—
12#16863122—21no—
13#16863122—21no—
14C2×C4⋊Q8×C67C2211yescharacteristic
15C2×C4×Q8×C67C2211yescharacteristic
16#400×C67—21no—
17#400×C67—21no—
18#391×C67—21no—
19#391×C67—21no—
20#391×C67C412yes—
21#382×C67C412yes—
22C4×C4⋊C4×C67C2211yescharacteristic
23C4⋊(C67⋊C8)—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)D414yes—
27C22⋊(C67⋊C8)—22no—
28C22⋊(C67⋊C8)—22no—
29#14523954—22no—
30#14523954—22no—
31#14523954—22no—
32C4×C67⋊C8—22no—
33C22×Q8×C67—21no—
34C22×Q8×C67C2×C411yescharacteristic
35C4⋊Q8×C67—22no—
36C4⋊Q8×C67—22no—
37C4×Q8×C67—22no—
38C4×Q8×C67—22no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67C2×C412yes—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C42×C67C2×C412yes—
46C2×C42×C67D411yescharacteristic
47C2×C42×C67D411yescharacteristic
48C2×C42×C67C2311yescharacteristic
49#95×C67—22no—
50#95×C67—22no—
51#95×C67—22no—
52#95×C67—22no—
53#95×C67—21no—
54C2×C67⋊C8—24no—
55C2×C67⋊C8—42no—
56C2×Q8×C67—22no—
57C2×Q8×C67—22no—
58C2×Q8×C67—41no—
59C2×Q8×C67—44no—
60C2×Q8×C67—41no—
61C2×Q8×C67—41no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22×C411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C4⋊C4×C67—22no—
72C4⋊C4×C67—22no—
73C4⋊C4×C67—22no—
74C4⋊C4×C67—24no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—42no—
82C42×C67C2×D412yes—
83C42×C67#4112yes—
84C23×C67C2×C22⋊C411yescharacteristic
85Q8×C67—44no—
86Q8×C67—82no—
87Q8×C67—82no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—44no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67#10112yes—
107C2×C4×C67C4×D412yes—
108C2×C4×C67#12011yescharacteristic
109C2×C4×C67#12112yes—
110C2×C4×C67#12311yescharacteristic
111C67⋊C8—44no—
112C67⋊C8—82no—
113C22×C67—21no—
114C22×C67—21no—
115C22×C67#38611yescharacteristic
116C22×C67C2×#10111yescharacteristic
117C22×C67#41311yescharacteristic
118C4×C67—21no—
119C4×C67—22no—
120C4×C67—22no—
121C4×C67—22no—
122C4×C67—41no—
123C4×C67—41no—
124C4×C67—41no—
125C4×C67—82no—
126C4×C67—82no—
127C4×C67#45412yes—
128#9456—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67#188811yescharacteristic
132C2×C67#190111yescharacteristic
133C2×C67#190911yescharacteristic
134#2983—671no—
135#2568D6711yescharacteristic
136#1556—672no—
137#1467—671no—
138#1285—672no—
139C67#945611yescharacteristic, complemented, Sylow
140C2×C4⋊Q8C2×D6711yescharacteristic
141C2×C4×Q8C2×D6711yescharacteristic
142#446—1344no—
143#432—672no—
144#432—672no—
145#432—1341no—
146#431—1341no—
147#400—21no—
148#400—21no—
149#391—21no—
150#391—21no—
151#391Dic6712yes—
152#382Dic6712yes—
153C4×C4⋊C4C2×D6711yescharacteristic
154#332—1344no—
155#324—1341no—
156#324—1341no—
157C22×Q8—21no—
158C22×Q8C2×Dic6711yescharacteristic
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×Dic6712yes—
166C2×C4⋊C4C2×Dic6711yescharacteristic
167C2×C4⋊C4C22⋊D6711yescharacteristic
168C2×C4⋊C4C22⋊D6711yescharacteristic
169C2×C42C2×Dic6712yes—
170C2×C42C22⋊D6711yescharacteristic
171C2×C42C22⋊D6711yescharacteristic
172C2×C42C22×D6711yescharacteristic
173C4⋊C8—674no—
174C4⋊C8—1342no—
175C4⋊C8—1342no—
176C4⋊C8—1342no—
177C22⋊C8—1342no—
178C22⋊C8—1342no—
179#97—1342no—
180#97—1342no—
181#97—1342no—
182C4×C8—1342no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—21no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—44no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4C22⋊Dic6711yescharacteristic
197C22×C4C22⋊Dic6711yescharacteristic
198C22×C4C22⋊Dic6711yescharacteristic
199C22×C4C22⋊Dic6711yescharacteristic
200C22×C4C22×Dic6711yescharacteristic
201C22×C4C2×C22⋊D6711yescharacteristic
202C22×C4C2×C22⋊D6711yescharacteristic
203C2×C8—2682no—
204C2×C8—1344no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—24no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C42D4×D6712yes—
217C42#1176162712yes—
218C23C2×C22⋊Dic6711yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—82no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4#1452396212yes—
241C2×C4D4×Dic6712yes—
242C2×C4#1452401111yescharacteristic
243C2×C4#1452401512yes—
244C2×C4#1452402411yescharacteristic
245C8—2684no—
246C8—5362no—
247C22—21no—
248C22—21no—
249C22#1686327611yescharacteristic
250C22C2×#1452396211yescharacteristic
251C22#1686338011yescharacteristic
252C4—21no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4#1686356812yes—
262C2—21no—
263C2—21no—
264C2#2256061311yescharacteristic
265C2#2256067811yescharacteristic
266C2#2256073211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2568C2
socleC22×C67#386
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2568D67
hypocenterC67#9456
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
139. C67#9456semidirect

Maximal subgroups

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

subgroupindex
#225692942
#225589262
#225585532
#225580652
C67×#25682
#945667