Finite Group Databasebeta

← #30751850

Subgroups — #30751850

Order 17152.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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#22568847C211yesmaximal, characteristic
2#22558894C212yesmaximal, complemented
3#22558523C211yesmaximal, characteristic
4#22558067C212yesmaximal, complemented
5C67×#2628C211yesmaximal, characteristic
6#16863457—24no—
7#16863449C2212yes—
8#16863420C2212yes—
9#16863360—21no—
10C4×#11761612—21no—
11#16863142—24no—
12#16863122—21no—
13#16863120—21no—
14C2×C4⋊Q8×C67C2211yescharacteristic
15C2×#117×C67C2211yescharacteristic
16#400×C67C412yes—
17#399×C67C412yes—
18#395×C67—21no—
19#393×C67—21no—
20#389×C67—21no—
21#382×C67C2211yescharacteristic
22#380×C67—21no—
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)D414yes—
27C22⋊(C67⋊C8)—24no—
28C22⋊(C67⋊C8)—22no—
29#14523954—22no—
30C4×C67⋊C8—22no—
31C4×C67⋊C8—22no—
32C22×Q8×C67—21no—
33C23×C4×C67D411yescharacteristic
34C4⋊Q8×C67—22no—
35C4⋊Q8×C67—22no—
36#117×C67—22no—
37#117×C67—22no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67C2×C412yes—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67—22no—
45C2×C22⋊C4×C67C2×C411yescharacteristic
46C2×C22⋊C4×C67D411yescharacteristic
47C2×C42×C67D411yescharacteristic
48C2×C42×C67C2311yescharacteristic
49#95×C67—22no—
50#95×C67—22no—
51#95×C67—21no—
52#95×C67C2×C412yes—
53#11761612—42no—
54#11761612—42no—
55C2×C67⋊C8—24no—
56C2×C67⋊C8—42no—
57C24×C67C22⋊C411yescharacteristic
58C2×Q8×C67—42no—
59C2×Q8×C67—82no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—22no—
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22×C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C4⋊C4×C67—22no—
73C4⋊C4×C67—22no—
74C4⋊C4×C67—24no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—82no—
79C22⋊C4×C67—22no—
80C22⋊C4×C67—22no—
81C22⋊C4×C67—82no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67C2×D412yes—
85C42×C67#4112yes—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67C2×C22⋊C411yescharacteristic
92Q8×C67—84no—
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—42no—
101C2×C4×C67—42no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67C4×D412yes—
111C2×C4×C67#12011yescharacteristic
112C2×C4×C67#12112yes—
113C2×C4×C67#12311yescharacteristic
114C2×C4×C67#13512yes—
115C2×C4×C67#13712yes—
116C67⋊C8—44no—
117C67⋊C8—82no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67—42no—
121C22×C67—41no—
122C22×C67—41no—
123C22×C67—41no—
124C22×C67—81no—
125C22×C67#38611yescharacteristic
126C22×C67#41611yescharacteristic
127C22×C67#41911yescharacteristic
128C4×C67—22no—
129C4×C67—22no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—41no—
133C4×C67—82no—
134C4×C67—82no—
135C4×C67#45212yes—
136C4×C67#48512yes—
137#9194—671nomaximal, Sylow
138C2×C67—21no—
139C2×C67—21no—
140C2×C67—81no—
141C2×C67#187311yescharacteristic
142C2×C67#191011yescharacteristic
143C2×C67#191511yescharacteristic
144#2921—671no—
145#2628D6711yescharacteristic
146#1549—672no—
147#1458—671no—
148#1286—672no—
149C67#919411yescharacteristic, complemented, Sylow
150C2×C4⋊Q8C2×D6711yescharacteristic
151C2×#117C2×D6711yescharacteristic
152#433—1344no—
153#431—672no—
154#423—672no—
155#407—1341no—
156C4×M16—1341no—
157#400Dic6712yes—
158#399Dic6712yes—
159#395—21no—
160#393—21no—
161#389—21no—
162#382C2×D6711yescharacteristic
163#380—21no—
164#332—1344no—
165#324—1341no—
166#323—1341no—
167C22×Q8—21no—
168C23×C4C22⋊D6711yescharacteristic
169C2×M16—1342no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×Dic6712yes—
178C2×C4⋊C4C2×Dic6711yescharacteristic
179C2×C4⋊C4C22⋊D6711yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic6711yescharacteristic
182C2×C22⋊C4C22⋊D6711yescharacteristic
183C2×C42C22⋊D6711yescharacteristic
184C2×C42C22×D6711yescharacteristic
185C4⋊C8—674no—
186C4⋊C8—1342no—
187C4⋊C8—1342no—
188C22⋊C8—1342no—
189C22⋊C8—1344no—
190#97—1342no—
191C4×C8—1342no—
192C4×C8—1342no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95C2×Dic6712yes—
197C24C22⋊Dic6711yescharacteristic
198C2×Q8—42no—
199C2×Q8—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊Dic6711yescharacteristic
207C22×C4C22⋊Dic6711yescharacteristic
208C22×C4C22⋊Dic6711yescharacteristic
209C22×C4C22×Dic6711yescharacteristic
210C22×C4C2×C22⋊D6711yescharacteristic
211C22×C4C2×C22⋊D6711yescharacteristic
212M16—2682no—
213M16—2682no—
214C2×C8—2682no—
215C2×C8—1344no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42D4×D6712yes—
229C42#1176162712yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic6711yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4D4×Dic6712yes—
255C2×C4#1452401111yescharacteristic
256C2×C4#1452401512yes—
257C2×C4#1452402411yescharacteristic
258C2×C4#1452406312yes—
259C2×C4#1452407412yes—
260C8—2684no—
261C8—5362no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1686327611yescharacteristic
270C22#1686339211yescharacteristic
271C22#1686340511yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#1686355612yes—
280C4#1686373412yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2256052211yescharacteristic
285C2#2256073711yescharacteristic
286C2#2256076211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2628C2
socleC22×C67#386
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2628D67
hypocenterC67#9194
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #22558894C2semidirect
4. #22558067C2semidirect
149. C67#9194semidirect

Maximal subgroups

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

subgroupindex
#225688472
#225588942
#225585232
#225580672
C67×#26282
#919467