Finite Group Databasebeta

← #30751830

Subgroups — #30751830

Order 17152.

436 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C4×#14523962C211yesmaximal, characteristic
3#22558811C212yesmaximal
4#22558073C212yesmaximal
5C67×#2634C211yesmaximal, characteristic
6#16863527—24no—
7#16863453—21no—
8#16863453C2212yes—
9#16863449—21no—
10C2×#14523962—21no—
11C4×#11761612C2212yes—
12#16863165—21no—
13#16863138—24no—
14C2×#125×C67C2211yescharacteristic
15C2×C4×Q8×C67C2211yescharacteristic
16#400×C67—21no—
17#400×C67C411yescharacteristic
18#400×C67C411yescharacteristic
19#398×C67C411yescharacteristic
20#398×C67C411yescharacteristic
21#395×C67—21no—
22#387×C67—21no—
23#387×C67—21no—
24#383×C67C2211yescharacteristic
25C2×#11761612—22no—
26C4⋊(C67⋊C8)—22no—
27C4⋊(C67⋊C8)—22no—
28C4⋊(C67⋊C8)—22no—
29#14523962—24no—
30C22⋊(C67⋊C8)—22no—
31#14523954—22no—
32#14523954—22no—
33#14523954D414yes—
34C4×C67⋊C8—22no—
35C4×C67⋊C8—22no—
36C22×Q8×C67D411yescharacteristic
37#125×C67—22no—
38#125×C67—22no—
39C4×Q8×C67—22no—
40C4×Q8×C67—22no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—21no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67—21no—
45C2×C4⋊C4×C67C2×C411yescharacteristic
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C4⋊C4×C67C2×C411yescharacteristic
48C2×C4⋊C4×C67C2×C411yescharacteristic
49C2×C4⋊C4×C67D411yescharacteristic
50C2×C42×C67D411yescharacteristic
51C2×C42×C67D411yescharacteristic
52C2×C42×C67C2311yescharacteristic
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58#95×C67—21no—
59#95×C67—21no—
60#95×C67—21no—
61#95×C67C2×C411yescharacteristic
62#95×C67C2×C411yescharacteristic
63#11761612—24no—
64#11761612—42no—
65C2×C67⋊C8—24no—
66C2×C67⋊C8—42no—
67C2×Q8×C67—22no—
68C2×Q8×C67—22no—
69C2×Q8×C67—41no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22×C411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80C22×C4×C67C2×D411yescharacteristic
81C4⋊C4×C67—22no—
82C4⋊C4×C67—22no—
83C4⋊C4×C67—22no—
84C4⋊C4×C67—22no—
85C4⋊C4×C67—22no—
86C4⋊C4×C67—42no—
87C4⋊C4×C67—42no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—42no—
90C4⋊C4×C67—41no—
91C42×C67—21no—
92C42×C67—21no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67C2×D412yes—
96C42×C67#4112yes—
97C23×C67C2×C22⋊C411yescharacteristic
98Q8×C67—44no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—42no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67#10412yes—
124C2×C4×C67C4×D412yes—
125C2×C4×C67#12011yescharacteristic
126C2×C4×C67#12112yes—
127C2×C4×C67#12311yescharacteristic
128C67⋊C8—44no—
129C67⋊C8—44no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67#38611yescharacteristic
133C22×C67C2×#10411yescharacteristic
134C22×C67#42111yescharacteristic
135C4×C67—22no—
136C4×C67—22no—
137C4×C67—21no—
138C4×C67—21no—
139C4×C67—22no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—41no—
143C4×C67—81no—
144C4×C67—81no—
145C4×C67—81no—
146C4×C67—81no—
147#9190—671nomaximal, Sylow
148C2×C67—21no—
149C2×C67—21no—
150C2×C67#187111yescharacteristic
151C2×C67#191211yescharacteristic
152C2×C67#191511yescharacteristic
153#2983—671no—
154#2634D6711yescharacteristic
155C4×#101—671no—
156#1528—672no—
157#1288—672no—
158C67#919011yescharacteristic, complemented, Sylow
159C2×#125C2×D6711yescharacteristic
160C2×C4×Q8C2×D6711yescharacteristic
161#446—1344no—
162#432—672no—
163#432—1341no—
164#431—1341no—
165C2×#101—1341no—
166C4×M16—672no—
167#400—21no—
168#400Dic6711yescharacteristic
169#400Dic6711yescharacteristic
170#398Dic6711yescharacteristic
171#398Dic6711yescharacteristic
172#395—21no—
173#387—21no—
174#387—21no—
175#383C2×D6711yescharacteristic
176#343—1341no—
177#330—1344no—
178C22×Q8C22⋊D6711yescharacteristic
179C2×M16—1342no—
180#125—22no—
181#125—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×Dic6711yescharacteristic
189C2×C4⋊C4C2×Dic6711yescharacteristic
190C2×C4⋊C4C2×Dic6711yescharacteristic
191C2×C4⋊C4C2×Dic6711yescharacteristic
192C2×C4⋊C4C22⋊D6711yescharacteristic
193C2×C42C22⋊D6711yescharacteristic
194C2×C42C22⋊D6711yescharacteristic
195C2×C42C22×D6711yescharacteristic
196C4⋊C8—1342no—
197C4⋊C8—1342no—
198C4⋊C8—1342no—
199#101—1344no—
200C22⋊C8—1342no—
201#97—674no—
202#97—1342no—
203#97—1342no—
204C4×C8—1342no—
205C4×C8—1342no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C2×Dic6711yescharacteristic
215#95C2×Dic6711yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4C22⋊Dic6711yescharacteristic
224C22×C4C22⋊Dic6711yescharacteristic
225C22×C4C22⋊Dic6711yescharacteristic
226C22×C4C22⋊Dic6711yescharacteristic
227C22×C4C22×Dic6711yescharacteristic
228C22×C4C2×C22⋊D6711yescharacteristic
229C22×C4C2×C22⋊D6711yescharacteristic
230M16—2682no—
231M16—1344no—
232C2×C8—2682no—
233C2×C8—1344no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42D4×D6712yes—
249C42#1176162712yes—
250C23C2×C22⋊Dic6711yescharacteristic
251Q8—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#1452397012yes—
277C2×C4D4×Dic6712yes—
278C2×C4#1452401111yescharacteristic
279C2×C4#1452401512yes—
280C2×C4#1452402411yescharacteristic
281C8—2684no—
282C8—2684no—
283C22—21no—
284C22—21no—
285C22#1686327611yescharacteristic
286C22C2×#1452397011yescharacteristic
287C22#1686341311yescharacteristic
288C4—22no—
289C4—22no—
290C4—21no—
291C4—21no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C2—21no—
301C2—21no—
302C2#2256051211yescharacteristic
303C2#2256074711yescharacteristic
304C2#2256076211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2634C2
socleC22×C67#386
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2634D67
hypocenterC67#9190
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2634

Complements

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

NH ≅ G/Ntype
158. C67#9190semidirect

Maximal subgroups

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

subgroupindex
#225692942
C4×#145239622
#225588112
#225580732
C67×#26342
#919067