Finite Group Databasebeta

← #30766667

Subgroups — #30766667

Order 17152.

340 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 87 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#22560205C211yesmaximal, characteristic
2#22558553C211yesmaximal, characteristic
3#22558236C212yesmaximal
4#22558215C212yesmaximal
5C67×#2634C211yesmaximal, characteristic
6#16863453—21no—
7#16863420—21no—
8C2×#14523962—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×#125×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#400×C67—21no—
15#400×C67C411yescharacteristic
16#400×C67C2211yescharacteristic
17#398×C67C411yescharacteristic
18#398×C67C2211yescharacteristic
19#395×C67—21no—
20#387×C67—21no—
21#387×C67—21no—
22#383×C67C411yescharacteristic
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—41no—
25C4⋊(C67⋊C8)—41no—
26C4⋊(C67⋊C8)—41no—
27#14523962—42no—
28C22⋊(C67⋊C8)—22no—
29#14523954—41no—
30C22×Q8×C67D411yescharacteristic
31#125×C67—41no—
32#125×C67—41no—
33C4×Q8×C67—41no—
34C4×Q8×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C4⋊C4×C67C2311yescharacteristic
44C2×C42×C67C2×C411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67C2×C411yescharacteristic
56#95×C67C2×C411yescharacteristic
57#11761612—42no—
58#11761612—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C2×Q8×C67—21no—
62C2×Q8×C67—21no—
63C2×Q8×C67—41no—
64C2×Q8×C67—41no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—41no—
84C4⋊C4×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C42×C67—21no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—21no—
93C23×C67C2×C22⋊C411yescharacteristic
94Q8×C67—42no—
95Q8×C67—42no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
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—41no—
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—42no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67#9911yescharacteristic
123C2×C4×C67#9911yescharacteristic
124C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C67#13711yescharacteristic
126C67⋊C8—82no—
127C67⋊C8—82no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67C2×#9911yescharacteristic
131C22×C67#41711yescharacteristic
132C22×C67#42111yescharacteristic
133C4×C67—21no—
134C4×C67—21no—
135C4×C67—21no—
136C4×C67—21no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—41no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—81no—
143C4×C67—81no—
144C4×C67—81no—
145C4×C67—81no—
146#11835—671nomaximal, Sylow
147C2×C67—21no—
148C2×C67—21no—
149C2×C67#151311yescharacteristic
150C2×C67#212311yescharacteristic
151C2×C67#213311yescharacteristic
152#2634D6711yescharacteristic
153#1801—671no—
154#1467—671no—
155#1353—672no—
156#1346—672no—
157C67#1183511yescharacteristic, complemented, Sylow
158C2×#125Dic6711yescharacteristic
159C2×C4×Q8C2×D6711yescharacteristic
160#432—1341no—
161#423—1341no—
162C2×#101—1341no—
163#400—21no—
164#400Dic6711yescharacteristic
165#400C2×D6711yescharacteristic
166#398Dic6711yescharacteristic
167#398C2×D6711yescharacteristic
168#395—21no—
169#387—21no—
170#387—21no—
171#383Dic6711yescharacteristic
172#343—672no—
173#324—672no—
174#324—1341no—
175C22×Q8C22⋊D6711yescharacteristic
176C2×M16—1342no—
177#125—41no—
178#125—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic6711yescharacteristic
186C2×C4⋊C4C2×Dic6711yescharacteristic
187C2×C4⋊C4C2×Dic6711yescharacteristic
188C2×C4⋊C4C22⋊D6711yescharacteristic
189C2×C4⋊C4C22×D6711yescharacteristic
190C2×C42C2×Dic6711yescharacteristic
191C2×C42C22⋊D6711yescharacteristic
192C2×C42C22⋊D6711yescharacteristic
193C4⋊C8—2681no—
194C4⋊C8—2681no—
195C4⋊C8—2681no—
196#101—2682no—
197C22⋊C8—1342no—
198#97—2681no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95C2×Dic6711yescharacteristic
208#95C2×Dic6711yescharacteristic
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C22⋊Dic6711yescharacteristic
218C22×C4C22⋊Dic6711yescharacteristic
219C22×C4C22⋊Dic6711yescharacteristic
220C22×C4C22⋊Dic6711yescharacteristic
221C22×C4C22×Dic6711yescharacteristic
222C22×C4C2×C22⋊D6711yescharacteristic
223C22×C4C2×C22⋊D6711yescharacteristic
224M16—2682no—
225M16—2682no—
226C2×C8—2682no—
227C2×C8—2682no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C23C2×C22⋊Dic6711yescharacteristic
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4#1452395811yescharacteristic
275C2×C4#1452395811yescharacteristic
276C2×C4#1452406811yescharacteristic
277C2×C4#1452407311yescharacteristic
278C8—5362no—
279C8—5362no—
280C22—21no—
281C22—21no—
282C22C2×#1452395811yescharacteristic
283C22#1686339711yescharacteristic
284C22#1686341311yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C2—21no—
299C2—21no—
300C2#2255875211yescharacteristic
301C2#2256186311yescharacteristic
302C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2634C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2634D67
hypocenterC67#11835
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2634

Complements

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

NH ≅ G/Ntype
157. C67#11835semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225585532
#225582362
#225582152
C67×#26342
#1183567