Finite Group Databasebeta

← #30766647

Subgroups — #30766647

Order 17152.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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#22558239C212yesmaximal
4#22558215C212yesmaximal
5C67×#2399C211yesmaximal, characteristic
6#16863453—21no—
7#16863420—21no—
8C2×#14523962—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×C4×Q8×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67C411yescharacteristic
16#398×C67C2211yescharacteristic
17#395×C67—21no—
18#395×C67C411yescharacteristic
19#395×C67C2211yescharacteristic
20C4×C4⋊C4×C67—21no—
21C4×C4⋊C4×C67—21no—
22#376×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×C67C2×C411yescharacteristic
31C22×Q8×C67D411yescharacteristic
32C4×Q8×C67—41no—
33C4×Q8×C67—41no—
34C4×Q8×C67—41no—
35C4×Q8×C67—41no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C4⋊C4×C67C2311yescharacteristic
45C2×C42×C67—21no—
46C2×C42×C67—21no—
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67D411yescharacteristic
49C2×C42×C67D411yescharacteristic
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—
65C2×Q8×C67—42no—
66C2×Q8×C67—41no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C4⋊C4×C67—42no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C42×C67—24no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—21no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67—41no—
96C42×C67—41no—
97C23×C67C2×C22⋊C411yescharacteristic
98Q8×C67—42no—
99Q8×C67—42no—
100Q8×C67—42no—
101Q8×C67—42no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—22no—
105C2×C4×C67—22no—
106C2×C4×C67—22no—
107C2×C4×C67—22no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
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—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67#9911yescharacteristic
130C2×C4×C67#9911yescharacteristic
131C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C67#13711yescharacteristic
133C67⋊C8—82no—
134C67⋊C8—82no—
135C22×C67—21no—
136C22×C67—21no—
137C22×C67C2×#9911yescharacteristic
138C22×C67#41711yescharacteristic
139C22×C67#42111yescharacteristic
140C4×C67—21no—
141C4×C67—21no—
142C4×C67—21no—
143C4×C67—42no—
144C4×C67—42no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—42no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—41no—
151C4×C67—81no—
152C4×C67#35612yes—
153#11831—671nomaximal, Sylow
154C2×C67—21no—
155C2×C67—21no—
156C2×C67#151311yescharacteristic
157C2×C67#212011yescharacteristic
158C2×C67C2×#35611yescharacteristic
159#2399D6711yescharacteristic
160#1801—671no—
161#1467—671no—
162#1354—672no—
163#1346—672no—
164C67#1183111yescharacteristic, complemented, Sylow
165C2×C4×Q8Dic6711yescharacteristic
166C2×C4×Q8C2×D6711yescharacteristic
167#432—1341no—
168#423—1341no—
169C2×#101—1341no—
170#398—21no—
171#398Dic6711yescharacteristic
172#398C2×D6711yescharacteristic
173#395—21no—
174#395Dic6711yescharacteristic
175#395C2×D6711yescharacteristic
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178#376Dic6711yescharacteristic
179#343—672no—
180#324—672no—
181#324—1341no—
182C22×Q8C2×Dic6711yescharacteristic
183C22×Q8C22⋊D6711yescharacteristic
184C2×M16—1342no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×Dic6711yescharacteristic
195C2×C4⋊C4C2×Dic6711yescharacteristic
196C2×C4⋊C4C22⋊D6711yescharacteristic
197C2×C4⋊C4C22×D6711yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×Dic6711yescharacteristic
201C2×C42C22⋊D6711yescharacteristic
202C2×C42C22⋊D6711yescharacteristic
203C4⋊C8—2681no—
204C4⋊C8—2681no—
205C4⋊C8—2681no—
206#101—2682no—
207C22⋊C8—1342no—
208#97—2681no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C2×Dic6711yescharacteristic
215#95C2×Dic6711yescharacteristic
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4C22⋊Dic6711yescharacteristic
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22×Dic6711yescharacteristic
231C22×C4C2×C22⋊D6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233M16—2682no—
234M16—2682no—
235C2×C8—2682no—
236C2×C8—2682no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C23C2×C22⋊Dic6711yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4#1452395811yescharacteristic
289C2×C4#1452395811yescharacteristic
290C2×C4#1452406811yescharacteristic
291C2×C4#1452407311yescharacteristic
292C8—5362no—
293C8—5362no—
294C22—21no—
295C22—21no—
296C22C2×#1452395811yescharacteristic
297C22#1686339711yescharacteristic
298C22#1686341311yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4#1686319512yes—
312C2—21no—
313C2—21no—
314C2#2255875211yescharacteristic
315C2#2256185011yescharacteristic
316C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2399C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2399D67
hypocenterC67#11831
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2399

Complements

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

NH ≅ G/Ntype
164. C67#11831semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225585532
#225582392
#225582152
C67×#23992
#1183167