Finite Group Databasebeta

← #30774476

Subgroups — #30774476

Order 17152.

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

normal subgroups only

This group has 484 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#22560197C211yesmaximal, characteristic
2C4×#14523960C211yesmaximal, characteristic
3#22558327C214yesmaximal
4C67×#2487C211yesmaximal, characteristic
5C2×#14523960C2212yes—
6C2×#14523958C2212yes—
7C4×#11761612—21no—
8#16863256—21no—
9#16863165—21no—
10#16863163—21no—
11C2×C4×D4×C67C2211yescharacteristic
12C22×C4⋊C4×C67C411yescharacteristic, complemented
13#400×C67—21no—
14#399×C67—21no—
15#389×C67—21no—
16#386×C67C2212yes—
17#385×C67C412yescomplemented
18#380×C67—21no—
19#380×C67C411yescharacteristic, complemented
20C2×#11761612—22no—
21C2×C4⋊Dic67—41no—
22C2×C22⋊Dic67—22no—
23#14523960—22no—
24#14523960Q814yes—
25#14523958—24no—
26#14523954—41no—
27C4×C67⋊C8—41no—
28C67⋊#95—41no—
29C22×D4×C67C2311yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67C2×C412yes—
32C23×C4×C67D411yescharacteristic
33C4×D4×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67—41no—
40C2×C4⋊C4×C67—41no—
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67C2×C412yes—
44C2×C22⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—22no—
48#95×C67—21no—
49#95×C67—22no—
50C22×Dic67—42no—
51#11761612—24no—
52#11761612—42no—
53C2×C67⋊C8—42no—
54C4⋊Dic67—82no—
55C22⋊Dic67—42no—
56C22⋊Dic67—42no—
57C24×C67C22×C411yescharacteristic
58C24×C67C2×D411yescharacteristic
59C2×D4×C67—41no—
60C2×D4×C67—41no—
61C2×D4×C67C2×Q812yes—
62C2×D4×C67#4112yes—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—22no—
68C22×C4×C67—22no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67—82no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C4⋊C4×C67—41no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—82no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—82no—
88C42×C67—21no—
89C42×C67—21no—
90C23×C67—21no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67C2×C22⋊C411yescharacteristic
99C23×C67C4×Q812yes—
100C23×C67#12411yescharacteristic
101C23×C67C4⋊Q811yescharacteristic
102D4×C67—42no—
103D4×C67—42no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—22no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67—81no—
123C2×C4×C67—84no—
124C2×C4×C67#9911yescharacteristic
125C2×C4×C67#9911yescharacteristic
126C2×C4×C67C22⋊Q812yes—
127C2×Dic67—82no—
128C2×Dic67—82no—
129C2×Dic67—82no—
130C67⋊C8—44no—
131C22×C67—21no—
132C22×C67—21no—
133C22×C67—21no—
134C22×C67—22no—
135C22×C67—21no—
136C22×C67—42no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—42no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67—41no—
143C22×C67—41no—
144C22×C67—81no—
145C22×C67#39111yescharacteristic
146C22×C67C2×#9911yescharacteristic
147C22×C67#41011yescharacteristic
148C4×C67—21no—
149C4×C67—22no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67—41no—
153C4×C67—82no—
154C4×C67—82no—
155C4×C67#35212yes—
156Dic67—162no—
157#13292—671nomaximal, Sylow
158C2×C67—21no—
159C2×C67—21no—
160C2×C67—41no—
161C2×C67—42no—
162C2×C67—41no—
163C2×C67#199611yescharacteristic
164C2×C67C2×#35211yescharacteristic
165C2×C67#212211yescharacteristic
166#2487D6711yescharacteristic
167#1799—671no—
168C4×#100—671no—
169#1390—674no—
170C67#1329211yescharacteristic, complemented, Sylow
171C2×C4×D4C2×D6711yescharacteristic
172C22×C4⋊C4Dic6711yescharacteristic, complemented
173C2×#100—672no—
174C2×#99—672no—
175C4×M16—1341no—
176#400—21no—
177#399—21no—
178#389—21no—
179#386C2×D6712yes—
180#385Dic6712yescomplemented
181#380—21no—
182#380—1341no—
183#380Dic6711yescharacteristic, complemented
184#343—1341no—
185#342—1341no—
186C22×D4C22×D6711yescharacteristic
187C23×C4C2×Dic6711yescharacteristic
188C23×C4C2×Dic6712yes—
189C23×C4C22⋊D6711yescharacteristic
190C2×M16—1342no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—2681no—
200C2×C4⋊C4C22⋊D6711yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—1342no—
203C2×C22⋊C4C2×Dic6712yes—
204C2×C22⋊C4C2×Dic6711yescharacteristic
205C2×C22⋊C4C22⋊D6711yescharacteristic
206C2×C42C22⋊D6711yescharacteristic
207#100—674no—
208#100—1342no—
209#99—1344no—
210#97—2681no—
211C4×C8—2681no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95—2681no—
216C24C22×Dic6711yescharacteristic
217C24C2×C22⋊D6711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4Q8×D6712yes—
221C2×D4#1176162912yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—82no—
232C22×C4—2682no—
233C22×C4C22⋊Dic6711yescharacteristic
234C22×C4C22⋊Dic6711yescharacteristic
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C2×C22⋊D6711yescharacteristic
238M16—2682no—
239M16—1344no—
240C2×C8—2682no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—82no—
247C4⋊C4—5362no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C22⋊C4—2682no—
253C22⋊C4—2682no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic6711yescharacteristic
265C23Q8×Dic6712yes—
266C23#1452403211yescharacteristic
267C23#1452404111yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—84no—
290C2×C4—5362no—
291C2×C4—5362no—
292C2×C4—5362no—
293C2×C4#1452395811yescharacteristic
294C2×C4#1452395811yescharacteristic
295C2×C4#1452401912yes—
296C8—2684no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#1686330311yescharacteristic
312C22C2×#1452395811yescharacteristic
313C22#1686337011yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—10722no—
322C4#1686318612yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2256119511yescharacteristic
329C2C2×#1686318611yescharacteristic
330C2#2256185911yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC22×D4C22×D67
FittingC67×#2487C2
socleC22×C67C2×#99
prime radical, cosocleC22×D4×C67C23
hypercenter#2487D67
hypocenterC67#13292
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
12. C22×C4⋊C4×C67C4semidirect
17. #385×C67C4semidirect
19. #380×C67C4semidirect
170. C67#13292semidirect
172. C22×C4⋊C4Dic67semidirect
180. #385Dic67semidirect
183. #380Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
C4×#145239602
#225583272
C67×#24872
#1329267