Finite Group Databasebeta

← #30766850

Subgroups — #30766850

Order 17152.

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

normal subgroups only

This group has 472 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#22560197C211yesmaximal, characteristic
2#22558530C211yesmaximal, characteristic
3#22558209C214yesmaximal, complemented
4C67×#2486C211yesmaximal, characteristic
5#16863449—21no—
6C2×#14523958—21no—
7#16863256—21no—
8#16863163C2212yes—
9#16863120—21no—
10#16863120C2212yes—
11C2×#123×C67—21no—
12C2×C4×D4×C67C2211yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic, complemented
14#398×C67C412yescomplemented
15#397×C67C2212yes—
16#389×C67—21no—
17#381×C67—21no—
18#380×C67—21no—
19#380×C67C411yescharacteristic, complemented
20C2×C4⋊Dic67—41no—
21C2×C22⋊Dic67—22no—
22C4⋊(C67⋊C8)—41no—
23#14523958—42no—
24C22⋊(C67⋊C8)—22no—
25C4×C67⋊C8—41no—
26C67⋊#95—41no—
27C22×D4×C67D411yescharacteristic
28C23×C4×C67—21no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31#123×C67—42no—
32#123×C67—42no—
33C4×D4×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—41no—
39C2×C4⋊C4×C67—41no—
40C2×C4⋊C4×C67C2×C412yes—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67C2311yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—22no—
48#95×C67—22no—
49#95×C67C2×C412yes—
50C22×Dic67—42no—
51C2×C67⋊C8—42no—
52C4⋊Dic67—82no—
53C22⋊Dic67—42no—
54C22⋊Dic67—42no—
55C24×C67C22⋊C411yescharacteristic
56C24×C67C2×D411yescharacteristic
57C2×D4×C67—41no—
58C2×D4×C67—41no—
59C2×D4×C67D812yes—
60C2×D4×C67SD1612yes—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C4⋊C4×C67—41no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—44no—
83C22⋊C4×C67—41no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—42no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
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×C67#10212yes—
99C23×C67C2×C22⋊C411yescharacteristic
100C23×C67C2×D811yescharacteristic
101C23×C67C2×SD1611yescharacteristic
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×C67#10212yes—
127C2×Dic67—82no—
128C2×Dic67—82no—
129C2×Dic67—82no—
130C67⋊C8—82no—
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×C67C2×#9911yescharacteristic
146C22×C67C2×#10211yescharacteristic
147C22×C67#42111yescharacteristic
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#11874—671nomaximal, Sylow
158C2×C67—21no—
159C2×C67—21no—
160C2×C67—41no—
161C2×C67—42no—
162C2×C67—41no—
163C2×C67#151911yescharacteristic
164C2×C67C2×#35211yescharacteristic
165C2×C67#212911yescharacteristic
166#2486D6711yescharacteristic
167#1799—671no—
168#1460—671no—
169#1344—674no—
170C67#1187411yescharacteristic, complemented, Sylow
171C2×#123—21no—
172C2×C4×D4C2×D6711yescharacteristic
173C22×C4⋊C4Dic6711yescharacteristic, complemented
174#431—1341no—
175C2×#99—1341no—
176#398Dic6712yescomplemented
177#397C2×D6712yes—
178#389—21no—
179#381—21no—
180#380—21no—
181#380—1341no—
182#380Dic6711yescharacteristic, complemented
183#342—672no—
184#323—672no—
185#323—1341no—
186C22×D4C22⋊D6711yescharacteristic
187C23×C4—21no—
188C23×C4C2×Dic6711yescharacteristic
189C23×C4C22⋊D6711yescharacteristic
190#123—42no—
191#123—42no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—2681no—
200C2×C4⋊C4C2×Dic6712yes—
201C2×C4⋊C4C2×Dic6711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1342no—
205C2×C22⋊C4C22⋊D6711yescharacteristic
206C2×C22⋊C4C22×D6711yescharacteristic
207C2×C42C22⋊D6711yescharacteristic
208C4⋊C8—2681no—
209#99—2682no—
210C22⋊C8—1342no—
211C4×C8—2681no—
212#95—22no—
213#95—22no—
214#95—2681no—
215#95C2×Dic6712yes—
216C24C22⋊Dic6711yescharacteristic
217C24C2×C22⋊D6711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1176161312yes—
221C2×D4#1176161612yes—
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—42no—
232C22×C4—42no—
233C22×C4—2682no—
234C22×C4C22⋊Dic6711yescharacteristic
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22×Dic6711yescharacteristic
238C22×C4C2×C22⋊D6711yescharacteristic
239C2×C8—2682no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—5362no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
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—
264C23#1452396412yes—
265C23C2×C22⋊Dic6711yescharacteristic
266C23C2×#1176161311yescharacteristic
267C23C2×#1176161611yescharacteristic
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#1452396412yes—
296C8—5362no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1452395811yescharacteristic
312C22C2×#1452396411yescharacteristic
313C22#1686341311yescharacteristic
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#2255877711yescharacteristic
329C2C2×#1686318611yescharacteristic
330C2#2256188411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2486C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2486D67
hypocenterC67#11874
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #22558209C2semidirect
13. C22×C4⋊C4×C67C4semidirect
14. #398×C67C4semidirect
19. #380×C67C4semidirect
170. C67#11874semidirect
173. C22×C4⋊C4Dic67semidirect
176. #398Dic67semidirect
182. #380Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585302
#225582092
C67×#24862
#1187467