Finite Group Databasebeta

← #30749377

Subgroups — #30749377

Order 17152.

530 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560197C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558088C212yesmaximal, complemented
4#22558076C212yesmaximal, complemented
5C67×#2349C211yesmaximal, characteristic
6C2×#14523958—21no—
7C2×C22⋊(C67⋊C8)C2212yes—
8#16863256C2212yes—
9#16863163—21no—
10#16863122—21no—
11#16863120—21no—
12C2×C4×D4×C67—21no—
13C2×C4×D4×C67C2211yescharacteristic
14#389×C67—21no—
15#385×C67—21no—
16#385×C67—21no—
17#382×C67C411yescharacteristic, complemented
18#380×C67C411yescharacteristic, complemented
19#380×C67C2211yescharacteristic
20C4×C4⋊C4×C67C411yescharacteristic, complemented
21C2×#95×C67C411yescharacteristic, complemented
22C2×#95×C67C2211yescharacteristic
23C22×C67⋊C8—22no—
24C2×C4⋊Dic67—22no—
25C2×C22⋊Dic67—22no—
26#14523958—42no—
27C22⋊(C67⋊C8)—22no—
28C22⋊(C67⋊C8)—22no—
29C67⋊#95—22no—
30C22×D4×C67D411yescharacteristic
31C23×C4×C67—21no—
32C23×C4×C67—21no—
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67C2311yescharacteristic
35C4×D4×C67—22no—
36C4×D4×C67—22no—
37C4×D4×C67—42no—
38C4×D4×C67—42no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—21no—
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67D411yescharacteristic
47C2×C22⋊C4×C67D411yescharacteristic
48C2×C42×C67—21no—
49C2×C42×C67C2×C411yescharacteristic
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67C2×C411yescharacteristic
54#95×C67C2×C411yescharacteristic
55#95×C67C2×C411yescharacteristic
56#95×C67C2×C411yescharacteristic
57C22×Dic67—42no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—82no—
60C4⋊Dic67—82no—
61C22⋊Dic67—42no—
62C22⋊Dic67—42no—
63C24×C67C22⋊C411yescharacteristic
64C24×C67C2×D411yescharacteristic
65C2×D4×C67—22no—
66C2×D4×C67—22no—
67C2×D4×C67D812yes—
68C2×D4×C67SD1612yes—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—42no—
76C22×C4×C67—42no—
77C22×C4×C67—42no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67—81no—
81C22×C4×C67—81no—
82C22×C4×C67—81no—
83C22×C4×C67—81no—
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22⋊C411yescharacteristic
86C22×C4×C67C22⋊C411yescharacteristic
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C22×C4×C67#4112yes—
90C22×C4×C67#4112yes—
91C4⋊C4×C67—22no—
92C4⋊C4×C67—42no—
93C4⋊C4×C67—42no—
94C4⋊C4×C67—81no—
95C4⋊C4×C67—81no—
96C22⋊C4×C67—22no—
97C22⋊C4×C67—22no—
98C22⋊C4×C67—42no—
99C22⋊C4×C67—41no—
100C22⋊C4×C67—41no—
101C42×C67—21no—
102C42×C67—21no—
103C42×C67—42no—
104C23×C67—21no—
105C23×C67—21no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67—41no—
109C23×C67—41no—
110C23×C67#10212yes—
111C23×C67C2×C22⋊C411yescharacteristic
112C23×C67#12312yes—
113C23×C67C2×D811yescharacteristic
114C23×C67C2×SD1611yescharacteristic
115C23×C67#13712yes—
116D4×C67—24no—
117D4×C67—24no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—22no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—42no—
126C2×C4×C67—42no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—81no—
141C2×C4×C67—81no—
142C2×C4×C67—82no—
143C2×C4×C67—82no—
144C2×C4×C67—84no—
145C2×C4×C67#10212yes—
146C2×C4×C67#11711yescharacteristic
147C2×C4×C67#11711yescharacteristic
148C2×C4×C67#12312yes—
149C2×Dic67—82no—
150C2×Dic67—82no—
151C2×Dic67—82no—
152C67⋊C8—82no—
153C22×C67—22no—
154C22×C67—21no—
155C22×C67—22no—
156C22×C67—21no—
157C22×C67—22no—
158C22×C67—41no—
159C22×C67—41no—
160C22×C67—42no—
161C22×C67—42no—
162C22×C67—42no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67#38111yescharacteristic
166C22×C67C2×#10211yescharacteristic
167C22×C67#41911yescharacteristic
168C22×C67#48012yes—
169C22×C67#48312yes—
170C4×C67—22no—
171C4×C67—22no—
172C4×C67—21no—
173C4×C67—21no—
174C4×C67—42no—
175C4×C67—41no—
176C4×C67—41no—
177C4×C67—81no—
178C4×C67—81no—
179C4×C67—81no—
180Dic67—162no—
181#8760—671nomaximal, Sylow
182C2×C67—22no—
183C2×C67—21no—
184C2×C67—21no—
185C2×C67—22no—
186C2×C67—42no—
187C2×C67#182311yescharacteristic
188C2×C67#182711yescharacteristic
189C2×C67#182811yescharacteristic
190#2349D6711yescharacteristic
191#1799—671no—
192#1459—671no—
193#1294—672no—
194#1289—672no—
195C67#876011yescharacteristic, complemented, Sylow
196C2×C4×D4—21no—
197C2×C4×D4C2×D6711yescharacteristic
198C2×#99—1341no—
199C2×C22⋊C8—672no—
200#389—21no—
201#385—21no—
202#385—21no—
203#382Dic6711yescharacteristic, complemented
204#380—672no—
205#380Dic6711yescharacteristic, complemented
206#380C2×D6711yescharacteristic
207C4×C4⋊C4Dic6711yescharacteristic, complemented
208C2×#95Dic6711yescharacteristic, complemented
209C2×#95C2×D6711yescharacteristic
210#342—1341no—
211#324—1341no—
212#323—1341no—
213C22×D4C22⋊D6711yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C2×Dic6711yescharacteristic
217C23×C4C22×D6711yescharacteristic
218C22×C8—1342no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—42no—
222C4×D4—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—1342no—
228C2×C4⋊C4C22⋊D6711yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1342no—
232C2×C22⋊C4C22⋊D6711yescharacteristic
233C2×C22⋊C4C22⋊D6711yescharacteristic
234C2×C42—21no—
235C2×C42C2×Dic6711yescharacteristic
236#99—2682no—
237C22⋊C8—1342no—
238C22⋊C8—1342no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—1342no—
243#95C2×Dic6711yescharacteristic
244#95C2×Dic6711yescharacteristic
245#95C2×Dic6711yescharacteristic
246#95C2×Dic6711yescharacteristic
247C24C22⋊Dic6711yescharacteristic
248C24C2×C22⋊D6711yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4#1176161312yes—
252C2×D4#1176161612yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—2682no—
269C22×C4C22⋊Dic6711yescharacteristic
270C22×C4C22⋊Dic6711yescharacteristic
271C22×C4C22⋊Dic6711yescharacteristic
272C22×C4C22×Dic6711yescharacteristic
273C22×C4C2×C22⋊D6711yescharacteristic
274C22×C4#1176162812yes—
275C22×C4#1176162812yes—
276C2×C8—2682no—
277C2×C8—5362no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—5362no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—2682no—
290C22⋊C4—2682no—
291C42—21no—
292C42—21no—
293C42—42no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#1452396412yes—
301C23C2×C22⋊Dic6711yescharacteristic
302C23#1452402612yes—
303C23C2×#1176161311yescharacteristic
304C23C2×#1176161611yescharacteristic
305C23#1452407412yes—
306D4—24no—
307D4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—84no—
335C2×C4—5362no—
336C2×C4—5362no—
337C2×C4—5362no—
338C2×C4#1452396412yes—
339C2×C4#1452400211yescharacteristic
340C2×C4#1452400211yescharacteristic
341C2×C4#1452402612yes—
342C8—5362no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22#1686325911yescharacteristic
356C22C2×#1452396411yescharacteristic
357C22#1686340511yescharacteristic
358C22#1686370912yes—
359C22#1686372612yes—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—10722no—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—22no—
375C2—42no—
376C2#2256029511yescharacteristic
377C2#2256031311yescharacteristic
378C2#2256031711yescharacteristic
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2349C2
socleC22×C67#381
prime radicalC22×D4×C67D4
cosocleC23×C4×C67C23
hypercenter#2349D67
hypocenterC67#8760
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22558088C2semidirect
4. #22558076C2semidirect
17. #382×C67C4semidirect
18. #380×C67C4semidirect
20. C4×C4⋊C4×C67C4semidirect
21. C2×#95×C67C4semidirect
195. C67#8760semidirect
203. #382Dic67semidirect
205. #380Dic67semidirect
207. C4×C4⋊C4Dic67semidirect
208. C2×#95Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585272
#225580882
#225580762
C67×#23492
#876067