Finite Group Databasebeta

← C2×#22558209 #30766002

Subgroups — C2×#22558209 #30766002

Order 17152.

668 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16863163C211yesmaximal, characteristic
2C2×#16863120C211yesmaximal, characteristic
3#22558209C214yesmaximal, direct product factor
4C2×#397×C67C211yesmaximal, characteristic
5C22×C22⋊Dic67—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863163—21no—
8#16863163C2212yes—
9#16863120—21no—
10#16863120C2212yes—
11C22×C4⋊C4×C67C411yescharacteristic, complemented
12C22×C22⋊C4×C67—21no—
13C22×C22⋊C4×C67C2211yescharacteristic
14#397×C67—21no—
15#397×C67—21no—
16#397×C67C412yescomplemented
17#397×C67C2212yes—
18C2×#95×C67—21no—
19C2×#95×C67C411yescharacteristic, complemented
20C23×Dic67—41no—
21C22×C67⋊C8—41no—
22C2×C22⋊Dic67—24no—
23C2×C22⋊Dic67—22no—
24C2×C22⋊Dic67—41no—
25C2×C22⋊Dic67—42no—
26C22⋊(C67⋊C8)—22no—
27C22⋊(C67⋊C8)—41no—
28C25×C67D411yescharacteristic
29C23×C4×C67—21no—
30C23×C4×C67—21no—
31C23×C4×C67C2×C411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—41no—
35C2×C4⋊C4×C67—41no—
36C2×C4⋊C4×C67C2×C412yescomplemented
37C2×C22⋊C4×C67—22no—
38C2×C22⋊C4×C67—22no—
39C2×C22⋊C4×C67—41no—
40C2×C22⋊C4×C67—41no—
41C2×C22⋊C4×C67—42no—
42C2×C22⋊C4×C67—42no—
43C2×C22⋊C4×C67C2×C411yescharacteristic
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
46C2×C22⋊C4×C67C2311yescharacteristic
47#95×C67—22no—
48#95×C67—21no—
49#95×C67—22no—
50#95×C67C2×C412yescomplemented
51C22×Dic67—44no—
52C22×Dic67—42no—
53C22×Dic67—82no—
54C22×Dic67—82no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—82no—
57C22⋊Dic67—24no—
58C22⋊Dic67—44no—
59C22⋊Dic67—42no—
60C24×C67—21no—
61C24×C67—21no—
62C24×C67—22no—
63C24×C67—42no—
64C24×C67—42no—
65C24×C67C22⋊C411yescharacteristic
66C24×C67C22⋊C411yescharacteristic
67C24×C67D812yes—
68C24×C67SD1612yes—
69C24×C67C2×D411yescharacteristic
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—22no—
75C22×C4×C67—22no—
76C22×C4×C67—42no—
77C22×C4×C67—42no—
78C22×C4×C67—42no—
79C22×C4×C67—42no—
80C22×C4×C67—41no—
81C22×C4×C67—41no—
82C22×C4×C67—42no—
83C22×C4×C67—42no—
84C22×C4×C67—42no—
85C22×C4×C67C22⋊C411yescharacteristic
86C22×C4×C67C22⋊C411yescharacteristic
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C4⋊C4×C67—42no—
90C4⋊C4×C67—42no—
91C22⋊C4×C67—41no—
92C22⋊C4×C67—41no—
93C22⋊C4×C67—41no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—44no—
96C22⋊C4×C67—42no—
97C22⋊C4×C67—42no—
98C23×C67—22no—
99C23×C67—22no—
100C23×C67—22no—
101C23×C67—22no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67—22no—
105C23×C67—22no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67—41no—
109C23×C67—41no—
110C23×C67—44no—
111C23×C67—44no—
112C23×C67—41no—
113C23×C67—41no—
114C23×C67—42no—
115C23×C67—42no—
116C23×C67—84no—
117C23×C67—81no—
118C23×C67—81no—
119C23×C67#9911yescharacteristic
120C23×C67#9911yescharacteristic
121C23×C67#10212yes—
122C23×C67#10212yes—
123C23×C67#10412yes—
124C23×C67C2×C22⋊C411yescharacteristic
125C23×C67C2×D811yescharacteristic
126C23×C67C2×SD1611yescharacteristic
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—42no—
132C2×C4×C67—42no—
133C2×C4×C67—42no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—42no—
139C2×C4×C67—42no—
140C2×C4×C67—42no—
141C2×C4×C67—42no—
142C2×C4×C67—42no—
143C2×C4×C67—42no—
144C2×C4×C67—81no—
145C2×C4×C67—84no—
146C2×C4×C67—82no—
147C2×C4×C67—82no—
148C2×Dic67—44no—
149C2×Dic67—88no—
150C2×Dic67—82no—
151C2×Dic67—82no—
152C67⋊C8—82no—
153C22×C67—24no—
154C22×C67—24no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—22no—
161C22×C67—21no—
162C22×C67—42no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—44no—
166C22×C67—44no—
167C22×C67—41no—
168C22×C67—41no—
169C22×C67—41no—
170C22×C67—41no—
171C22×C67—82no—
172C22×C67—82no—
173C22×C67—81no—
174C22×C67—81no—
175C22×C67#32712yes—
176C22×C67#32712yes—
177C22×C67#35212yes—
178C22×C67#35412yes—
179C22×C67C2×#9911yescharacteristic
180C22×C67C2×#10211yescharacteristic
181C22×C67C2×#10411yescharacteristic
182C4×C67—41no—
183C4×C67—41no—
184C4×C67—41no—
185C4×C67—41no—
186C4×C67—82no—
187C4×C67—82no—
188Dic67—84no—
189C2×#1344—671nomaximal, Sylow
190C2×C67—21no—
191C2×C67—22no—
192C2×C67—21no—
193C2×C67—41no—
194C2×C67—42no—
195C2×C67—41no—
196C2×C67#134414yescomplemented
197C2×C67C2×#32711yescharacteristic
198C2×C67C2×#35211yescharacteristic
199C2×C67C2×#35411yescharacteristic
200C2×#397D6711yescharacteristic
201C2×#342—671no—
202C2×#323—671no—
203#1344—674no—
204C67C2×#134411yescharacteristic, complemented, Sylow
205C22×C4⋊C4Dic6711yescharacteristic, complemented
206C22×C22⋊C4—21no—
207C22×C22⋊C4—1341no—
208C22×C22⋊C4C2×D6711yescharacteristic
209C2×C22⋊C8—1341no—
210#397—21no—
211#397—21no—
212#397Dic6712yescomplemented
213#397C2×D6712yes—
214C2×#95—21no—
215C2×#95Dic6711yescharacteristic, complemented
216#342—672no—
217#342—1341no—
218#323—672no—
219#323—1341no—
220C25C22⋊D6711yescharacteristic
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—2681no—
224C23×C4C2×Dic6711yescharacteristic
225C23×C4C22⋊D6711yescharacteristic
226C22×C8—2681no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4C2×Dic6712yescomplemented
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—2681no—
238C2×C22⋊C4—2682no—
239C2×C22⋊C4—1344no—
240C2×C22⋊C4—1342no—
241C2×C22⋊C4C2×Dic6711yescharacteristic
242C2×C22⋊C4C22⋊D6711yescharacteristic
243C2×C22⋊C4C22⋊D6711yescharacteristic
244C2×C22⋊C4C22×D6711yescharacteristic
245C22⋊C8—2681no—
246C22⋊C8—1342no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95C2×Dic6712yescomplemented
251C24—21no—
252C24—21no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24C22⋊Dic6711yescharacteristic
257C24C22⋊Dic6711yescharacteristic
258C24#1176161312yes—
259C24#1176161612yes—
260C24C2×C22⋊D6711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—2682no—
277C22×C4—2684no—
278C22×C4—5362no—
279C22×C4—5362no—
280C22×C4C22⋊Dic6711yescharacteristic
281C22×C4C22⋊Dic6711yescharacteristic
282C22×C4C22×Dic6711yescharacteristic
283C22×C4C2×C22⋊D6711yescharacteristic
284C2×C8—2682no—
285C2×C8—5362no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—2684no—
296C22⋊C4—2682no—
297C22⋊C4—1344no—
298C23—22no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—44no—
311C23—44no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—84no—
317C23—81no—
318C23—81no—
319C23#1452395811yescharacteristic
320C23#1452395811yescharacteristic
321C23#1452396412yes—
322C23#1452396412yes—
323C23#1452397012yes—
324C23C2×C22⋊Dic6711yescharacteristic
325C23C2×#1176161311yescharacteristic
326C23C2×#1176161611yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—2684no—
349C2×C4—5368no—
350C2×C4—5362no—
351C2×C4—5362no—
352C8—5362no—
353C22—24no—
354C22—24no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—44no—
366C22—44no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—82no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22#1686313012yes—
376C22#1686313012yes—
377C22#1686318612yes—
378C22#1686319112yes—
379C22C2×#1452395811yescharacteristic
380C22C2×#1452396411yescharacteristic
381C22C2×#1452397011yescharacteristic
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—82no—
387C4—82no—
388C4—5364no—
389C2—21no—
390C2—22no—
391C2—21no—
392C2—41no—
393C2—42no—
394C2—41no—
395C2#2255820914yesdirect product factor
396C2C2×#1686313011yescharacteristic
397C2C2×#1686318611yescharacteristic
398C2C2×#1686319111yescharacteristic
subgroupG/H
centerC23#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC2×#397×C67C2
socleC23×C67#99
prime radicalC25×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenterC2×#397D67
hypocenterC67C2×#1344
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. #22558209C2direct (N × H)
11. C22×C4⋊C4×C67C4semidirect
16. #397×C67C4semidirect
19. C2×#95×C67C4semidirect
36. C2×C4⋊C4×C67C2×C4semidirect
50. #95×C67C2×C4semidirect
196. C2×C67#1344semidirect
204. C67C2×#1344semidirect
205. C22×C4⋊C4Dic67semidirect
212. #397Dic67semidirect
215. C2×#95Dic67semidirect
230. C2×C4⋊C4C2×Dic67semidirect
250. #95C2×Dic67semidirect
395. C2#22558209direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168631632
C2×#168631202
#225582092
C2×#397×C672
C2×#134467