Finite Group Databasebeta

← C2×#22587915 #31980025

Subgroups — C2×#22587915 #31980025

Order 18176.

694 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986350C211yesmaximal, characteristic, complemented
2C2×#17986289C211yesmaximal, characteristic
3C2×#17986279C211yesmaximal, characteristic
4#22587915C218yesmaximal, direct product factor
5C2×#17985750C211yesmaximal, characteristic, complemented
6C2×#17985747C211yesmaximal, characteristic, complemented
7C2×#17985746C211yesmaximal, characteristic, complemented
8C2×#436×C71C211yesmaximal, characteristic
9C2×#14556128C2211yescharacteristic
10C2×#14556128C2211yescharacteristic
11#17986350—22no—
12#17986350C2214yescomplemented
13#17986289—22no—
14#17986289C2214yes—
15#17986279—22no—
16#17986279C2214yes—
17C2×#14556061—21no—
18C2×#14556059—21no—
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×#14556053—21no—
22C2×#14556053—21no—
23C2×#14556049—21no—
24C2×#14556049—21no—
25#17985750C2214yescomplemented
26#17985747C2214yescomplemented
27#17985746C2214yescomplemented
28C22×M16×C71—21no—
29C2×C4×D4×C71C2211yescharacteristic
30#436×C71—22no—
31#436×C71C2214yes—
32C2×C4⋊C8×C71C2211yescharacteristic
33C2×C22⋊C8×C71—21no—
34C2×C4×C8×C71C2211yescharacteristic
35C22×Dic142—21no—
36C22×Dic142—21no—
37C22×C71⋊C8—21no—
38C22×C71⋊C8—21no—
39#14556128—21no—
40#14556128—21no—
41#14556128—21no—
42#14556128—21no—
43#14556128—21no—
44#14556128—21no—
45#14556128C2312yes—
46#14556128C2312yes—
47C2×C4⋊Dic71—21no—
48C2×C4⋊Dic71—21no—
49C2×C4⋊Dic71—21no—
50C2×C4⋊Dic71—21no—
51#14556061—24no—
52#14556059—24no—
53C4⋊(C71⋊C8)—21no—
54C4⋊(C71⋊C8)—21no—
55C4⋊(C71⋊C8)C2312yes—
56C4⋊(C71⋊C8)C2312yes—
57#14556053—24no—
58#14556053—24no—
59#14556049—24no—
60#14556049—24no—
61C22×D4×C71D411yescharacteristic
62C23×C4×C71—21no—
63C2×M16×C71—24no—
64C2×M16×C71—44no—
65C22×C8×C71—21no—
66C22×C8×C71D411yescharacteristic
67C22×C8×C71D411yescharacteristic
68C4×D4×C71—21no—
69C4×D4×C71—21no—
70C4×D4×C71D412yes—
71C4×D4×C71C2312yes—
72C2×C4⋊C4×C71D411yescharacteristic
73C2×C22⋊C4×C71—21no—
74C2×C42×C71C2311yescharacteristic
75C4⋊C8×C71—21no—
76C4⋊C8×C71C2312yes—
77C22⋊C8×C71—24no—
78C4×C8×C71D412yes—
79C4×C8×C71C2312yes—
80C2×Dic142—22no—
81C2×Dic142—22no—
82C2×Dic142—42no—
83C2×Dic142—42no—
84C2×Dic142—41no—
85C2×Dic142—41no—
86C2×Dic142—82no—
87C22×Dic71—41no—
88C22×Dic71—41no—
89C2×C71⋊C8—22no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8—42no—
92C2×C71⋊C8—42no—
93C4⋊Dic71—22no—
94C4⋊Dic71—22no—
95C4⋊Dic71—22no—
96C4⋊Dic71—22no—
97C4⋊Dic71—41no—
98C4⋊Dic71—41no—
99C4⋊Dic71—41no—
100C4⋊Dic71—41no—
101C24×C71—21no—
102C2×D4×C71—21no—
103C2×D4×C71—22no—
104C2×D4×C71—22no—
105C2×D4×C71C2×D412yes—
106C22×C4×C71—21no—
107C22×C4×C71—22no—
108C22×C4×C71—22no—
109C22×C4×C71—42no—
110C22×C4×C71—42no—
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2×D411yescharacteristic
113C22×C4×C71#4111yescharacteristic
114M16×C71—48no—
115C2×C8×C71—22no—
116C2×C8×C71—22no—
117C2×C8×C71—22no—
118C2×C8×C71—42no—
119C2×C8×C71C2×D412yes—
120C2×C8×C71C2×D412yes—
121C4⋊C4×C71—21no—
122C4⋊C4×C71C2×D412yes—
123C22⋊C4×C71—22no—
124C22⋊C4×C71—22no—
125C42×C71C2×D411yescharacteristic
126C42×C71C2×D411yescharacteristic
127C42×C71#4111yescharacteristic
128C42×C71C2411yescharacteristic
129C23×C71—22no—
130C23×C71—42no—
131C23×C71—42no—
132C23×C71—42no—
133C23×C71#12111yescharacteristic
134D4×C71—22no—
135D4×C71—22no—
136D4×C71—42no—
137C2×C4×C71—22no—
138C2×C4×C71—22no—
139C2×C4×C71—42no—
140C2×C4×C71—42no—
141C2×C4×C71—42no—
142C2×C4×C71—44no—
143C2×C4×C71#12111yescharacteristic
144C2×C4×C71#12111yescharacteristic
145C2×C4×C71#12111yescharacteristic
146C2×C4×C71#13512yes—
147C2×C4×C71AGL(1,ℤ/8)12yes—
148C2×C4×C71#13712yes—
149C2×C4×C71#13712yes—
150C2×C4×C71C22×D411yescharacteristic
151C2×C4×C71C22×D411yescharacteristic
152C2×C4×C71C2×#4111yescharacteristic
153C2×Dic71—42no—
154C2×Dic71—42no—
155C2×Dic71—82no—
156C2×Dic71—82no—
157Dic142—42no—
158Dic142—42no—
159Dic142—82no—
160Dic142—82no—
161Dic142—82no—
162C8×C71—22no—
163C8×C71—22no—
164C8×C71—42no—
165C71⋊C8—42no—
166C71⋊C8—42no—
167C22×C71—42no—
168C22×C71—42no—
169C22×C71—42no—
170C22×C71—84no—
171C22×C71#46312yes—
172C22×C71#47012yes—
173C22×C71#47112yes—
174C22×C71C2×#12111yescharacteristic
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—42no—
178C4×C71#46312yes—
179C4×C71#47012yes—
180C4×C71C2×#13511yescharacteristic
181C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
182C4×C71C2×#13711yescharacteristic
183C4×C71C2×#13711yescharacteristic
184Dic71—82no—
185Dic71—82no—
186C2×#1703—711nomaximal, Sylow
187C2×C71—82no—
188C2×C71#170314yescomplemented
189C2×C71C2×#46311yescharacteristic
190C2×C71C2×#47011yescharacteristic
191C2×C71C2×#47111yescharacteristic
192C2×#487—711no—
193C2×#476—711no—
194C2×#474—711no—
195C2×#436D7111yescharacteristic
196#1703—718no—
197C2×#334—711no—
198C2×#333—711no—
199C2×#332—711no—
200C71C2×#170311yescharacteristic, complemented, Sylow
201C22×M16—21no—
202C2×C4⋊Q8—711no—
203C2×C4⋊Q8—711no—
204C2×C4×D4C2×D7111yescharacteristic
205#487—714no—
206#487—1422no—
207#476—714no—
208#476—1422no—
209#474—714no—
210#474—1422no—
211#436—22no—
212#436C2×D7114yes—
213C2×#107—1421no—
214C2×#106—1421no—
215C2×C4⋊C8—711no—
216C2×C4⋊C8—711no—
217C2×C4⋊C8C2×D7111yescharacteristic
218C2×#103—1421no—
219C2×#103—1421no—
220C2×#102—1421no—
221C2×#102—1421no—
222C2×C22⋊C8—21no—
223C2×C4×C8C2×D7111yescharacteristic
224#334—714no—
225#333—714no—
226#332—714no—
227C22×Q8—1421no—
228C22×Q8—1421no—
229C22×D4D28411yescharacteristic
230C23×C4—21no—
231C2×M16—24no—
232C2×M16—44no—
233C22×C8—21no—
234C22×C8—1421no—
235C22×C8—1421no—
236C22×C8C22⋊D7111yescharacteristic
237C22×C8C22⋊D7111yescharacteristic
238C4⋊Q8—712no—
239C4⋊Q8—712no—
240C4⋊Q8—1421no—
241C4⋊Q8—1421no—
242C4⋊Q8—1421no—
243C4⋊Q8—1421no—
244C4⋊Q8—1421no—
245C4⋊Q8—1421no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4D28412yes—
249C4×D4C22×D7112yes—
250C2×C4⋊C4—1421no—
251C2×C4⋊C4—1421no—
252C2×C4⋊C4—1421no—
253C2×C4⋊C4—1421no—
254C2×C4⋊C4D28411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C42C22×D7111yescharacteristic
257#107—1424no—
258#106—1424no—
259C4⋊C8—21no—
260C4⋊C8—712no—
261C4⋊C8—712no—
262C4⋊C8—1421no—
263C4⋊C8—1421no—
264C4⋊C8C22×D7112yes—
265#103—1424no—
266#103—1424no—
267#102—1424no—
268#102—1424no—
269C22⋊C8—24no—
270C4×C8C22⋊D7112yes—
271C4×C8C22×D7112yes—
272C24—21no—
273C2×Q8—2841no—
274C2×Q8—2841no—
275C2×Q8—2842no—
276C2×Q8—2842no—
277C2×Q8—5682no—
278C2×Q8—1422no—
279C2×Q8—1422no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4C2×D28412yes—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—2841no—
290C22×C4—2841no—
291C22×C4C2×C22⋊D7111yescharacteristic
292C22×C4C2×D28411yescharacteristic
293C22×C4#1176586511yescharacteristic
294M16—48no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—2842no—
300C2×C8—2842no—
301C2×C8—1422no—
302C2×C8—1422no—
303C2×C8C2×C22⋊D7112yes—
304C2×C8C2×C22⋊D7112yes—
305C4⋊C4—21no—
306C4⋊C4—2841no—
307C4⋊C4—2841no—
308C4⋊C4—2841no—
309C4⋊C4—2841no—
310C4⋊C4—1422no—
311C4⋊C4—1422no—
312C4⋊C4—1422no—
313C4⋊C4—1422no—
314C4⋊C4C2×D28412yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C42C2×C22⋊D7111yescharacteristic
318C42C2×D28411yescharacteristic
319C42#1176586511yescharacteristic
320C42C23×D7111yescharacteristic
321C23—22no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23#1455610211yescharacteristic
326Q8—2842no—
327Q8—2842no—
328Q8—5682no—
329Q8—5682no—
330Q8—5682no—
331D4—22no—
332D4—22no—
333D4—42no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—2842no—
341C2×C4—2842no—
342C2×C4—5682no—
343C2×C4—5682no—
344C2×C4#1455610211yescharacteristic
345C2×C4#1455610211yescharacteristic
346C2×C4#1455610211yescharacteristic
347C2×C4#1455614812yes—
348C2×C4#1455615412yes—
349C2×C4#1455616112yes—
350C2×C4#1455616112yes—
351C2×C4C22×C22⋊D7111yescharacteristic
352C2×C4C22×D28411yescharacteristic
353C2×C4C2×#1176586511yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—2842no—
358C8—2842no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—84no—
363C22#1798621412yes—
364C22#1798625712yes—
365C22#1798626212yes—
366C22C2×#1455610211yescharacteristic
367C4—21no—
368C4—21no—
369C4—42no—
370C4—5682no—
371C4—5682no—
372C4#1798621412yes—
373C4#1798625712yes—
374C4C2×#1455614811yescharacteristic
375C4C2×#1455615411yescharacteristic
376C4C2×#1455616111yescharacteristic
377C4C2×#1455616111yescharacteristic
378C2—82no—
379C2#2258791514yesdirect product factor
380C2C2×#1798621411yescharacteristic
381C2C2×#1798625711yescharacteristic
382C2C2×#1798626211yescharacteristic
subgroupG/H
centerC23#14556102
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#436×C71C2
socleC23×C71#121
prime radicalC22×D4×C71D4
hypercenterC2×#436D71
hypocenterC71C2×#1703
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×#17986350C2semidirect
4. #22587915C2direct (N × H)
5. C2×#17985750C2semidirect
6. C2×#17985747C2semidirect
7. C2×#17985746C2semidirect
12. #17986350C22semidirect
25. #17985750C22semidirect
26. #17985747C22semidirect
27. #17985746C22semidirect
188. C2×C71#1703semidirect
200. C71C2×#1703semidirect
379. C2#22587915direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179863502
C2×#179862892
C2×#179862792
#225879152
C2×#179857502
C2×#179857472
C2×#179857462
C2×#436×C712
C2×#170371