Finite Group Databasebeta

← C2×#22586343 #31907645

Subgroups — C2×#22586343 #31907645

Order 18176.

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

normal subgroups only

This group has 824 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×#17985768C211yesmaximal, characteristic, complemented
2C2×#17985725C211yesmaximal, characteristic, complemented
3#22586343C214yesmaximal, direct product factor
4C2×#394×C71C211yesmaximal, characteristic
5C22×#11765849—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985768—21no—
8#17985768C2212yescomplemented
9#17985725—21no—
10#17985725C2212yescomplemented
11C23×D4×C71C411yescharacteristic
12C22×C4⋊C4×C71C2211yescharacteristic
13C22×C22⋊C4×C71—21no—
14C22×C22⋊C4×C71—21no—
15#394×C71—21no—
16#394×C71—21no—
17#394×C71C412yes—
18#394×C71C2212yes—
19C2×#95×C71C411yescharacteristic
20C2×#11765849—24no—
21C2×#11765849—22no—
22C2×#11765849—41no—
23C2×#11765849—42no—
24C22×C71⋊C8—41no—
25C22×C71⋊C8—41no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28C25×C71—21no—
29C22×D4×C71—21no—
30C22×D4×C71—41no—
31C22×D4×C71—41no—
32C22×D4×C71—44no—
33C22×D4×C71C2×C412yes—
34C23×C4×C71—21no—
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C2×C4⋊C4×C71—41no—
39C2×C4⋊C4×C71—41no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71—22no—
48C2×C22⋊C4×C71—44no—
49C2×C22⋊C4×C71—42no—
50C2×C22⋊C4×C71—42no—
51#95×C71—21no—
52#95×C71C2×C412yes—
53#11765849—24no—
54#11765849—44no—
55#11765849—42no—
56C2×C71⋊C8—44no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—82no—
60C24×C71—24no—
61C24×C71—22no—
62C24×C71—42no—
63C24×C71—42no—
64C24×C71—42no—
65C24×C71—41no—
66C24×C71—41no—
67C24×C71—42no—
68C24×C71—42no—
69C24×C71C22⋊C411yescharacteristic
70C2×D4×C71—42no—
71C2×D4×C71—42no—
72C2×D4×C71—44no—
73C2×D4×C71—44no—
74C2×D4×C71—48no—
75C2×D4×C71—44no—
76C2×D4×C71—44no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—22no—
85C22×C4×C71—42no—
86C22×C4×C71—42no—
87C22×C4×C71—42no—
88C22×C4×C71—42no—
89C22×C4×C71—42no—
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71D812yes—
94C22×C4×C71SD1612yes—
95C22×C4×C71C22×C411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71C2×D411yescharacteristic
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—41no—
102C22⋊C4×C71—44no—
103C22⋊C4×C71—44no—
104C22⋊C4×C71—44no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—42no—
107C23×C71—22no—
108C23×C71—22no—
109C23×C71—24no—
110C23×C71—42no—
111C23×C71—44no—
112C23×C71—48no—
113C23×C71—48no—
114C23×C71—42no—
115C23×C71—42no—
116C23×C71—44no—
117C23×C71—42no—
118C23×C71—42no—
119C23×C71—42no—
120C23×C71—84no—
121C23×C71—84no—
122C23×C71—82no—
123C23×C71—82no—
124C23×C71—82no—
125C23×C71—82no—
126C23×C71#9911yescharacteristic
127C23×C71#9911yescharacteristic
128C23×C71#10212yes—
129C23×C71#10412yes—
130C23×C71C2×C22⋊C411yescharacteristic
131D4×C71—44no—
132D4×C71—44no—
133D4×C71—48no—
134C2×C4×C71—22no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—22no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—42no—
147C2×C4×C71—42no—
148C2×C4×C71—42no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—42no—
154C2×C4×C71—42no—
155C2×C4×C71—42no—
156C2×C4×C71—81no—
157C2×C4×C71—84no—
158C2×C4×C71#10212yes—
159C2×C4×C71C2×D811yescharacteristic
160C2×C4×C71C2×SD1611yescharacteristic
161C71⋊C8—44no—
162C71⋊C8—82no—
163C22×C71—24no—
164C22×C71—24no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—44no—
170C22×C71—44no—
171C22×C71—44no—
172C22×C71—82no—
173C22×C71—82no—
174C22×C71—88no—
175C22×C71—88no—
176C22×C71—82no—
177C22×C71—82no—
178C22×C71#32712yes—
179C22×C71#32712yes—
180C22×C71AΣL(1,𝔽16)12yes—
181C22×C71#35512yes—
182C22×C71C2×#9911yescharacteristic
183C22×C71C2×#10211yescharacteristic
184C22×C71C2×#10411yescharacteristic
185C4×C71—21no—
186C4×C71—22no—
187C4×C71—21no—
188C4×C71—41no—
189C4×C71—41no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—82no—
193C2×#1345—711nomaximal, Sylow
194C2×C71—21no—
195C2×C71—21no—
196C2×C71—22no—
197C2×C71—84no—
198C2×C71#134514yescomplemented
199C2×C71C2×#32711yescharacteristic
200C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
201C2×C71C2×#35511yescharacteristic
202C2×#394D7111yescharacteristic
203C2×#343—711no—
204C2×#324—711no—
205#1345—714no—
206C71C2×#134511yescharacteristic, complemented, Sylow
207C23×D4Dic7111yescharacteristic
208C22×M16—1421no—
209C22×C4⋊C4C2×D7111yescharacteristic
210C22×C22⋊C4—21no—
211C22×C22⋊C4—21no—
212C2×C22⋊C8—1421no—
213#394—21no—
214#394—21no—
215#394Dic7112yes—
216#394C2×D7112yes—
217C2×#95Dic7111yescharacteristic
218#343—712no—
219#343—1421no—
220#324—712no—
221#324—1421no—
222C25—21no—
223C22×D4—21no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—44no—
227C22×D4C2×Dic7112yes—
228C23×C4—21no—
229C23×C4C2×Dic7111yescharacteristic
230C23×C4C22⋊D7111yescharacteristic
231C23×C4C22⋊D7111yescharacteristic
232C2×M16—2841no—
233C2×M16—2842no—
234C2×M16—1424no—
235C2×M16—1422no—
236C22×C8—2841no—
237C22×C8—2841no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C2×Dic7111yescharacteristic
241C2×C4⋊C4C22⋊D7111yescharacteristic
242C2×C4⋊C4C22⋊D7111yescharacteristic
243C2×C4⋊C4C22×D7111yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C22⋊C8—2841no—
252C22⋊C8—1422no—
253#95—21no—
254#95C2×Dic7112yes—
255C24—24no—
256C24—22no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24C22⋊Dic7111yescharacteristic
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—44no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C22⋊Dic7111yescharacteristic
286C22×C4C22⋊Dic7111yescharacteristic
287C22×C4C22⋊Dic7111yescharacteristic
288C22×C4#1176585012yes—
289C22×C4#1176585312yes—
290C22×C4C22×Dic7111yescharacteristic
291C22×C4C2×C22⋊D7111yescharacteristic
292C22×C4C2×C22⋊D7111yescharacteristic
293M16—2844no—
294M16—2842no—
295M16—1424no—
296C2×C8—2842no—
297C2×C8—2844no—
298C2×C8—2842no—
299C2×C8—5682no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C23—22no—
310C23—22no—
311C23—24no—
312C23—42no—
313C23—44no—
314C23—42no—
315C23—42no—
316C23—48no—
317C23—48no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—44no—
322C23—82no—
323C23—84no—
324C23—84no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23#1455604311yescharacteristic
329C23#1455604311yescharacteristic
330C23#1455604912yes—
331C23#1455605512yes—
332C23C2×C22⋊Dic7111yescharacteristic
333D4—44no—
334D4—44no—
335D4—48no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—84no—
360C2×C4#1455604912yes—
361C2×C4C2×#1176585011yescharacteristic
362C2×C4C2×#1176585311yescharacteristic
363C8—2844no—
364C8—5682no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—24no—
370C22—24no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—82no—
375C22—82no—
376C22—88no—
377C22—82no—
378C22—82no—
379C22—88no—
380C22#1798573312yes—
381C22#1798573312yes—
382C22#1798578612yes—
383C22#1798579612yes—
384C22C2×#1455604311yescharacteristic
385C22C2×#1455604911yescharacteristic
386C22C2×#1455605511yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—82no—
395C2—21no—
396C2—21no—
397C2—22no—
398C2—84no—
399C2#2258634314yesdirect product factor
400C2C2×#1798573311yescharacteristic
401C2C2×#1798578611yescharacteristic
402C2C2×#1798579611yescharacteristic
subgroupG/H
centerC23#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC2×#394×C71C2
socleC23×C71#99
prime radicalC23×D4×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenterC2×#394D71
hypocenterC71C2×#1345
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#17985768C2semidirect
2. C2×#17985725C2semidirect
3. #22586343C2direct (N × H)
8. #17985768C22semidirect
10. #17985725C22semidirect
198. C2×C71#1345semidirect
206. C71C2×#1345semidirect
399. C2#22586343direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179857682
C2×#179857252
#225863432
C2×#394×C712
C2×#134571