Finite Group Databasebeta

← C2×#22586361 #31907723

Subgroups — C2×#22586361 #31907723

Order 18176.

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

normal subgroups only

This group has 776 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×#17985766C211yesmaximal, characteristic
2C2×#17985723C211yesmaximal, characteristic
3#22586361C214yesmaximal, direct product factor
4C2×#393×C71C211yesmaximal, characteristic
5C22×C22⋊Dic71—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985766—21no—
8#17985766C2212yes—
9#17985723—21no—
10#17985723C2212yes—
11C23×Q8×C71C411yescharacteristic, complemented
12C22×C22⋊C4×C71—21no—
13C22×C22⋊C4×C71C2211yescharacteristic
14#393×C71—21no—
15#393×C71—21no—
16#393×C71C412yescomplemented
17#393×C71C2212yes—
18C2×#95×C71—21no—
19C2×#95×C71C411yescharacteristic, complemented
20C23×Dic71—41no—
21C22×C71⋊C8—41no—
22C2×C22⋊Dic71—24no—
23C2×C22⋊Dic71—22no—
24C2×C22⋊Dic71—41no—
25C2×C22⋊Dic71—42no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28C25×C71D411yescharacteristic
29C22×Q8×C71—21no—
30C22×Q8×C71—44no—
31C22×Q8×C71—41no—
32C22×Q8×C71—41no—
33C22×Q8×C71C2×C412yescomplemented
34C23×C4×C71—21no—
35C23×C4×C71—21no—
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71—22no—
40C2×C22⋊C4×C71—41no—
41C2×C22⋊C4×C71—41no—
42C2×C22⋊C4×C71—44no—
43C2×C22⋊C4×C71C2×C411yescharacteristic
44C2×C22⋊C4×C71D411yescharacteristic
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71C2311yescharacteristic
47#95×C71—22no—
48#95×C71—21no—
49#95×C71—22no—
50#95×C71C2×C412yescomplemented
51C22×Dic71—44no—
52C22×Dic71—42no—
53C22×Dic71—82no—
54C22×Dic71—82no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—82no—
57C22⋊Dic71—24no—
58C22⋊Dic71—44no—
59C22⋊Dic71—42no—
60C24×C71—21no—
61C24×C71—21no—
62C24×C71—22no—
63C24×C71—42no—
64C24×C71—42no—
65C24×C71C22⋊C411yescharacteristic
66C24×C71C22⋊C411yescharacteristic
67C24×C71SD1612yes—
68C24×C71Q1612yes—
69C24×C71C2×D411yescharacteristic
70C2×Q8×C71—44no—
71C2×Q8×C71—44no—
72C2×Q8×C71—42no—
73C2×Q8×C71—42no—
74C2×Q8×C71—48no—
75C2×Q8×C71—44no—
76C2×Q8×C71—44no—
77C22×C4×C71—24no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—22no—
82C22×C4×C71—21no—
83C22×C4×C71—22no—
84C22×C4×C71—42no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—42no—
88C22×C4×C71—42no—
89C22×C4×C71—42no—
90C22×C4×C71—42no—
91C22×C4×C71—42no—
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22⋊C411yescharacteristic
94C22×C4×C71C22×C411yescharacteristic
95C22×C4×C71C2×D411yescharacteristic
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—44no—
101C22⋊C4×C71—44no—
102C23×C71—22no—
103C23×C71—22no—
104C23×C71—22no—
105C23×C71—22no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—22no—
109C23×C71—22no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—44no—
116C23×C71—44no—
117C23×C71—41no—
118C23×C71—42no—
119C23×C71—42no—
120C23×C71—81no—
121C23×C71—84no—
122C23×C71—81no—
123C23×C71#9911yescharacteristic
124C23×C71#9911yescharacteristic
125C23×C71#10312yes—
126C23×C71#10312yes—
127C23×C71#10412yes—
128C23×C71C2×C22⋊C411yescharacteristic
129C23×C71C2×SD1611yescharacteristic
130C23×C71C2×Q1611yescharacteristic
131Q8×C71—44no—
132Q8×C71—44no—
133Q8×C71—48no—
134C2×C4×C71—24no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—42no—
140C2×C4×C71—44no—
141C2×C4×C71—42no—
142C2×C4×C71—44no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—42no—
151C2×C4×C71—42no—
152C2×C4×C71—42no—
153C2×C4×C71—81no—
154C2×C4×C71—84no—
155C2×Dic71—44no—
156C2×Dic71—82no—
157C2×Dic71—88no—
158C2×Dic71—82no—
159C71⋊C8—82no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—24no—
163C22×C71—24no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—22no—
168C22×C71—21no—
169C22×C71—42no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71—44no—
175C22×C71—44no—
176C22×C71—41no—
177C22×C71—41no—
178C22×C71—81no—
179C22×C71—81no—
180C22×C71—82no—
181C22×C71—82no—
182C22×C71#32812yes—
183C22×C71#32812yes—
184C22×C71AΣL(1,𝔽16)12yes—
185C22×C71#35412yes—
186C22×C71C2×#9911yescharacteristic
187C22×C71C2×#10311yescharacteristic
188C22×C71C2×#10411yescharacteristic
189C4×C71—44no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—41no—
193C4×C71—41no—
194C4×C71—82no—
195Dic71—84no—
196C2×#1351—711nomaximal, Sylow
197C2×C71—21no—
198C2×C71—21no—
199C2×C71—22no—
200C2×C71—41no—
201C2×C71—42no—
202C2×C71—41no—
203C2×C71#135114yescomplemented
204C2×C71C2×#32811yescharacteristic
205C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
206C2×C71C2×#35411yescharacteristic
207C2×#393D7111yescharacteristic
208C2×#342—711no—
209C2×#323—711no—
210#1351—714no—
211C71C2×#135111yescharacteristic, complemented, Sylow
212C23×Q8Dic7111yescharacteristic, complemented
213C22×C22⋊C4—21no—
214C22×C22⋊C4—1421no—
215C22×C22⋊C4C2×D7111yescharacteristic
216C2×C22⋊C8—1421no—
217#393—21no—
218#393—21no—
219#393Dic7112yescomplemented
220#393C2×D7112yes—
221C2×#95—21no—
222C2×#95Dic7111yescharacteristic, complemented
223#342—712no—
224#342—1421no—
225#323—712no—
226#323—1421no—
227C25C22⋊D7111yescharacteristic
228C22×Q8—21no—
229C22×Q8—44no—
230C22×Q8—41no—
231C22×Q8—41no—
232C22×Q8C2×Dic7112yescomplemented
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—2841no—
236C23×C4C2×Dic7111yescharacteristic
237C23×C4C22⋊D7111yescharacteristic
238C22×C8—2841no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—2841no—
245C2×C22⋊C4—2842no—
246C2×C22⋊C4—1424no—
247C2×C22⋊C4—1422no—
248C2×C22⋊C4C2×Dic7111yescharacteristic
249C2×C22⋊C4C22⋊D7111yescharacteristic
250C2×C22⋊C4C22⋊D7111yescharacteristic
251C2×C22⋊C4C22×D7111yescharacteristic
252C22⋊C8—2841no—
253C22⋊C8—1422no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95C2×Dic7112yescomplemented
258C24—21no—
259C24—21no—
260C24—22no—
261C24—42no—
262C24—42no—
263C24C22⋊Dic7111yescharacteristic
264C24C22⋊Dic7111yescharacteristic
265C24#1176585212yes—
266C24#1176585512yes—
267C24C2×C22⋊D7111yescharacteristic
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—48no—
273C2×Q8—44no—
274C2×Q8—44no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—2842no—
291C22×C4—2844no—
292C22×C4—5682no—
293C22×C4—5682no—
294C22×C4C22⋊Dic7111yescharacteristic
295C22×C4C22⋊Dic7111yescharacteristic
296C22×C4C22×Dic7111yescharacteristic
297C22×C4C2×C22⋊D7111yescharacteristic
298C2×C8—2842no—
299C2×C8—5682no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—2844no—
307C22⋊C4—2842no—
308C22⋊C4—1424no—
309C23—22no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—44no—
323C23—44no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—81no—
328C23—81no—
329C23—84no—
330C23#1455604311yescharacteristic
331C23#1455604311yescharacteristic
332C23#1455605212yes—
333C23#1455605212yes—
334C23#1455605512yes—
335C23C2×C22⋊Dic7111yescharacteristic
336C23C2×#1176585211yescharacteristic
337C23C2×#1176585511yescharacteristic
338Q8—44no—
339Q8—44no—
340Q8—48no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4—2844no—
363C2×C4—5682no—
364C2×C4—5682no—
365C2×C4—5688no—
366C8—5682no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—24no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—44no—
384C22—44no—
385C22—81no—
386C22—81no—
387C22—82no—
388C22—82no—
389C22#1798573612yes—
390C22#1798573612yes—
391C22#1798578612yes—
392C22#1798579412yes—
393C22C2×#1455604311yescharacteristic
394C22C2×#1455605211yescharacteristic
395C22C2×#1455605511yescharacteristic
396C4—44no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—5684no—
403C2—21no—
404C2—21no—
405C2—22no—
406C2—41no—
407C2—42no—
408C2—41no—
409C2#2258636114yesdirect product factor
410C2C2×#1798573611yescharacteristic
411C2C2×#1798578611yescharacteristic
412C2C2×#1798579411yescharacteristic
subgroupG/H
centerC23#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC2×#393×C71C2
socleC23×C71#99
prime radicalC25×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenterC2×#393D71
hypocenterC71C2×#1351
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
3. #22586361C2direct (N × H)
11. C23×Q8×C71C4semidirect
16. #393×C71C4semidirect
19. C2×#95×C71C4semidirect
33. C22×Q8×C71C2×C4semidirect
50. #95×C71C2×C4semidirect
203. C2×C71#1351semidirect
211. C71C2×#1351semidirect
212. C23×Q8Dic71semidirect
219. #393Dic71semidirect
222. C2×#95Dic71semidirect
232. C22×Q8C2×Dic71semidirect
257. #95C2×Dic71semidirect
409. C2#22586361direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179857662
C2×#179857232
#225863612
C2×#393×C712
C2×#135171