Finite Group Databasebeta

← C2×#22586364 #31907727

Subgroups — C2×#22586364 #31907727

Order 18176.

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

normal subgroups only

This group has 712 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#22586364C214yesmaximal, direct product factor
4C2×#399×C71C211yesmaximal, characteristic
5C22×C22⋊Dic71—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985766—21no—
8#17985766C2212yes—
9#17985723—21no—
10#17985723C2212yes—
11C22×C4⋊C4×C71—21no—
12C22×C4⋊C4×C71C411yescharacteristic, complemented
13C22×C22⋊C4×C71—21no—
14C22×C22⋊C4×C71C2211yescharacteristic
15#399×C71—21no—
16#399×C71—21no—
17#399×C71C412yescomplemented
18#399×C71C2212yes—
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
29C23×C4×C71—21no—
30C23×C4×C71—21no—
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C2×C4⋊C4×C71—22no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—44no—
37C2×C4⋊C4×C71—41no—
38C2×C4⋊C4×C71—41no—
39C2×C4⋊C4×C71C2×C412yescomplemented
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—41no—
43C2×C22⋊C4×C71—41no—
44C2×C22⋊C4×C71—44no—
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71C2311yescharacteristic
49#95×C71—21no—
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
70C22×C4×C71—24no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—21no—
76C22×C4×C71—22no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71—42no—
82C22×C4×C71—42no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—42no—
90C4⋊C4×C71—44no—
91C4⋊C4×C71—44no—
92C4⋊C4×C71—42no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—44no—
98C22⋊C4×C71—44no—
99C23×C71—22no—
100C23×C71—22no—
101C23×C71—22no—
102C23×C71—22no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—22no—
106C23×C71—22no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—44no—
113C23×C71—44no—
114C23×C71—41no—
115C23×C71—42no—
116C23×C71—42no—
117C23×C71—81no—
118C23×C71—84no—
119C23×C71—81no—
120C23×C71#9911yescharacteristic
121C23×C71#9911yescharacteristic
122C23×C71#10312yes—
123C23×C71#10312yes—
124C23×C71#10412yes—
125C23×C71C2×C22⋊C411yescharacteristic
126C23×C71C2×SD1611yescharacteristic
127C23×C71C2×Q1611yescharacteristic
128C2×C4×C71—24no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—42no—
134C2×C4×C71—48no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
146C2×C4×C71—81no—
147C2×C4×C71—82no—
148C2×C4×C71—82no—
149C2×Dic71—44no—
150C2×Dic71—82no—
151C2×Dic71—88no—
152C2×Dic71—82no—
153C71⋊C8—82no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—24no—
157C22×C71—24no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—21no—
163C22×C71—42no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—44no—
169C22×C71—44no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—81no—
173C22×C71—81no—
174C22×C71—82no—
175C22×C71—82no—
176C22×C71#32812yes—
177C22×C71#32812yes—
178C22×C71#35212yes—
179C22×C71#35312yes—
180C22×C71C2×#9911yescharacteristic
181C22×C71C2×#10311yescharacteristic
182C22×C71C2×#10411yescharacteristic
183C4×C71—44no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—82no—
189Dic71—84no—
190C2×#1352—711nomaximal, Sylow
191C2×C71—21no—
192C2×C71—21no—
193C2×C71—22no—
194C2×C71—41no—
195C2×C71—42no—
196C2×C71—41no—
197C2×C71#135214yescomplemented
198C2×C71C2×#32811yescharacteristic
199C2×C71C2×#35211yescharacteristic
200C2×C71C2×#35311yescharacteristic
201C2×#399D7111yescharacteristic
202C2×#342—711no—
203C2×#323—711no—
204#1352—714no—
205C71C2×#135211yescharacteristic, complemented, Sylow
206C22×C4⋊C4—21no—
207C22×C4⋊C4Dic7111yescharacteristic, complemented
208C22×C22⋊C4—21no—
209C22×C22⋊C4—1421no—
210C22×C22⋊C4C2×D7111yescharacteristic
211C2×C22⋊C8—1421no—
212#399—21no—
213#399—21no—
214#399Dic7112yescomplemented
215#399C2×D7112yes—
216C2×#95Dic7111yescharacteristic, complemented
217#342—712no—
218#342—1421no—
219#323—712no—
220#323—1421no—
221C25C22⋊D7111yescharacteristic
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—2841no—
225C23×C4C2×Dic7111yescharacteristic
226C23×C4C22⋊D7111yescharacteristic
227C22×C8—2841no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×Dic7112yescomplemented
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—2841no—
241C2×C22⋊C4—2842no—
242C2×C22⋊C4—1424no—
243C2×C22⋊C4—1422no—
244C2×C22⋊C4C2×Dic7111yescharacteristic
245C2×C22⋊C4C22⋊D7111yescharacteristic
246C2×C22⋊C4C22⋊D7111yescharacteristic
247C2×C22⋊C4C22×D7111yescharacteristic
248C22⋊C8—2841no—
249C22⋊C8—1422no—
250#95—21no—
251#95C2×Dic7112yescomplemented
252C24—21no—
253C24—21no—
254C24—22no—
255C24—42no—
256C24—42no—
257C24C22⋊Dic7111yescharacteristic
258C24C22⋊Dic7111yescharacteristic
259C24#1176585212yes—
260C24#1176585512yes—
261C24C2×C22⋊D7111yescharacteristic
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—2842no—
278C22×C4—2844no—
279C22×C4—5682no—
280C22×C4—5682no—
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22⋊Dic7111yescharacteristic
283C22×C4C22×Dic7111yescharacteristic
284C22×C4C2×C22⋊D7111yescharacteristic
285C2×C8—2842no—
286C2×C8—5682no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—2844no—
298C22⋊C4—2842no—
299C22⋊C4—1424no—
300C23—22no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—44no—
314C23—44no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23#1455604311yescharacteristic
322C23#1455604311yescharacteristic
323C23#1455605212yes—
324C23#1455605212yes—
325C23#1455605512yes—
326C23C2×C22⋊Dic7111yescharacteristic
327C23C2×#1176585211yescharacteristic
328C23C2×#1176585511yescharacteristic
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—2844no—
351C2×C4—5682no—
352C2×C4—5682no—
353C2×C4—5688no—
354C8—5682no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—24no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—44no—
372C22—44no—
373C22—81no—
374C22—81no—
375C22—82no—
376C22—82no—
377C22#1798573612yes—
378C22#1798573612yes—
379C22#1798578912yes—
380C22#1798579212yes—
381C22C2×#1455604311yescharacteristic
382C22C2×#1455605211yescharacteristic
383C22C2×#1455605511yescharacteristic
384C4—44no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—82no—
390C4—5684no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—42no—
396C2—41no—
397C2#2258636414yesdirect product factor
398C2C2×#1798573611yescharacteristic
399C2C2×#1798578911yescharacteristic
400C2C2×#1798579211yescharacteristic
subgroupG/H
centerC23#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC2×#399×C71C2
socleC23×C71#99
prime radicalC25×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenterC2×#399D71
hypocenterC71C2×#1352
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
3. #22586364C2direct (N × H)
12. C22×C4⋊C4×C71C4semidirect
17. #399×C71C4semidirect
19. C2×#95×C71C4semidirect
39. C2×C4⋊C4×C71C2×C4semidirect
50. #95×C71C2×C4semidirect
197. C2×C71#1352semidirect
205. C71C2×#1352semidirect
207. C22×C4⋊C4Dic71semidirect
214. #399Dic71semidirect
216. C2×#95Dic71semidirect
234. C2×C4⋊C4C2×Dic71semidirect
251. #95C2×Dic71semidirect
397. C2#22586364direct (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
#225863642
C2×#399×C712
C2×#135271