Finite Group Databasebeta

← C2×#22587347 #31969680

Subgroups — C2×#22587347 #31969680

Order 18176.

1028 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 203 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×#17986630C211yesmaximal, characteristic
2C2×#17986329C212yesmaximal
3#22587347C218yesmaximal, direct product factor
4C2×#17985735C212yesmaximal
5C2×#17985725C211yesmaximal, characteristic
6C2×#398×C71C211yesmaximal, characteristic
7#17986630—24no—
8#17986630—22no—
9#17986630C2214yes—
10C2×#14556127—22no—
11C2×#14556119—22no—
12C2×#14556107—21no—
13C2×#14556107C2212yes—
14C2×#14556087—21no—
15#17986329—24no—
16#17986329C2218yes—
17C2×#14556059—22no—
18C2×#14556051—22no—
19C2×C22⋊(C71⋊C8)C2212yes—
20#17985735C2218yes—
21#17985725C2214yes—
22C22×C4⋊C4×C71—21no—
23C22×C4⋊C4×C71C2211yescharacteristic
24C22×C4⋊C4×C71C2211yescharacteristic
25#398×C71—22no—
26#398×C71C2214yes—
27C2×#95×C71—21no—
28C2×#95×C71C2211yescharacteristic
29C22×Dic142—21no—
30C22×Dic142D412yes—
31C22×C71⋊C8—22no—
32#14556127—28no—
33#14556127—28no—
34#14556119—28no—
35#14556119—28no—
36#14556107—24no—
37#14556107—22no—
38#14556107—22no—
39#14556107—24no—
40#14556107D414yes—
41#14556107C2314yes—
42#14556087—24no—
43#14556087—24no—
44C2×C4⋊Dic71—21no—
45C2×C4⋊Dic71—21no—
46C2×C4⋊Dic71—21no—
47C2×C4⋊Dic71D412yes—
48C2×Dic71⋊C4—22no—
49C2×Dic71⋊C4—22no—
50C2×C22⋊Dic71—22no—
51C2×C4×Dic71—22no—
52#14556059—28no—
53#14556051—28no—
54C22⋊(C71⋊C8)—22no—
55C22⋊(C71⋊C8)C2314yes—
56C23×C4×C71—21no—
57C23×C4×C71—21no—
58C23×C4×C71D411yescharacteristic
59C23×C4×C71D411yescharacteristic
60C23×C4×C71C2311yescharacteristic
61C2×C4⋊C4×C71—22no—
62C2×C4⋊C4×C71—22no—
63C2×C4⋊C4×C71—22no—
64C2×C4⋊C4×C71—22no—
65C2×C4⋊C4×C71—24no—
66C2×C4⋊C4×C71—21no—
67C2×C4⋊C4×C71—44no—
68C2×C4⋊C4×C71D412yes—
69C2×C4⋊C4×C71C2312yes—
70C2×C4⋊C4×C71C2312yes—
71#95×C71—24no—
72#95×C71—21no—
73#95×C71C2312yes—
74C2×Dic142—24no—
75C2×Dic142—22no—
76C2×Dic142—44no—
77C2×Dic142—82no—
78C2×Dic142C2×D414yes—
79C22×Dic71—22no—
80C22×Dic71—22no—
81C2×C71⋊C8—24no—
82C2×C71⋊C8—44no—
83C4⋊Dic71—24no—
84C4⋊Dic71—24no—
85C4⋊Dic71—24no—
86C4⋊Dic71—22no—
87C4⋊Dic71C2×D414yes—
88Dic71⋊C4—28no—
89Dic71⋊C4—24no—
90Dic71⋊C4—24no—
91C22⋊Dic71—24no—
92C22⋊Dic71—24no—
93C4×Dic71—28no—
94C24×C71C2×D411yescharacteristic
95C22×C4×C71—21no—
96C22×C4×C71—21no—
97C22×C4×C71—22no—
98C22×C4×C71—24no—
99C22×C4×C71—21no—
100C22×C4×C71—22no—
101C22×C4×C71—21no—
102C22×C4×C71—22no—
103C22×C4×C71—42no—
104C22×C4×C71—42no—
105C22×C4×C71—42no—
106C22×C4×C71—42no—
107C22×C4×C71—41no—
108C22×C4×C71—41no—
109C22×C4×C71—42no—
110C22×C4×C71—42no—
111C22×C4×C71SD1614yes—
112C22×C4×C71C2×D412yes—
113C22×C4×C71C2×D412yes—
114C22×C4×C71C2×D412yes—
115C22×C4×C71C2×D412yes—
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71C2411yescharacteristic
118C4⋊C4×C71—24no—
119C4⋊C4×C71—24no—
120C4⋊C4×C71—24no—
121C4⋊C4×C71—42no—
122C4⋊C4×C71—48no—
123C23×C71—22no—
124C23×C71—22no—
125C23×C71#12011yescharacteristic
126C23×C71#12011yescharacteristic
127C23×C71C2×SD1612yes—
128C23×C71#13712yes—
129C23×C71C22×D411yescharacteristic
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—24no—
133C2×C4×C71—24no—
134C2×C4×C71—24no—
135C2×C4×C71—24no—
136C2×C4×C71—22no—
137C2×C4×C71—42no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—48no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
146C2×C4×C71—42no—
147C2×C4×C71—42no—
148C2×C4×C71—81no—
149C2×C4×C71—82no—
150C2×C4×C71—82no—
151C2×C4×C71#12012yes—
152C2×C4×C71C2×SD1614yes—
153C2×C4×C71C22×D412yes—
154C2×Dic71—24no—
155C2×Dic71—24no—
156C2×Dic71—44no—
157C2×Dic71—44no—
158Dic142—44no—
159Dic142—82no—
160Dic142—84no—
161C71⋊C8—44no—
162C22×C71—24no—
163C22×C71—24no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71#44812yes—
169C22×C71#44812yes—
170C22×C71#45612yes—
171C22×C71#45812yes—
172C22×C71C2×#12011yescharacteristic
173C22×C71C22×SD1611yescharacteristic
174C22×C71C2×#13711yescharacteristic
175C4×C71—24no—
176C4×C71—22no—
177C4×C71—22no—
178C4×C71—44no—
179C4×C71—42no—
180C4×C71—82no—
181Dic71—44no—
182Dic71—44no—
183C2×#1611—711nomaximal, Sylow
184C2×C71—21no—
185C2×C71—21no—
186C2×C71—22no—
187C2×C71#161114yescomplemented
188C2×C71C2×#44811yescharacteristic
189C2×C71C2×#45611yescharacteristic
190C2×C71C2×#45811yescharacteristic
191C2×#544—711no—
192C2×#483—712no—
193C2×#398D7111yescharacteristic
194#1611—718no—
195C2×#328—712no—
196C2×#324—711no—
197C71C2×#161111yescharacteristic, complemented, Sylow
198#544—714no—
199#544—1424no—
200#544—1422no—
201C2×C4⋊Q8—1422no—
202C2×#125—1422no—
203C2×C22⋊Q8—712no—
204C2×C22⋊Q8—1421no—
205C2×#117—1421no—
206C22×C4⋊C4—21no—
207C22×C4⋊C4C2×D7111yescharacteristic
208C22×C4⋊C4C2×D7111yescharacteristic
209#483—718no—
210#483—1424no—
211C2×#106—1422no—
212C2×#103—1422no—
213C2×C22⋊C8—712no—
214#398—22no—
215#398C2×D7114yes—
216C2×#95—21no—
217C2×#95C2×D7111yescharacteristic
218#328—718no—
219#324—714no—
220C22×Q8—712no—
221C22×Q8—1421no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C22⋊D7111yescharacteristic
225C23×C4C22⋊D7111yescharacteristic
226C23×C4C22×D7111yescharacteristic
227C22×C8—1422no—
228C4⋊Q8—1428no—
229C4⋊Q8—1428no—
230#125—1428no—
231#125—1428no—
232C22⋊Q8—714no—
233C22⋊Q8—714no—
234C22⋊Q8—1424no—
235C22⋊Q8—1422no—
236C22⋊Q8—1422no—
237C22⋊Q8—1424no—
238#117—1424no—
239#117—1424no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—712no—
248C2×C4⋊C4—1421no—
249C2×C4⋊C4—1421no—
250C2×C4⋊C4—1421no—
251C2×C4⋊C4—1422no—
252C2×C4⋊C4—1422no—
253C2×C4⋊C4C22⋊D7112yes—
254C2×C4⋊C4C22×D7112yes—
255C2×C4⋊C4C22×D7112yes—
256C2×C22⋊C4—1422no—
257C2×C42—1422no—
258#106—1428no—
259#103—1428no—
260C22⋊C8—714no—
261C22⋊C8—1422no—
262#95—24no—
263#95—21no—
264#95C22×D7112yes—
265C24C2×C22⋊D7111yescharacteristic
266C2×Q8—2844no—
267C2×Q8—5682no—
268C2×Q8—714no—
269C2×Q8—1424no—
270C2×Q8—1422no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—1422no—
288C22×C4—1422no—
289C22×C4#1176585314yes—
290C22×C4D4×D7112yes—
291C22×C4D4×D7112yes—
292C22×C4C2×C22⋊D7112yes—
293C22×C4C2×C22⋊D7112yes—
294C22×C4C2×C22⋊D7111yescharacteristic
295C22×C4C23×D7111yescharacteristic
296C2×C8—2844no—
297C2×C8—1424no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—714no—
304C4⋊C4—1428no—
305C4⋊C4—1424no—
306C4⋊C4—1424no—
307C4⋊C4—1424no—
308C4⋊C4—1424no—
309C4⋊C4—1424no—
310C4⋊C4—1422no—
311C22⋊C4—1424no—
312C22⋊C4—1424no—
313C42—1428no—
314C23—22no—
315C23—22no—
316C23#1455609611yescharacteristic
317C23#1455609611yescharacteristic
318C23C2×#1176585312yes—
319C23#1455615812yes—
320C23C22×C22⋊D7111yescharacteristic
321Q8—2844no—
322Q8—5682no—
323Q8—5684no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—2844no—
346C2×C4—2844no—
347C2×C4—1424no—
348C2×C4—1424no—
349C2×C4#1455609612yes—
350C2×C4C2×#1176585314yes—
351C2×C4C2×D4×D7112yes—
352C8—2844no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—24no—
359C22#1798613712yes—
360C22#1798613712yes—
361C22#1798618112yes—
362C22#1798618812yes—
363C22C2×#1455609611yescharacteristic
364C22C22×#1176585311yescharacteristic
365C22C2×#1455615811yescharacteristic
366C4—22no—
367C4—22no—
368C4—24no—
369C4—44no—
370C4—42no—
371C4—82no—
372C4—2844no—
373C4—2844no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2#2258734714yesdirect product factor
378C2C2×#1798613711yescharacteristic
379C2C2×#1798618111yescharacteristic
380C2C2×#1798618811yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#398×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
hypercenterC2×#398D71
hypocenterC71C2×#1611
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. #22587347C2direct (N × H)
187. C2×C71#1611semidirect
197. C71C2×#1611semidirect
377. C2#22587347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179866302
C2×#179863292
#225873472
C2×#179857352
C2×#179857252
C2×#398×C712
C2×#161171