Finite Group Databasebeta

← #31889205

Subgroups — #31889205

Order 18176.

998 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 159 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
1#22588332C211yesmaximal, characteristic, complemented
2#22587021C212yesmaximal, complemented
3#22586765C212yesmaximal, complemented
4#22586661C211yesmaximal, characteristic, complemented
5C2×#547×C71C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556045—21no—
9#17985768—21no—
10#17985748—24no—
11D4⋊(C71⋊C8)—24no—
12#17985723—22no—
13C22×#41×C71C412yes—
14#547×C71—24no—
15#547×C71—24no—
16C2×C4⋊Q8×C71C2211yescharacteristic
17C2×#123×C71—22no—
18C2×C22⋊Q8×C71C412yes—
19C2×#121×C71—21no—
20C2×C4×D4×C71C2211yescharacteristic
21C2×C4×D4×C71C2211yescharacteristic
22C22×C4⋊C4×C71—21no—
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27#14556045—42no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—22no—
30C2×#41×C71—24no—
31C2×#41×C71—24no—
32C2×#41×C71—48no—
33C22×Q8×C71C2×C412yes—
34C22×D4×C71—22no—
35C22×D4×C71C2×C411yescharacteristic
36C22×D4×C71D411yescharacteristic
37C23×C4×C71—22no—
38C23×C4×C71—21no—
39C23×C4×C71C2×C412yes—
40C4⋊Q8×C71—22no—
41C4⋊Q8×C71D414yes—
42#123×C71—216no—
43C22⋊Q8×C71—24no—
44C22⋊Q8×C71—24no—
45#121×C71—28no—
46C4×D4×C71—22no—
47C4×D4×C71—22no—
48C4×D4×C71—22no—
49C4×D4×C71D414yes—
50C2×C4⋊C4×C71—28no—
51C2×C4⋊C4×C71—22no—
52C2×C4⋊C4×C71—21no—
53C2×C4⋊C4×C71—21no—
54C2×C4⋊C4×C71C2×C411yescharacteristic
55C2×C4⋊C4×C71D411yescharacteristic
56C2×C22⋊C4×C71—22no—
57C2×C22⋊C4×C71—21no—
58C2×C22⋊C4×C71D412yes—
59C2×C42×C71C2311yescharacteristic
60#11765849—42no—
61#11765849—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C24×C71—21no—
65C24×C71C22⋊C412yes—
66#41×C71—416no—
67#41×C71—48no—
68#41×C71—48no—
69C2×Q8×C71—22no—
70C2×Q8×C71—44no—
71C2×Q8×C71C22⋊C414yes—
72C2×D4×C71—24no—
73C2×D4×C71—28no—
74C2×D4×C71—21no—
75C2×D4×C71—22no—
76C2×D4×C71—22no—
77C2×D4×C71—21no—
78C2×D4×C71—21no—
79C2×D4×C71—48no—
80C2×D4×C71C22⋊C412yes—
81C22×C4×C71—28no—
82C22×C4×C71—21no—
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—22no—
86C22×C4×C71—24no—
87C22×C4×C71—44no—
88C22×C4×C71—48no—
89C22×C4×C71—42no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71C22⋊C412yes—
93C22×C4×C71C22×C411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71C2×D411yescharacteristic
96C4⋊C4×C71—28no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71—24no—
100C4⋊C4×C71—24no—
101C4⋊C4×C71C22⋊C412yes—
102C22⋊C4×C71—28no—
103C22⋊C4×C71—24no—
104C22⋊C4×C71—24no—
105C42×C71C2×D412yes—
106C42×C71C2×D412yes—
107C23×C71—24no—
108C23×C71—22no—
109C23×C71—22no—
110C23×C71—22no—
111C23×C71—22no—
112C23×C71—42no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71C2×C22⋊C411yescharacteristic
118Q8×C71—44no—
119Q8×C71—44no—
120D4×C71—24no—
121D4×C71—22no—
122D4×C71—416no—
123D4×C71—42no—
124D4×C71—42no—
125D4×C71#10014yescomplemented
126C2×C4×C71—22no—
127C2×C4×C71—22no—
128C2×C4×C71—22no—
129C2×C4×C71—24no—
130C2×C4×C71—41no—
131C2×C4×C71—44no—
132C2×C4×C71—44no—
133C2×C4×C71—416no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—44no—
137C2×C4×C71—44no—
138C2×C4×C71—44no—
139C2×C4×C71—42no—
140C2×C4×C71C2×C22⋊C411yescharacteristic
141C2×C4×C71C2×C22⋊C411yescharacteristic
142C2×C4×C71#12012yes—
143C2×C4×C71#12012yes—
144C2×C4×C71#13512yes—
145C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C71#13711yescharacteristic
147C71⋊C8—82no—
148C71⋊C8—82no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—24no—
152C22×C71—44no—
153C22×C71—44no—
154C22×C71—42no—
155C22×C71—42no—
156C22×C71—42no—
157C22×C71—42no—
158C22×C71—42no—
159C22×C71—82no—
160C22×C71—82no—
161C22×C71#37911yescharacteristic
162C22×C71#41611yescharacteristic
163C22×C71#41711yescharacteristic
164C4×C71—22no—
165C4×C71—21no—
166C4×C71—42no—
167C4×C71—44no—
168C4×C71—42no—
169C4×C71C2×#10012yes—
170C4×C71#44912yes—
171C4×C71#45212yes—
172#8393—711nomaximal, Sylow
173C2×C71—21no—
174C2×C71—21no—
175C2×C71—44no—
176C2×C71—82no—
177C2×C71#178311yescharacteristic
178C2×C71#178811yescharacteristic
179C2×C71#179611yescharacteristic
180C2×#547D7111yescharacteristic
181#1800—711no—
182#1548—712no—
183#1486—712no—
184#1460—711no—
185C71#839311yescharacteristic, complemented, Sylow
186C22×#41Dic7112yes—
187#547—24no—
188#547—24no—
189C2×C4⋊Q8C2×D7111yescharacteristic
190C2×#123—22no—
191C2×C22⋊Q8Dic7112yes—
192C2×#121—21no—
193C2×C4×D4C2×D7111yescharacteristic
194C2×C4×D4C2×D7111yescharacteristic
195C22×C4⋊C4—21no—
196#431—712no—
197#423—712no—
198C2×#100—1421no—
199#343—1421no—
200#333—1424no—
201D4⋊C8—1424no—
202#323—1422no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—48no—
206C22×Q8C2×Dic7112yes—
207C22×D4—22no—
208C22×D4C2×Dic7111yescharacteristic
209C22×D4C22⋊D7111yescharacteristic
210C23×C4—22no—
211C23×C4—21no—
212C23×C4C2×Dic7112yes—
213C2×M16—1422no—
214C4⋊Q8—22no—
215C4⋊Q8C22⋊D7114yes—
216#123—216no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219#121—28no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4C22⋊D7114yes—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic7111yescharacteristic
229C2×C4⋊C4C22⋊D7111yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4C22⋊D7112yes—
233C2×C42C22×D7111yescharacteristic
234C4⋊C8—1422no—
235C4⋊C8—1422no—
236C4⋊C8—1422no—
237#100—2842no—
238C22⋊C8—1422no—
239C4×C8—1422no—
240C24—21no—
241C24C22⋊Dic7112yes—
242#41—416no—
243#41—48no—
244#41—48no—
245C2×Q8—22no—
246C2×Q8—44no—
247C2×Q8C22⋊Dic7114yes—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—48no—
256C2×D4C22⋊Dic7112yes—
257C22×C4—28no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C22⋊Dic7112yes—
269C22×C4C22×Dic7111yescharacteristic
270C22×C4C2×C22⋊D7111yescharacteristic
271C22×C4C2×C22⋊D7111yescharacteristic
272M16—2842no—
273M16—2842no—
274C2×C8—2842no—
275C2×C8—2842no—
276C4⋊C4—28no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4C22⋊Dic7112yes—
282C22⋊C4—28no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C42C2×C22⋊D7112yes—
286C42C2×C22⋊D7112yes—
287C23—24no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic7111yescharacteristic
298Q8—44no—
299Q8—44no—
300D4—24no—
301D4—22no—
302D4—416no—
303D4—42no—
304D4—42no—
305D4#1455604514yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4C2×C22⋊Dic7111yescharacteristic
321C2×C4C2×C22⋊Dic7111yescharacteristic
322C2×C4#1455609712yes—
323C2×C4#1455609712yes—
324C2×C4#1455614812yes—
325C2×C4#1455615311yescharacteristic
326C2×C4#1455615811yescharacteristic
327C8—5682no—
328C8—5682no—
329C22—21no—
330C22—21no—
331C22—24no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22#1798585511yescharacteristic
342C22#1798599511yescharacteristic
343C22#1798600011yescharacteristic
344C4—21no—
345C4—22no—
346C4—42no—
347C4—44no—
348C4—42no—
349C4C2×#1455604512yes—
350C4#1798614412yes—
351C4#1798616112yes—
352C2—21no—
353C2—21no—
354C2—44no—
355C2—82no—
356C2#2258826111yescharacteristic
357C2#2258828011yescharacteristic
358C2#2258831511yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
Fitting, prime radicalC2×#547×C71C2
socleC22×C71#379
cosocleC2×C42×C71C23
hypercenterC2×#547D71
hypocenterC71#8393
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #22588332C2semidirect
2. #22587021C2semidirect
3. #22586765C2semidirect
4. #22586661C2semidirect
125. D4×C71#100semidirect
185. C71#8393semidirect
305. D4#14556045semidirect

Maximal subgroups

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

subgroupindex
#225883322
#225870212
#225867652
#225866612
C2×#547×C712
#839371