Finite Group Databasebeta

← #31897178

Subgroups — #31897178

Order 18176.

484 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 115 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#22588378C211yesmaximal, characteristic
2#22588116C211yesmaximal, characteristic
3#22586860C212yesmaximal
4#22586704C212yesmaximal
5C2×#467×C71C211yesmaximal, characteristic
6C2×#14556064—21no—
7#17985963C2212yes—
8C2×C22⋊(C71⋊C8)C2211yescharacteristic
9C2×#14556038—21no—
10C4×C22⋊Dic71C2211yescharacteristic
11#17985764—21no—
12#17985753—21no—
13#17985736—24no—
14#17985725—24no—
15C22×Q16×C71C411yescharacteristic, complemented
16C23×C8×C71C2211yescharacteristic
17C2×C22⋊Q8×C71C2212yes—
18#467×C71—21no—
19#467×C71—21no—
20#467×C71—21no—
21#467×C71—21no—
22C2×#107×C71C411yescharacteristic, complemented
23C2×#103×C71C412yescomplemented
24C2×#11765849—22no—
25C22×C71⋊C8—21no—
26C2×C22⋊Dic71—21no—
27C2×C4×Dic71—21no—
28#14556064—42no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—21no—
32C22⋊(C71⋊C8)D412yes—
33#14556038—42no—
34C71⋊#95D412yes—
35C22×Q8×C71C2×C412yes—
36C23×C4×C71C2311yescharacteristic
37C2×Q16×C71—21no—
38C2×Q16×C71—21no—
39C2×Q16×C71—42no—
40C22×C8×C71—42no—
41C22×C8×C71C2×C411yescharacteristic
42C22×C8×C71C2×C411yescharacteristic
43C22×C8×C71D411yescharacteristic
44C22×C8×C71D411yescharacteristic
45C22×C8×C71D411yescharacteristic
46C22×C8×C71D411yescharacteristic
47C22⋊Q8×C71—22no—
48C22⋊Q8×C71—22no—
49C22⋊Q8×C71—22no—
50C22⋊Q8×C71—22no—
51C2×C4⋊C4×C71—22no—
52C2×C4⋊C4×C71C2×C412yes—
53C2×C22⋊C4×C71—22no—
54#107×C71—21no—
55#107×C71—21no—
56#103×C71—22no—
57#103×C71—22no—
58C22×Dic71—22no—
59#11765849—42no—
60#11765849—42no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—42no—
65C22⋊Dic71—42no—
66C4×Dic71—41no—
67C4×Dic71—41no—
68C24×C71#4111yescharacteristic
69C2×Q8×C71—22no—
70C2×Q8×C71—22no—
71C2×Q8×C71—42no—
72C2×Q8×C71—42no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4111yescharacteristic
83Q16×C71—42no—
84Q16×C71—42no—
85C2×C8×C71—22no—
86C2×C8×C71—22no—
87C2×C8×C71—22no—
88C2×C8×C71—41no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C2×C8×C71C22⋊C411yescharacteristic
92C2×C8×C71C22⋊C411yescharacteristic
93C2×C8×C71C22⋊C411yescharacteristic
94C2×C8×C71C22⋊C411yescharacteristic
95C4⋊C4×C71—22no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—42no—
98C4⋊C4×C71—42no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C23×C71—22no—
102C23×C71—22no—
103C23×C71—41no—
104C23×C71C4×D411yescharacteristic
105C23×C71#12111yescharacteristic
106C23×C71#12411yescharacteristic
107Q8×C71—42no—
108Q8×C71—42no—
109Q8×C71—82no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—44no—
122C2×C4×C71—82no—
123C2×C4×C71C2×C22⋊C411yescharacteristic
124C2×C4×C71C4×D411yescharacteristic
125C2×C4×C71#12111yescharacteristic
126C2×C4×C71#12711yescharacteristic
127C2×Dic71—42no—
128C2×Dic71—42no—
129C2×Dic71—42no—
130C8×C71—21no—
131C8×C71—21no—
132C8×C71—21no—
133C8×C71#9912yes—
134C71⋊C8—42no—
135C71⋊C8—82no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—44no—
144C22×C71#39011yescharacteristic
145C22×C71#44412yes—
146C22×C71#47312yes—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—21no—
150C4×C71—82no—
151C4×C71—82no—
152C4×C71C2×#9911yescharacteristic
153C4×C71#41011yescharacteristic
154Dic71—82no—
155#9817—711nomaximal, Sylow
156C2×C71—22no—
157C2×C71—22no—
158C2×C71—41no—
159C2×C71#196411yescharacteristic
160C2×C71#197511yescharacteristic
161C2×C71#197911yescharacteristic
162C2×#467D7111yescharacteristic
163#1810—711no—
164#1752—711no—
165#1508—712no—
166#1472—712no—
167C71#981711yescharacteristic, complemented, Sylow
168C22×Q16Dic7111yescharacteristic, complemented
169C23×C8C2×D7111yescharacteristic
170C2×C22⋊Q8C2×D7112yes—
171#467—21no—
172#467—21no—
173#467—21no—
174#467—21no—
175C2×#108—1421no—
176C2×#107Dic7111yescharacteristic, complemented
177C2×#103Dic7112yescomplemented
178#407—712no—
179C2×C22⋊C8—711no—
180C2×#97—1421no—
181C4×C22⋊C4—711no—
182#341—1421no—
183#336—1421no—
184#328—1424no—
185#324—1424no—
186C22×Q8C2×Dic7112yes—
187C23×C4C22×D7111yescharacteristic
188C2×Q16—21no—
189C2×Q16—21no—
190C2×Q16—42no—
191C2×M16—1422no—
192C22×C8—42no—
193C22×C8—1421no—
194C22×C8C2×Dic7111yescharacteristic
195C22×C8C2×Dic7111yescharacteristic
196C22×C8C22⋊D7111yescharacteristic
197C22×C8C22⋊D7111yescharacteristic
198C22×C8C22⋊D7111yescharacteristic
199C22×C8C22⋊D7111yescharacteristic
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C2×Dic7112yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1421no—
208C2×C42—1421no—
209#108—2842no—
210#107—21no—
211#107—21no—
212#103—22no—
213#103—22no—
214C22⋊C8—712no—
215C22⋊C8—1422no—
216C22⋊C8—1422no—
217C22⋊C8—1421no—
218#97—2842no—
219#95—712no—
220C24#1176586411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—1422no—
230C22×C4C22×Dic7111yescharacteristic
231C22×C4D4×D7111yescharacteristic
232C22×C4D4×D7111yescharacteristic
233C22×C4C2×C22⋊D7111yescharacteristic
234C22×C4C2×C22⋊D7111yescharacteristic
235C22×C4#1176586411yescharacteristic
236Q16—42no—
237Q16—42no—
238M16—2842no—
239M16—2842no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—2841no—
247C2×C8—2841no—
248C2×C8—2842no—
249C2×C8—1422no—
250C2×C8C22⋊Dic7111yescharacteristic
251C2×C8C22⋊Dic7111yescharacteristic
252C2×C8C22⋊Dic7111yescharacteristic
253C2×C8C22⋊Dic7111yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—2842no—
261C42—2841no—
262C42—2841no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23D4×Dic7111yescharacteristic
267C23#1455609911yescharacteristic
268C23#1455611511yescharacteristic
269Q8—42no—
270Q8—42no—
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—2842no—
286C2×C4—2842no—
287C2×C4—2842no—
288C2×C4C2×C22⋊Dic7111yescharacteristic
289C2×C4D4×Dic7111yescharacteristic
290C2×C4#1455609911yescharacteristic
291C2×C4#1455612411yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—2842no—
296C8—5682no—
297C8#1455604312yes—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—44no—
306C22#1798590211yescharacteristic
307C22#1798612112yes—
308C22#1798627112yes—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—82no—
314C4—5682no—
315C4C2×#1455604311yescharacteristic
316C4#1798597311yescharacteristic
317C2—22no—
318C2—22no—
319C2—41no—
320C2#2258917711yescharacteristic
321C2#2258923411yescharacteristic
322C2#2258925611yescharacteristic
subgroupG/H
centerC22#17985902
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#467×C71C2
socleC22×C71#390
prime radicalC24×C71#41
cosocleC23×C4×C71C23
hypercenterC2×#467D71
hypocenterC71#9817
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
15. C22×Q16×C71C4semidirect
22. C2×#107×C71C4semidirect
23. C2×#103×C71C4semidirect
167. C71#9817semidirect
168. C22×Q16Dic71semidirect
176. C2×#107Dic71semidirect
177. C2×#103Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883782
#225881162
#225868602
#225867042
C2×#467×C712
#981771