Finite Group Databasebeta

← #31915775

Subgroups — #31915775

Order 18176.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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×#17985796C214yesmaximal, complemented
2#22588336C211yesmaximal, characteristic, complemented
3C4×#14556047C211yesmaximal, characteristic, complemented
4C71×#2548C211yesmaximal, characteristic
5#17986023—21no—
6C2×#14556047C2212yes—
7C2×#14556047C2212yes—
8C4×#11765849—21no—
9#17985796—28no—
10#17985768—21no—
11#17985768—21no—
12C2×#127×C71C411yescharacteristic
13C2×#124×C71C2212yes—
14C2×C4×Q8×C71C2211yescharacteristic
15#394×C71—21no—
16#394×C71—21no—
17#390×C71—21no—
18#390×C71—21no—
19#390×C71C412yes—
20C4×C4⋊C4×C71C411yescharacteristic
21C2×#11765849—22no—
22C2×#11765849—22no—
23C4⋊(C71⋊C8)—41no—
24C4⋊(C71⋊C8)—41no—
25#14556047—24no—
26#14556047—22no—
27#14556047D414yescomplemented
28#14556039—41no—
29C4×C71⋊C8—41no—
30C22×Q8×C71C2311yescharacteristic
31C22×D4×C71—21no—
32C22×D4×C71—21no—
33C22×D4×C71C2×C412yes—
34#127×C71—41no—
35#127×C71—41no—
36#124×C71—24no—
37#124×C71—24no—
38C4×Q8×C71—41no—
39C4×Q8×C71—41no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71—22no—
48C2×C42×C71C2×C412yes—
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#11765849—24no—
54#11765849—42no—
55#11765849—42no—
56#11765849—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C24×C71—22no—
60C2×Q8×C71—41no—
61C2×Q8×C71—41no—
62C2×Q8×C71C2×D412yes—
63C2×Q8×C71#4112yes—
64C2×D4×C71—24no—
65C2×D4×C71—42no—
66C2×D4×C71—42no—
67C2×D4×C71—42no—
68C2×D4×C71—42no—
69C2×D4×C71—42no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—82no—
73C22×C4×C71—22no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—42no—
85C22⋊C4×C71—44no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—82no—
89C22⋊C4×C71—82no—
90C42×C71—21no—
91C42×C71—21no—
92C42×C71—21no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—24no—
97C23×C71—42no—
98C23×C71—42no—
99C23×C71—42no—
100C23×C71—42no—
101C23×C71—42no—
102C23×C71—42no—
103C23×C71—42no—
104C23×C71C2×C22⋊C411yescharacteristic
105Q8×C71—42no—
106Q8×C71—42no—
107D4×C71—84no—
108D4×C71—82no—
109D4×C71—84no—
110D4×C71—82no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—22no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71#9911yescharacteristic
128C2×C4×C71#9911yescharacteristic
129C2×C4×C71C4×D412yes—
130C2×C4×C71#12112yes—
131C2×C4×C71#12411yescharacteristic
132C2×C4×C71#12711yescharacteristic
133C71⋊C8—44no—
134C71⋊C8—82no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—82no—
138C22×C71—82no—
139C22×C71—82no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71—82no—
143C22×C71—82no—
144C22×C71#39011yescharacteristic
145C22×C71C2×#9911yescharacteristic
146C22×C71#41011yescharacteristic
147C4×C71—21no—
148C4×C71—22no—
149C4×C71—21no—
150C4×C71—22no—
151C4×C71—21no—
152C4×C71—41no—
153C4×C71—41no—
154C4×C71—41no—
155C4×C71—82no—
156C4×C71#35512yescomplemented
157#13226—711nomaximal, Sylow
158C2×C71—21no—
159C2×C71—21no—
160C2×C71—162no—
161C2×C71#196411yescharacteristic
162C2×C71C2×#35511yescharacteristic
163C2×C71#213411yescharacteristic
164#2548D7111yescharacteristic
165C2×#355—714no—
166#1801—711no—
167C4×#101—711no—
168C71#1322611yescharacteristic, complemented, Sylow
169C2×#127Dic7111yescharacteristic
170C2×#124C2×D7112yes—
171C2×C4×Q8C2×D7111yescharacteristic
172#423—1421no—
173C2×#101—712no—
174C2×#101—712no—
175C4×M16—1421no—
176#394—21no—
177#394—21no—
178#390—21no—
179#390—21no—
180#390Dic7112yes—
181C4×C4⋊C4Dic7111yescharacteristic
182#355—1428no—
183#343—1421no—
184#343—1421no—
185C22×Q8C22×D7111yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×Dic7112yes—
189C2×M16—1422no—
190C2×M16—1422no—
191#127—41no—
192#127—41no—
193#124—24no—
194#124—24no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic7111yescharacteristic
199C2×C4⋊C4C22⋊D7111yescharacteristic
200C2×C4⋊C4C22⋊D7111yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C2×Dic7112yes—
206C2×C42C2×Dic7111yescharacteristic
207C2×C42C22⋊D7111yescharacteristic
208C2×C42C22⋊D7111yescharacteristic
209C4⋊C8—2841no—
210C4⋊C8—2841no—
211#101—714no—
212#101—1424no—
213#101—1422no—
214#97—2841no—
215C4×C8—2841no—
216#95—21no—
217C24—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D4×D7112yes—
221C2×Q8#1176586412yes—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C22×C4—22no—
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22⋊Dic7111yescharacteristic
234C22×C4C22⋊Dic7111yescharacteristic
235C22×C4C22⋊Dic7111yescharacteristic
236C22×C4C22×Dic7111yescharacteristic
237C22×C4C2×C22⋊D7111yescharacteristic
238C22×C4C2×C22⋊D7111yescharacteristic
239M16—2842no—
240M16—2842no—
241M16—2842no—
242M16—1424no—
243C2×C8—2842no—
244C2×C8—2842no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—24no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊Dic7111yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#1455604311yescharacteristic
292C2×C4#1455604311yescharacteristic
293C2×C4D4×Dic7112yes—
294C2×C4#1455609912yes—
295C2×C4#1455611511yescharacteristic
296C2×C4#1455612411yescharacteristic
297C8—2844no—
298C8—5682no—
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#1798590211yescharacteristic
309C22C2×#1455604311yescharacteristic
310C22#1798597311yescharacteristic
311C4—21no—
312C4—22no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4#1798579612yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2#2258917711yescharacteristic
325C2C2×#1798579611yescharacteristic
326C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2548C2
socleC22×C71C2×#99
prime radicalC2×#127×C71C4
cosocleC22×Q8×C71C23
hypercenter#2548D71
hypocenterC71#13226
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C2×#17985796C2semidirect
2. #22588336C2semidirect
3. C4×#14556047C2semidirect
27. #14556047D4semidirect
156. C4×C71#355semidirect
168. C71#13226semidirect
320. C4#17985796semidirect

Maximal subgroups

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

subgroupindex
C2×#179857962
#225883362
C4×#145560472
C71×#25482
#1322671