Finite Group Databasebeta

← #31893198

Subgroups — #31893198

Order 18176.

756 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 143 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×#17985845C211yesmaximal, characteristic, complemented
2#22588408C211yesmaximal, characteristic, complemented
3C2×#17985733C212yesmaximal
4#22586218C212yesmaximal, complemented
5C71×#2854C211yesmaximal, characteristic
6C22×C4×Dic71—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985845—24no—
9C2×C71⋊#95—21no—
10C2×C71⋊#95C2212yes—
11#17985754—22no—
12#17985733—24no—
13C2×#123×C71—21no—
14C2×#121×C71C2211yescharacteristic
15C22×C42×C71C2211yescharacteristic
16#390×C71C412yescomplemented
17#389×C71C412yescomplemented
18#388×C71—22no—
19#386×C71—21no—
20#380×C71C2211yescharacteristic
21C23×Dic71—22no—
22C22×C71⋊C8—22no—
23C2×C4×Dic71—24no—
24C2×C4×Dic71—28no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C71⋊#95—24no—
28C71⋊#95—22no—
29C71⋊#95D414yes—
30C22×D4×C71—21no—
31C22×D4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35#123×C71—44no—
36#121×C71—22no—
37#121×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C412yes—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71—21no—
45C2×C42×C71—22no—
46C2×C42×C71—22no—
47C2×C42×C71C2×C412yes—
48C2×C42×C71D412yes—
49#95×C71—22no—
50#95×C71—21no—
51C22×Dic71—28no—
52C22×Dic71—24no—
53C22×Dic71—44no—
54C22×Dic71—42no—
55C22×Dic71—42no—
56C2×C71⋊C8—44no—
57C2×C71⋊C8—42no—
58C4×Dic71—216no—
59C24×C71—21no—
60C24×C71C2×D411yescharacteristic
61C2×D4×C71—22no—
62C2×D4×C71—42no—
63C2×D4×C71—41no—
64C2×D4×C71—41no—
65C2×D4×C71—42no—
66C2×D4×C71—42no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—24no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—21no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D412yes—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4112yes—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—44no—
86C22⋊C4×C71—44no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—44no—
89C22⋊C4×C71—42no—
90C42×C71—24no—
91C42×C71—24no—
92C23×C71—22no—
93C23×C71—22no—
94C23×C71—41no—
95C23×C71—42no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71—42no—
99C23×C71C2×C22⋊C411yescharacteristic
100C23×C71#12011yescharacteristic
101C23×C71#12311yescharacteristic
102C23×C71#13512yes—
103C23×C71AGL(1,ℤ/8)12yes—
104D4×C71—82no—
105D4×C71—82no—
106D4×C71—82no—
107C2×C4×C71—24no—
108C2×C4×C71—24no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—42no—
122C2×C4×C71—41no—
123C2×C4×C71—42no—
124C2×C4×C71—41no—
125C2×C4×C71—42no—
126C2×C4×C71#10414yes—
127C2×C4×C71C4×D412yes—
128C2×C4×C71#12112yes—
129C2×Dic71—28no—
130C2×Dic71—44no—
131C2×Dic71—48no—
132C2×Dic71—48no—
133C2×Dic71—44no—
134C71⋊C8—82no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—24no—
140C22×C71—24no—
141C22×C71—81no—
142C22×C71—81no—
143C22×C71—81no—
144C22×C71—82no—
145C22×C71—82no—
146C22×C71#38611yescharacteristic
147C22×C71#41012yes—
148C22×C71#41611yescharacteristic
149C22×C71#41811yescharacteristic
150C22×C71C2×#10412yes—
151C22×C71#44912yes—
152C22×C71#48212yes—
153C4×C71—24no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—42no—
157C4×C71—81no—
158C4×C71—82no—
159Dic71—48no—
160#9149—711nomaximal, Sylow
161C2×C71—22no—
162C2×C71—22no—
163C2×C71—161no—
164C2×C71#150012yes—
165C2×C71#150512yes—
166C2×C71#187511yescharacteristic
167C2×C71#189611yescharacteristic
168C2×C71#189911yescharacteristic
169#2854D7111yescharacteristic
170C2×#376—711no—
171#1817—711no—
172C2×#327—712no—
173#1294—712no—
174C71#914911yescharacteristic, complemented, Sylow
175C2×#123—21no—
176C2×#121C2×D7111yescharacteristic
177C22×C42—1421no—
178C22×C42C2×D7111yescharacteristic
179C2×C22⋊C8—712no—
180#390Dic7112yescomplemented
181#389Dic7112yescomplemented
182#388—22no—
183#386—21no—
184#380C2×D7111yescharacteristic
185#376—1424no—
186C2×#95—712no—
187C2×#95—1421no—
188#336—1422no—
189#327—1424no—
190C22×D4—21no—
191C22×D4C2×Dic7111yescharacteristic
192C23×C4—1422no—
193C23×C4C22⋊D7111yescharacteristic
194C23×C4C22⋊D7111yescharacteristic
195C23×C4C22×D7111yescharacteristic
196C22×C8—1422no—
197#123—44no—
198#121—22no—
199#121—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C2×Dic7111yescharacteristic
202C2×C4⋊C4C2×Dic7112yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—1424no—
210C2×C42—1428no—
211C2×C42C2×Dic7112yes—
212C2×C42C22⋊D7112yes—
213C22⋊C8—1422no—
214C22⋊C8—1422no—
215#95—22no—
216#95—21no—
217#95—714no—
218#95—1424no—
219#95—1422no—
220C24—21no—
221C24C2×C22⋊D7111yescharacteristic
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—2844no—
239C22×C4—2842no—
240C22×C4—2842no—
241C22×C4—1428no—
242C22×C4—1424no—
243C22×C4C22⋊Dic7112yes—
244C22×C4C22⋊Dic7112yes—
245C22×C4C22×Dic7111yescharacteristic
246C22×C4D4×D7112yes—
247C22×C4C2×C22⋊D7111yescharacteristic
248C22×C4#1176586412yes—
249C2×C8—2844no—
250C2×C8—2842no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C42—24no—
259C42—24no—
260C42—14216no—
261C23—22no—
262C23—22no—
263C23—41no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23C2×C22⋊Dic7111yescharacteristic
269C23#1455609611yescharacteristic
270C23#1455610911yescharacteristic
271C23#1455614812yes—
272C23#1455615412yes—
273D4—82no—
274D4—82no—
275D4—82no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—2844no—
296C2×C4—2848no—
297C2×C4—2848no—
298C2×C4—2844no—
299C2×C4—1428no—
300C2×C4#1455605514yes—
301C2×C4D4×Dic7112yes—
302C2×C4#1455610012yes—
303C8—5682no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—81no—
311C22—81no—
312C22—82no—
313C22—82no—
314C22—81no—
315C22#1798587911yescharacteristic
316C22#1798597312yes—
317C22#1798599511yescharacteristic
318C22#1798600411yescharacteristic
319C22C2×#1455605512yes—
320C22#1798614212yes—
321C22#1798632012yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—82no—
327C4—81no—
328C4—2848no—
329C2—22no—
330C2—22no—
331C2—161no—
332C2#2258682612yes—
333C2#2258684812yes—
334C2#2258866311yescharacteristic
335C2#2258878411yescharacteristic
336C2#2258879911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2854C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#121×C71C22
cosocleC23×C4×C71C23
hypercenter#2854D71
hypocenterC71#9149
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#17985845C2semidirect
2. #22588408C2semidirect
4. #22586218C2semidirect
16. #390×C71C4semidirect
17. #389×C71C4semidirect
174. C71#9149semidirect
180. #390Dic71semidirect
181. #389Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858452
#225884082
C2×#179857332
#225862182
C71×#28542
#914971