Finite Group Databasebeta

← #31895402

Subgroups — #31895402

Order 18176.

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

normal subgroups only

This group has 684 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
1#22597369C211yesmaximal, characteristic
2#22588264C212yesmaximal, complemented
3#22586678C211yesmaximal, characteristic
4#22586246C212yesmaximal, complemented
5C71×#2409C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8C2×#14556047—24no—
9#17985963C2212yes—
10#17985963C2212yes—
11C2×C22⋊(C71⋊C8)—21no—
12#17985725—21no—
13#17985723—21no—
14C23×Q8×C71C2211yescharacteristic
15C2×C22⋊Q8×C71—21no—
16C2×C4×D4×C71C2211yescharacteristic
17#395×C71—21no—
18#393×C71—21no—
19#391×C71—21no—
20#386×C71C412yes—
21#385×C71C412yes—
22#380×C71C2211yescharacteristic
23C2×#11765849—22no—
24C2×#11765849D414yes—
25C22×C71⋊C8—22no—
26C4⋊(C71⋊C8)—22no—
27#14556047—48no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33#14556039—22no—
34C22×Q8×C71—21no—
35C22×Q8×C71—22no—
36C22×Q8×C71—21no—
37C22×Q8×C71—22no—
38C22×Q8×C71—44no—
39C22×D4×C71C2×C411yescharacteristic
40C23×C4×C71C2×C412yes—
41C23×C4×C71D411yescharacteristic
42C23×C4×C71C2311yescharacteristic
43C22⋊Q8×C71—44no—
44C4×D4×C71—22no—
45C4×D4×C71—22no—
46C2×C4⋊C4×C71—22no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C22⋊C4×C71—22no—
50C2×C22⋊C4×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71C2×C412yes—
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54#95×C71—22no—
55#95×C71—22no—
56#95×C71—21no—
57#11765849—44no—
58#11765849—44no—
59C2×C71⋊C8—24no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C22×C411yescharacteristic
64C2×Q8×C71—24no—
65C2×Q8×C71—28no—
66C2×Q8×C71—42no—
67C2×Q8×C71—44no—
68C2×Q8×C71—44no—
69C2×Q8×C71—48no—
70C2×Q8×C71—41no—
71C2×Q8×C71—42no—
72C2×Q8×C71—41no—
73C2×D4×C71—22no—
74C2×D4×C71—22no—
75C2×D4×C71—41no—
76C22×C4×C71—22no—
77C22×C4×C71—24no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—22no—
81C22×C4×C71—44no—
82C22×C4×C71—42no—
83C22×C4×C71—42no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C2×D412yes—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71#4112yes—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—44no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—44no—
97C22⋊C4×C71—42no—
98C42×C71—21no—
99C42×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71#10112yescomplemented
109C23×C71C2×C22⋊C411yescharacteristic
110C23×C71C4×D412yes—
111Q8×C71—48no—
112Q8×C71—44no—
113Q8×C71—44no—
114D4×C71—44no—
115C2×C4×C71—22no—
116C2×C4×C71—24no—
117C2×C4×C71—24no—
118C2×C4×C71—28no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—41no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—44no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—81no—
136C2×C4×C71#12011yescharacteristic
137C2×C4×C71#12112yes—
138C2×C4×C71#12311yescharacteristic
139C71⋊C8—44no—
140C71⋊C8—82no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—22no—
147C22×C71—42no—
148C22×C71—42no—
149C22×C71—42no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—81no—
153C22×C71#38611yescharacteristic
154C22×C71C2×#10111yescharacteristic
155C22×C71#41311yescharacteristic
156C4×C71—22no—
157C4×C71—21no—
158C4×C71—44no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—82no—
163C4×C71B(4,𝔽2)12yes—
164#9505—711nomaximal, Sylow
165C2×C71—21no—
166C2×C71—21no—
167C2×C71—42no—
168C2×C71—42no—
169C2×C71#188911yescharacteristic
170C2×C71#190111yescharacteristic
171C2×C71#191511yescharacteristic
172#2973—711no—
173#2409D7111yescharacteristic
174#1784—712no—
175#1465—711no—
176#1305—712no—
177C71#950511yescharacteristic, complemented, Sylow
178C23×Q8C2×D7111yescharacteristic
179C2×C22⋊Q8—21no—
180C2×C4×D4C2×D7111yescharacteristic
181#435—1424no—
182#432—1421no—
183C2×#101—1424no—
184#407—712no—
185#407—712no—
186C2×C22⋊C8—1421no—
187#395—21no—
188#393—21no—
189#391—21no—
190#386Dic7112yes—
191#385Dic7112yes—
192#380C2×D7111yescharacteristic
193#324—1421no—
194#323—1421no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×D4C2×Dic7111yescharacteristic
201C23×C4C2×Dic7112yes—
202C23×C4C22⋊D7111yescharacteristic
203C23×C4C22×D7111yescharacteristic
204C2×M16—714no—
205C2×M16—1422no—
206C22×C8—1422no—
207C22⋊Q8—44no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22⋊D7111yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic7111yescharacteristic
215C2×C22⋊C4C2×Dic7112yes—
216C2×C22⋊C4C22⋊D7111yescharacteristic
217C2×C42C22⋊D7111yescharacteristic
218C4⋊C8—1422no—
219#101—2848no—
220C22⋊C8—1422no—
221C22⋊C8—1422no—
222C22⋊C8—1422no—
223C22⋊C8—1422no—
224C22⋊C8—1422no—
225#97—1422no—
226#95—22no—
227#95—22no—
228#95—21no—
229C24C22⋊Dic7111yescharacteristic
230C24C22×Dic7111yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22⋊Dic7111yescharacteristic
253C22×C4C22⋊Dic7111yescharacteristic
254C22×C4D4×D7112yes—
255C22×C4C2×C22⋊D7111yescharacteristic
256C22×C4C2×C22⋊D7111yescharacteristic
257C22×C4#1176586412yes—
258M16—2844no—
259M16—2844no—
260C2×C8—2842no—
261C2×C8—5682no—
262C2×C8—1424no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1455604712yescomplemented
281C23C2×C22⋊Dic7111yescharacteristic
282C23D4×Dic7112yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4#1455609611yescharacteristic
309C2×C4#1455610012yes—
310C2×C4#1455610911yescharacteristic
311C8—2844no—
312C8—5682no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1798587911yescharacteristic
326C22C2×#1455604711yescharacteristic
327C22#1798598311yescharacteristic
328C4—22no—
329C4—21no—
330C4—44no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4#1798618512yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#2258874911yescharacteristic
341C2#2258880911yescharacteristic
342C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2409C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC23×C4×C71C23
hypercenter#2409D71
hypocenterC71#9505
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #22588264C2semidirect
4. #22586246C2semidirect
108. C23×C71#101semidirect
177. C71#9505semidirect
280. C23#14556047semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225882642
#225866782
#225862462
C71×#24092
#950571