Finite Group Databasebeta

← #31908148

Subgroups — #31908148

Order 18176.

448 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588420C211yesmaximal, characteristic, complemented
2#22586670C211yesmaximal, characteristic, complemented
3#22586346C212yesmaximal, complemented
4#22586343C212yesmaximal, complemented
5C71×#2618C211yesmaximal, characteristic
6#17985963—21no—
7#17985963—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C2×#123×C71—21no—
13C2×#121×C71C411yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#398×C71C411yescharacteristic
16#398×C71C2211yescharacteristic
17#396×C71—21no—
18#396×C71—21no—
19#394×C71C411yescharacteristic
20#394×C71C2211yescharacteristic
21#386×C71—21no—
22#380×C71C411yescharacteristic
23C2×#11765849—22no—
24C2×#11765849—41no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—41no—
27C22⋊(C71⋊C8)—41no—
28C22⋊(C71⋊C8)—41no—
29C22×D4×C71—21no—
30C22×D4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34#123×C71—44no—
35#121×C71—41no—
36#121×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—41no—
41C2×C4⋊C4×C71—41no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71C2311yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71C2×C411yescharacteristic
56#95×C71C2×C411yescharacteristic
57#11765849—42no—
58#11765849—42no—
59#11765849—82no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C24×C71—21no—
63C24×C71C22⋊C411yescharacteristic
64C2×D4×C71—41no—
65C2×D4×C71—42no—
66C2×D4×C71—42no—
67C2×D4×C71—41no—
68C2×D4×C71—41no—
69C2×D4×C71—81no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—22no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—41no—
81C22×C4×C71—81no—
82C22×C4×C71—81no—
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—42no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—81no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—42no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—81no—
102C22⋊C4×C71—81no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71#9911yescharacteristic
115C23×C71#9911yescharacteristic
116C23×C71C2×C22⋊C411yescharacteristic
117D4×C71—82no—
118D4×C71—82no—
119D4×C71—82no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—81no—
146C2×C4×C71—81no—
147C2×C4×C71—81no—
148C2×C4×C71AGL(1,ℤ/8)12yes—
149C71⋊C8—82no—
150C71⋊C8—82no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—42no—
160C22×C71—42no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71—81no—
166C22×C71—81no—
167C22×C71—81no—
168C22×C71C2×#9911yescharacteristic
169C22×C71#41811yescharacteristic
170C22×C71#42111yescharacteristic
171C4×C71—22no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71—41no—
175C4×C71—41no—
176C4×C71—41no—
177C4×C71—81no—
178C4×C71—81no—
179C4×C71—81no—
180#11805—711nomaximal, Sylow
181C2×C71—21no—
182C2×C71—21no—
183C2×C71—41no—
184C2×C71—41no—
185C2×C71—161no—
186C2×C71#151011yescharacteristic
187C2×C71#212111yescharacteristic
188C2×C71#213311yescharacteristic
189#2618D7111yescharacteristic
190#1821—711no—
191#1463—711no—
192#1346—712no—
193#1345—712no—
194C71#1180511yescharacteristic, complemented, Sylow
195C2×#123—21no—
196C2×#121Dic7111yescharacteristic
197C22×C4⋊C4C2×D7111yescharacteristic
198#407—1421no—
199#407—1421no—
200#398Dic7111yescharacteristic
201#398C2×D7111yescharacteristic
202#396—21no—
203#396—21no—
204#394Dic7111yescharacteristic
205#394C2×D7111yescharacteristic
206#386—21no—
207#380Dic7111yescharacteristic
208#343—712no—
209#343—1421no—
210#324—712no—
211#324—1421no—
212C22×D4—21no—
213C22×D4C2×Dic7111yescharacteristic
214C23×C4C2×Dic7111yescharacteristic
215C23×C4C22⋊D7111yescharacteristic
216C23×C4C22⋊D7111yescharacteristic
217C2×M16—2841no—
218C2×M16—1422no—
219#123—44no—
220#121—41no—
221#121—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C2×Dic7111yescharacteristic
228C2×C4⋊C4C2×Dic7111yescharacteristic
229C2×C4⋊C4C22⋊D7111yescharacteristic
230C2×C4⋊C4C22⋊D7111yescharacteristic
231C2×C4⋊C4C22×D7111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C22⋊C8—2841no—
239C22⋊C8—2841no—
240C22⋊C8—2841no—
241C22⋊C8—1422no—
242#95—21no—
243#95—21no—
244#95C2×Dic7111yescharacteristic
245#95C2×Dic7111yescharacteristic
246C24—21no—
247C24C22⋊Dic7111yescharacteristic
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C22⋊Dic7111yescharacteristic
268C22×C4C22⋊Dic7111yescharacteristic
269C22×C4C22⋊Dic7111yescharacteristic
270C22×C4C22×Dic7111yescharacteristic
271C22×C4C2×C22⋊D7111yescharacteristic
272C22×C4C2×C22⋊D7111yescharacteristic
273M16—2842no—
274M16—2842no—
275M16—5682no—
276C2×C8—2842no—
277C2×C8—2842no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#1455604311yescharacteristic
304C23#1455604311yescharacteristic
305C23C2×C22⋊Dic7111yescharacteristic
306D4—82no—
307D4—82no—
308D4—82no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#1455615412yes—
338C8—5682no—
339C8—5682no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C2×#1455604311yescharacteristic
358C22#1798600411yescharacteristic
359C22#1798601611yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—161no—
374C2#2258686911yescharacteristic
375C2#2258998611yescharacteristic
376C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2618C2
socleC22×C71C2×#99
prime radicalC2×#121×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2618D71
hypocenterC71#11805
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. #22588420C2semidirect
2. #22586670C2semidirect
3. #22586346C2semidirect
4. #22586343C2semidirect
194. C71#11805semidirect

Maximal subgroups

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

subgroupindex
#225884202
#225866702
#225863462
#225863432
C71×#26182
#1180571