Finite Group Databasebeta

← #31908375

Subgroups — #31908375

Order 18176.

438 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 91 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
2C2×#17985725C211yesmaximal, characteristic, complemented
3#22586367C212yesmaximal, complemented
4#22586343C212yesmaximal, complemented
5C71×#2591C211yesmaximal, characteristic
6#17985963—21no—
7C2×C22⋊(C71⋊C8)—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C2×#121×C71C411yescharacteristic
13C22×C4⋊C4×C71C2211yescharacteristic
14#400×C71C411yescharacteristic
15#400×C71C2211yescharacteristic
16#394×C71C411yescharacteristic
17#394×C71C2211yescharacteristic
18#388×C71—21no—
19#388×C71—21no—
20#386×C71—21no—
21#386×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8—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#121×C71—41no—
35#121×C71—41no—
36C2×C4⋊C4×C71—41no—
37C2×C4⋊C4×C71—41no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71C2311yescharacteristic
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C42×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71C2×C411yescharacteristic
55#95×C71C2×C411yescharacteristic
56#11765849—42no—
57#11765849—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—82no—
61C24×C71—21no—
62C24×C71C22⋊C411yescharacteristic
63C2×D4×C71—41no—
64C2×D4×C71—42no—
65C2×D4×C71—42no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71—81no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—22no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—41no—
80C22×C4×C71—81no—
81C22×C4×C71—81no—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C22⋊C4×C71—42no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—81no—
98C22⋊C4×C71—81no—
99C42×C71—81no—
100C23×C71—22no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71#9911yescharacteristic
112C23×C71#9911yescharacteristic
113C23×C71C2×C22⋊C411yescharacteristic
114D4×C71—82no—
115D4×C71—82no—
116D4×C71—82no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
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—42no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
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—81no—
143C2×C4×C71—81no—
144C2×C4×C71—81no—
145C2×C4×C71#13512yes—
146C71⋊C8—82no—
147C71⋊C8—82no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—42no—
156C22×C71—42no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71—81no—
160C22×C71—81no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71#35512yes—
165C22×C71C2×#9911yescharacteristic
166C22×C71#41611yescharacteristic
167C22×C71#42111yescharacteristic
168C4×C71—22no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71—81no—
175C4×C71—81no—
176C4×C71—81no—
177#11852—711nomaximal, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—22no—
181C2×C71—41no—
182C2×C71—161no—
183C2×C71#151611yescharacteristic
184C2×C71#212111yescharacteristic
185C2×C71C2×#35511yescharacteristic
186#2591D7111yescharacteristic
187#1821—711no—
188C2×#324—711no—
189#1353—712no—
190#1345—712no—
191C71#1185211yescharacteristic, complemented, Sylow
192C2×#121Dic7111yescharacteristic
193C22×C4⋊C4C2×D7111yescharacteristic
194#407—1421no—
195C2×C22⋊C8—1421no—
196#400Dic7111yescharacteristic
197#400C2×D7111yescharacteristic
198#394Dic7111yescharacteristic
199#394C2×D7111yescharacteristic
200#388—21no—
201#388—21no—
202#386—21no—
203#386—21no—
204C4×C22⋊C4Dic7111yescharacteristic
205#343—712no—
206#343—1421no—
207#324—712no—
208#324—1421no—
209C22×D4—21no—
210C22×D4C2×Dic7111yescharacteristic
211C23×C4C2×Dic7111yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C23×C4C22⋊D7111yescharacteristic
214C2×M16—1422no—
215C22×C8—2841no—
216#121—41no—
217#121—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C2×Dic7111yescharacteristic
221C2×C4⋊C4C2×Dic7111yescharacteristic
222C2×C4⋊C4C22⋊D7111yescharacteristic
223C2×C4⋊C4C22⋊D7111yescharacteristic
224C2×C4⋊C4C22×D7111yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C22⋊C8—2841no—
233C22⋊C8—2841no—
234C22⋊C8—2841no—
235C22⋊C8—1422no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×Dic7111yescharacteristic
241#95C2×Dic7111yescharacteristic
242C24—21no—
243C24C22⋊Dic7111yescharacteristic
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—81no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22⋊Dic7111yescharacteristic
265C22×C4C22⋊Dic7111yescharacteristic
266C22×C4C22×Dic7111yescharacteristic
267C22×C4C2×C22⋊D7111yescharacteristic
268C22×C4C2×C22⋊D7111yescharacteristic
269M16—2842no—
270M16—2842no—
271C2×C8—2842no—
272C2×C8—2842no—
273C2×C8—5682no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C42—81no—
286C23—22no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1455604311yescharacteristic
298C23#1455604311yescharacteristic
299C23C2×C22⋊Dic7111yescharacteristic
300D4—82no—
301D4—82no—
302D4—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4#1455614812yes—
332C8—5682no—
333C8—5682no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#1798579612yes—
351C22C2×#1455604311yescharacteristic
352C22#1798599511yescharacteristic
353C22#1798601611yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—161no—
368C2#2258689611yescharacteristic
369C2#2258998611yescharacteristic
370C2C2×#1798579611yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2591C2
socleC22×C71C2×#99
prime radicalC2×#121×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2591D71
hypocenterC71#11852
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. #22588420C2semidirect
2. C2×#17985725C2semidirect
3. #22586367C2semidirect
4. #22586343C2semidirect
191. C71#11852semidirect

Maximal subgroups

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

subgroupindex
#225884202
C2×#179857252
#225863672
#225863432
C71×#25912
#1185271