Finite Group Databasebeta

← #31908263

Subgroups — #31908263

Order 18176.

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

normal subgroups only

This group has 420 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#22588332C211yesmaximal, characteristic
2#22586678C211yesmaximal, characteristic
3#22586367C212yesmaximal, complemented
4#22586346C212yesmaximal, complemented
5C71×#2395C211yesmaximal, characteristic
6#17986023—21no—
7C2×#14556045—21no—
8#17985963—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C4×D4×C71C2211yescharacteristic
13C2×#117×C71C411yescharacteristic
14#400×C71C2211yescharacteristic
15#399×C71C411yescharacteristic
16#398×C71C2211yescharacteristic
17#397×C71—21no—
18#397×C71C411yescharacteristic
19#394×C71—21no—
20C4×C4⋊C4×C71—21no—
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—22no—
24C2×#11765849—41no—
25C4⋊(C71⋊C8)—41no—
26C4⋊(C71⋊C8)—41no—
27#14556045—42no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—41no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C4×D4×C71—41no—
34C4×D4×C71—41no—
35#117×C71—41no—
36#117×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C42×C71—21no—
49C2×C42×C71—21no—
50C2×C42×C71D411yescharacteristic
51#95×C71—21no—
52#95×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×C71C22⋊C411yescharacteristic
63C24×C71C22⋊C411yescharacteristic
64C2×D4×C71—21no—
65C2×D4×C71—21no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C22×C4×C71—21no—
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—21no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—42no—
94C22⋊C4×C71—42no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—42no—
98C42×C71—41no—
99C42×C71—41no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71C2×C22⋊C411yescharacteristic
111D4×C71—42no—
112D4×C71—42no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—42no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
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—81no—
138C2×C4×C71#9911yescharacteristic
139C2×C4×C71#9911yescharacteristic
140C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C71#13711yescharacteristic
142C71⋊C8—82no—
143C71⋊C8—82no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—42no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—81no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71C2×#9911yescharacteristic
157C22×C71#41711yescharacteristic
158C22×C71#42111yescharacteristic
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—42no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—81no—
167C4×C71—81no—
168C4×C71—81no—
169C4×C71#35212yes—
170#11830—711nomaximal, Sylow
171C2×C71—21no—
172C2×C71—21no—
173C2×C71—81no—
174C2×C71—81no—
175C2×C71#151311yescharacteristic
176C2×C71C2×#35211yescharacteristic
177C2×C71#213211yescharacteristic
178#2395D7111yescharacteristic
179#1800—711no—
180#1465—711no—
181#1353—712no—
182#1346—712no—
183C71#1183011yescharacteristic, complemented, Sylow
184C2×C4×D4C2×D7111yescharacteristic
185C2×#117Dic7111yescharacteristic
186#423—1421no—
187C2×#100—1421no—
188#407—1421no—
189#400C2×D7111yescharacteristic
190#399Dic7111yescharacteristic
191#398C2×D7111yescharacteristic
192#397—21no—
193#397Dic7111yescharacteristic
194#394—21no—
195C4×C4⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4Dic7111yescharacteristic
198#343—712no—
199#324—712no—
200#323—1421no—
201C22×D4C22⋊D7111yescharacteristic
202C23×C4C2×Dic7111yescharacteristic
203C23×C4C22⋊D7111yescharacteristic
204C2×M16—2841no—
205C2×M16—1422no—
206C4×D4—41no—
207C4×D4—41no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×Dic7111yescharacteristic
213C2×C4⋊C4C2×Dic7111yescharacteristic
214C2×C4⋊C4C22×D7111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×Dic7111yescharacteristic
220C2×C22⋊C4C22⋊D7111yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22⋊D7111yescharacteristic
224C4⋊C8—2841no—
225C4⋊C8—2841no—
226#100—2842no—
227C22⋊C8—2841no—
228C22⋊C8—1422no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×Dic7111yescharacteristic
234#95C2×Dic7111yescharacteristic
235C24C22⋊Dic7111yescharacteristic
236C24C22⋊Dic7111yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22⋊Dic7111yescharacteristic
253C22×C4C22×Dic7111yescharacteristic
254C22×C4C2×C22⋊D7111yescharacteristic
255C22×C4C2×C22⋊D7111yescharacteristic
256M16—2842no—
257M16—2842no—
258M16—5682no—
259C2×C8—2842no—
260C2×C8—2842no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42—41no—
277C42—41no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊Dic7111yescharacteristic
289D4—42no—
290D4—42no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4#1455604311yescharacteristic
317C2×C4#1455604311yescharacteristic
318C2×C4#1455615311yescharacteristic
319C2×C4#1455615811yescharacteristic
320C8—5682no—
321C8—5682no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1455604311yescharacteristic
335C22#1798600011yescharacteristic
336C22#1798601611yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4#1798578912yes—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—81no—
352C2#2258688311yescharacteristic
353C2C2×#1798578911yescharacteristic
354C2#2259002511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2395C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C4⋊C4×C71C23
hypercenter#2395D71
hypocenterC71#11830
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2395

Complements

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

NH ≅ G/Ntype
3. #22586367C2semidirect
4. #22586346C2semidirect
183. C71#11830semidirect

Maximal subgroups

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

subgroupindex
#225883322
#225866782
#225863672
#225863462
C71×#23952
#1183071