Finite Group Databasebeta

← #31885737

Subgroups — #31885737

Order 18176.

478 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588483C211yesmaximal, characteristic, complemented
2#22588457C211yesmaximal, characteristic, complemented
3#22586163C212yesmaximal, complemented
4#22586161C212yesmaximal
5C71×#2293C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7#17985754—21no—
8#17985754—21no—
9#17985754—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C4×D4×C71C411yescharacteristic
13C2×#117×C71—21no—
14#390×C71C411yescharacteristic
15#390×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C2211yescharacteristic
18#388×C71—21no—
19#388×C71—21no—
20C4×C4⋊C4×C71C411yescharacteristic
21C4×C22⋊C4×C71—21no—
22C43×C71C2211yescharacteristic
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—42no—
26C22×D4×C71C2×C411yescharacteristic
27C23×C4×C71—21no—
28C4×D4×C71—22no—
29C4×D4×C71—22no—
30#117×C71—42no—
31#117×C71—42no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71C2×C411yescharacteristic
35C2×C4⋊C4×C71C2×C411yescharacteristic
36C2×C4⋊C4×C71C2×C411yescharacteristic
37C2×C22⋊C4×C71—21no—
38C2×C22⋊C4×C71—21no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C42×C71—21no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71C2311yescharacteristic
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53C2×C71⋊C8—42no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C24×C71—21no—
60C2×D4×C71—21no—
61C2×D4×C71—41no—
62C2×D4×C71—41no—
63C2×D4×C71M1612yes—
64C22×C4×C71—24no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22×C411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C4⋊C4×C71—21no—
78C4⋊C4×C71—41no—
79C4⋊C4×C71—41no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71M1612yes—
84C22⋊C4×C71—24no—
85C22⋊C4×C71—42no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—41no—
88C22⋊C4×C71—41no—
89C22⋊C4×C71—42no—
90C42×C71—21no—
91C42×C71—21no—
92C42×C71—22no—
93C42×C71—41no—
94C42×C71—41no—
95C42×C71—41no—
96C42×C71—41no—
97C42×C71—42no—
98C42×C71D812yes—
99C42×C71SD1612yes—
100C23×C71—22no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71C2×C22⋊C411yescharacteristic
108D4×C71—42no—
109D4×C71—42no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—48no—
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#10212yes—
137C2×C4×C71#10212yes—
138C2×C4×C71C2×M1611yescharacteristic
139C2×C4×C71C2×M1611yescharacteristic
140C2×C4×C71C2×D811yescharacteristic
141C2×C4×C71C2×SD1611yescharacteristic
142C2×C4×C71#13512yes—
143C71⋊C8—82no—
144C71⋊C8—82no—
145C22×C71—42no—
146C22×C71—42no—
147C22×C71—42no—
148C22×C71—81no—
149C22×C71—81no—
150C22×C71—81no—
151C22×C71—81no—
152C22×C71#40711yescharacteristic
153C22×C71#41011yescharacteristic
154C22×C71#41311yescharacteristic
155C22×C71C2×#10211yescharacteristic
156C22×C71#41611yescharacteristic
157C22×C71#42111yescharacteristic
158C22×C71#42111yescharacteristic
159C4×C71—22no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—22no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—22no—
166C4×C71—42no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—81no—
170C4×C71—81no—
171#7642—711nomaximal, Sylow
172C2×C71—82no—
173C2×C71#146411yescharacteristic
174C2×C71#148611yescharacteristic
175C2×C71#148811yescharacteristic
176C2×C71#151711yescharacteristic
177C2×C71#152111yescharacteristic
178C2×C71#153511yescharacteristic
179C2×C71#154011yescharacteristic
180#2293D7111yescharacteristic
181#1839—711no—
182#1831—711no—
183#1271—712no—
184#1270—712no—
185C71#764211yescharacteristic, complemented, Sylow
186C2×C4×D4Dic7111yescharacteristic
187C2×#117—21no—
188C2×C4⋊C8—1421no—
189#390Dic7111yescharacteristic
190#390C2×D7111yescharacteristic
191#389Dic7111yescharacteristic
192#389C2×D7111yescharacteristic
193#388—21no—
194#388—21no—
195C4×C4⋊C4Dic7111yescharacteristic
196C4×C22⋊C4—21no—
197C43C2×D7111yescharacteristic
198#336—712no—
199#336—712no—
200#336—1421no—
201#336—1421no—
202#336—1421no—
203C22×D4C2×Dic7111yescharacteristic
204C23×C4—21no—
205C22×C8—1422no—
206C22×C8—1422no—
207C4×D4—22no—
208C4×D4—22no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×Dic7111yescharacteristic
214C2×C4⋊C4C2×Dic7111yescharacteristic
215C2×C4⋊C4C2×Dic7111yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42C2×Dic7111yescharacteristic
223C2×C42C2×Dic7111yescharacteristic
224C2×C42C22⋊D7111yescharacteristic
225C2×C42C22⋊D7111yescharacteristic
226C2×C42C22⋊D7111yescharacteristic
227C2×C42C22⋊D7111yescharacteristic
228C2×C42C22×D7111yescharacteristic
229C4⋊C8—2842no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4#1176584912yes—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22⋊Dic7111yescharacteristic
247C22×C4C22⋊Dic7111yescharacteristic
248C22×C4C22×Dic7111yescharacteristic
249C22×C4C2×C22⋊D7111yescharacteristic
250C22×C4C2×C22⋊D7111yescharacteristic
251C2×C8—2842no—
252C2×C8—2842no—
253C2×C8—2842no—
254C2×C8—2842no—
255C2×C8—2842no—
256C2×C8—2842no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4#1176584912yes—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42#1176585012yes—
279C42#1176585312yes—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊Dic7111yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—48no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4#1455604912yes—
317C2×C4#1455604912yes—
318C2×C4C2×#1176584911yescharacteristic
319C2×C4C2×#1176584911yescharacteristic
320C2×C4C2×#1176585011yescharacteristic
321C2×C4C2×#1176585311yescharacteristic
322C2×C4#1455614812yes—
323C8—5682no—
324C8—5682no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#1798596311yescharacteristic
333C22#1798597311yescharacteristic
334C22#1798598311yescharacteristic
335C22C2×#1455604911yescharacteristic
336C22#1798599511yescharacteristic
337C22#1798601611yescharacteristic
338C22#1798601611yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—42no—
348C4—41no—
349C4—81no—
350C4—81no—
351C2—82no—
352C2#2258667411yescharacteristic
353C2#2258676511yescharacteristic
354C2#2258677411yescharacteristic
355C2#2258690011yescharacteristic
356C2#2258691711yescharacteristic
357C2#2258697111yescharacteristic
358C2#2258698911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2293C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2293D71
hypocenterC71#7642
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. #22588483C2semidirect
2. #22588457C2semidirect
3. #22586163C2semidirect
185. C71#7642semidirect

Maximal subgroups

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

subgroupindex
#225884832
#225884572
#225861632
#225861612
C71×#22932
#764271