Finite Group Databasebeta

← #31908611

Subgroups — #31908611

Order 18176.

440 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 103 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#22586658C211yesmaximal, characteristic
3#22586370C212yesmaximal, complemented
4#22586367C212yesmaximal, complemented
5C71×#2564C211yesmaximal, characteristic
6#17986023—21no—
7C2×#14556045—21no—
8C2×C22⋊(C71⋊C8)—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C22⋊Q8×C71C411yescharacteristic
13C2×C4×D4×C71C2211yescharacteristic
14#400×C71C2211yescharacteristic
15#399×C71C411yescharacteristic
16#395×C71C2211yescharacteristic
17#393×C71C411yescharacteristic
18#387×C71—21no—
19#386×C71—21no—
20#385×C71—21no—
21#380×C71—21no—
22C2×#95×C71C411yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8—41no—
25C4⋊(C71⋊C8)—41no—
26C4⋊(C71⋊C8)—41no—
27#14556045—42no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—41no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×D4×C71D411yescharacteristic
32C23×C4×C71—21no—
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C22⋊Q8×C71—41no—
36C22⋊Q8×C71—41no—
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50#95×C71—21no—
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—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C22⋊C411yescharacteristic
64C2×Q8×C71—41no—
65C2×Q8×C71—81no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71SD1612yes—
69C2×D4×C71Q1612yes—
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—21no—
77C22×C4×C71—21no—
78C22×C4×C71—42no—
79C22×C4×C71—42no—
80C22×C4×C71—81no—
81C22×C4×C71—81no—
82C22×C4×C71—81no—
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—42no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—81no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—81no—
99C42×C71—21no—
100C42×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71#10312yes—
110C23×C71#10312yes—
111C23×C71C2×C22⋊C411yescharacteristic
112Q8×C71—82no—
113D4×C71—42no—
114D4×C71—42no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—22no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—41no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
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—81no—
139C2×C4×C71—81no—
140C2×C4×C71—84no—
141C2×C4×C71#9911yescharacteristic
142C2×C4×C71#9911yescharacteristic
143C2×C4×C71C2×SD1611yescharacteristic
144C2×C4×C71C2×Q1611yescharacteristic
145C71⋊C8—82no—
146C71⋊C8—82no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—42no—
152C22×C71—42no—
153C22×C71—42no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—81no—
159C22×C71C2×#9911yescharacteristic
160C22×C71C2×#10311yescharacteristic
161C22×C71#42111yescharacteristic
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—42no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—81no—
171C4×C71—81no—
172C4×C71—81no—
173#11906—711nomaximal, Sylow
174C2×C71—21no—
175C2×C71—21no—
176C2×C71—42no—
177C2×C71—42no—
178C2×C71#152011yescharacteristic
179C2×C71#212111yescharacteristic
180C2×C71#213211yescharacteristic
181#2564D7111yescharacteristic
182#1800—711no—
183#1459—711no—
184#1354—712no—
185#1353—712no—
186C71#1190611yescharacteristic, complemented, Sylow
187C2×C22⋊Q8Dic7111yescharacteristic
188C2×C4×D4C2×D7111yescharacteristic
189#423—1421no—
190C2×#100—1421no—
191C2×C22⋊C8—1421no—
192#400C2×D7111yescharacteristic
193#399Dic7111yescharacteristic
194#395C2×D7111yescharacteristic
195#393Dic7111yescharacteristic
196#387—21no—
197#386—21no—
198#385—21no—
199#380—21no—
200C2×#95Dic7111yescharacteristic
201#343—712no—
202#324—712no—
203#323—1421no—
204C22×Q8C2×Dic7111yescharacteristic
205C22×D4C22⋊D7111yescharacteristic
206C23×C4—21no—
207C23×C4C2×Dic7111yescharacteristic
208C23×C4C22⋊D7111yescharacteristic
209C2×M16—1422no—
210C22×C8—2841no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×Dic7111yescharacteristic
219C2×C4⋊C4C22×D7111yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4C2×Dic7111yescharacteristic
224C2×C22⋊C4C22⋊D7111yescharacteristic
225C2×C42C22⋊D7111yescharacteristic
226C4⋊C8—2841no—
227C4⋊C8—2841no—
228#100—2842no—
229C22⋊C8—2841no—
230C22⋊C8—1422no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C2×Dic7111yescharacteristic
237#95C2×Dic7111yescharacteristic
238C24C22⋊Dic7111yescharacteristic
239C24C22⋊Dic7111yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4#1176585212yes—
245C2×D4#1176585512yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C22⋊Dic7111yescharacteristic
260C22×C4C22⋊Dic7111yescharacteristic
261C22×C4C22×Dic7111yescharacteristic
262C22×C4C2×C22⋊D7111yescharacteristic
263C22×C4C2×C22⋊D7111yescharacteristic
264M16—2842no—
265M16—2842no—
266C2×C8—2842no—
267C2×C8—2842no—
268C2×C8—5682no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#1455605212yes—
291C23#1455605212yes—
292C23C2×C22⋊Dic7111yescharacteristic
293Q8—82no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
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—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—84no—
322C2×C4#1455604311yescharacteristic
323C2×C4#1455604311yescharacteristic
324C2×C4C2×#1176585211yescharacteristic
325C2×C4C2×#1176585511yescharacteristic
326C8—5682no—
327C8—5682no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22C2×#1455604311yescharacteristic
341C22C2×#1455605211yescharacteristic
342C22#1798601611yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#2258691211yescharacteristic
359C2#2258998611yescharacteristic
360C2#2259002511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2564C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C4⋊C4×C71C23
hypercenter#2564D71
hypocenterC71#11906
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #22586370C2semidirect
4. #22586367C2semidirect
186. C71#11906semidirect

Maximal subgroups

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

subgroupindex
#225883322
#225866582
#225863702
#225863672
C71×#25642
#1190671