Finite Group Databasebeta

← #31908335

Subgroups — #31908335

Order 18176.

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

normal subgroups only

This group has 432 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#22588417C211yesmaximal, characteristic, complemented
2#22586654C211yesmaximal, characteristic, complemented
3#22586361C212yesmaximal, complemented
4#22586337C212yesmaximal
5C71×#2610C211yesmaximal, characteristic
6#17986052—21no—
7#17985862—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×#124×C71C411yescharacteristic, complemented
13C2×#117×C71C2211yescharacteristic
14#395×C71C411yescharacteristic, complemented
15#394×C71C411yescharacteristic, complemented
16#393×C71C2211yescharacteristic
17#392×C71C2211yescharacteristic
18#390×C71—21no—
19#388×C71—21no—
20#388×C71—21no—
21#386×C71—21no—
22#376×C71C411yescharacteristic, complemented
23C2×C22⋊Dic71—22no—
24C4⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—41no—
27C71⋊#95—41no—
28C71⋊#95—41no—
29C22×Q8×C71C2×C411yescharacteristic
30C22×D4×C71—21no—
31C22×D4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33#124×C71—41no—
34#124×C71—41no—
35#117×C71—41no—
36#117×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71C2311yescharacteristic
48C2×C42×C71—21no—
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71D411yescharacteristic
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71C2×C411yescharacteristic
55#95×C71C2×C411yescharacteristic
56C22×Dic71—42no—
57C2×C71⋊C8—42no—
58C22⋊Dic71—42no—
59C22⋊Dic71—42no—
60C24×C71—21no—
61C24×C71C2×D411yescharacteristic
62C2×Q8×C71—41no—
63C2×Q8×C71—81no—
64C2×D4×C71—41no—
65C2×D4×C71—42no—
66C2×D4×C71—42no—
67C2×D4×C71—81no—
68C2×D4×C71—81no—
69C22×C4×C71—21no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—41no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—81no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—42no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—81no—
93C22⋊C4×C71—81no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—21no—
98C42×C71—42no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71C2×C22⋊C411yescharacteristic
111C23×C71#13512yes—
112Q8×C71—82no—
113D4×C71—84no—
114D4×C71—82no—
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—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
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—41no—
138C2×C4×C71—41no—
139C2×C4×C71#9911yescharacteristic
140C2×C4×C71#9911yescharacteristic
141C2×Dic71—82no—
142C2×Dic71—82no—
143C2×Dic71—82no—
144C71⋊C8—82no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—22no—
148C22×C71—42no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—81no—
152C22×C71—81no—
153C22×C71—81no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71—81no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71C2×#9911yescharacteristic
160C22×C71#41611yescharacteristic
161C22×C71#42111yescharacteristic
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—42no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—81no—
171C4×C71—81no—
172Dic71—162no—
173#11844—711nomaximal, Sylow
174C2×C71—21no—
175C2×C71—21no—
176C2×C71—42no—
177C2×C71—161no—
178C2×C71#151611yescharacteristic
179C2×C71#212211yescharacteristic
180C2×C71#212611yescharacteristic
181#2610D7111yescharacteristic
182#1820—711no—
183#1458—711no—
184#1351—712no—
185#1343—712no—
186C71#1184411yescharacteristic, complemented, Sylow
187C2×#124Dic7111yescharacteristic, complemented
188C2×#117C2×D7111yescharacteristic
189#431—1421no—
190#395Dic7111yescharacteristic, complemented
191#394Dic7111yescharacteristic, complemented
192#393C2×D7111yescharacteristic
193#392C2×D7111yescharacteristic
194#390—21no—
195#388—21no—
196#388—21no—
197#386—21no—
198#381—1421no—
199#376Dic7111yescharacteristic, complemented
200#342—712no—
201#342—1421no—
202#324—1421no—
203#323—712no—
204C22×Q8C2×Dic7111yescharacteristic
205C22×D4—21no—
206C22×D4C2×Dic7111yescharacteristic
207C23×C4C22⋊D7111yescharacteristic
208#124—41no—
209#124—41no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
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⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—1422no—
222C2×C22⋊C4C22⋊D7111yescharacteristic
223C2×C22⋊C4C22×D7111yescharacteristic
224C2×C42—21no—
225C2×C42C2×Dic7111yescharacteristic
226C2×C42C22⋊D7111yescharacteristic
227C4⋊C8—2841no—
228C22⋊C8—1422no—
229C4×C8—2841no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—2841no—
234#95—2841no—
235#95C2×Dic7111yescharacteristic
236#95C2×Dic7111yescharacteristic
237C24—21no—
238C24C2×C22⋊D7111yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—81no—
245C2×D4—81no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—2842no—
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4C22×Dic7111yescharacteristic
259C22×C4C2×C22⋊D7111yescharacteristic
260C2×C8—2842no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—2842no—
274C22⋊C4—2842no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—42no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic7111yescharacteristic
292C23#1455614812yes—
293Q8—82no—
294D4—84no—
295D4—82no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
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—41no—
319C2×C4—41no—
320C2×C4—5682no—
321C2×C4—5682no—
322C2×C4—5682no—
323C2×C4#1455604311yescharacteristic
324C2×C4#1455604311yescharacteristic
325C8—5682no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C2×#1455604311yescharacteristic
341C22#1798599511yescharacteristic
342C22#1798601611yescharacteristic
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—11362no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—161no—
358C2#2258689611yescharacteristic
359C2#2258999011yescharacteristic
360C2#2259000511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2610C2
socleC22×C71C2×#99
prime radical#392×C71C22
cosocleC2×C22⋊C4×C71C23
hypercenter#2610D71
hypocenterC71#11844
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #22588417C2semidirect
2. #22586654C2semidirect
3. #22586361C2semidirect
12. C2×#124×C71C4semidirect
14. #395×C71C4semidirect
15. #394×C71C4semidirect
22. #376×C71C4semidirect
186. C71#11844semidirect
187. C2×#124Dic71semidirect
190. #395Dic71semidirect
191. #394Dic71semidirect
199. #376Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884172
#225866542
#225863612
#225863372
C71×#26102
#1184471