Finite Group Databasebeta

← #31908223

Subgroups — #31908223

Order 18176.

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

normal subgroups only

This group has 434 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#22588328C211yesmaximal, characteristic
2#22586681C211yesmaximal, characteristic
3#22586364C212yesmaximal, complemented
4#22586340C212yesmaximal, complemented
5C71×#2395C211yesmaximal, characteristic
6#17986056—21no—
7C2×#14556043—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C2×C4×D4×C71C2211yescharacteristic
13C2×#117×C71C411yescharacteristic, complemented
14#400×C71C411yescharacteristic, complemented
15#399×C71C2211yescharacteristic
16#398×C71C411yescharacteristic, complemented
17#397×C71—21no—
18#397×C71C2211yescharacteristic
19#394×C71—21no—
20C4×C4⋊C4×C71—21no—
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic, complemented
23C2×C4⋊Dic71—41no—
24C2×C22⋊Dic71—22no—
25C4⋊(C71⋊C8)—41no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28#14556039—41no—
29C71⋊#95—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×C71C2×C411yescharacteristic
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×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
57C22×Dic71—42no—
58C2×C71⋊C8—42no—
59C4⋊Dic71—82no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C2×D411yescharacteristic
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×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C22⋊C4×C71—41no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—42no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
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—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71C2×C22⋊C411yescharacteristic
110C23×C71AGL(1,ℤ/8)11yescharacteristic
111C23×C71#13711yescharacteristic
112D4×C71—42no—
113D4×C71—42no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
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—81no—
140C2×C4×C71#9911yescharacteristic
141C2×C4×C71#9911yescharacteristic
142C2×Dic71—82no—
143C2×Dic71—82no—
144C2×Dic71—82no—
145C71⋊C8—82no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71—81no—
160C22×C71C2×#9911yescharacteristic
161C22×C71#41711yescharacteristic
162C22×C71#42111yescharacteristic
163C4×C71—21no—
164C4×C71—42no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—81no—
170C4×C71—81no—
171C4×C71—81no—
172C4×C71#35212yes—
173Dic71—162no—
174#11822—711nomaximal, Sylow
175C2×C71—21no—
176C2×C71—21no—
177C2×C71—41no—
178C2×C71—41no—
179C2×C71—81no—
180C2×C71#151311yescharacteristic
181C2×C71C2×#35211yescharacteristic
182C2×C71#212611yescharacteristic
183#2395D7111yescharacteristic
184#1799—711no—
185#1466—711no—
186#1352—712no—
187#1344—712no—
188C71#1182211yescharacteristic, complemented, Sylow
189C2×C4×D4C2×D7111yescharacteristic
190C2×#117Dic7111yescharacteristic, complemented
191#432—1421no—
192C2×#99—1421no—
193#400Dic7111yescharacteristic, complemented
194#399C2×D7111yescharacteristic
195#398Dic7111yescharacteristic, complemented
196#397—21no—
197#397C2×D7111yescharacteristic
198#394—21no—
199#380—1421no—
200C4×C4⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4Dic7111yescharacteristic, complemented
203#342—712no—
204#323—712no—
205#323—1421no—
206C22×D4C22⋊D7111yescharacteristic
207C23×C4C2×Dic7111yescharacteristic
208C23×C4C22⋊D7111yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—2841no—
216C2×C4⋊C4C2×Dic7111yescharacteristic
217C2×C4⋊C4C2×Dic7111yescharacteristic
218C2×C4⋊C4C2×Dic7111yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1422no—
224C2×C22⋊C4C22⋊D7111yescharacteristic
225C2×C22⋊C4C22×D7111yescharacteristic
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C22⋊D7111yescharacteristic
229C4⋊C8—2841no—
230#99—2842no—
231C22⋊C8—1422no—
232#97—2841no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—2841no—
238#95C2×Dic7111yescharacteristic
239#95C2×Dic7111yescharacteristic
240C24C22⋊Dic7111yescharacteristic
241C24C2×C22⋊D7111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
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—2842no—
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4C22⋊Dic7111yescharacteristic
259C22×C4C22⋊Dic7111yescharacteristic
260C22×C4C22×Dic7111yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262C2×C8—2842no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—5682no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—2842no—
277C22⋊C4—2842no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—41no—
282C42—41no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊Dic7111yescharacteristic
293C23#1455615311yescharacteristic
294C23#1455615811yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
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—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—5682no—
324C2×C4—5682no—
325C2×C4—5682no—
326C2×C4#1455604311yescharacteristic
327C2×C4#1455604311yescharacteristic
328C8—5682no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#1455604311yescharacteristic
344C22#1798600011yescharacteristic
345C22#1798601611yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—11362no—
356C4#1798578912yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2#2258688311yescharacteristic
363C2C2×#1798578911yescharacteristic
364C2#2259000511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2395C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2395D71
hypocenterC71#11822
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. #22586364C2semidirect
4. #22586340C2semidirect
13. C2×#117×C71C4semidirect
14. #400×C71C4semidirect
16. #398×C71C4semidirect
22. C4×C22⋊C4×C71C4semidirect
188. C71#11822semidirect
190. C2×#117Dic71semidirect
193. #400Dic71semidirect
195. #398Dic71semidirect
202. C4×C22⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866812
#225863642
#225863402
C71×#23952
#1182271