Finite Group Databasebeta

← #31908576

Subgroups — #31908576

Order 18176.

580 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 107 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#22586661C211yesmaximal, characteristic
3#22586361C214yesmaximal, complemented
4C71×#2409C211yesmaximal, characteristic
5#17986052—21no—
6C2×#14556043—21no—
7#17985859—21no—
8#17985766C2212yes—
9#17985723—21no—
10#17985723C2212yes—
11C23×Q8×C71C411yescharacteristic, complemented
12C2×C22⋊Q8×C71—21no—
13C2×C4×D4×C71C2211yescharacteristic
14#395×C71C412yescomplemented
15#393×C71C2212yes—
16#391×C71—21no—
17#386×C71—21no—
18#385×C71—21no—
19#380×C71C411yescharacteristic, complemented
20C2×C4⋊Dic71—41no—
21C2×C22⋊Dic71—22no—
22C4⋊(C71⋊C8)—41no—
23#14556043—42no—
24C22⋊(C71⋊C8)—22no—
25C4×C71⋊C8—41no—
26C71⋊#95—41no—
27C22×Q8×C71—21no—
28C22×Q8×C71—41no—
29C22×Q8×C71—41no—
30C22×Q8×C71—44no—
31C22×Q8×C71C2×C412yes—
32C22×D4×C71D411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C22⋊Q8×C71—44no—
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71D411yescharacteristic
45C2×C22⋊C4×C71C2311yescharacteristic
46C2×C42×C71D411yescharacteristic
47#95×C71—22no—
48#95×C71—22no—
49#95×C71C2×C412yes—
50C22×Dic71—42no—
51C2×C71⋊C8—42no—
52C4⋊Dic71—82no—
53C22⋊Dic71—42no—
54C22⋊Dic71—42no—
55C24×C71C22⋊C411yescharacteristic
56C24×C71C2×D411yescharacteristic
57C2×Q8×C71—24no—
58C2×Q8×C71—44no—
59C2×Q8×C71—42no—
60C2×Q8×C71—42no—
61C2×Q8×C71—44no—
62C2×Q8×C71—48no—
63C2×Q8×C71—44no—
64C2×Q8×C71—41no—
65C2×Q8×C71—41no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71SD1612yes—
69C2×D4×C71Q1612yes—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—22no—
76C22×C4×C71—44no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71—42no—
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—44no—
88C22⋊C4×C71—41no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—44no—
91C22⋊C4×C71—42no—
92C42×C71—21no—
93C42×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71#10312yes—
103C23×C71C2×C22⋊C411yescharacteristic
104C23×C71C2×SD1611yescharacteristic
105C23×C71C2×Q1611yescharacteristic
106Q8×C71—44no—
107Q8×C71—44no—
108Q8×C71—44no—
109Q8×C71—44no—
110D4×C71—42no—
111D4×C71—42no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—22no—
115C2×C4×C71—24no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—48no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—81no—
132C2×C4×C71#9911yescharacteristic
133C2×C4×C71#9911yescharacteristic
134C2×C4×C71#10312yes—
135C2×Dic71—82no—
136C2×Dic71—82no—
137C2×Dic71—82no—
138C71⋊C8—82no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—22no—
143C22×C71—21no—
144C22×C71—42no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—42no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—81no—
153C22×C71C2×#9911yescharacteristic
154C22×C71C2×#10311yescharacteristic
155C22×C71#42111yescharacteristic
156C4×C71—21no—
157C4×C71—22no—
158C4×C71—44no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—82no—
163C4×C71AΣL(1,𝔽16)12yes—
164Dic71—162no—
165#11898—711nomaximal, Sylow
166C2×C71—21no—
167C2×C71—21no—
168C2×C71—41no—
169C2×C71—42no—
170C2×C71—41no—
171C2×C71#152011yescharacteristic
172C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
173C2×C71#212911yescharacteristic
174#2409D7111yescharacteristic
175#1799—711no—
176#1460—711no—
177#1351—714no—
178C71#1189811yescharacteristic, complemented, Sylow
179C23×Q8Dic7111yescharacteristic, complemented
180C2×C22⋊Q8—21no—
181C2×C4×D4C2×D7111yescharacteristic
182#431—1421no—
183C2×#99—1421no—
184#395Dic7112yescomplemented
185#393C2×D7112yes—
186#391—21no—
187#386—21no—
188#385—21no—
189#380—1421no—
190#380Dic7111yescharacteristic, complemented
191#342—712no—
192#323—712no—
193#323—1421no—
194C22×Q8—21no—
195C22×Q8—41no—
196C22×Q8—41no—
197C22×Q8—44no—
198C22×Q8C2×Dic7112yes—
199C22×D4C22⋊D7111yescharacteristic
200C23×C4—21no—
201C23×C4C2×Dic7111yescharacteristic
202C23×C4C22⋊D7111yescharacteristic
203C22⋊Q8—44no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—2841no—
209C2×C4⋊C4C2×Dic7111yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1422no—
213C2×C22⋊C4C22⋊D7111yescharacteristic
214C2×C22⋊C4C22×D7111yescharacteristic
215C2×C42C22⋊D7111yescharacteristic
216C4⋊C8—2841no—
217#99—2842no—
218C22⋊C8—1422no—
219C4×C8—2841no—
220#95—22no—
221#95—22no—
222#95—2841no—
223#95C2×Dic7112yes—
224C24C22⋊Dic7111yescharacteristic
225C24C2×C22⋊D7111yescharacteristic
226C2×Q8—24no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4#1176585212yes—
238C2×D4#1176585512yes—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—2842no—
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22⋊Dic7111yescharacteristic
253C22×C4C22×Dic7111yescharacteristic
254C22×C4C2×C22⋊D7111yescharacteristic
255C2×C8—2842no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—5682no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—2842no—
265C22⋊C4—2842no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23#1455605212yes—
277C23C2×C22⋊Dic7111yescharacteristic
278C23C2×#1176585211yescharacteristic
279C23C2×#1176585511yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—44no—
283Q8—44no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—5682no—
307C2×C4—5682no—
308C2×C4—5682no—
309C2×C4#1455604311yescharacteristic
310C2×C4#1455604311yescharacteristic
311C2×C4#1455605212yes—
312C8—5682no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#1455604311yescharacteristic
328C22C2×#1455605211yescharacteristic
329C22#1798601611yescharacteristic
330C4—21no—
331C4—22no—
332C4—44no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—11362no—
338C4#1798578612yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2#2258691211yescharacteristic
345C2C2×#1798578611yescharacteristic
346C2#2259001511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2409C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2409D71
hypocenterC71#11898
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #22586361C2semidirect
11. C23×Q8×C71C4semidirect
14. #395×C71C4semidirect
19. #380×C71C4semidirect
178. C71#11898semidirect
179. C23×Q8Dic71semidirect
184. #395Dic71semidirect
190. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866612
#225863612
C71×#24092
#1189871