Finite Group Databasebeta

← #31886014

Subgroups — #31886014

Order 18176.

640 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 111 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#22596685C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic, complemented
3#22586160C214yesmaximal, complemented
4C71×#1749C211yesmaximal, characteristic, complemented
5C2×#14556125C411yescharacteristic
6C2×#14556102—21no—
7C2×#14556102—21no—
8C2×#14556102C412yes—
9C2×C22⋊(C71⋊C8)—21no—
10#17985901—21no—
11#17985901—21no—
12#17985901C2212yes—
13#17985869C411yescharacteristic
14#17985754—21no—
15#17985754C2212yes—
16C22×C42×C71C2211yescharacteristic
17C2×C22⋊C8×C71—21no—
18C2×#97×C71—21no—
19#336×C71C2212yes—
20C22×D284—21no—
21C22×D284—21no—
22C22×D284C2×C412yes—
23C22×C22⋊D71—22no—
24C22×C71⋊C8—22no—
25#14556125—42no—
26#14556102—44no—
27#14556102—44no—
28#14556102—44no—
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71C2×C412yes—
32C2×(C2×D71)⋊C4—22no—
33C2×(C2×D71)⋊C4—22no—
34C2×(C2×D71)⋊C4—22no—
35C22⋊(C71⋊C8)—42no—
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C22×C8×C71—22no—
40C2×C42×C71—42no—
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71C2311yescharacteristic
45C22⋊C8×C71—24no—
46#97×C71—42no—
47C23×D71—22no—
48C2×D284—24no—
49C2×D284—42no—
50C2×D284—42no—
51C2×D284—42no—
52C2×D284—42no—
53C2×D284—42no—
54C2×D284—42no—
55C2×D284—42no—
56C2×D284—42no—
57C2×C22⋊D71—44no—
58C2×C22⋊D71—44no—
59C2×C22⋊D71—44no—
60C22×Dic71—22no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C4⋊Dic71—42no—
65C4⋊Dic71—42no—
66C4⋊Dic71—42no—
67(C2×D71)⋊C4—44no—
68(C2×D71)⋊C4—44no—
69(C2×D71)⋊C4—44no—
70C24×C71C22⋊C411yescharacteristic
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C2×C8×C71—24no—
87C2×C8×C71—42no—
88C2×C8×C71—42no—
89C42×C71—22no—
90C42×C71—22no—
91C42×C71—41no—
92C42×C71—41no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71#10012yes—
100C23×C71C2×C22⋊C411yescharacteristic
101C22×D71—42no—
102C22×D71—42no—
103C22×D71—42no—
104C22×D71—42no—
105C22×D71—42no—
106C22×D71—42no—
107C22×D71—42no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
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—44no—
122C2×C4×C71—44no—
123C2×C4×C71—44no—
124C2×C4×C71#9912yes—
125C2×C4×C71#10412yes—
126C2×C4×C71#13112yes—
127C2×C4×C71AGL(1,ℤ/8)12yes—
128C2×C4×C71AGL(1,ℤ/8)12yes—
129C22⋊D71—88no—
130C2×Dic71—42no—
131C2×Dic71—42no—
132C2×Dic71—42no—
133D284—84no—
134D284—84no—
135D284—84no—
136C8×C71—44no—
137C71⋊C8—82no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71#40811yescharacteristic
146C22×C71C2×#9911yescharacteristic
147C22×C71C2×#10011yescharacteristic
148C22×C71#41811yescharacteristic
149C22×C71#41811yescharacteristic
150C22×C71C2×#10411yescharacteristic
151C22×C71#42111yescharacteristic
152C2×D71—82no—
153C2×D71—82no—
154C2×D71—82no—
155C2×D71—82no—
156C2×D71—82no—
157C2×D71—82no—
158C2×D71—82no—
159C4×C71—22no—
160C4×C71—22no—
161C4×C71—22no—
162C4×C71—22no—
163C4×C71—22no—
164C4×C71—41no—
165Dic71—82no—
166#7704—711nomaximal, Sylow
167C2×C71—42no—
168C2×C71#147111yescharacteristic
169C2×C71#148411yescharacteristic
170C2×C71#149611yescharacteristic
171C2×C71#150711yescharacteristic
172C2×C71#151011yescharacteristic
173C2×C71#153011yescharacteristic
174C2×C71#155511yescharacteristic
175D71—162no—
176#2856—711no—
177#1817—711no—
178#1749D7111yescharacteristic, complemented
179#1270—714no—
180C71#770411yescharacteristic, complemented, Sylow
181C2×#127—711no—
182C2×#121—712no—
183C2×#121—1421no—
184C2×#121—1421no—
185C22×C42C2×D7111yescharacteristic
186C2×C22⋊C8—21no—
187C2×C22⋊C8—1421no—
188C2×#97—21no—
189#390—712no—
190#390—1421no—
191#390—1421no—
192#384—711no—
193#336—712no—
194#336—1421no—
195#336C2×D7112yes—
196C22×D4—712no—
197C22×D4—1422no—
198C22×D4—1421no—
199C22×D4—1421no—
200C23×C4C4×D7111yescharacteristic
201C23×C4D28411yescharacteristic
202C23×C4C22⋊D7111yescharacteristic
203C22×C8—22no—
204C22×C8—1422no—
205#127—2842no—
206#121—2844no—
207#121—2844no—
208#121—2844no—
209C2×C4⋊C4—712no—
210C2×C4⋊C4—1421no—
211C2×C4⋊C4—1421no—
212C2×C22⋊C4—1422no—
213C2×C22⋊C4—1422no—
214C2×C22⋊C4—1422no—
215C2×C42—42no—
216C2×C42C4×D7111yescharacteristic
217C2×C42D28411yescharacteristic
218C2×C42C22⋊D7111yescharacteristic
219C2×C42C22×D7111yescharacteristic
220C22⋊C8—24no—
221C22⋊C8—2842no—
222#97—42no—
223C24—1422no—
224C24(C2×D71)⋊C411yescharacteristic
225C2×D4—2842no—
226C2×D4—2842no—
227C2×D4—2842no—
228C2×D4—2842no—
229C2×D4—2842no—
230C2×D4—2842no—
231C2×D4—2844no—
232C2×D4—2844no—
233C2×D4—2844no—
234C2×D4—2842no—
235C2×D4—2842no—
236C2×D4—1424no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—1422no—
247C22×C4(C2×D71)⋊C411yescharacteristic
248C22×C4(C2×D71)⋊C411yescharacteristic
249C22×C4(C2×D71)⋊C411yescharacteristic
250C22×C4C2×C4×D7111yescharacteristic
251C22×C4C2×C22⋊D7111yescharacteristic
252C22×C4C2×D28411yescharacteristic
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—2842no—
257C2×C8—2842no—
258C2×C8—2842no—
259C4⋊C4—2842no—
260C4⋊C4—2842no—
261C4⋊C4—2842no—
262C22⋊C4—2844no—
263C22⋊C4—2844no—
264C22⋊C4—2844no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—2842no—
276C23—2842no—
277C23—2842no—
278C23—2842no—
279C23—2842no—
280C23—2842no—
281C23—2842no—
282C23#1455604412yes—
283C23C2×(C2×D71)⋊C411yescharacteristic
284D4—5684no—
285D4—5684no—
286D4—5684no—
287D4—5688no—
288C2×C4—21no—
289C2×C4—21no—
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—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—2842no—
305C2×C4—2842no—
306C2×C4—2842no—
307C2×C4#1455604212yes—
308C2×C4#1455605612yes—
309C2×C4#1455613612yes—
310C2×C4#1455615312yes—
311C2×C4#1455615612yes—
312C8—44no—
313C8—5682no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—5682no—
322C22—5682no—
323C22—5682no—
324C22—5682no—
325C22—5682no—
326C22—5682no—
327C22—5682no—
328C22#1798596711yescharacteristic
329C22C2×#1455604211yescharacteristic
330C22C2×#1455604411yescharacteristic
331C22#1798600311yescharacteristic
332C22#1798600511yescharacteristic
333C22C2×#1455605611yescharacteristic
334C22#1798601511yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—5682no—
342C2—42no—
343C2—11362no—
344C2#2258670111yescharacteristic
345C2#2258675811yescharacteristic
346C2#2258680811yescharacteristic
347C2#2258685511yescharacteristic
348C2#2258687011yescharacteristic
349C2#2258695011yescharacteristic
350C2#2258705311yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1749C2
socleC23×C71C2×C22⋊C4
prime radical#22596685C2
cosocleC2×C42×C71C23
hypercenter#1749D71
hypocenterC71#7704
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #22588408C2semidirect
3. #22586160C2semidirect
4. C71×#1749C2semidirect
178. #1749D71semidirect
180. C71#7704semidirect

Maximal subgroups

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

subgroupindex
#225966852
#225884082
#225861602
C71×#17492
#770471