Finite Group Databasebeta

← #31886020

Subgroups — #31886020

Order 18176.

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

normal subgroups only

This group has 426 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#22596707C211yesmaximal, characteristic
2#22588089C211yesmaximal, characteristic, complemented
3#22586162C212yesmaximal, complemented
4#22586160C212yesmaximal
5C71×#1831C211yesmaximal, characteristic, complemented
6C2×#14556122—21no—
7C2×#14556039—21no—
8#17985901C411yescharacteristic
9#17985901C411yescharacteristic
10#17985901C2211yescharacteristic
11#17985897C411yescharacteristic
12#17985897C411yescharacteristic
13#17985897C2211yescharacteristic
14#17985890—21no—
15#17985890—21no—
16#17985890—21no—
17#17985754—21no—
18#17985754C2212yes—
19C2×C4⋊C8×C71—21no—
20C43×C71C2211yescharacteristic
21#336×C71—21no—
22#336×C71C2212yes—
23C22×D284C2×C411yescharacteristic
24C22×C71⋊C8—22no—
25#14556122—44no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×Dic71⋊C4—21no—
30C2×Dic71⋊C4—21no—
31C2×Dic71⋊C4—21no—
32C2×(C2×D71)⋊C4—21no—
33C2×(C2×D71)⋊C4—21no—
34C2×(C2×D71)⋊C4—21no—
35C2×(C2×D71)⋊C4—21no—
36C2×(C2×D71)⋊C4—21no—
37C2×(C2×D71)⋊C4—21no—
38#14556039—24no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C22×C8×C71—22no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71C2311yescharacteristic
50C4⋊C8×C71—42no—
51C23×D71—21no—
52C2×D284—22no—
53C2×D284—41no—
54C2×D284—41no—
55C22×Dic71—21no—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C2×C71⋊C8—24no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—22no—
62C4⋊Dic71—41no—
63C4⋊Dic71—41no—
64Dic71⋊C4—42no—
65Dic71⋊C4—42no—
66Dic71⋊C4—42no—
67(C2×D71)⋊C4—42no—
68(C2×D71)⋊C4—42no—
69(C2×D71)⋊C4—42no—
70(C2×D71)⋊C4—42no—
71(C2×D71)⋊C4—41no—
72(C2×D71)⋊C4—42no—
73(C2×D71)⋊C4—41no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C2×C8×C71—42no—
82C2×C8×C71—42no—
83C2×C8×C71—42no—
84C42×C71—22no—
85C42×C71—22no—
86C42×C71—21no—
87C42×C71—22no—
88C42×C71—21no—
89C42×C71—41no—
90C42×C71—41no—
91C42×C71—41no—
92C23×C71C2×C22⋊C411yescharacteristic
93C22×D71—41no—
94C22×D71—41no—
95C22×D71—41no—
96C22×D71—41no—
97C22×D71—41no—
98C22×D71—41no—
99C22×D71—41no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
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#10412yes—
118C2×C4×C71#13112yes—
119C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C71AGL(1,ℤ/8)12yes—
121C2×C4×C71#13711yescharacteristic
122C2×Dic71—41no—
123C2×Dic71—41no—
124C2×Dic71—41no—
125C2×Dic71—41no—
126C2×Dic71—41no—
127C2×Dic71—41no—
128C2×Dic71—41no—
129C2×Dic71—41no—
130C2×Dic71—41no—
131D284—82no—
132C8×C71—82no—
133C71⋊C8—44no—
134C22×C71#40811yescharacteristic
135C22×C71#41011yescharacteristic
136C22×C71#41311yescharacteristic
137C22×C71#41711yescharacteristic
138C22×C71#41811yescharacteristic
139C22×C71C2×#10411yescharacteristic
140C22×C71#42111yescharacteristic
141C2×D71—81no—
142C2×D71—81no—
143C2×D71—81no—
144C2×D71—81no—
145C2×D71—81no—
146C2×D71—81no—
147C2×D71—81no—
148C4×C71—22no—
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—41no—
154C4×C71—41no—
155C4×C71—41no—
156Dic71—81no—
157Dic71—81no—
158Dic71—81no—
159#7706—711nomaximal, Sylow
160C2×C71#147311yescharacteristic
161C2×C71#148311yescharacteristic
162C2×C71#149611yescharacteristic
163C2×C71#150511yescharacteristic
164C2×C71#151211yescharacteristic
165C2×C71#152911yescharacteristic
166C2×C71#155711yescharacteristic
167D71—161no—
168#2860—711no—
169#1831D7111yescharacteristic, complemented
170#1745—711no—
171#1271—712no—
172#1270—712no—
173C71#770611yescharacteristic, complemented, Sylow
174C2×#126—1421no—
175C2×C4⋊C8—21no—
176C2×#97—1421no—
177#390—711no—
178#390—711no—
179#390—711no—
180#389—711no—
181#389—711no—
182#389—711no—
183#388—1421no—
184#388—1421no—
185#388—1421no—
186C43C2×D7111yescharacteristic
187#336—21no—
188#336—712no—
189#336—1421no—
190#336C2×D7112yes—
191C22×D4—711no—
192C22×C8—22no—
193C22×C8—1422no—
194#126—2844no—
195C2×C4⋊C4—711no—
196C2×C4⋊C4—711no—
197C2×C4⋊C4—711no—
198C2×C4⋊C4—1421no—
199C2×C4⋊C4—1421no—
200C2×C4⋊C4—1421no—
201C2×C22⋊C4—1421no—
202C2×C22⋊C4—1421no—
203C2×C22⋊C4—1421no—
204C2×C22⋊C4—1421no—
205C2×C22⋊C4—1421no—
206C2×C22⋊C4—1421no—
207C2×C42C4×D7111yescharacteristic
208C2×C42C4×D7111yescharacteristic
209C2×C42D28411yescharacteristic
210C2×C42D28411yescharacteristic
211C2×C42C22⋊D7111yescharacteristic
212C2×C42C22⋊D7111yescharacteristic
213C2×C42C22×D7111yescharacteristic
214C4⋊C8—42no—
215#97—1424no—
216#95—1421no—
217#95—1421no—
218#95—1421no—
219C24—1421no—
220C2×D4—2841no—
221C2×D4—2841no—
222C2×D4—1422no—
223C22×C4—1421no—
224C22×C4—1421no—
225C22×C4—1421no—
226C22×C4(C2×D71)⋊C411yescharacteristic
227C22×C4(C2×D71)⋊C411yescharacteristic
228C22×C4(C2×D71)⋊C411yescharacteristic
229C22×C4(C2×D71)⋊C411yescharacteristic
230C22×C4C2×C4×D7111yescharacteristic
231C22×C4C2×C22⋊D7111yescharacteristic
232C22×C4C2×D28411yescharacteristic
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—2842no—
237C2×C8—2842no—
238C2×C8—1424no—
239C4⋊C4—2841no—
240C4⋊C4—2841no—
241C4⋊C4—2842no—
242C4⋊C4—2842no—
243C4⋊C4—2842no—
244C4⋊C4—1422no—
245C22⋊C4—2842no—
246C22⋊C4—2842no—
247C22⋊C4—2842no—
248C22⋊C4—2842no—
249C22⋊C4—2841no—
250C22⋊C4—2842no—
251C22⋊C4—2841no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—22no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42—41no—
260C23—2841no—
261C23—2841no—
262C23—2841no—
263C23—2841no—
264C23—2841no—
265C23—2841no—
266C23—2841no—
267C23C2×(C2×D71)⋊C411yescharacteristic
268D4—5682no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—2841no—
287C2×C4—2841no—
288C2×C4—2841no—
289C2×C4—2841no—
290C2×C4—2841no—
291C2×C4—2841no—
292C2×C4—2841no—
293C2×C4—2841no—
294C2×C4—2841no—
295C2×C4#1455605412yes—
296C2×C4#1455613412yes—
297C2×C4#1455615411yescharacteristic
298C2×C4#1455615612yes—
299C2×C4#1455615911yescharacteristic
300C8—82no—
301C8—2844no—
302C22—5681no—
303C22—5681no—
304C22—5681no—
305C22—5681no—
306C22—5681no—
307C22—5681no—
308C22—5681no—
309C22#1798596511yescharacteristic
310C22#1798597211yescharacteristic
311C22#1798598111yescharacteristic
312C22#1798599811yescharacteristic
313C22#1798600511yescharacteristic
314C22C2×#1455605411yescharacteristic
315C22#1798601711yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—5681no—
325C4—5681no—
326C4—5681no—
327C2—11361no—
328C2#2258670711yescharacteristic
329C2#2258675111yescharacteristic
330C2#2258681011yescharacteristic
331C2#2258684911yescharacteristic
332C2#2258687611yescharacteristic
333C2#2258694811yescharacteristic
334C2#2258706011yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1831C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1831D71
hypocenterC71#7706
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
2. #22588089C2semidirect
3. #22586162C2semidirect
5. C71×#1831C2semidirect
169. #1831D71semidirect
173. C71#7706semidirect

Maximal subgroups

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

subgroupindex
#225967072
#225880892
#225861622
#225861602
C71×#18312
#770671