Finite Group Databasebeta

← #31908507

Subgroups — #31908507

Order 18176.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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#22593339C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic, complemented
3#22586344C214yesmaximal, complemented
4C71×#1461C211yesmaximal, characteristic, complemented
5C2×#14556125C411yescharacteristic
6C2×#14556117—21no—
7#17986023—21no—
8C2×#14556047—21no—
9#17985916C412yes—
10#17985916C2212yes—
11#17985901—21no—
12#17985901—21no—
13#17985900—21no—
14C4×C4⋊Dic71C411yescharacteristic
15#17985768C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#431×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C22×D284—21no—
21C22×D284—21no—
22C22×D284C2×C412yes—
23C2×#11765849—22no—
24#14556125—41no—
25#14556125—41no—
26#14556117—42no—
27#14556117—42no—
28C2×C4⋊Dic71—21no—
29C2×(C2×D71)⋊C4—22no—
30C2×(C2×D71)⋊C4—22no—
31C2×(C2×D71)⋊C4—22no—
32C2×(C2×D71)⋊C4—22no—
33C2×C4×Dic71—21no—
34C4⋊(C71⋊C8)—41no—
35C4⋊(C71⋊C8)—41no—
36#14556047—42no—
37C71⋊#95C2×C412yes—
38C22×Q8×C71D411yescharacteristic
39C4×Q8×C71—41no—
40C4×Q8×C71—41no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C4⋊C8×C71—41no—
48C22⋊C8×C71—22no—
49C4×C8×C71—41no—
50C23×D71—22no—
51C2×D284—42no—
52C2×D284—42no—
53C2×D284—42no—
54C2×D284—42no—
55C2×D284—42no—
56C2×D284—41no—
57C2×D284—41no—
58C2×D284—42no—
59C2×D284—82no—
60C22×Dic71—22no—
61#11765849—42no—
62#11765849—42no—
63C2×C71⋊C8—42no—
64C4⋊Dic71—42no—
65(C2×D71)⋊C4—44no—
66(C2×D71)⋊C4—42no—
67(C2×D71)⋊C4—42no—
68(C2×D71)⋊C4—82no—
69(C2×D71)⋊C4—82no—
70C4×Dic71—42no—
71C2×Q8×C71—41no—
72C2×Q8×C71—41no—
73C2×Q8×C71D812yes—
74C2×Q8×C71SD1612yes—
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C2×C8×C71—42no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C42×C71—21no—
87C42×C71—21no—
88C42×C71—21no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—21no—
92C23×C71C2×C22⋊C411yescharacteristic
93C22×D71—42no—
94C22×D71—42no—
95C22×D71—42no—
96C22×D71—42no—
97C22×D71—42no—
98C22×D71—42no—
99C22×D71—42no—
100Q8×C71—42no—
101Q8×C71—42no—
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—22no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71#9911yescharacteristic
116C2×C4×C71#9911yescharacteristic
117C2×C4×C71#10212yes—
118C2×C4×C71#10212yes—
119C2×C4×C71C2×D811yescharacteristic
120C2×C4×C71C2×SD1611yescharacteristic
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124D284—82no—
125D284—84no—
126D284—84no—
127D284—82no—
128C8×C71—82no—
129C71⋊C8—82no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71C2×#9911yescharacteristic
133C22×C71C2×#10211yescharacteristic
134C22×C71#42111yescharacteristic
135C2×D71—82no—
136C2×D71—82no—
137C2×D71—82no—
138C2×D71—82no—
139C2×D71—82no—
140C2×D71—82no—
141C2×D71—82no—
142C4×C71—21no—
143C4×C71—21no—
144C4×C71—22no—
145C4×C71—22no—
146C4×C71—21no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71#35512yes—
151Dic71—82no—
152#11882—711nomaximal, Sylow
153C2×C71—21no—
154C2×C71—21no—
155C2×C71#151911yescharacteristic
156C2×C71#212211yescharacteristic
157C2×C71C2×#35511yescharacteristic
158D71—162no—
159#2548—711no—
160#1801—711no—
161#1461D7111yescharacteristic, complemented
162#1345—714no—
163C71#1188211yescharacteristic, complemented, Sylow
164C2×#127—711no—
165C2×#124—1421no—
166C2×C4×Q8C2×D7111yescharacteristic
167#431—21no—
168#423—1421no—
169C2×#101—1421no—
170#394—712no—
171#394—712no—
172#390—1421no—
173#390—1421no—
174#390—1421no—
175C4×C4⋊C4—711no—
176#343—712no—
177#324—21no—
178#324C2×D7112yes—
179C22×Q8D28411yescharacteristic
180C22×D4—712no—
181C22×D4—1421no—
182C22×D4—1421no—
183C2×M16—1422no—
184#127—2841no—
185#127—2841no—
186#124—2842no—
187#124—2842no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—1421no—
191C2×C4⋊C4C4×D7111yescharacteristic
192C2×C4⋊C4C22⋊D7111yescharacteristic
193C2×C4⋊C4C22×D7111yescharacteristic
194C2×C22⋊C4—1422no—
195C2×C22⋊C4—1422no—
196C2×C22⋊C4—1422no—
197C2×C22⋊C4—1422no—
198C2×C42—1421no—
199C2×C42C4×D7111yescharacteristic
200C2×C42D28411yescharacteristic
201C2×C42C22⋊D7111yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—2841no—
204C4⋊C8—2841no—
205#101—2842no—
206C22⋊C8—22no—
207C4×C8—41no—
208#95—712no—
209C24—1422no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8D56812yes—
213C2×Q8#1176585412yes—
214C2×D4—2842no—
215C2×D4—2842no—
216C2×D4—2842no—
217C2×D4—2842no—
218C2×D4—2842no—
219C2×D4—2841no—
220C2×D4—2841no—
221C2×D4—2842no—
222C2×D4—5682no—
223C22×C4—1422no—
224C22×C4(C2×D71)⋊C411yescharacteristic
225C22×C4(C2×D71)⋊C411yescharacteristic
226C22×C4(C2×D71)⋊C411yescharacteristic
227C22×C4(C2×D71)⋊C411yescharacteristic
228C22×C4C2×C4×D7111yescharacteristic
229C22×C4C2×C22⋊D7111yescharacteristic
230C22×C4C2×D28411yescharacteristic
231M16—2842no—
232M16—2842no—
233C2×C8—42no—
234C2×C8—2842no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—2842no—
239C22⋊C4—2844no—
240C22⋊C4—2842no—
241C22⋊C4—2842no—
242C22⋊C4—5682no—
243C22⋊C4—5682no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—2842no—
251C23—2842no—
252C23—2842no—
253C23—2842no—
254C23—2842no—
255C23—2842no—
256C23—2842no—
257C23—2842no—
258C23C2×(C2×D71)⋊C411yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—5684no—
262D4—5682no—
263D4—5684no—
264D4—5682no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—2842no—
281C2×C4#1455604211yescharacteristic
282C2×C4#1455604211yescharacteristic
283C2×C4#1455605012yes—
284C2×C4#1455605012yes—
285C2×C4C2×D56811yescharacteristic
286C2×C4C2×#1176585411yescharacteristic
287C8—82no—
288C8—5682no—
289C22—21no—
290C22—21no—
291C22—5682no—
292C22—5682no—
293C22—5682no—
294C22—5682no—
295C22—5682no—
296C22—5682no—
297C22—5682no—
298C22C2×#1455604211yescharacteristic
299C22C2×#1455605011yescharacteristic
300C22#1798601511yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—5682no—
310C4#1798579512yes—
311C2—21no—
312C2—21no—
313C2—11362no—
314C2#2258690911yescharacteristic
315C2#2258998911yescharacteristic
316C2C2×#1798579511yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1461C2
socleC22×C71C2×#99
prime radicalC2×#14556125C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1461D71
hypocenterC71#11882
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #22588336C2semidirect
3. #22586344C2semidirect
4. C71×#1461C2semidirect
161. #1461D71semidirect
163. C71#11882semidirect

Maximal subgroups

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

subgroupindex
#225933392
#225883362
#225863442
C71×#14612
#1188271