Finite Group Databasebeta

← #31893197

Subgroups — #31893197

Order 18176.

756 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 139 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#22596671C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic, complemented
3C2×#17985732C212yesmaximal
4#22586219C212yesmaximal, complemented
5C2×#376×C71C211yesmaximal, characteristic, complemented
6C2×#14556111—21no—
7C2×#14556102C2211yescharacteristic
8C2×C22⋊(C71⋊C8)C2212yes—
9#17985901C412yescomplemented
10#17985897C412yescomplemented
11#17985890—22no—
12#17985878—21no—
13#17985859C2211yescharacteristic
14#17985754—22no—
15#17985732—24no—
16C22×C42×C71—21no—
17C22×C42×C71C2211yescharacteristic
18#376×C71—24no—
19C2×#95×C71—21no—
20C2×#95×C71C2212yes—
21C22×D284C2×C411yescharacteristic
22C22×C22⋊D71—21no—
23C22×C71⋊C8—22no—
24#14556111—44no—
25#14556102—22no—
26#14556102—22no—
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C412yes—
29C2×Dic71⋊C4—22no—
30C2×C22⋊Dic71—21no—
31C2×(C2×D71)⋊C4—22no—
32C2×(C2×D71)⋊C4—22no—
33C2×(C2×D71)⋊C4—21no—
34C22⋊(C71⋊C8)—22no—
35C22⋊(C71⋊C8)—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C23×C4×C71—22no—
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C23×C4×C71C2311yescharacteristic
42C2×C42×C71—28no—
43C2×C42×C71—24no—
44C2×C42×C71—22no—
45C2×C42×C71—22no—
46C2×C42×C71C2×C412yes—
47C2×C42×C71D412yes—
48#95×C71—24no—
49#95×C71—22no—
50#95×C71D414yes—
51C23×D71—21no—
52C2×D284—22no—
53C2×D284—42no—
54C2×C22⋊D71—41no—
55C2×C22⋊D71—41no—
56C2×C22⋊D71—42no—
57C2×C22⋊D71—42no—
58C22×Dic71—21no—
59C22×Dic71—22no—
60C2×C71⋊C8—44no—
61C2×C71⋊C8—42no—
62C4⋊Dic71—22no—
63C4⋊Dic71—42no—
64Dic71⋊C4—44no—
65C22⋊Dic71—42no—
66(C2×D71)⋊C4—44no—
67(C2×D71)⋊C4—44no—
68(C2×D71)⋊C4—42no—
69C24×C71C2×D411yescharacteristic
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—24no—
73C22×C4×C71—28no—
74C22×C4×C71—21no—
75C22×C4×C71—24no—
76C22×C4×C71—21no—
77C22×C4×C71—22no—
78C22×C4×C71—22no—
79C22×C4×C71—42no—
80C22×C4×C71—44no—
81C22×C4×C71—42no—
82C22×C4×C71—42no—
83C22×C4×C71C22⋊C412yes—
84C22×C4×C71C22⋊C412yes—
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D412yes—
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71#4112yes—
89C42×C71—216no—
90C42×C71—24no—
91C42×C71—24no—
92C23×C71—22no—
93C23×C71—22no—
94C23×C71C2×C22⋊C411yescharacteristic
95C23×C71#12011yescharacteristic
96C23×C71#12311yescharacteristic
97C23×C71#13512yes—
98C23×C71AGL(1,ℤ/8)12yes—
99C22×D71—41no—
100C22×D71—42no—
101C22×D71—41no—
102C22×D71—41no—
103C22×D71—42no—
104C2×C4×C71—24no—
105C2×C4×C71—24no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—28no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—44no—
117C2×C4×C71—48no—
118C2×C4×C71—48no—
119C2×C4×C71—42no—
120C2×C4×C71—44no—
121C2×C4×C71—42no—
122C2×C4×C71#10414yes—
123C2×C4×C71C4×D412yes—
124C2×C4×C71#12112yes—
125C22⋊D71—82no—
126C22⋊D71—82no—
127C2×Dic71—41no—
128C2×Dic71—42no—
129C2×Dic71—41no—
130C2×Dic71—42no—
131C2×Dic71—41no—
132C2×Dic71—42no—
133D284—82no—
134C71⋊C8—82no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—24no—
140C22×C71—24no—
141C22×C71#38611yescharacteristic
142C22×C71#41012yes—
143C22×C71#41611yescharacteristic
144C22×C71#41811yescharacteristic
145C22×C71C2×#10412yes—
146C22×C71#44912yes—
147C22×C71#48212yes—
148C2×D71—81no—
149C2×D71—81no—
150C2×D71—81no—
151C2×D71—82no—
152C2×D71—82no—
153C4×C71—24no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—48no—
157C4×C71—42no—
158Dic71—81no—
159Dic71—82no—
160#9149—711nomaximal, Sylow
161C2×C71—22no—
162C2×C71—22no—
163C2×C71#150012yes—
164C2×C71#150512yes—
165C2×C71#187511yescharacteristic
166C2×C71#189611yescharacteristic
167C2×C71#189911yescharacteristic
168D71—161no—
169#2854—711no—
170C2×#376D7111yescharacteristic, complemented
171#1817—711no—
172C2×#327—712no—
173#1294—712no—
174C71#914911yescharacteristic, complemented, Sylow
175C2×#123—1421no—
176C2×#121—711no—
177C22×C42—21no—
178C22×C42C2×D7111yescharacteristic
179C2×C22⋊C8—712no—
180#390—712no—
181#389—712no—
182#388—1422no—
183#386—1421no—
184#380—711no—
185#376—24no—
186C2×#95—21no—
187C2×#95C2×D7112yes—
188#336—1422no—
189#327—1424no—
190C22×D4—711no—
191C22×D4—1421no—
192C23×C4—22no—
193C23×C4C22⋊D7111yescharacteristic
194C23×C4C22⋊D7111yescharacteristic
195C23×C4C22×D7111yescharacteristic
196C22×C8—1422no—
197#123—2844no—
198#121—1422no—
199#121—1422no—
200C2×C4⋊C4—712no—
201C2×C4⋊C4—711no—
202C2×C4⋊C4—1422no—
203C2×C22⋊C4—1421no—
204C2×C22⋊C4—1422no—
205C2×C22⋊C4—1422no—
206C2×C22⋊C4—1421no—
207C2×C42—24no—
208C2×C42—28no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C4×D7112yes—
212C2×C42D28412yes—
213C22⋊C8—1422no—
214C22⋊C8—1422no—
215#95—24no—
216#95—22no—
217#95—1422no—
218#95—1421no—
219#95C22⋊D7114yes—
220C24—1421no—
221C24C2×C22⋊D7111yescharacteristic
222C2×D4—2842no—
223C2×D4—2841no—
224C2×D4—2841no—
225C2×D4—2842no—
226C2×D4—2842no—
227C2×D4—1422no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—1422no—
242C22×C4—1421no—
243C22×C4(C2×D71)⋊C412yes—
244C22×C4(C2×D71)⋊C412yes—
245C22×C4C2×C4×D7111yescharacteristic
246C22×C4C2×C22⋊D7112yes—
247C22×C4C2×D28411yescharacteristic
248C22×C4#1176586512yes—
249C2×C8—2844no—
250C2×C8—2842no—
251C4⋊C4—2842no—
252C4⋊C4—2844no—
253C4⋊C4—1422no—
254C22⋊C4—2844no—
255C22⋊C4—2842no—
256C22⋊C4—2844no—
257C22⋊C4—2842no—
258C42—216no—
259C42—24no—
260C42—24no—
261C23—22no—
262C23—22no—
263C23—2841no—
264C23—2842no—
265C23—2841no—
266C23—2841no—
267C23—2842no—
268C23C2×(C2×D71)⋊C411yescharacteristic
269C23#1455609711yescharacteristic
270C23#1455611111yescharacteristic
271C23#1455614812yes—
272C23#1455615312yes—
273D4—5682no—
274D4—5682no—
275D4—5682no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—2841no—
295C2×C4—2842no—
296C2×C4—2841no—
297C2×C4—2842no—
298C2×C4—2841no—
299C2×C4—2842no—
300C2×C4#1455605614yes—
301C2×C4C4×C22⋊D7112yes—
302C2×C4#1455610212yes—
303C8—5682no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—5681no—
311C22—5681no—
312C22—5682no—
313C22—5682no—
314C22—5681no—
315C22#1798587811yescharacteristic
316C22#1798597212yes—
317C22#1798599311yescharacteristic
318C22#1798600311yescharacteristic
319C22C2×#1455605612yes—
320C22#1798614412yes—
321C22#1798632312yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—42no—
327C4—5682no—
328C4—5681no—
329C2—22no—
330C2—22no—
331C2—11361no—
332C2#2258682412yes—
333C2#2258684712yes—
334C2#2258866211yescharacteristic
335C2#2258878311yescharacteristic
336C2#2258879811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#376×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#14556102C22
cosocleC23×C4×C71C23
hypercenterC2×#376D71
hypocenterC71#9149
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
2. #22588408C2semidirect
4. #22586219C2semidirect
5. C2×#376×C71C2semidirect
9. #17985901C4semidirect
10. #17985897C4semidirect
170. C2×#376D71semidirect
174. C71#9149semidirect

Maximal subgroups

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

subgroupindex
#225966712
#225884082
C2×#179857322
#225862192
C2×#376×C712
#914971