Finite Group Databasebeta

← #31893232

Subgroups — #31893232

Order 18176.

692 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 147 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×#382×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×C4⋊C4×C71—21no—
17C22×C42×C71C2211yescharacteristic
18#382×C71—24no—
19C2×#95×C71—21no—
20C2×#95×C71C2212yes—
21C22×D284C2×C411yescharacteristic
22C22×C22⋊D71—21no—
23C22×C71⋊C8—22no—
24#14556111—42no—
25#14556111—42no—
26#14556102—22no—
27#14556102—22no—
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×C4⋊Dic71C2×C412yes—
30C2×Dic71⋊C4—22no—
31C2×C22⋊Dic71—21no—
32C2×(C2×D71)⋊C4—22no—
33C2×(C2×D71)⋊C4—22no—
34C2×(C2×D71)⋊C4—21no—
35C22⋊(C71⋊C8)—22no—
36C22⋊(C71⋊C8)—22no—
37C71⋊#95—22no—
38C71⋊#95—21no—
39C23×C4×C71—22no—
40C23×C4×C71D411yescharacteristic
41C23×C4×C71D411yescharacteristic
42C23×C4×C71C2311yescharacteristic
43C2×C4⋊C4×C71—24no—
44C2×C4⋊C4×C71—42no—
45C2×C4⋊C4×C71—42no—
46C2×C42×C71—22no—
47C2×C42×C71—22no—
48C2×C42×C71C2×C412yes—
49C2×C42×C71D412yes—
50#95×C71—24no—
51#95×C71—22no—
52#95×C71D414yes—
53C23×D71—21no—
54C2×D284—22no—
55C2×D284—42no—
56C2×C22⋊D71—41no—
57C2×C22⋊D71—41no—
58C2×C22⋊D71—41no—
59C2×C22⋊D71—41no—
60C2×C22⋊D71—41no—
61C2×C22⋊D71—41no—
62C22×Dic71—21no—
63C22×Dic71—22no—
64C2×C71⋊C8—44no—
65C2×C71⋊C8—42no—
66C4⋊Dic71—22no—
67C4⋊Dic71—42no—
68Dic71⋊C4—44no—
69C22⋊Dic71—41no—
70C22⋊Dic71—41no—
71(C2×D71)⋊C4—44no—
72(C2×D71)⋊C4—44no—
73(C2×D71)⋊C4—42no—
74C24×C71C2×D411yescharacteristic
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—24no—
82C22×C4×C71—42no—
83C22×C4×C71—42no—
84C22×C4×C71—44no—
85C22×C4×C71—44no—
86C22×C4×C71—42no—
87C22×C4×C71—42no—
88C22×C4×C71C22⋊C412yes—
89C22×C4×C71C22⋊C412yes—
90C22×C4×C71D812yes—
91C22×C4×C71SD1612yes—
92C22×C4×C71C22×C411yescharacteristic
93C22×C4×C71C2×D412yes—
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71#4112yes—
96C4⋊C4×C71—48no—
97C42×C71—24no—
98C42×C71—24no—
99C23×C71—22no—
100C23×C71—22no—
101C23×C71C2×C22⋊C411yescharacteristic
102C23×C71#12011yescharacteristic
103C23×C71#12311yescharacteristic
104C23×C71C2×D811yescharacteristic
105C23×C71C2×SD1611yescharacteristic
106C23×C71AGL(1,ℤ/8)11yescharacteristic
107C23×C71#13711yescharacteristic
108C22×D71—41no—
109C22×D71—42no—
110C22×D71—41no—
111C22×D71—41no—
112C22×D71—42no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—22no—
123C2×C4×C71—22no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—44no—
129C2×C4×C71—44no—
130C2×C4×C71—44no—
131C2×C4×C71—84no—
132C2×C4×C71—84no—
133C2×C4×C71#10214yes—
134C2×C4×C71C4×D412yes—
135C2×C4×C71#12112yes—
136C22⋊D71—81no—
137C22⋊D71—81no—
138C22⋊D71—81no—
139C22⋊D71—81no—
140C2×Dic71—41no—
141C2×Dic71—42no—
142C2×Dic71—42no—
143C2×Dic71—41no—
144C2×Dic71—41no—
145C2×Dic71—42no—
146D284—82no—
147C71⋊C8—82no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71#38611yescharacteristic
157C22×C71#41012yes—
158C22×C71C2×#10211yescharacteristic
159C22×C71#41711yescharacteristic
160C22×C71#42112yes—
161C22×C71#44711yescharacteristic
162C22×C71#44811yescharacteristic
163C22×C71#48011yescharacteristic
164C22×C71#48111yescharacteristic
165C2×D71—81no—
166C2×D71—81no—
167C2×D71—81no—
168C2×D71—82no—
169C2×D71—82no—
170C4×C71—24no—
171C4×C71—22no—
172C4×C71—22no—
173C4×C71—42no—
174C4×C71—84no—
175Dic71—81no—
176Dic71—82no—
177#9156—711nomaximal, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—21no—
181C2×C71—21no—
182C2×C71#151212yes—
183C2×C71#151712yes—
184C2×C71#187411yescharacteristic
185C2×C71#189711yescharacteristic
186C2×C71#189911yescharacteristic
187D71—161no—
188#2854—711no—
189C2×#382D7111yescharacteristic, complemented
190#1817—711no—
191C2×#327—712no—
192#1294—712no—
193C71#915611yescharacteristic, complemented, Sylow
194C2×#123—1421no—
195C2×#121—711no—
196C22×C4⋊C4—21no—
197C22×C42C2×D7111yescharacteristic
198C2×C22⋊C8—712no—
199#390—712no—
200#389—712no—
201#388—1422no—
202#386—1421no—
203#382—24no—
204#380—711no—
205C2×#95—21no—
206C2×#95C2×D7112yes—
207#336—1422no—
208#327—1424no—
209C22×D4—711no—
210C22×D4—1421no—
211C23×C4—22no—
212C23×C4C22⋊D7111yescharacteristic
213C23×C4C22⋊D7111yescharacteristic
214C23×C4C22×D7111yescharacteristic
215C22×C8—1422no—
216#123—2842no—
217#123—2842no—
218#121—1422no—
219#121—1422no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—712no—
224C2×C4⋊C4—711no—
225C2×C4⋊C4—1422no—
226C2×C22⋊C4—1421no—
227C2×C22⋊C4—1422no—
228C2×C22⋊C4—1422no—
229C2×C22⋊C4—1421no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D7112yes—
233C2×C42D28412yes—
234C22⋊C8—1422no—
235C22⋊C8—1422no—
236#95—24no—
237#95—22no—
238#95—1422no—
239#95—1421no—
240#95C22⋊D7114yes—
241C24—1421no—
242C24C2×C22⋊D7111yescharacteristic
243C2×D4—2842no—
244C2×D4—2841no—
245C2×D4—2841no—
246C2×D4—2841no—
247C2×D4—2841no—
248C2×D4—2841no—
249C2×D4—2841no—
250C2×D4—1422no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—1422no—
265C22×C4—1421no—
266C22×C4(C2×D71)⋊C412yes—
267C22×C4(C2×D71)⋊C412yes—
268C22×C4#1176585012yes—
269C22×C4#1176585212yes—
270C22×C4C2×C4×D7111yescharacteristic
271C22×C4C2×C22⋊D7112yes—
272C22×C4C2×D28411yescharacteristic
273C22×C4#1176586512yes—
274C2×C8—2844no—
275C2×C8—2842no—
276C4⋊C4—48no—
277C4⋊C4—2842no—
278C4⋊C4—2844no—
279C4⋊C4—1422no—
280C22⋊C4—2844no—
281C22⋊C4—2841no—
282C22⋊C4—2841no—
283C22⋊C4—2844no—
284C22⋊C4—2842no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—2841no—
290C23—2842no—
291C23—2841no—
292C23—2841no—
293C23—2842no—
294C23C2×(C2×D71)⋊C411yescharacteristic
295C23#1455609711yescharacteristic
296C23#1455611111yescharacteristic
297C23C2×#1176585011yescharacteristic
298C23C2×#1176585211yescharacteristic
299C23#1455615411yescharacteristic
300C23#1455615911yescharacteristic
301D4—5682no—
302D4—5681no—
303D4—5681no—
304D4—5681no—
305D4—5681no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—2841no—
327C2×C4—2842no—
328C2×C4—2842no—
329C2×C4—2841no—
330C2×C4—2841no—
331C2×C4—2842no—
332C2×C4#1455604814yes—
333C2×C4C4×C22⋊D7112yes—
334C2×C4#1455610212yes—
335C8—5682no—
336C22—22no—
337C22—22no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—5682no—
345C22—5681no—
346C22—5681no—
347C22—5681no—
348C22—5682no—
349C22#1798587811yescharacteristic
350C22#1798597212yes—
351C22C2×#1455604811yescharacteristic
352C22#1798599811yescharacteristic
353C22#1798601712yes—
354C22#1798613311yescharacteristic
355C22#1798613811yescharacteristic
356C22#1798631211yescharacteristic
357C22#1798631711yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—84no—
363C4—5682no—
364C4—5681no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—11361no—
370C2#2258687612yes—
371C2#2258689912yes—
372C2#2258865711yescharacteristic
373C2#2258878811yescharacteristic
374C2#2258879811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#382×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#14556102C22
cosocleC23×C4×C71C23
hypercenterC2×#382D71
hypocenterC71#9156
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. #22588408C2semidirect
4. #22586219C2semidirect
5. C2×#382×C71C2semidirect
9. #17985901C4semidirect
10. #17985897C4semidirect
189. C2×#382D71semidirect
193. C71#9156semidirect

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×#382×C712
#915671