Finite Group Databasebeta

← #31890791

Subgroups — #31890791

Order 18176.

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

normal subgroups only

This group has 830 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
1C2×#17985859C211yesmaximal, characteristic
2#22588250C211yesmaximal, characteristic
3#22586182C214yesmaximal, complemented
4C71×#1779C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6C22×C4⋊Dic71C411yescharacteristic
7C22×C22⋊Dic71C412yes—
8C2×C22⋊(C71⋊C8)—21no—
9C2×C22⋊(C71⋊C8)—21no—
10C2×C22⋊(C71⋊C8)C2212yes—
11#17985859—21no—
12#17985859—21no—
13#17985859—21no—
14#17985859—21no—
15C2×C71⋊#95C2212yes—
16C24×C4×C71C2211yescharacteristic
17C2×C22⋊C8×C71—21no—
18C2×C22⋊C8×C71—21no—
19C2×C22⋊C8×C71C2212yes—
20C23×Dic71C2×C412yes—
21C23×Dic71C2×C412yes—
22C22×C71⋊C8—22no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—42no—
28C2×C4⋊Dic71—42no—
29C2×C22⋊Dic71—22no—
30C2×C22⋊Dic71—22no—
31C2×C22⋊Dic71—44no—
32C22⋊(C71⋊C8)—24no—
33C22⋊(C71⋊C8)—42no—
34C22⋊(C71⋊C8)—42no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C25×C71C2×C411yescharacteristic
38C23×C4×C71—24no—
39C23×C4×C71—42no—
40C23×C4×C71—42no—
41C23×C4×C71C2×C411yescharacteristic
42C23×C4×C71D411yescharacteristic
43C23×C4×C71D411yescharacteristic
44C23×C4×C71D411yescharacteristic
45C23×C4×C71D411yescharacteristic
46C23×C4×C71C2311yescharacteristic
47C22×C8×C71—22no—
48C22⋊C8×C71—24no—
49C22⋊C8×C71—42no—
50C22⋊C8×C71—42no—
51C22×Dic71—22no—
52C22×Dic71—22no—
53C22×Dic71—24no—
54C22×Dic71—24no—
55C22×Dic71—42no—
56C22×Dic71—42no—
57C22×Dic71—42no—
58C22×Dic71—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C4⋊Dic71—42no—
63C4⋊Dic71—42no—
64C4⋊Dic71—42no—
65C4⋊Dic71—42no—
66C22⋊Dic71—44no—
67C22⋊Dic71—44no—
68C24×C71—22no—
69C24×C71—22no—
70C24×C71—22no—
71C24×C71—22no—
72C24×C71—22no—
73C24×C71—41no—
74C24×C71—41no—
75C24×C71C22⋊C411yescharacteristic
76C24×C71C22⋊C411yescharacteristic
77C24×C71C22×C411yescharacteristic
78C22×C4×C71—22no—
79C22×C4×C71—22no—
80C22×C4×C71—22no—
81C22×C4×C71—22no—
82C22×C4×C71—22no—
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—22no—
88C22×C4×C71—22no—
89C22×C4×C71—41no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71—41no—
93C22×C4×C71—41no—
94C22×C4×C71—41no—
95C22×C4×C71—416no—
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C2×D411yescharacteristic
99C22×C4×C71C2×D411yescharacteristic
100C22×C4×C71#4114yes—
101C2×C8×C71—42no—
102C2×C8×C71—42no—
103C2×C8×C71—42no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—28no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71—21no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71—44no—
122C23×C71—44no—
123C23×C71—44no—
124C23×C71—44no—
125C23×C71—44no—
126C23×C71—44no—
127C23×C71#10012yes—
128C23×C71#10112yes—
129C23×C71C2×C22⋊C411yescharacteristic
130C23×C71#11712yes—
131C23×C71#13112yes—
132C23×C71#13112yes—
133C2×C4×C71—28no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—44no—
143C2×C4×C71—44no—
144C2×C4×C71—44no—
145C2×C4×C71—44no—
146C2×C4×C71—44no—
147C2×C4×C71#9912yes—
148C2×C4×C71#9912yes—
149C2×C4×C71#12312yes—
150C2×C4×C71#12312yes—
151C2×Dic71—42no—
152C2×Dic71—42no—
153C2×Dic71—42no—
154C2×Dic71—42no—
155C2×Dic71—42no—
156C2×Dic71—42no—
157C2×Dic71—84no—
158C2×Dic71—84no—
159C8×C71—82no—
160C71⋊C8—82no—
161C22×C71—22no—
162C22×C71—22no—
163C22×C71—22no—
164C22×C71—22no—
165C22×C71—22no—
166C22×C71—22no—
167C22×C71—22no—
168C22×C71—22no—
169C22×C71—22no—
170C22×C71—22no—
171C22×C71—22no—
172C22×C71—22no—
173C22×C71—416no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71—41no—
177C22×C71—41no—
178C22×C71—41no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71#38111yescharacteristic
183C22×C71#40811yescharacteristic
184C22×C71#40811yescharacteristic
185C22×C71C2×#9911yescharacteristic
186C22×C71C2×#9911yescharacteristic
187C22×C71C2×#10011yescharacteristic
188C22×C71C2×#10111yescharacteristic
189C22×C71#43314yes—
190C4×C71—22no—
191C4×C71—22no—
192C4×C71—22no—
193C4×C71—22no—
194Dic71—82no—
195Dic71—82no—
196#8704—711nomaximal, Sylow
197C2×C71—24no—
198C2×C71—42no—
199C2×C71—42no—
200C2×C71#147111yescharacteristic
201C2×C71#147111yescharacteristic
202C2×C71#147211yescharacteristic
203C2×C71#147211yescharacteristic
204C2×C71#181811yescharacteristic
205C2×C71#181911yescharacteristic
206C2×C71#182111yescharacteristic
207C2×#380—711no—
208#1779—711no—
209#1779D7111yescharacteristic
210#1280—714no—
211C71#870411yescharacteristic, complemented, Sylow
212C24×C4C2×D7111yescharacteristic
213C22×C4⋊C4—711no—
214C22×C4⋊C4—711no—
215C22×C22⋊C4—712no—
216C2×C22⋊C8—21no—
217C2×C22⋊C8—21no—
218C2×C22⋊C8—712no—
219C2×C22⋊C8—1421no—
220C2×C22⋊C8—1421no—
221C2×C22⋊C8C2×D7112yes—
222#380—1421no—
223#380—1421no—
224#380—1421no—
225#380—1421no—
226C2×#95—712no—
227C25C4×D7111yescharacteristic
228C23×C4—24no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—712no—
232C23×C4—712no—
233C23×C4C4×D7111yescharacteristic
234C23×C4D28411yescharacteristic
235C23×C4D28411yescharacteristic
236C23×C4C22⋊D7111yescharacteristic
237C23×C4C22⋊D7111yescharacteristic
238C23×C4C22×D7111yescharacteristic
239C22×C8—22no—
240C22×C8—1422no—
241C2×C4⋊C4—2842no—
242C2×C4⋊C4—2842no—
243C2×C4⋊C4—1421no—
244C2×C4⋊C4—1421no—
245C2×C4⋊C4—1421no—
246C2×C4⋊C4—1421no—
247C2×C22⋊C4—2844no—
248C2×C22⋊C4—1422no—
249C2×C22⋊C4—1422no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—2842no—
254C22⋊C8—2842no—
255C22⋊C8—1424no—
256#95—1422no—
257#95—1422no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24(C2×D71)⋊C411yescharacteristic
266C24(C2×D71)⋊C411yescharacteristic
267C24C2×C4×D7111yescharacteristic
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—416no—
286C22×C4—2842no—
287C22×C4—2842no—
288C22×C4—2842no—
289C22×C4—2842no—
290C22×C4—1422no—
291C22×C4—1422no—
292C22×C4—1424no—
293C22×C4—1424no—
294C22×C4(C2×D71)⋊C411yescharacteristic
295C22×C4(C2×D71)⋊C411yescharacteristic
296C22×C4C2×C22⋊D7111yescharacteristic
297C22×C4C2×D28411yescharacteristic
298C22×C4#1176586414yes—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—2842no—
303C2×C8—2842no—
304C2×C8—2842no—
305C4⋊C4—2842no—
306C4⋊C4—2842no—
307C4⋊C4—2842no—
308C4⋊C4—2842no—
309C22⋊C4—2844no—
310C22⋊C4—2844no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—28no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23#1455604412yes—
335C23#1455604612yes—
336C23C2×(C2×D71)⋊C411yescharacteristic
337C23#1455608412yes—
338C23#1455613412yes—
339C23#1455613612yes—
340C2×C4—28no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—2842no—
355C2×C4—2842no—
356C2×C4—2842no—
357C2×C4—2842no—
358C2×C4—2842no—
359C2×C4—2842no—
360C2×C4—5684no—
361C2×C4—5684no—
362C2×C4#1455604212yes—
363C2×C4#1455604212yes—
364C2×C4#1455610912yes—
365C2×C4#1455611212yes—
366C8—82no—
367C8—5682no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—416no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22#1798586011yescharacteristic
390C22#1798596511yescharacteristic
391C22#1798596711yescharacteristic
392C22C2×#1455604211yescharacteristic
393C22C2×#1455604211yescharacteristic
394C22C2×#1455604411yescharacteristic
395C22C2×#1455604611yescharacteristic
396C22#1798605714yes—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—5682no—
402C4—5682no—
403C2—24no—
404C2—42no—
405C2—42no—
406C2#2258669911yescharacteristic
407C2#2258670111yescharacteristic
408C2#2258670311yescharacteristic
409C2#2258670511yescharacteristic
410C2#2258840911yescharacteristic
411C2#2258841211yescharacteristic
412C2#2258841811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1779C2
socleC23×C71C2×C22⋊C4
prime radicalC25×C71C2×C4
cosocleC23×C4×C71C23
hypercenter#1779D71
hypocenterC71#8704
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #22586182C2semidirect
211. C71#8704semidirect

Maximal subgroups

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

subgroupindex
C2×#179858592
#225882502
#225861822
C71×#17792
#870471