Finite Group Databasebeta

← #31886563

Subgroups — #31886563

Order 18176.

404 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 87 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#22589898C211yesmaximal, characteristic, complemented
2#22589811C211yesmaximal, characteristic, complemented
3#22589743C211yesmaximal, characteristic
4#22586163C211yesmaximal, characteristic, complemented
5#22586163C211yesmaximal, characteristic, complemented
6#22586161C211yesmaximal, characteristic
7C71×#2727C211yesmaximal, characteristic
8C2×#14556061—21no—
9C2×#14556049—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18C2×#126×C71—21no—
19#399×C71—21no—
20#396×C71—21no—
21#394×C71—21no—
22#390×C71C2211yescharacteristic
23#389×C71—21no—
24#389×C71C2211yescharacteristic
25#389×C71C2211yescharacteristic
26#388×C71—21no—
27C22×C71⋊C8—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33#14556061—42no—
34#14556049—42no—
35C22×D4×C71D411yescharacteristic
36#126×C71—42no—
37#126×C71—42no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C42×C71—21no—
54C2×C42×C71C2311yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58C22×Dic71—41no—
59C2×C71⋊C8—22no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C4⋊Dic71—42no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C24×C71—21no—
74C2×D4×C71—21no—
75C2×D4×C71—21no—
76C2×D4×C71—41no—
77C2×D4×C71—41no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C4⋊C4×C71—21no—
87C4⋊C4×C71—21no—
88C4⋊C4×C71—21no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—81no—
98C4⋊C4×C71D812yes—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—81no—
106C22⋊C4×C71—81no—
107C42×C71—41no—
108C42×C71—42no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71#12011yescharacteristic
117D4×C71—42no—
118D4×C71—81no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—22no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71C2×D811yescharacteristic
141C2×C4×C71#13511yescharacteristic
142C2×C4×C71#13511yescharacteristic
143C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C71#13711yescharacteristic
146C2×Dic71—81no—
147C2×Dic71—81no—
148C2×Dic71—81no—
149C71⋊C8—42no—
150C71⋊C8—81no—
151C71⋊C8—81no—
152C22×C71—81no—
153C22×C71—81no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71—81no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71#44711yescharacteristic
160C22×C71#44911yescharacteristic
161C22×C71#45211yescharacteristic
162C22×C71#45411yescharacteristic
163C22×C71#45511yescharacteristic
164C22×C71#45611yescharacteristic
165C22×C71#45811yescharacteristic
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—42no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71—81no—
176C4×C71—81no—
177Dic71—161no—
178#7799—711nomaximal, Sylow
179C2×C71—161no—
180C2×C71#161011yescharacteristic
181C2×C71#161211yescharacteristic
182C2×C71#161511yescharacteristic
183C2×C71#162511yescharacteristic
184C2×C71#164111yescharacteristic
185C2×C71#165011yescharacteristic
186C2×C71#166411yescharacteristic
187#2727D7111yescharacteristic
188#2099—711no—
189#2083—711no—
190#2071—711no—
191#1271—711no—
192#1271—711no—
193#1270—711no—
194C71#779911yescharacteristic, complemented, Sylow
195C2×#126—21no—
196C2×#107—1421no—
197C2×#102—1421no—
198#399—21no—
199#396—21no—
200#394—21no—
201#390C2×D7111yescharacteristic
202#389—21no—
203#389C2×D7111yescharacteristic
204#389C2×D7111yescharacteristic
205#388—21no—
206#384—711no—
207#340—1421no—
208#340—1421no—
209#340—1421no—
210#340—1421no—
211#336—711no—
212#336—711no—
213#336—711no—
214C22×D4C22⋊D7111yescharacteristic
215C22×C8—1421no—
216C22×C8—1421no—
217C22×C8—1421no—
218#126—42no—
219#126—42no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—1421no—
225C2×C4⋊C4—1421no—
226C2×C4⋊C4—1421no—
227C2×C4⋊C4C22⋊D7111yescharacteristic
228C2×C4⋊C4C22⋊D7111yescharacteristic
229C2×C4⋊C4C22⋊D7111yescharacteristic
230C2×C4⋊C4C22⋊D7111yescharacteristic
231C2×C4⋊C4C22⋊D7111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42C22×D7111yescharacteristic
240#107—2842no—
241#102—2842no—
242#95—21no—
243#95—21no—
244#95—21no—
245C24—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2841no—
256C22×C4C2×C22⋊D7111yescharacteristic
257C22×C4C2×C22⋊D7111yescharacteristic
258C22×C4C2×C22⋊D7111yescharacteristic
259C2×C8—2841no—
260C2×C8—2841no—
261C2×C8—2841no—
262C2×C8—2841no—
263C2×C8—2841no—
264C2×C8—2841no—
265C2×C8—2841no—
266C2×C8—2841no—
267C2×C8—1422no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—2842no—
281C4⋊C4—2841no—
282C4⋊C4—2841no—
283C4⋊C4—2841no—
284C4⋊C4—2841no—
285C4⋊C4#1176585012yes—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C42—41no—
295C42—42no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#1455609711yescharacteristic
304D4—42no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—5681no—
328C2×C4—5681no—
329C2×C4—5681no—
330C2×C4C2×#1176585011yescharacteristic
331C2×C4#1455614811yescharacteristic
332C2×C4#1455614811yescharacteristic
333C2×C4#1455615311yescharacteristic
334C2×C4#1455615411yescharacteristic
335C2×C4#1455615811yescharacteristic
336C8—2842no—
337C8—5681no—
338C8—5681no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1798613311yescharacteristic
347C22#1798614411yescharacteristic
348C22#1798616111yescharacteristic
349C22#1798617211yescharacteristic
350C22#1798617711yescharacteristic
351C22#1798618211yescharacteristic
352C22#1798618911yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—11361no—
365C2—161no—
366C2#2258734311yescharacteristic
367C2#2258735411yescharacteristic
368C2#2258737111yescharacteristic
369C2#2258742911yescharacteristic
370C2#2258753311yescharacteristic
371C2#2258759611yescharacteristic
372C2#2258767411yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2727C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2727D71
hypocenterC71#7799
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #22589898C2semidirect
2. #22589811C2semidirect
4. #22586163C2semidirect
5. #22586163C2semidirect
194. C71#7799semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225898112
#225897432
#225861632
#225861612
C71×#27272
#779971