Finite Group Databasebeta

← #31886715

Subgroups — #31886715

Order 18176.

446 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 91 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#22589903C211yesmaximal, characteristic, complemented
2#22589878C211yesmaximal, characteristic, complemented
3#22589873C211yesmaximal, characteristic
4#22586166C211yesmaximal, characteristic, complemented
5#22586163C211yesmaximal, characteristic, complemented
6#22586161C211yesmaximal, characteristic
7C71×#2732C211yesmaximal, characteristic
8C2×#14556061—21no—
9C2×#14556059—21no—
10C2×#14556052—21no—
11C2×#14556049—21no—
12#17985869C2211yescharacteristic
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18C2×C22⋊Q8×C71—21no—
19#399×C71—21no—
20#398×C71—21no—
21#396×C71—21no—
22#391×C71C2211yescharacteristic
23#390×C71C2211yescharacteristic
24#389×C71C2211yescharacteristic
25#386×C71—21no—
26#385×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#14556059—42no—
35#14556052—42no—
36#14556049—42no—
37C22×Q8×C71D411yescharacteristic
38C22×D4×C71D411yescharacteristic
39C23×C4×C71—21no—
40C22⋊Q8×C71—42no—
41C22⋊Q8×C71—42no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C42×C71C2311yescharacteristic
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60C22×Dic71—41no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—22no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8—41no—
69C2×C71⋊C8—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C4⋊Dic71—41no—
74C4⋊Dic71—42no—
75C24×C71—21no—
76C2×Q8×C71—21no—
77C2×Q8×C71—41no—
78C2×Q8×C71—41no—
79C2×Q8×C71SD1612yes—
80C2×D4×C71—21no—
81C2×D4×C71—41no—
82C2×D4×C71—41no—
83C2×D4×C71Q1612yes—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—42no—
90C22×C4×C71—81no—
91C22×C4×C71—81no—
92C22×C4×C71C2×D411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—42no—
101C4⋊C4×C71—42no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
104C4⋊C4×C71—81no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—41no—
107C22⋊C4×C71—41no—
108C22⋊C4×C71—41no—
109C22⋊C4×C71—41no—
110C22⋊C4×C71—81no—
111C22⋊C4×C71—81no—
112C42×C71—41no—
113C23×C71—22no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71#12011yescharacteristic
121Q8×C71—42no—
122Q8×C71—81no—
123D4×C71—42no—
124D4×C71—81no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—22no—
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×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—84no—
147C2×C4×C71C2×SD1611yescharacteristic
148C2×C4×C71C2×Q1611yescharacteristic
149C2×C4×C71#13511yescharacteristic
150C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C71#13711yescharacteristic
152C2×C4×C71#13711yescharacteristic
153C2×Dic71—81no—
154C2×Dic71—81no—
155C2×Dic71—81no—
156C71⋊C8—42no—
157C71⋊C8—42no—
158C71⋊C8—81no—
159C22×C71—42no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71—81no—
166C22×C71#45011yescharacteristic
167C22×C71#45111yescharacteristic
168C22×C71#45211yescharacteristic
169C22×C71#45311yescharacteristic
170C22×C71#45311yescharacteristic
171C22×C71#45411yescharacteristic
172C22×C71#45811yescharacteristic
173C4×C71—21no—
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—42no—
180C4×C71—81no—
181C4×C71—81no—
182C4×C71—81no—
183C4×C71—81no—
184Dic71—161no—
185#7823—711nomaximal, Sylow
186C2×C71—82no—
187C2×C71#161311yescharacteristic
188C2×C71#161411yescharacteristic
189C2×C71#161511yescharacteristic
190C2×C71#163111yescharacteristic
191C2×C71#163211yescharacteristic
192C2×C71#163711yescharacteristic
193C2×C71#165611yescharacteristic
194#2732D7111yescharacteristic
195#2100—711no—
196#2095—711no—
197#2094—711no—
198#1272—711no—
199#1271—711no—
200#1270—711no—
201C71#782311yescharacteristic, complemented, Sylow
202C2×C22⋊Q8—21no—
203C2×#107—1421no—
204C2×#106—1421no—
205C2×#103—1421no—
206C2×#102—1421no—
207#399—21no—
208#398—21no—
209#396—21no—
210#391C2×D7111yescharacteristic
211#390C2×D7111yescharacteristic
212#389C2×D7111yescharacteristic
213#386—21no—
214#385—21no—
215#384—711no—
216#340—1421no—
217#340—1421no—
218#336—711no—
219#336—711no—
220#336—711no—
221C22×Q8C22⋊D7111yescharacteristic
222C22×D4C22⋊D7111yescharacteristic
223C23×C4—21no—
224C22×C8—1421no—
225C22×C8—1421no—
226C22×C8—1421no—
227C22⋊Q8—42no—
228C22⋊Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1421no—
234C2×C4⋊C4—1421no—
235C2×C4⋊C4—1421no—
236C2×C4⋊C4C22⋊D7111yescharacteristic
237C2×C4⋊C4C22⋊D7111yescharacteristic
238C2×C4⋊C4C22⋊D7111yescharacteristic
239C2×C4⋊C4C22⋊D7111yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42C22×D7111yescharacteristic
246#107—2842no—
247#106—2842no—
248#103—2842no—
249#102—2842no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254C24—21no—
255C2×Q8—21no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8#1176585312yes—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4#1176585512yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—2841no—
272C22×C4C2×C22⋊D7111yescharacteristic
273C22×C4C2×C22⋊D7111yescharacteristic
274C22×C4C2×C22⋊D7111yescharacteristic
275C2×C8—2841no—
276C2×C8—2841no—
277C2×C8—2841no—
278C2×C8—2841no—
279C2×C8—2841no—
280C2×C8—2841no—
281C2×C8—2841no—
282C2×C8—1422no—
283C2×C8—1422no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—81no—
294C4⋊C4—2841no—
295C4⋊C4—2841no—
296C4⋊C4—2841no—
297C4⋊C4—2841no—
298C4⋊C4—2842no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C42—41no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23#1455609711yescharacteristic
315Q8—42no—
316Q8—81no—
317D4—42no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—84no—
341C2×C4—5681no—
342C2×C4—5681no—
343C2×C4—5681no—
344C2×C4C2×#1176585311yescharacteristic
345C2×C4C2×#1176585511yescharacteristic
346C2×C4#1455614811yescharacteristic
347C2×C4#1455615411yescharacteristic
348C2×C4#1455615811yescharacteristic
349C2×C4#1455615911yescharacteristic
350C8—2842no—
351C8—2842no—
352C8—5681no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1798615011yescharacteristic
361C22#1798615511yescharacteristic
362C22#1798616111yescharacteristic
363C22#1798616711yescharacteristic
364C22#1798616711yescharacteristic
365C22#1798617211yescharacteristic
366C22#1798618911yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—42no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—11361no—
379C2—82no—
380C2#2258736011yescharacteristic
381C2#2258736511yescharacteristic
382C2#2258737111yescharacteristic
383C2#2258746711yescharacteristic
384C2#2258747411yescharacteristic
385C2#2258750511yescharacteristic
386C2#2258763311yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2732C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2732D71
hypocenterC71#7823
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
1. #22589903C2semidirect
2. #22589878C2semidirect
4. #22586166C2semidirect
5. #22586163C2semidirect
201. C71#7823semidirect

Maximal subgroups

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

subgroupindex
#225899032
#225898782
#225898732
#225861662
#225861632
#225861612
C71×#27322
#782371