Finite Group Databasebeta

← #32026577

Subgroups — #32026577

Order 18176.

766 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596930C211yesmaximal, characteristic
2#22596929C211yesmaximal, characteristic
3C2×#17986035C211yesmaximal, characteristic
4C2×#17986027C211yesmaximal, characteristic
5#22589854C214yesmaximal, complemented
6#22589849C214yesmaximal, complemented
7#22588078C211yesmaximal, characteristic
8C2×#17985763C211yesmaximal, characteristic
9C71×#1742C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C2×#14556087C2211yescharacteristic
14#17986042—22no—
15#17986041—22no—
16#17986041C2214yes—
17#17986039C2214yes—
18#17986037—22no—
19#17986035—22no—
20#17986035—22no—
21#17986035—22no—
22C2×#14556061C2212yes—
23C2×#14556059C2212yes—
24#17986027—22no—
25#17986027C2212yes—
26#17986027C2212yes—
27C2×Dic71⋊C8—22no—
28#17985767C2214yes—
29#17985763C2212yes—
30#17985763C2212yes—
31C22×M16×C71C2211yescharacteristic
32C22×M16×C71C2211yescharacteristic
33C2×#117×C71C2211yescharacteristic
34#343×C71C2214yes—
35C2×#11765849—21no—
36C2×#11765849—22no—
37C2×#11765849Q814yes—
38C2×#11765849C2312yes—
39C22×C71⋊C8—21no—
40C22×C71⋊C8—22no—
41C22×C71⋊C8—22no—
42C22×C71⋊C8D412yes—
43C22×C71⋊C8D412yes—
44C22×C71⋊C8C2311yescharacteristic
45C22×C71⋊C8C2311yescharacteristic
46#14556087—21no—
47#14556087—21no—
48#14556087—21no—
49#14556087—21no—
50#14556087—21no—
51#14556087—21no—
52#14556087D412yes—
53#14556087C2312yes—
54C2×C4⋊Dic71D412yes—
55C2×C4⋊Dic71C2312yes—
56C2×C22⋊Dic71—21no—
57C2×C22⋊Dic71—21no—
58C2×C4×Dic71—21no—
59C2×C4×Dic71—21no—
60#14556062—24no—
61#14556061—24no—
62#14556061—24no—
63#14556060—24no—
64#14556059—24no—
65#14556059—24no—
66Dic71⋊C8—24no—
67Dic71⋊C8—24no—
68C23×C4×C71C2311yescharacteristic
69C2×M16×C71—21no—
70C2×M16×C71—22no—
71C2×M16×C71—21no—
72C2×M16×C71—22no—
73C2×M16×C71—22no—
74C2×M16×C71Q814yes—
75C2×M16×C71C2312yes—
76C2×M16×C71C2312yes—
77C22×C8×C71—21no—
78C22×C8×C71—21no—
79#117×C71—21no—
80#117×C71—21no—
81#117×C71—21no—
82#117×C71—21no—
83C2×C4⋊C4×C71C2312yes—
84C2×C22⋊C4×C71—21no—
85C2×C42×C71—21no—
86C22×Dic71—22no—
87C22×Dic71—22no—
88#11765849—24no—
89#11765849—22no—
90#11765849C2×Q814yes—
91C2×C71⋊C8—24no—
92C2×C71⋊C8—24no—
93C2×C71⋊C8—24no—
94C2×C71⋊C8—22no—
95C2×C71⋊C8—21no—
96C2×C71⋊C8—21no—
97C2×C71⋊C8—21no—
98C2×C71⋊C8—21no—
99C2×C71⋊C8C2×D412yes—
100C2×C71⋊C8C2×D412yes—
101C4⋊Dic71—22no—
102C4⋊Dic71—22no—
103C4⋊Dic71—22no—
104C4⋊Dic71C2×D414yes—
105C22⋊Dic71—22no—
106C22⋊Dic71—22no—
107C22⋊Dic71—41no—
108C22⋊Dic71—41no—
109C4×Dic71—22no—
110C4×Dic71—22no—
111C4×Dic71—41no—
112C4×Dic71—41no—
113C24×C71#4111yescharacteristic
114C22×C4×C71—22no—
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71C2×D411yescharacteristic
118C22×C4×C71C2×D411yescharacteristic
119C22×C4×C71C2×Q812yes—
120C22×C4×C71C2×Q812yes—
121C22×C4×C71#4112yes—
122C22×C4×C71#4111yescharacteristic
123C22×C4×C71#4112yes—
124C22×C4×C71C2411yescharacteristic
125M16×C71—24no—
126M16×C71—22no—
127M16×C71—22no—
128M16×C71—22no—
129M16×C71—42no—
130M16×C71C2×Q814yes—
131C2×C8×C71—24no—
132C2×C8×C71—22no—
133C2×C8×C71—22no—
134C2×C8×C71—42no—
135C4⋊C4×C71—22no—
136C4⋊C4×C71—22no—
137C22⋊C4×C71—41no—
138C22⋊C4×C71—41no—
139C42×C71—41no—
140C42×C71—41no—
141C23×C71—42no—
142C23×C71#12111yescharacteristic
143C23×C71#12111yescharacteristic
144C23×C71#12512yes—
145C23×C71#12512yes—
146C23×C71C2×#4111yescharacteristic
147C2×C4×C71—42no—
148C2×C4×C71—82no—
149C2×C4×C71#12111yescharacteristic
150C2×C4×C71#12111yescharacteristic
151C2×C4×C71C22⋊Q812yes—
152C2×C4×C71C22⋊Q812yes—
153C2×C4×C71C22⋊Q812yes—
154C2×C4×C71C22⋊Q812yes—
155C2×C4×C71#12312yes—
156C2×C4×C71#12312yes—
157C2×C4×C71#12512yes—
158C2×C4×C71#12512yes—
159C2×C4×C71C4⋊Q812yes—
160C2×C4×C71C4⋊Q812yes—
161C2×C4×C71C22×D411yescharacteristic
162C2×C4×C71C22×D411yescharacteristic
163C2×C4×C71C22×Q811yescharacteristic
164C2×C4×C71C22×Q811yescharacteristic
165C2×C4×C71C2×#4111yescharacteristic
166C2×C4×C71C2×#4111yescharacteristic
167C2×Dic71—24no—
168C2×Dic71—44no—
169C2×Dic71—42no—
170C2×Dic71—82no—
171C8×C71—24no—
172C8×C71—42no—
173C71⋊C8—24no—
174C71⋊C8—22no—
175C71⋊C8—22no—
176C22×C71—22no—
177C22×C71—22no—
178C22×C71—22no—
179C22×C71—42no—
180C22×C71—42no—
181C22×C71#39812yes—
182C22×C71#39812yes—
183C22×C71C2×#12111yescharacteristic
184C22×C71C2×#12511yescharacteristic
185C22×C71C2×#12511yescharacteristic
186C4×C71—82no—
187C4×C71#39812yes—
188C4×C71#39812yes—
189C4×C71C2×C22⋊Q811yescharacteristic
190C4×C71C2×C22⋊Q811yescharacteristic
191C4×C71C2×#12311yescharacteristic
192C4×C71C2×C4⋊Q811yescharacteristic
193Dic71—44no—
194Dic71—82no—
195#23674—711nomaximal, Sylow
196C2×C71—21no—
197C2×C71—21no—
198C2×C71—22no—
199C2×C71—22no—
200C2×C71C2×#39811yescharacteristic
201C2×C71#306712yes—
202#2911—711no—
203#2911—711no—
204C2×#427—711no—
205C2×#424—711no—
206#2090—714no—
207#2089—714no—
208#1742—711no—
209#1742D7111yescharacteristic
210C2×#341—711no—
211C71#2367411yescharacteristic, complemented, Sylow
212C22×M16—711no—
213C22×M16C2×D7111yescharacteristic
214C22×M16C2×D7111yescharacteristic
215C23×C8—711no—
216C2×#117—711no—
217C2×#117—711no—
218C2×#117C2×D7111yescharacteristic
219#428—714no—
220#428—714no—
221#428—1422no—
222#428—1422no—
223#427—1422no—
224#427—1422no—
225#427—1422no—
226#427—1422no—
227C2×#107—712no—
228C2×#106—712no—
229#424—712no—
230#424—712no—
231#424—1422no—
232C2×C4⋊C8—1422no—
233#343—714no—
234#343C2×D7114yes—
235#341—712no—
236#341—712no—
237C23×C4C22×D7111yescharacteristic
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—714no—
244C2×M16—712no—
245C2×M16—1421no—
246C2×M16—1422no—
247C2×M16Dic14214yes—
248C2×M16C22×D7112yes—
249C2×M16C22×D7112yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—712no—
253C22×C8—712no—
254C22×C8—711no—
255C22×C8—711no—
256C22×C8—1421no—
257C22×C8—1422no—
258C22×C8—1422no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—712no—
264#117—712no—
265#117—1421no—
266#117—1421no—
267#117—1421no—
268#117—1421no—
269#117—1421no—
270#117—1421no—
271C2×C4⋊C4—712no—
272C2×C4⋊C4—712no—
273C2×C4⋊C4C22×D7112yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1421no—
276C2×C22⋊C4—1421no—
277C2×C42—21no—
278C2×C42—1421no—
279C2×C42—1421no—
280#107—1424no—
281#107—1424no—
282#107—1424no—
283#106—1424no—
284#106—1424no—
285#106—1424no—
286C4⋊C8—1424no—
287C4⋊C8—1424no—
288C24#1176586511yescharacteristic
289C22×C4—22no—
290C22×C4—1422no—
291C22×C4—1422no—
292C22×C4D4×D7111yescharacteristic
293C22×C4D4×D7111yescharacteristic
294C22×C4D4×D7111yescharacteristic
295C22×C4D4×D7111yescharacteristic
296C22×C4Q8×D7112yes—
297C22×C4C2×Dic14212yes—
298C22×C4#1176586412yes—
299C22×C4#1176586512yes—
300C22×C4#1176586511yescharacteristic
301C22×C4C23×D7111yescharacteristic
302M16—22no—
303M16—24no—
304M16—22no—
305M16—22no—
306M16—42no—
307M16—714no—
308M16—1422no—
309M16—1424no—
310M16C2×Dic14214yes—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—712no—
316C2×C8—712no—
317C2×C8—1422no—
318C2×C8—1421no—
319C2×C8—1421no—
320C2×C8—1421no—
321C2×C8—1421no—
322C2×C8—1424no—
323C2×C8—1424no—
324C2×C8—1424no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—714no—
328C4⋊C4—1422no—
329C4⋊C4—1422no—
330C4⋊C4—1422no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—2841no—
334C22⋊C4—2841no—
335C22⋊C4—1422no—
336C22⋊C4—1422no—
337C42—41no—
338C42—41no—
339C42—2841no—
340C42—2841no—
341C42—1422no—
342C42—1422no—
343C23—42no—
344C23#1455610111yescharacteristic
345C23#1455610111yescharacteristic
346C23#1455611812yes—
347C23#1455612012yes—
348C23C2×#1176586511yescharacteristic
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—2844no—
352C2×C4—2842no—
353C2×C4—5682no—
354C2×C4—1424no—
355C2×C4#1455610111yescharacteristic
356C2×C4#1455610111yescharacteristic
357C2×C4#1455610512yes—
358C2×C4#1455610512yes—
359C2×C4#1455610612yes—
360C2×C4#1455610612yes—
361C2×C4#1455611012yes—
362C2×C4#1455611012yes—
363C2×C4#1455611812yes—
364C2×C4#1455612012yes—
365C2×C4#1455612712yes—
366C2×C4#1455612712yes—
367C2×C4C2×D4×D7111yescharacteristic
368C2×C4C2×D4×D7111yescharacteristic
369C2×C4C2×Q8×D7111yescharacteristic
370C2×C4C22×Dic14211yescharacteristic
371C2×C4C2×#1176586411yescharacteristic
372C2×C4C2×#1176586511yescharacteristic
373C8—24no—
374C8—42no—
375C8—1424no—
376C8—1422no—
377C8—1422no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22#1798593312yes—
384C22#1798593312yes—
385C22C2×#1455610111yescharacteristic
386C22C2×#1455611811yescharacteristic
387C22C2×#1455612011yescharacteristic
388C4—82no—
389C4—2844no—
390C4—5682no—
391C4#1798593312yes—
392C4#1798593312yes—
393C4C2×#1455610511yescharacteristic
394C4C2×#1455610611yescharacteristic
395C4C2×#1455611011yescharacteristic
396C4C2×#1455612711yescharacteristic
397C2—22no—
398C2—22no—
399C2—21no—
400C2—21no—
401C2C2×#1798593311yescharacteristic
402C2#2259805912yes—
subgroupG/H
centerC2×C4#14556101
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#1742C2
socleC22×C71C2×#121
prime radicalC24×C71#41
hypercenter#1742D71
hypocenterC71#23674
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
5. #22589854C2semidirect
6. #22589849C2semidirect
211. C71#23674semidirect

Maximal subgroups

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

subgroupindex
#225969302
#225969292
C2×#179860352
C2×#179860272
#225898542
#225898492
#225880782
C2×#179857632
C71×#17422
#2367471