Finite Group Databasebeta

← #31967654

Subgroups — #31967654

Order 18176.

658 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986130C211yesmaximal, characteristic, complemented
2#22597052C211yesmaximal, characteristic
3#22587231C214yesmaximal, complemented
4#22587177C214yesmaximal, complemented
5#22587111C211yesmaximal, characteristic, complemented
6#22586769C211yesmaximal, characteristic, complemented
7#22586739C211yesmaximal, characteristic
8#22586624C211yesmaximal, characteristic, complemented
9C71×#3103C211yesmaximal, characteristic
10C2×#14556135—21no—
11#17986130—21no—
12#17986130—21no—
13#17986130C2212yes—
14#17986130C2212yes—
15#17986077—21no—
16#17986077C2212yes—
17D4×C71⋊C8—21no—
18D4×C71⋊C8C2212yes—
19#17986052C2211yescharacteristic
20#17986023C2211yescharacteristic
21C2×C4⋊(C71⋊C8)—21no—
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23#17985966—21no—
24C2×#14556039—21no—
25C2×C4×C71⋊C8C2211yescharacteristic
26#17985750C2214yes—
27Q8⋊(C71⋊C8)C2214yes—
28D4⋊(C71⋊C8)C2214yes—
29#17985721C2212yes—
30C8×C71⋊C8C2212yes—
31#532×C71C2211yescharacteristic
32C2×C4×Q8×C71C2211yescharacteristic
33#481×C71C412yes—
34#475×C71C412yes—
35#460×C71C412yes—
36#448×C71C412yes—
37C4×SD16×C71C2214yes—
38#431×C71C2211yescharacteristic
39#14556135—44no—
40C2×#11765849—22no—
41C2×#11765849—21no—
42C22×C71⋊C8—22no—
43C22×C71⋊C8—21no—
44C22×C71⋊C8D411yescharacteristic
45C22×C71⋊C8D411yescharacteristic
46C4⋊(C71⋊C8)—22no—
47C4⋊(C71⋊C8)—22no—
48C4⋊(C71⋊C8)—21no—
49C4⋊(C71⋊C8)—21no—
50C4⋊(C71⋊C8)—21no—
51C4⋊(C71⋊C8)C2312yes—
52C4⋊(C71⋊C8)C2312yes—
53C22⋊(C71⋊C8)—22no—
54C22⋊(C71⋊C8)—22no—
55C22⋊(C71⋊C8)—21no—
56#14556039—22no—
57#14556039—22no—
58C4×C71⋊C8D411yescharacteristic
59C4×C71⋊C8D411yescharacteristic
60C4×C71⋊C8C2312yes—
61C4×C71⋊C8C2311yescharacteristic
62C4×C71⋊C8C2311yescharacteristic
63C2×#41×C71—21no—
64C22×Q8×C71C2×C411yescharacteristic
65C2×SD16×C71C2×C414yes—
66C4⋊Q8×C71C2×C411yescharacteristic
67#127×C71C2×C411yescharacteristic
68#124×C71—21no—
69#121×C71—21no—
70#121×C71C2×C412yes—
71C4×Q8×C71—21no—
72C4×Q8×C71—21no—
73C4×Q8×C71C2×C412yes—
74C4×Q8×C71C2312yes—
75C4×D4×C71—21no—
76C4×D4×C71C2312yes—
77C2×C4⋊C4×C71—21no—
78C2×C4⋊C4×C71C2×C411yescharacteristic
79C2×C42×C71—21no—
80C2×C42×C71C2311yescharacteristic
81#106×C71C2×C414yes—
82C4⋊C8×C71C2×C412yes—
83#103×C71C2×C414yes—
84#102×C71C2×C414yes—
85C22⋊C8×C71C2×C412yes—
86C4×C8×C71C2312yes—
87#11765849—42no—
88#11765849—44no—
89C2×C71⋊C8—22no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8—22no—
92C2×C71⋊C8—22no—
93C2×C71⋊C8—42no—
94C2×C71⋊C8—44no—
95C2×C71⋊C8SD1614yescomplemented
96C2×C71⋊C8C2×D411yescharacteristic
97C2×C71⋊C8C2×D411yescharacteristic
98C2×C71⋊C8C2×D411yescharacteristic
99C2×C71⋊C8C2×D411yescharacteristic
100#41×C71—42no—
101#41×C71—42no—
102C2×Q8×C71—21no—
103C2×Q8×C71—21no—
104C2×Q8×C71—42no—
105C2×Q8×C71C22×C412yes—
106C2×D4×C71—22no—
107C2×D4×C71—21no—
108C2×D4×C71—21no—
109C2×D4×C71C22×C412yes—
110C22×C4×C71—21no—
111C22×C4×C71—21no—
112C22×C4×C71—22no—
113C22×C4×C71C22×C411yescharacteristic
114C22×C4×C71C2×D411yescharacteristic
115C22×C4×C71#4111yescharacteristic
116SD16×C71—44no—
117C2×C8×C71C22×C414yes—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—21no—
120C4⋊C4×C71—21no—
121C4⋊C4×C71—22no—
122C4⋊C4×C71C22×C412yes—
123C4⋊C4×C71C22×C412yes—
124C22⋊C4×C71—22no—
125C22⋊C4×C71—22no—
126C42×C71—22no—
127C42×C71—22no—
128C42×C71C22×C411yescharacteristic
129C42×C71C2×D411yescharacteristic
130C42×C71#4111yescharacteristic
131C42×C71C2411yescharacteristic
132C23×C71—22no—
133C23×C71C4×D411yescharacteristic
134Q8×C71—22no—
135Q8×C71—22no—
136Q8×C71—42no—
137Q8×C71—42no—
138D4×C71—24no—
139D4×C71—42no—
140D4×C71—44no—
141D4×C71—42no—
142C2×C4×C71—22no—
143C2×C4×C71—22no—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—41no—
147C2×C4×C71—42no—
148C2×C4×C71—42no—
149C2×C4×C71—42no—
150C2×C4×C71—42no—
151C2×C4×C71C4×D411yescharacteristic
152C2×C4×C71C4×D411yescharacteristic
153C2×C4×C71C4×D411yescharacteristic
154C2×C4×C71C2×SD1612yes—
155C2×C4×C71#13512yes—
156C2×C4×C71C23×C411yescharacteristic
157C2×C4×C71C22×D411yescharacteristic
158C2×C4×C71C2×#4111yescharacteristic
159C8×C71—24no—
160C71⋊C8—21no—
161C71⋊C8—21no—
162C71⋊C8—42no—
163C71⋊C8—42no—
164C71⋊C8—42no—
165C71⋊C8C2×SD1612yes—
166C71⋊C8C2×SD1612yes—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—42no—
170C22×C71—42no—
171C22×C71—42no—
172C22×C71C4×SD1612yes—
173C22×C71C2×C4×D411yescharacteristic
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—42no—
178C4×C71—42no—
179C4×C71—42no—
180C4×C71C4×SD1612yes—
181C4×C71#56812yes—
182C4×C71C22×SD1611yescharacteristic
183C4×C71C2×#13511yescharacteristic
184#17929—711nomaximal, Sylow
185C2×C71—22no—
186C2×C71—82no—
187C2×C71#293011yescharacteristic
188C2×C71C2×C4×SD1611yescharacteristic
189C2×C71#295511yescharacteristic
190#3103D7111yescharacteristic
191C2×#446—711no—
192#2931—711no—
193#1590—714no—
194#1582—714no—
195#1570—711no—
196#1487—711no—
197#1480—711no—
198#1450—711no—
199C71#1792911yescharacteristic, complemented, Sylow
200C2×#131—1421no—
201#532C2×D7111yescharacteristic
202C2×C4×Q8C2×D7111yescharacteristic
203#481Dic7112yes—
204#475Dic7112yes—
205#460Dic7112yes—
206#448Dic7112yes—
207#446—712no—
208#446—712no—
209#446—1421no—
210#446—1421no—
211C4×SD16C2×D7114yes—
212#436—712no—
213#436—1421no—
214C8×D4—712no—
215C8×D4—1421no—
216#431—711no—
217#431C2×D7111yescharacteristic
218#423—711no—
219C2×C4⋊C8—711no—
220C2×C4⋊C8—1421no—
221#408—1421no—
222C2×#97—1421no—
223C2×C4×C8—711no—
224#334—714no—
225Q8⋊C8—714no—
226D4⋊C8—714no—
227#322—712no—
228C82—712no—
229C2×#41—21no—
230C22×Q8C2×Dic7111yescharacteristic
231C2×SD16C2×Dic7114yes—
232#131—2844no—
233C2×M16—1422no—
234C2×M16—1421no—
235C22×C8—711no—
236C22×C8—711no—
237C22×C8—1422no—
238C22×C8—1421no—
239C4⋊Q8C2×Dic7111yescharacteristic
240#127C2×Dic7111yescharacteristic
241#124—21no—
242#121—21no—
243#121C2×Dic7112yes—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8C2×Dic7112yes—
247C4×Q8C22×D7112yes—
248C4×D4—21no—
249C4×D4C22×D7112yes—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic7111yescharacteristic
252C2×C42—21no—
253C2×C42C22×D7111yescharacteristic
254#106C2×Dic7114yes—
255C4⋊C8—712no—
256C4⋊C8—712no—
257C4⋊C8—1422no—
258C4⋊C8—1422no—
259C4⋊C8—1421no—
260C4⋊C8—1421no—
261C4⋊C8—1421no—
262C4⋊C8C2×Dic7112yes—
263#103C2×Dic7114yes—
264#102C2×Dic7114yes—
265C22⋊C8—1422no—
266C22⋊C8—1422no—
267C22⋊C8—1421no—
268C22⋊C8C2×Dic7112yes—
269#97—1422no—
270#97—1422no—
271C4×C8—711no—
272C4×C8—711no—
273C4×C8—711no—
274C4×C8—711no—
275C4×C8—712no—
276C4×C8C22×D7112yes—
277#41—42no—
278#41—42no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—42no—
282C2×Q8C22×Dic7112yes—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4C22×Dic7112yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4C22×Dic7111yescharacteristic
291C22×C4D4×D7111yescharacteristic
292C22×C4#1176586411yescharacteristic
293SD16—44no—
294M16—2842no—
295M16—2844no—
296C2×C8—2842no—
297C2×C8—2844no—
298C2×C8—711no—
299C2×C8—711no—
300C2×C8—714no—
301C2×C8—711no—
302C2×C8—711no—
303C2×C8—1422no—
304C2×C8—1422no—
305C2×C8—1422no—
306C2×C8—1422no—
307C2×C8C22×Dic7114yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4C22×Dic7112yes—
313C4⋊C4C22×Dic7112yes—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C42—22no—
317C42—22no—
318C42C22×Dic7111yescharacteristic
319C42D4×D7111yescharacteristic
320C42#1176586411yescharacteristic
321C42C23×D7111yescharacteristic
322C23—22no—
323C23D4×Dic7111yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—42no—
327Q8—42no—
328D4—24no—
329D4—42no—
330D4—44no—
331D4—42no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4D4×Dic7111yescharacteristic
342C2×C4D4×Dic7111yescharacteristic
343C2×C4D4×Dic7111yescharacteristic
344C2×C4SD16×D7112yes—
345C2×C4#1455614912yes—
346C2×C4C23×Dic7111yescharacteristic
347C2×C4C2×D4×D7111yescharacteristic
348C2×C4C2×#1176586411yescharacteristic
349C8—24no—
350C8—2842no—
351C8—2842no—
352C8—2842no—
353C8—712no—
354C8—712no—
355C8—1421no—
356C8—1421no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22SD16×Dic7112yes—
363C22C2×D4×Dic7111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4SD16×Dic7112yes—
371C4#1798676612yes—
372C4C2×SD16×D7111yescharacteristic
373C4C2×#1455614911yescharacteristic
374C2—22no—
375C2—82no—
376C2#2259704411yescharacteristic
377C2C2×SD16×Dic7111yescharacteristic
378C2#2259724311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#3103C2
socleC22×C71C2×C4×D4
prime radical#532×C71C22
cosocleC42×C71C24
hypercenter#3103D71
hypocenterC71#17929
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. C2×#17986130C2semidirect
3. #22587231C2semidirect
4. #22587177C2semidirect
5. #22587111C2semidirect
6. #22586769C2semidirect
8. #22586624C2semidirect
95. C2×C71⋊C8SD16semidirect
199. C71#17929semidirect

Maximal subgroups

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

subgroupindex
C2×#179861302
#225970522
#225872312
#225871772
#225871112
#225867692
#225867392
#225866242
C71×#31032
#1792971