Finite Group Databasebeta

← #31890769

Subgroups — #31890769

Order 18176.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597316C211yesmaximal, characteristic
2C4×#14556061C211yesmaximal, characteristic, complemented
3#22587101C211yesmaximal, characteristic, complemented
4#22586865C211yesmaximal, characteristic
5#22586196C211yesmaximal, characteristic, complemented
6#22586195C211yesmaximal, characteristic, complemented
7C71×#1509C211yesmaximal, characteristic
8#17986130C411yescharacteristic, complemented
9Q8×C71⋊C8C411yescharacteristic, complemented
10#17986077C411yescharacteristic, complemented
11#17986071—21no—
12D4×C71⋊C8—21no—
13D4×C71⋊C8C411yescharacteristic, complemented
14#17986060—21no—
15#17986056C2211yescharacteristic
16#17986056C2211yescharacteristic
17C2×#14556061—21no—
18#17985955—21no—
19C2×C4×C71⋊C8C2211yescharacteristic
20C4×C4⋊Dic71C2211yescharacteristic
21#17985761—21no—
22#17985751—21no—
23#17985749—21no—
24#17985736—21no—
25#17985733—21no—
26#529×C71C2211yescharacteristic
27#432×C71C2211yescharacteristic
28C4×C4⋊C4×C71C2211yescharacteristic
29#328×C71—21no—
30#327×C71—21no—
31C2×#11765849—21no—
32C22×C71⋊C8—21no—
33C22×C71⋊C8D411yescharacteristic
34C22×C71⋊C8D411yescharacteristic
35C2×C4⋊Dic71—21no—
36C2×C4×Dic71—21no—
37#14556061—24no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)C2×C411yescharacteristic
42C4⋊(C71⋊C8)C2×C411yescharacteristic
43C22⋊(C71⋊C8)—21no—
44C22⋊(C71⋊C8)—21no—
45C22⋊(C71⋊C8)—21no—
46C22⋊(C71⋊C8)—21no—
47#14556039—21no—
48#14556039—21no—
49#14556039—21no—
50C4×C71⋊C8—21no—
51C4×C71⋊C8C2×C411yescharacteristic
52C4×C71⋊C8C2×C411yescharacteristic
53C4×C71⋊C8D411yescharacteristic
54C4×C71⋊C8D411yescharacteristic
55C71⋊#95—21no—
56#126×C71—21no—
57#125×C71Q811yescharacteristic
58#124×C71Q811yescharacteristic
59#123×C71—21no—
60C22⋊Q8×C71D411yescharacteristic
61#121×C71D411yescharacteristic
62C4×Q8×C71C2×C411yescharacteristic
63C4×D4×C71—21no—
64C4×D4×C71C2×C411yescharacteristic
65#117×C71—21no—
66C2×C4⋊C4×C71—21no—
67C2×C42×C71—21no—
68C2×C42×C71C2311yescharacteristic
69C4⋊C8×C71—21no—
70C22⋊C8×C71—21no—
71#97×C71—21no—
72#95×C71—21no—
73C22×Dic71—41no—
74#11765849—42no—
75C2×C71⋊C8—22no—
76C2×C71⋊C8—21no—
77C2×C71⋊C8—21no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8—42no—
80C2×C71⋊C8—41no—
81C2×C71⋊C8C22⋊C411yescharacteristic
82C2×C71⋊C8C22⋊C411yescharacteristic
83C2×C71⋊C8C22⋊C411yescharacteristic
84C2×C71⋊C8C22⋊C411yescharacteristic
85C4⋊Dic71—22no—
86C4⋊Dic71—41no—
87C4×Dic71—81no—
88C2×Q8×C71C4⋊C411yescharacteristic
89C2×D4×C71—21no—
90C2×D4×C71C4⋊C411yescharacteristic
91C22×C4×C71—21no—
92C22×C4×C71—41no—
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71#4111yescharacteristic
96C2×C8×C71—41no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71—22no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71C4⋊C411yescharacteristic
104C4⋊C4×C71C4⋊C411yescharacteristic
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71—21no—
109C22⋊C4×C71—21no—
110C42×C71—21no—
111C42×C71—81no—
112C42×C71C22×C411yescharacteristic
113C42×C71C2×D411yescharacteristic
114C42×C71C2×Q811yescharacteristic
115C42×C71#4111yescharacteristic
116C23×C71—21no—
117C23×C71#12111yescharacteristic
118Q8×C71—41no—
119D4×C71—41no—
120D4×C71—41no—
121D4×C71—41no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—42no—
133C2×C4×C71—81no—
134C2×C4×C71—81no—
135C2×C4×C71C2×C22⋊C411yescharacteristic
136C2×C4×C71C2×C4⋊C411yescharacteristic
137C2×C4×C71#11711yescharacteristic
138C2×C4×C71#12111yescharacteristic
139C2×C4×C71C22⋊Q811yescharacteristic
140C2×C4×C71C22⋊Q811yescharacteristic
141C2×Dic71—42no—
142C2×Dic71—81no—
143C2×Dic71—81no—
144C8×C71—81no—
145C71⋊C8—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C71⋊C8—41no—
149C71⋊C8—41no—
150C71⋊C8—81no—
151C71⋊C8#10412yes—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71#38011yescharacteristic
158C22×C71#47211yescharacteristic
159C22×C71#47311yescharacteristic
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—82no—
169C4×C71C2×#10411yescharacteristic
170C4×C71#42111yescharacteristic
171Dic71—82no—
172#8697—711nomaximal, Sylow
173C2×C71—21no—
174C2×C71—21no—
175C2×C71—81no—
176C2×C71#180611yescharacteristic
177C2×C71#181211yescharacteristic
178C2×C71#181611yescharacteristic
179#2966—711no—
180C4×#107—711no—
181#1568—711no—
182#1509—711no—
183#1509D7111yescharacteristic
184#1285—711no—
185#1285—711no—
186C71#869711yescharacteristic, complemented, Sylow
187#529C2×D7111yescharacteristic
188#446—711no—
189C8×Q8—711no—
190#436—711no—
191#435—1421no—
192C8×D4—711no—
193C8×D4—1421no—
194#433—1421no—
195#432—711no—
196#432—711no—
197#432C2×D7111yescharacteristic
198C2×#107—1421no—
199#405—1421no—
200C2×C4×C8—711no—
201C4×C4⋊C4—711no—
202C4×C4⋊C4C2×D7111yescharacteristic
203#340—1421no—
204#335—1421no—
205#334—1421no—
206#328—21no—
207#328—1421no—
208#327—21no—
209#327—1421no—
210C2×M16—1421no—
211C22×C8—711no—
212C22×C8—711no—
213C22×C8—1421no—
214#126—21no—
215#125Dic14211yescharacteristic
216#124Dic14211yescharacteristic
217#123—21no—
218C22⋊Q8C22⋊D7111yescharacteristic
219#121C22⋊D7111yescharacteristic
220C4×Q8C4×D7111yescharacteristic
221C4×D4—21no—
222C4×D4C4×D7111yescharacteristic
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1421no—
226C2×C42—21no—
227C2×C42—1421no—
228C2×C42C22×D7111yescharacteristic
229#107—1424no—
230C4⋊C8—21no—
231C4⋊C8—711no—
232C4⋊C8—711no—
233C4⋊C8—1421no—
234C4⋊C8—1421no—
235C4⋊C8—1421no—
236C22⋊C8—21no—
237C22⋊C8—1421no—
238C22⋊C8—1421no—
239C22⋊C8—1421no—
240C22⋊C8—1421no—
241#97—21no—
242#97—1421no—
243#97—1421no—
244#97—1421no—
245C4×C8—711no—
246C4×C8—711no—
247C4×C8—711no—
248C4×C8—711no—
249C4×C8—1421no—
250#95—21no—
251#95—1421no—
252C2×Q8Dic71⋊C411yescharacteristic
253C2×D4—21no—
254C2×D4Dic71⋊C411yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—2841no—
258C22×C4D4×D7111yescharacteristic
259C22×C4C2×C22⋊D7111yescharacteristic
260C22×C4#1176586411yescharacteristic
261M16—2842no—
262C2×C8—41no—
263C2×C8—2842no—
264C2×C8—2841no—
265C2×C8—2841no—
266C2×C8—711no—
267C2×C8—711no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8—1421no—
271C2×C8—1421no—
272C2×C8—1422no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—2841no—
280C4⋊C4—1422no—
281C4⋊C4Dic71⋊C411yescharacteristic
282C4⋊C4Dic71⋊C411yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—5681no—
291C42C2×C4×D7111yescharacteristic
292C42D4×D7111yescharacteristic
293C42C2×Dic14211yescharacteristic
294C42#1176586411yescharacteristic
295C23—21no—
296C23#1455610011yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—2842no—
315C2×C4—5681no—
316C2×C4—5681no—
317C2×C4C22⋊C4×D7111yescharacteristic
318C2×C4C2×Dic71⋊C411yescharacteristic
319C2×C4#1455608411yescharacteristic
320C2×C4#1455610011yescharacteristic
321C2×C4#1455610511yescharacteristic
322C2×C4#1455610511yescharacteristic
323C8—81no—
324C8—2841no—
325C8—2841no—
326C8—5681no—
327C8—712no—
328C8—1421no—
329C8—1421no—
330C8—1421no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1798585711yescharacteristic
337C22#1798626711yescharacteristic
338C22#1798627211yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—5682no—
349C4#104×D7111yescharacteristic
350C4#1798601411yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2258835811yescharacteristic
355C2#2258838411yescharacteristic
356C2#2258840211yescharacteristic
subgroupG/H
centerC22#17985857
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1509C2
socleC22×C71#380
prime radical#121×C71D4
cosocleC2×C42×C71C23
hypercenter#1509D71
hypocenterC71#8697
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#14556061C2semidirect
3. #22587101C2semidirect
5. #22586196C2semidirect
6. #22586195C2semidirect
8. #17986130C4semidirect
9. Q8×C71⋊C8C4semidirect
10. #17986077C4semidirect
13. D4×C71⋊C8C4semidirect
186. C71#8697semidirect

Maximal subgroups

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

subgroupindex
#225973162
C4×#145560612
#225871012
#225868652
#225861962
#225861952
C71×#15092
#869771