Finite Group Databasebeta

← #31885760

Subgroups — #31885760

Order 18176.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556107C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic
3#22586165C212yesmaximal, complemented
4#22586162C212yesmaximal, complemented
5C71×#1797C211yesmaximal, characteristic
6C2×#14556107C411yescharacteristic
7C2×C4×Dic142C411yescharacteristic
8C2×C22⋊(C71⋊C8)—21no—
9#17985905C2211yescharacteristic
10#17985897C2211yescharacteristic
11#17985883—21no—
12#17985872—21no—
13#17985859C411yescharacteristic
14C4×Dic71⋊C4—21no—
15C4×C4⋊Dic71C411yescharacteristic
16C4×C22⋊Dic71—21no—
17#17985754—21no—
18#17985754C2212yes—
19C22×C42×C71C2211yescharacteristic
20C2×C4⋊C8×C71—21no—
21C2×C22⋊C8×C71—21no—
22#336×C71C2212yes—
23C22×Dic142C2×C411yescharacteristic
24C22×C71⋊C8—22no—
25#14556107—22no—
26#14556107—22no—
27C4×Dic142—42no—
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×C4⋊Dic71C2×C411yescharacteristic
30C2×C4⋊Dic71C2×C411yescharacteristic
31C2×Dic71⋊C4—21no—
32C2×Dic71⋊C4—21no—
33C2×Dic71⋊C4—21no—
34C2×C22⋊Dic71—21no—
35C2×C22⋊Dic71—21no—
36C2×C4×Dic71—21no—
37C2×C4×Dic71—21no—
38C22⋊(C71⋊C8)—42no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C71⋊#95—21no—
43C71⋊#95—21no—
44C23×C4×C71C2×C411yescharacteristic
45C23×C4×C71D411yescharacteristic
46C23×C4×C71D411yescharacteristic
47C22×C8×C71—22no—
48C2×C42×C71—42no—
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52C2×C42×C71C2311yescharacteristic
53C4⋊C8×C71—42no—
54C22⋊C8×C71—42no—
55C2×Dic142—21no—
56C2×Dic142—41no—
57C2×Dic142—41no—
58C2×Dic142M1612yes—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C22×Dic71—21no—
62C22×Dic71—21no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C4⋊Dic71—21no—
67C4⋊Dic71—41no—
68C4⋊Dic71—41no—
69C4⋊Dic71M1612yes—
70Dic71⋊C4—24no—
71Dic71⋊C4—42no—
72Dic71⋊C4—42no—
73C22⋊Dic71—24no—
74C22⋊Dic71—42no—
75C4×Dic71—42no—
76C4×Dic71—42no—
77C24×C71C22⋊C411yescharacteristic
78C22×C4×C71—22no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—22no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71SD1612yes—
91C22×C4×C71Q1612yes—
92C22×C4×C71C22×C411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C2×C8×C71—42no—
96C2×C8×C71—42no—
97C2×C8×C71—42no—
98C42×C71—22no—
99C42×C71—22no—
100C42×C71—41no—
101C42×C71—41no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71#10312yes—
109C23×C71C2×C22⋊C411yescharacteristic
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—44no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71#9912yes—
129C2×C4×C71#10312yes—
130C2×C4×C71C2×M1611yescharacteristic
131C2×C4×C71C2×M1611yescharacteristic
132C2×C4×C71C2×SD1611yescharacteristic
133C2×C4×C71C2×Q1611yescharacteristic
134C2×C4×C71#13512yes—
135C2×Dic71—22no—
136C2×Dic71—22no—
137C2×Dic71—41no—
138C2×Dic71—41no—
139C2×Dic71—41no—
140C2×Dic71—41no—
141C2×Dic71—41no—
142C2×Dic71—41no—
143C2×Dic71—41no—
144C2×Dic71—41no—
145C2×Dic71—41no—
146C2×Dic71—41no—
147Dic142—42no—
148Dic142—42no—
149C8×C71—82no—
150C71⋊C8—82no—
151C22×C71—22no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71#40711yescharacteristic
159C22×C71C2×#9911yescharacteristic
160C22×C71#41311yescharacteristic
161C22×C71C2×#10311yescharacteristic
162C22×C71#41611yescharacteristic
163C22×C71#42111yescharacteristic
164C22×C71#42111yescharacteristic
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—22no—
168C4×C71—22no—
169C4×C71—21no—
170C4×C71—22no—
171C4×C71—21no—
172C4×C71—41no—
173Dic71—42no—
174Dic71—42no—
175Dic71—81no—
176Dic71—81no—
177#7648—711nomaximal, Sylow
178C2×C71—42no—
179C2×C71#146511yescharacteristic
180C2×C71#148711yescharacteristic
181C2×C71#148811yescharacteristic
182C2×C71#151611yescharacteristic
183C2×C71#152011yescharacteristic
184C2×C71#153511yescharacteristic
185C2×C71#154211yescharacteristic
186C4×C22⋊Q8—711no—
187#1817—711no—
188#1797D7111yescharacteristic
189#1272—712no—
190#1271—712no—
191C71#764811yescharacteristic, complemented, Sylow
192C2×C22⋊Q8—711no—
193C2×C4×Q8—711no—
194C22×C42C2×D7111yescharacteristic
195C2×C4⋊C8—21no—
196C2×C22⋊C8—21no—
197C2×C22⋊C8—1421no—
198#391—711no—
199#389—711no—
200#387—1421no—
201#385—1421no—
202#380—711no—
203C4×C4⋊C4—711no—
204C4×C4⋊C4—1421no—
205C4×C22⋊C4—1421no—
206#336—712no—
207#336—1421no—
208#336C2×D7112yes—
209C22×Q8—711no—
210C23×C4C4×D7111yescharacteristic
211C23×C4D28411yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C22×C8—22no—
214C22×C8—1422no—
215C22⋊Q8—1422no—
216C22⋊Q8—1422no—
217C4×Q8—2842no—
218C2×C4⋊C4—711no—
219C2×C4⋊C4—711no—
220C2×C4⋊C4—711no—
221C2×C4⋊C4—1421no—
222C2×C4⋊C4—1421no—
223C2×C4⋊C4—1421no—
224C2×C22⋊C4—1421no—
225C2×C22⋊C4—1421no—
226C2×C42—42no—
227C2×C42—1421no—
228C2×C42—1421no—
229C2×C42C4×D7111yescharacteristic
230C2×C42D28411yescharacteristic
231C2×C42C22⋊D7111yescharacteristic
232C2×C42C22×D7111yescharacteristic
233C4⋊C8—42no—
234C22⋊C8—42no—
235C22⋊C8—2842no—
236#95—1421no—
237#95—1421no—
238#95—1421no—
239#95—1421no—
240#95—1421no—
241C24(C2×D71)⋊C411yescharacteristic
242C2×Q8—2841no—
243C2×Q8—2841no—
244C2×Q8—712no—
245C2×Q8—1421no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1421no—
256C22×C4—1421no—
257C22×C4—1421no—
258C22×C4—1421no—
259C22×C4(C2×D71)⋊C411yescharacteristic
260C22×C4(C2×D71)⋊C411yescharacteristic
261C22×C4(C2×D71)⋊C411yescharacteristic
262C22×C4#1176585312yes—
263C22×C4#1176585512yes—
264C22×C4C2×C4×D7111yescharacteristic
265C22×C4C2×C22⋊D7111yescharacteristic
266C22×C4C2×D28411yescharacteristic
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—2842no—
271C2×C8—2842no—
272C2×C8—2842no—
273C4⋊C4—2841no—
274C4⋊C4—2841no—
275C4⋊C4—2842no—
276C4⋊C4—2842no—
277C4⋊C4—712no—
278C4⋊C4—1424no—
279C4⋊C4—1421no—
280C22⋊C4—2842no—
281C22⋊C4—1424no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—2842no—
287C42—2842no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1455605112yes—
295C23C2×(C2×D71)⋊C411yescharacteristic
296Q8—2842no—
297Q8—2842no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—2841no—
317C2×C4—2841no—
318C2×C4—2841no—
319C2×C4—2841no—
320C2×C4—2841no—
321C2×C4—2841no—
322C2×C4—2841no—
323C2×C4—2841no—
324C2×C4—2841no—
325C2×C4—2841no—
326C2×C4—1422no—
327C2×C4—1422no—
328C2×C4#1455604212yes—
329C2×C4#1455605112yes—
330C2×C4M16×D7111yescharacteristic
331C2×C4M16×D7111yescharacteristic
332C2×C4C2×#1176585311yescharacteristic
333C2×C4C2×#1176585511yescharacteristic
334C2×C4#1455615012yes—
335C8—82no—
336C8—5682no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1798596211yescharacteristic
345C22C2×#1455604211yescharacteristic
346C22#1798598111yescharacteristic
347C22C2×#1455605111yescharacteristic
348C22#1798599611yescharacteristic
349C22#1798601511yescharacteristic
350C22#1798601711yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—2842no—
360C4—2842no—
361C4—5681no—
362C4—5681no—
363C2—42no—
364C2#2258667611yescharacteristic
365C2#2258676811yescharacteristic
366C2#2258677511yescharacteristic
367C2#2258689711yescharacteristic
368C2#2258691111yescharacteristic
369C2#2258696911yescharacteristic
370C2#2258699811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1797C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1797D71
hypocenterC71#7648
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #22586165C2semidirect
4. #22586162C2semidirect
191. C71#7648semidirect

Maximal subgroups

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

subgroupindex
C4×#145561072
#225884082
#225861652
#225861622
C71×#17972
#764871