Finite Group Databasebeta

← #31885815

Subgroups — #31885815

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596679C211yesmaximal, characteristic
2#22588320C211yesmaximal, characteristic
3#22586165C212yesmaximal, complemented
4#22586164C212yesmaximal, complemented
5C4×C22⋊C8×C71C211yesmaximal, characteristic
6C2×#14556107C411yescharacteristic
7C2×C4⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9#17985905C411yescharacteristic
10#17985905C2211yescharacteristic
11#17985897C411yescharacteristic
12#17985897C2211yescharacteristic
13#17985883—21no—
14#17985883—21no—
15#17985872—21no—
16#17985862—21no—
17#17985859C411yescharacteristic
18#17985754C2212yes—
19C22×C42×C71C2211yescharacteristic
20C2×C22⋊C8×C71—21no—
21C2×C4×C8×C71—21no—
22#336×C71C2212yes—
23C22×Dic142C2×C411yescharacteristic
24C22×C71⋊C8—22no—
25#14556107—42no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×Dic71⋊C4—21no—
30C2×Dic71⋊C4—21no—
31C2×Dic71⋊C4—21no—
32C2×C22⋊Dic71—21no—
33C2×C22⋊Dic71—21no—
34C4⋊(C71⋊C8)—42no—
35C22⋊(C71⋊C8)—42no—
36C71⋊#95—21no—
37C71⋊#95—21no—
38C71⋊#95—21no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C71⋊#95—21no—
43C23×C4×C71C2×C411yescharacteristic
44C23×C4×C71D411yescharacteristic
45C23×C4×C71D411yescharacteristic
46C22×C8×C71—22no—
47C2×C42×C71—42no—
48C2×C42×C71C2×C411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71C2311yescharacteristic
52C22⋊C8×C71—24no—
53C4×C8×C71—42no—
54C2×Dic142—22no—
55C2×Dic142—41no—
56C2×Dic142—41no—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C4⋊Dic71—22no—
65C4⋊Dic71—41no—
66C4⋊Dic71—41no—
67Dic71⋊C4—42no—
68Dic71⋊C4—42no—
69Dic71⋊C4—42no—
70C22⋊Dic71—42no—
71C22⋊Dic71—42no—
72C24×C71C22⋊C411yescharacteristic
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71SD1612yes—
85C22×C4×C71Q1612yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C2×C8×C71—24no—
90C2×C8×C71—42no—
91C2×C8×C71—42no—
92C42×C71—22no—
93C42×C71—22no—
94C42×C71—41no—
95C42×C71—41no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71#10312yes—
103C23×C71C2×C22⋊C411yescharacteristic
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
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—42no—
117C2×C4×C71—42no—
118C2×C4×C71—44no—
119C2×C4×C71—44no—
120C2×C4×C71#9912yes—
121C2×C4×C71#10312yes—
122C2×C4×C71#10412yes—
123C2×C4×C71#13112yes—
124C2×C4×C71C2×SD1611yescharacteristic
125C2×C4×C71C2×Q1611yescharacteristic
126C2×C4×C71#13512yes—
127C2×Dic71—41no—
128C2×Dic71—41no—
129C2×Dic71—41no—
130C2×Dic71—41no—
131C2×Dic71—41no—
132C2×Dic71—41no—
133C2×Dic71—41no—
134C2×Dic71—41no—
135C2×Dic71—41no—
136C2×Dic71—41no—
137C2×Dic71—41no—
138C2×Dic71—41no—
139Dic142—82no—
140C8×C71—44no—
141C71⋊C8—82no—
142C22×C71—22no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71#40811yescharacteristic
150C22×C71C2×#9911yescharacteristic
151C22×C71#41311yescharacteristic
152C22×C71C2×#10311yescharacteristic
153C22×C71#41611yescharacteristic
154C22×C71C2×#10411yescharacteristic
155C22×C71#42111yescharacteristic
156C4×C71—22no—
157C4×C71—22no—
158C4×C71—22no—
159C4×C71—22no—
160C4×C71—21no—
161C4×C71—22no—
162C4×C71—21no—
163Dic71—81no—
164Dic71—81no—
165Dic71—81no—
166Dic71—81no—
167#7661—711nomaximal, Sylow
168C2×C71—42no—
169C2×C71#146811yescharacteristic
170C2×C71#148211yescharacteristic
171C2×C71#149311yescharacteristic
172C2×C71#150111yescharacteristic
173C2×C71#152011yescharacteristic
174C2×C71#152911yescharacteristic
175C2×C71#154211yescharacteristic
176#2855—711no—
177#1797—711no—
178C4×C22⋊C8D7111yescharacteristic
179#1272—712no—
180#1271—712no—
181C71#766111yescharacteristic, complemented, Sylow
182C2×C22⋊Q8—711no—
183C22×C42C2×D7111yescharacteristic
184C2×C4⋊C8—1421no—
185C2×C22⋊C8—21no—
186C2×C22⋊C8—1421no—
187C2×C4×C8—21no—
188#391—711no—
189#391—711no—
190#389—711no—
191#389—711no—
192#387—1421no—
193#387—1421no—
194#385—1421no—
195#381—1421no—
196#380—711no—
197#336—712no—
198#336C2×D7112yes—
199C22×Q8—711no—
200C23×C4C4×D7111yescharacteristic
201C23×C4D28411yescharacteristic
202C23×C4C22⋊D7111yescharacteristic
203C22×C8—22no—
204C22×C8—1422no—
205C22⋊Q8—2842no—
206C2×C4⋊C4—711no—
207C2×C4⋊C4—711no—
208C2×C4⋊C4—711no—
209C2×C4⋊C4—1421no—
210C2×C4⋊C4—1421no—
211C2×C4⋊C4—1421no—
212C2×C22⋊C4—1421no—
213C2×C22⋊C4—1421no—
214C2×C42—42no—
215C2×C42C4×D7111yescharacteristic
216C2×C42D28411yescharacteristic
217C2×C42C22⋊D7111yescharacteristic
218C2×C42C22×D7111yescharacteristic
219C4⋊C8—2842no—
220C22⋊C8—24no—
221C22⋊C8—2842no—
222C4×C8—42no—
223#95—1421no—
224#95—1421no—
225#95—1421no—
226#95—1421no—
227#95—1421no—
228#95—1421no—
229#95—1421no—
230C24(C2×D71)⋊C411yescharacteristic
231C2×Q8—2841no—
232C2×Q8—2841no—
233C2×Q8—1422no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1421no—
243C22×C4—1421no—
244C22×C4—1421no—
245C22×C4—1421no—
246C22×C4(C2×D71)⋊C411yescharacteristic
247C22×C4(C2×D71)⋊C411yescharacteristic
248C22×C4(C2×D71)⋊C411yescharacteristic
249C22×C4#1176585412yes—
250C22×C4Dic28412yes—
251C22×C4C2×C4×D7111yescharacteristic
252C22×C4C2×C22⋊D7111yescharacteristic
253C22×C4C2×D28411yescharacteristic
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—2842no—
258C2×C8—2842no—
259C2×C8—2842no—
260C4⋊C4—2841no—
261C4⋊C4—2841no—
262C4⋊C4—2842no—
263C4⋊C4—2842no—
264C4⋊C4—2842no—
265C4⋊C4—1422no—
266C22⋊C4—2842no—
267C22⋊C4—2842no—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#1455605312yes—
279C23C2×(C2×D71)⋊C411yescharacteristic
280Q8—5682no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—2841no—
298C2×C4—2841no—
299C2×C4—2841no—
300C2×C4—2841no—
301C2×C4—2841no—
302C2×C4—2841no—
303C2×C4—2841no—
304C2×C4—2841no—
305C2×C4—2841no—
306C2×C4—2841no—
307C2×C4—2841no—
308C2×C4—2841no—
309C2×C4#1455604212yes—
310C2×C4#1455605312yes—
311C2×C4#1455605612yes—
312C2×C4#1455613412yes—
313C2×C4C2×#1176585411yescharacteristic
314C2×C4C2×Dic28411yescharacteristic
315C2×C4#1455614812yes—
316C8—44no—
317C8—5682no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#1798596511yescharacteristic
326C22C2×#1455604211yescharacteristic
327C22#1798598111yescharacteristic
328C22C2×#1455605311yescharacteristic
329C22#1798599311yescharacteristic
330C22C2×#1455605611yescharacteristic
331C22#1798601511yescharacteristic
332C4—21no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—5681no—
340C4—5681no—
341C4—5681no—
342C4—5681no—
343C2—42no—
344C2#2258668611yescharacteristic
345C2#2258674611yescharacteristic
346C2#2258679611yescharacteristic
347C2#2258682911yescharacteristic
348C2#2258691311yescharacteristic
349C2#2258694511yescharacteristic
350C2#2258699511yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×C22⋊C8×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC4×C22⋊C8D71
hypocenterC71#7661
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #22586165C2semidirect
4. #22586164C2semidirect
181. C71#7661semidirect

Maximal subgroups

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

subgroupindex
#225966792
#225883202
#225861652
#225861642
C4×C22⋊C8×C712
#766171