Finite Group Databasebeta

← #31885882

Subgroups — #31885882

Order 18176.

488 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596679C211yesmaximal, characteristic
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
3#22586165C212yesmaximal, complemented
4#22586164C212yesmaximal, complemented
5C71×#1749C211yesmaximal, characteristic
6C2×#14556107C411yescharacteristic
7C2×C22⋊(C71⋊C8)—21no—
8C2×C4×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×#97×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—
34C22⋊(C71⋊C8)—24no—
35C4×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—
53#97×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—24no—
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—22no—
78C22×C4×C71—22no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C2×C8×C71—24no—
89C2×C8×C71—42no—
90C2×C8×C71—42no—
91C42×C71—22no—
92C42×C71—22no—
93C42×C71—41no—
94C42×C71—41no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71#10412yes—
102C23×C71C2×C22⋊C411yescharacteristic
103C2×C4×C71—21no—
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—44no—
117C2×C4×C71—44no—
118C2×C4×C71—44no—
119C2×C4×C71#9912yes—
120C2×C4×C71#10412yes—
121C2×C4×C71#13112yes—
122C2×C4×C71#13512yes—
123C2×C4×C71#13712yes—
124C2×Dic71—41no—
125C2×Dic71—41no—
126C2×Dic71—41no—
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—
136Dic142—82no—
137C8×C71—44no—
138C71⋊C8—44no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71—22no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71#40811yescharacteristic
147C22×C71C2×#9911yescharacteristic
148C22×C71#41311yescharacteristic
149C22×C71#41611yescharacteristic
150C22×C71#41911yescharacteristic
151C22×C71C2×#10411yescharacteristic
152C22×C71C2×#10411yescharacteristic
153C4×C71—22no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—22no—
157C4×C71—22no—
158C4×C71—41no—
159Dic71—81no—
160Dic71—81no—
161Dic71—81no—
162Dic71—81no—
163#7675—711nomaximal, Sylow
164C2×C71—42no—
165C2×C71#146811yescharacteristic
166C2×C71#148511yescharacteristic
167C2×C71#149411yescharacteristic
168C2×C71#150111yescharacteristic
169C2×C71#150811yescharacteristic
170C2×C71#152411yescharacteristic
171C2×C71#155411yescharacteristic
172#2855—711no—
173#1749D7111yescharacteristic
174C4×C22⋊C8—711no—
175#1272—712no—
176#1271—712no—
177C71#767511yescharacteristic, complemented, Sylow
178C2×C22⋊Q8—711no—
179C22×C42C2×D7111yescharacteristic
180C2×C22⋊C8—21no—
181C2×C22⋊C8—1421no—
182C2×#97—21no—
183C2×C4×C8—1421no—
184#391—711no—
185#391—711no—
186#389—711no—
187#389—711no—
188#387—1421no—
189#387—1421no—
190#385—1421no—
191#381—1421no—
192#380—711no—
193#336—712no—
194#336C2×D7112yes—
195C22×Q8—711no—
196C23×C4C4×D7111yescharacteristic
197C23×C4D28411yescharacteristic
198C23×C4C22⋊D7111yescharacteristic
199C22×C8—22no—
200C22×C8—1422no—
201C22⋊Q8—2842no—
202C2×C4⋊C4—711no—
203C2×C4⋊C4—711no—
204C2×C4⋊C4—711no—
205C2×C4⋊C4—1421no—
206C2×C4⋊C4—1421no—
207C2×C4⋊C4—1421no—
208C2×C22⋊C4—1421no—
209C2×C22⋊C4—1421no—
210C2×C42—42no—
211C2×C42C4×D7111yescharacteristic
212C2×C42D28411yescharacteristic
213C2×C42C22⋊D7111yescharacteristic
214C2×C42C22×D7111yescharacteristic
215C22⋊C8—24no—
216C22⋊C8—1424no—
217#97—42no—
218C4×C8—2842no—
219#95—1421no—
220#95—1421no—
221#95—1421no—
222#95—1421no—
223#95—1421no—
224#95—1421no—
225#95—1421no—
226C24(C2×D71)⋊C411yescharacteristic
227C2×Q8—2841no—
228C2×Q8—2841no—
229C2×Q8—1422no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—1421no—
240C22×C4—1421no—
241C22×C4—1421no—
242C22×C4—1421no—
243C22×C4(C2×D71)⋊C411yescharacteristic
244C22×C4(C2×D71)⋊C411yescharacteristic
245C22×C4(C2×D71)⋊C411yescharacteristic
246C22×C4C2×C4×D7111yescharacteristic
247C22×C4C2×C22⋊D7111yescharacteristic
248C22×C4C2×D28411yescharacteristic
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—2842no—
253C2×C8—2842no—
254C2×C8—1424no—
255C4⋊C4—2841no—
256C4⋊C4—2841no—
257C4⋊C4—2842no—
258C4⋊C4—2842no—
259C4⋊C4—2842no—
260C4⋊C4—1422no—
261C22⋊C4—2842no—
262C22⋊C4—2842no—
263C42—22no—
264C42—22no—
265C42—41no—
266C42—41no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#1455605412yes—
274C23C2×(C2×D71)⋊C411yescharacteristic
275Q8—5682no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
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—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—2841no—
293C2×C4—2841no—
294C2×C4—2841no—
295C2×C4—2841no—
296C2×C4—2841no—
297C2×C4—2841no—
298C2×C4—2841no—
299C2×C4—2841no—
300C2×C4—2841no—
301C2×C4—2841no—
302C2×C4—2841no—
303C2×C4—2841no—
304C2×C4#1455604212yes—
305C2×C4#1455605612yes—
306C2×C4#1455613612yes—
307C2×C4#1455614812yes—
308C2×C4#1455616112yes—
309C8—44no—
310C8—2844no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22#1798596711yescharacteristic
319C22C2×#1455604211yescharacteristic
320C22#1798598111yescharacteristic
321C22#1798599311yescharacteristic
322C22#1798600911yescharacteristic
323C22C2×#1455605411yescharacteristic
324C22C2×#1455605611yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—41no—
331C4—5681no—
332C4—5681no—
333C4—5681no—
334C4—5681no—
335C2—42no—
336C2#2258668911yescharacteristic
337C2#2258676211yescharacteristic
338C2#2258679811yescharacteristic
339C2#2258682911yescharacteristic
340C2#2258686111yescharacteristic
341C2#2258692611yescharacteristic
342C2#2258705111yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1749C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1749D71
hypocenterC71#7675
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #22586165C2semidirect
4. #22586164C2semidirect
177. C71#7675semidirect

Maximal subgroups

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

subgroupindex
#225966792
C4×C22⋊(C71⋊C8)2
#225861652
#225861642
C71×#17492
#767571