Finite Group Databasebeta

← #31892509

Subgroups — #31892509

Order 18176.

660 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#17985929C211yesmaximal, characteristic
2#22590958C211yesmaximal, characteristic
3#22588427C212yesmaximal
4#22586220C212yesmaximal
5C2×#324×C71C211yesmaximal, characteristic
6C2×#14556087—21no—
7C22×C4⋊Dic71C2211yescharacteristic
8C22×Dic71⋊C4—21no—
9#17985929—22no—
10#17985929C414yescomplemented
11#17985898—24no—
12#17985859—21no—
13#17985859C2212yes—
14C2×C71⋊#95—21no—
15C2×C71⋊#95C2211yescharacteristic
16#17985762—24no—
17C22×C4⋊C4×C71C2211yescharacteristic
18C2×C22⋊C8×C71C2212yes—
19#324×C71—22no—
20C23×Dic71—21no—
21C23×Dic71—21no—
22#14556087—42no—
23#14556087—42no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71—22no—
26C2×C4⋊Dic71—22no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—22no—
29C2×C4⋊Dic71C2×C412yes—
30C2×C4⋊Dic71D414yes—
31C2×Dic71⋊C4—24no—
32C2×Dic71⋊C4—42no—
33C2×Dic71⋊C4—42no—
34C2×C22⋊Dic71—22no—
35C2×C4×Dic71—22no—
36C71⋊#95—22no—
37C71⋊#95—24no—
38C71⋊#95—21no—
39C71⋊#95—22no—
40C71⋊#95C2×C412yes—
41C23×C4×C71D411yescharacteristic
42C23×C4×C71D411yescharacteristic
43C23×C4×C71C2311yescharacteristic
44C22×C8×C71—22no—
45C2×C4⋊C4×C71—22no—
46C2×C4⋊C4×C71—22no—
47C2×C4⋊C4×C71C2×C412yes—
48C2×C4⋊C4×C71D412yes—
49C22⋊C8×C71—22no—
50C22⋊C8×C71—22no—
51C22×Dic71—24no—
52C22×Dic71—22no—
53C22×Dic71—22no—
54C22×Dic71—24no—
55C22×Dic71—42no—
56C22×Dic71—41no—
57C22×Dic71—41no—
58C22×Dic71—41no—
59C22×Dic71—41no—
60C22×Dic71—41no—
61C22×Dic71—41no—
62C22×Dic71—41no—
63C22×Dic71—41no—
64C4⋊Dic71—24no—
65C4⋊Dic71—44no—
66C4⋊Dic71—42no—
67C4⋊Dic71—44no—
68C4⋊Dic71—82no—
69Dic71⋊C4—48no—
70C22⋊Dic71—42no—
71C22⋊Dic71—42no—
72C4×Dic71—44no—
73C24×C71C2×D411yescharacteristic
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—42no—
81C22×C4×C71—42no—
82C22×C4×C71C22⋊C412yes—
83C22×C4×C71C22⋊C412yes—
84C22×C4×C71D812yes—
85C22×C4×C71SD1612yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D412yes—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71#4112yes—
90C2×C8×C71—44no—
91C2×C8×C71—42no—
92C4⋊C4×C71—24no—
93C4⋊C4×C71—24no—
94C23×C71—22no—
95C23×C71—22no—
96C23×C71C2×C22⋊C411yescharacteristic
97C23×C71#12011yescharacteristic
98C23×C71#12311yescharacteristic
99C23×C71C2×D811yescharacteristic
100C23×C71C2×SD1611yescharacteristic
101C23×C71AGL(1,ℤ/8)11yescharacteristic
102C23×C71#13711yescharacteristic
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—24no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—81no—
117C2×C4×C71#10214yes—
118C2×C4×C71C4×D412yes—
119C2×C4×C71#12112yes—
120C2×Dic71—24no—
121C2×Dic71—44no—
122C2×Dic71—44no—
123C2×Dic71—42no—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—42no—
128C2×Dic71—44no—
129C2×Dic71—44no—
130C2×Dic71—44no—
131C2×Dic71—81no—
132C2×Dic71—81no—
133C2×Dic71—81no—
134C2×Dic71—81no—
135C8×C71—82no—
136C22×C71—22no—
137C22×C71—22no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71#38611yescharacteristic
145C22×C71C2×#10211yescharacteristic
146C22×C71#41711yescharacteristic
147C22×C71#44711yescharacteristic
148C22×C71#44811yescharacteristic
149C22×C71#45612yes—
150C22×C71#45812yes—
151C22×C71#48011yescharacteristic
152C22×C71#48111yescharacteristic
153C4×C71—24no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—42no—
157Dic71—44no—
158Dic71—84no—
159Dic71—82no—
160#9027—711nomaximal, Sylow
161C2×C71—21no—
162C2×C71—21no—
163C2×C71—21no—
164C2×C71—21no—
165C2×C71#161012yes—
166C2×C71#161112yes—
167C2×C71#187411yescharacteristic
168C2×C71#188611yescharacteristic
169C2×C71#189011yescharacteristic
170C2×#398—711no—
171#2316—711no—
172#1823—712no—
173C2×#324D7111yescharacteristic
174#1294—712no—
175C71#902711yescharacteristic, complemented, Sylow
176C2×#117—1421no—
177C22×C4⋊C4—711no—
178C22×C4⋊C4—1421no—
179C22×C4⋊C4C2×D7111yescharacteristic
180C2×C22⋊C8C2×D7112yes—
181#398—714no—
182#398—1422no—
183#389—1424no—
184#380—712no—
185#380—1421no—
186C2×#95—711no—
187C2×#95—1421no—
188#340—1424no—
189#324—22no—
190C23×C4—1421no—
191C23×C4—1421no—
192C23×C4D28411yescharacteristic
193C23×C4D28411yescharacteristic
194C23×C4C22×D7111yescharacteristic
195C22×C8—22no—
196#117—2842no—
197#117—2842no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—2842no—
201C2×C4⋊C4—2842no—
202C2×C4⋊C4—714no—
203C2×C4⋊C4—712no—
204C2×C4⋊C4—1422no—
205C2×C4⋊C4—1424no—
206C2×C4⋊C4—1421no—
207C2×C4⋊C4—1422no—
208C2×C4⋊C4—1422no—
209C2×C4⋊C4—1422no—
210C2×C4⋊C4C4×D7112yes—
211C2×C4⋊C4C22⋊D7112yes—
212C2×C22⋊C4—1422no—
213C2×C42—1422no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—712no—
217#95—1422no—
218#95—1424no—
219#95—1422no—
220#95—1421no—
221C24C2×D28411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—2842no—
231C22×C4—2841no—
232C22×C4—2841no—
233C22×C4—2841no—
234C22×C4—2841no—
235C22×C4—2841no—
236C22×C4—2841no—
237C22×C4—2841no—
238C22×C4—2841no—
239C22×C4—1422no—
240C22×C4—1424no—
241C22×C4—1424no—
242C22×C4—1422no—
243C22×C4(C2×D71)⋊C412yes—
244C22×C4(C2×D71)⋊C412yes—
245C22×C4D56812yes—
246C22×C4#1176585412yes—
247C22×C4C2×C4×D7111yescharacteristic
248C22×C4D4×D7112yes—
249C22×C4C2×C22⋊D7111yescharacteristic
250C22×C4#1176586412yes—
251C2×C8—44no—
252C2×C8—42no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—2844no—
256C4⋊C4—2844no—
257C4⋊C4—2848no—
258C4⋊C4—2842no—
259C4⋊C4—5682no—
260C4⋊C4—1424no—
261C22⋊C4—2842no—
262C22⋊C4—2842no—
263C42—2844no—
264C23—22no—
265C23—22no—
266C23C2×(C2×D71)⋊C411yescharacteristic
267C23#1455609811yescharacteristic
268C23#1455611211yescharacteristic
269C23C2×D56811yescharacteristic
270C23C2×#1176585411yescharacteristic
271C23#1455615611yescharacteristic
272C23#1455616111yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—2842no—
288C2×C4—2844no—
289C2×C4—2844no—
290C2×C4—2842no—
291C2×C4—2842no—
292C2×C4—2844no—
293C2×C4—2844no—
294C2×C4—2844no—
295C2×C4—2842no—
296C2×C4—2842no—
297C2×C4—5681no—
298C2×C4—5681no—
299C2×C4—5681no—
300C2×C4—5681no—
301C2×C4—1424no—
302C2×C4#1455605014yes—
303C2×C4(C22⋊D71)⋊C412yes—
304C2×C4#1455610012yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22#1798587611yescharacteristic
315C22C2×#1455605011yescharacteristic
316C22#1798600111yescharacteristic
317C22#1798613511yescharacteristic
318C22#1798614011yescharacteristic
319C22#1798618412yes—
320C22#1798619012yes—
321C22#1798631311yescharacteristic
322C22#1798631811yescharacteristic
323C4—22no—
324C4—22no—
325C4—24no—
326C4—42no—
327C4—2844no—
328C4—5684no—
329C4—5682no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#2258734512yes—
335C2#2258735012yes—
336C2#2258865511yescharacteristic
337C2#2258873111yescharacteristic
338C2#2258875111yescharacteristic
subgroupG/H
centerC23#14556098
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#324×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#324D71
hypocenterC71#9027
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
10. #17985929C4semidirect
175. C71#9027semidirect

Maximal subgroups

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

subgroupindex
C2×#179859292
#225909582
#225884272
#225862202
C2×#324×C712
#902771