Finite Group Databasebeta

← #31892489

Subgroups — #31892489

Order 18176.

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

normal subgroups only

This group has 616 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×#17985938C211yesmaximal, characteristic
2#22590958C211yesmaximal, characteristic
3#22588431C212yesmaximal
4#22586220C212yesmaximal
5C2×#324×C71C211yesmaximal, characteristic
6C2×#14556087—21no—
7C22×C4⋊Dic71C2211yescharacteristic
8#17985938—22no—
9#17985938C414yescomplemented
10#17985898—24no—
11#17985859—21no—
12#17985859C2212yes—
13C2×C71⋊#95—21no—
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×C22⋊Dic71—22no—
32C2×C4×Dic71—22no—
33C71⋊#95—22no—
34C71⋊#95—24no—
35C71⋊#95—24no—
36C71⋊#95—21no—
37C71⋊#95—22no—
38C71⋊#95C2×C412yes—
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C23×C4×C71C2311yescharacteristic
42C22×C8×C71—22no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71C2×C412yes—
46C2×C4⋊C4×C71D412yes—
47C22⋊C8×C71—22no—
48C22⋊C8×C71—22no—
49C22×Dic71—22no—
50C22×Dic71—22no—
51C22×Dic71—24no—
52C22×Dic71—42no—
53C22×Dic71—42no—
54C22×Dic71—44no—
55C22×Dic71—41no—
56C22×Dic71—41no—
57C22×Dic71—41no—
58C22×Dic71—41no—
59C4⋊Dic71—24no—
60C4⋊Dic71—44no—
61C4⋊Dic71—42no—
62C4⋊Dic71—44no—
63C4⋊Dic71—82no—
64C22⋊Dic71—42no—
65C22⋊Dic71—42no—
66C4×Dic71—44no—
67C24×C71C2×D411yescharacteristic
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71SD1612yes—
79C22×C4×C71Q1612yes—
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D412yes—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71#4112yes—
84C2×C8×C71—44no—
85C2×C8×C71—42no—
86C4⋊C4×C71—24no—
87C4⋊C4×C71—24no—
88C23×C71—22no—
89C23×C71—22no—
90C23×C71C2×C22⋊C411yescharacteristic
91C23×C71#12011yescharacteristic
92C23×C71#12311yescharacteristic
93C23×C71C2×SD1611yescharacteristic
94C23×C71C2×Q1611yescharacteristic
95C23×C71AGL(1,ℤ/8)11yescharacteristic
96C23×C71#13711yescharacteristic
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—24no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—81no—
111C2×C4×C71#10314yes—
112C2×C4×C71C4×D412yes—
113C2×C4×C71#12112yes—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—44no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—44no—
120C2×Dic71—44no—
121C2×Dic71—44no—
122C2×Dic71—82no—
123C2×Dic71—82no—
124C2×Dic71—82no—
125C2×Dic71—82no—
126C8×C71—82no—
127C22×C71—22no—
128C22×C71—22no—
129C22×C71—22no—
130C22×C71—22no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71#38611yescharacteristic
136C22×C71C2×#10311yescharacteristic
137C22×C71#41711yescharacteristic
138C22×C71#45011yescharacteristic
139C22×C71#45111yescharacteristic
140C22×C71#45412yes—
141C22×C71#45812yes—
142C22×C71#48311yescharacteristic
143C22×C71#48411yescharacteristic
144C4×C71—24no—
145C4×C71—22no—
146C4×C71—22no—
147C4×C71—42no—
148Dic71—82no—
149Dic71—84no—
150Dic71—82no—
151#9023—711nomaximal, Sylow
152C2×C71—21no—
153C2×C71—21no—
154C2×C71—21no—
155C2×C71—21no—
156C2×C71#161612yes—
157C2×C71#161712yes—
158C2×C71#187211yescharacteristic
159C2×C71#188811yescharacteristic
160C2×C71#189011yescharacteristic
161C2×#400—711no—
162#2316—711no—
163#1824—712no—
164C2×#324D7111yescharacteristic
165#1294—712no—
166C71#902311yescharacteristic, complemented, Sylow
167C2×#117—1421no—
168C22×C4⋊C4—711no—
169C22×C4⋊C4C2×D7111yescharacteristic
170C2×C22⋊C8C2×D7112yes—
171#400—714no—
172#400—1422no—
173#389—1424no—
174#380—712no—
175#380—1421no—
176C2×#95—711no—
177C2×#95—1421no—
178C2×#95—1421no—
179#340—1424no—
180#324—22no—
181C23×C4—1421no—
182C23×C4—1421no—
183C23×C4D28411yescharacteristic
184C23×C4D28411yescharacteristic
185C23×C4C22×D7111yescharacteristic
186C22×C8—22no—
187#117—2842no—
188#117—2842no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—714no—
192C2×C4⋊C4—712no—
193C2×C4⋊C4—1422no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4—1422no—
196C2×C4⋊C4—1422no—
197C2×C4⋊C4—1422no—
198C2×C4⋊C4C4×D7112yes—
199C2×C4⋊C4C22⋊D7112yes—
200C2×C22⋊C4—1422no—
201C2×C42—1422no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#95—712no—
205#95—1422no—
206#95—1424no—
207#95—1424no—
208#95—1422no—
209#95—1421no—
210C24C2×D28411yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—2842no—
220C22×C4—2842no—
221C22×C4—2844no—
222C22×C4—2841no—
223C22×C4—2841no—
224C22×C4—2841no—
225C22×C4—2841no—
226C22×C4—1422no—
227C22×C4—1422no—
228C22×C4—1424no—
229C22×C4(C2×D71)⋊C412yes—
230C22×C4(C2×D71)⋊C412yes—
231C22×C4#1176585412yes—
232C22×C4Dic28412yes—
233C22×C4C2×C4×D7111yescharacteristic
234C22×C4D4×D7112yes—
235C22×C4C2×C22⋊D7111yescharacteristic
236C22×C4#1176586412yes—
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—2844no—
242C4⋊C4—2844no—
243C4⋊C4—2842no—
244C4⋊C4—5682no—
245C4⋊C4—1424no—
246C22⋊C4—2842no—
247C22⋊C4—2842no—
248C42—2844no—
249C23—22no—
250C23—22no—
251C23C2×(C2×D71)⋊C411yescharacteristic
252C23#1455609811yescharacteristic
253C23#1455611211yescharacteristic
254C23C2×#1176585411yescharacteristic
255C23C2×Dic28411yescharacteristic
256C23#1455615611yescharacteristic
257C23#1455616111yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—2844no—
275C2×C4—2842no—
276C2×C4—2844no—
277C2×C4—2844no—
278C2×C4—2844no—
279C2×C4—2842no—
280C2×C4—5682no—
281C2×C4—5682no—
282C2×C4—5682no—
283C2×C4—5682no—
284C2×C4#1455605314yes—
285C2×C4(C22⋊D71)⋊C412yes—
286C2×C4#1455610012yes—
287C8—82no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1798587611yescharacteristic
297C22C2×#1455605311yescharacteristic
298C22#1798600111yescharacteristic
299C22#1798615211yescharacteristic
300C22#1798615711yescharacteristic
301C22#1798617412yes—
302C22#1798619012yes—
303C22#1798633011yescharacteristic
304C22#1798633511yescharacteristic
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—5682no—
310C4—5682no—
311C4—5684no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2258737912yes—
317C2#2258738412yes—
318C2#2258864511yescharacteristic
319C2#2258874111yescharacteristic
320C2#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#9023
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
9. #17985938C4semidirect
166. C71#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#179859382
#225909582
#225884312
#225862202
C2×#324×C712
#902371