Finite Group Databasebeta

← #31893175

Subgroups — #31893175

Order 18176.

694 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 135 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
2C2×#17985845C211yesmaximal, characteristic
3C2×#17985737C212yesmaximal
4#22586220C212yesmaximal
5C71×#1817C211yesmaximal, characteristic
6C2×#14556107C2211yescharacteristic
7C22×C4×Dic71—21no—
8#17985905C412yescomplemented
9#17985897C412yescomplemented
10#17985883—22no—
11#17985872—21no—
12#17985862—21no—
13#17985859C2211yescharacteristic
14#17985845—24no—
15C2×C71⋊#95—21no—
16C2×C71⋊#95C2212yes—
17#17985737—24no—
18C22×C42×C71C2211yescharacteristic
19C2×C22⋊C8×C71C2212yes—
20#336×C71—22no—
21C22×Dic142C2×C411yescharacteristic
22C23×Dic71—22no—
23#14556107—22no—
24#14556107—22no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—21no—
29C2×C22⋊Dic71—21no—
30C2×C22⋊Dic71—21no—
31C2×C4×Dic71—24no—
32C2×C4×Dic71—28no—
33C71⋊#95—24no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—22no—
38C71⋊#95—21no—
39C71⋊#95D414yes—
40C23×C4×C71D411yescharacteristic
41C23×C4×C71D411yescharacteristic
42C23×C4×C71C2311yescharacteristic
43C22×C8×C71—22no—
44C2×C42×C71—22no—
45C2×C42×C71—22no—
46C2×C42×C71C2×C412yes—
47C2×C42×C71D412yes—
48C22⋊C8×C71—22no—
49C22⋊C8×C71—22no—
50C2×Dic142—22no—
51C2×Dic142—42no—
52C22×Dic71—28no—
53C22×Dic71—24no—
54C22×Dic71—21no—
55C22×Dic71—22no—
56C22×Dic71—21no—
57C22×Dic71—44no—
58C22×Dic71—42no—
59C22×Dic71—42no—
60C4⋊Dic71—22no—
61C4⋊Dic71—42no—
62Dic71⋊C4—44no—
63Dic71⋊C4—42no—
64C22⋊Dic71—41no—
65C22⋊Dic71—41no—
66C22⋊Dic71—42no—
67C4×Dic71—216no—
68C24×C71C2×D411yescharacteristic
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—24no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D412yes—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4112yes—
83C2×C8×C71—44no—
84C2×C8×C71—42no—
85C42×C71—24no—
86C42×C71—24no—
87C23×C71—22no—
88C23×C71—22no—
89C23×C71C2×C22⋊C411yescharacteristic
90C23×C71#12011yescharacteristic
91C23×C71#12311yescharacteristic
92C23×C71#13512yes—
93C23×C71#13712yes—
94C2×C4×C71—24no—
95C2×C4×C71—24no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71#10414yes—
108C2×C4×C71C4×D412yes—
109C2×C4×C71#12112yes—
110C2×Dic71—28no—
111C2×Dic71—44no—
112C2×Dic71—48no—
113C2×Dic71—48no—
114C2×Dic71—44no—
115C2×Dic71—41no—
116C2×Dic71—41no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122Dic142—82no—
123C8×C71—82no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—24no—
129C22×C71—24no—
130C22×C71#38611yescharacteristic
131C22×C71#41012yes—
132C22×C71#41611yescharacteristic
133C22×C71#41911yescharacteristic
134C22×C71C2×#10412yes—
135C22×C71#45212yes—
136C22×C71#48512yes—
137C4×C71—24no—
138C4×C71—22no—
139C4×C71—22no—
140C4×C71—42no—
141Dic71—48no—
142Dic71—81no—
143Dic71—81no—
144Dic71—82no—
145#9145—711nomaximal, Sylow
146C2×C71—22no—
147C2×C71—22no—
148C2×C71#150012yes—
149C2×C71#150612yes—
150C2×C71#187311yescharacteristic
151C2×C71#189611yescharacteristic
152C2×C71#189911yescharacteristic
153#2855—711no—
154C2×#376—711no—
155#1817D7111yescharacteristic
156C2×#328—712no—
157#1294—712no—
158C71#914511yescharacteristic, complemented, Sylow
159C2×C22⋊Q8—711no—
160C22×C42—1421no—
161C22×C42C2×D7111yescharacteristic
162C2×C22⋊C8C2×D7112yes—
163#391—712no—
164#389—712no—
165#387—1422no—
166#385—1421no—
167#381—1421no—
168#380—711no—
169#376—1424no—
170C2×#95—712no—
171C2×#95—1421no—
172#336—22no—
173#328—1424no—
174C22×Q8—711no—
175C23×C4—1422no—
176C23×C4D28411yescharacteristic
177C23×C4D28411yescharacteristic
178C23×C4C22×D7111yescharacteristic
179C22×C8—22no—
180C22⋊Q8—1422no—
181C22⋊Q8—1422no—
182C2×C4⋊C4—712no—
183C2×C4⋊C4—711no—
184C2×C4⋊C4—1422no—
185C2×C4⋊C4—1421no—
186C2×C22⋊C4—1421no—
187C2×C22⋊C4—1421no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—1424no—
191C2×C42—1428no—
192C2×C42C4×D7112yes—
193C2×C42C22⋊D7112yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—714no—
197#95—1424no—
198#95—1422no—
199#95—1422no—
200#95—1422no—
201#95—1422no—
202#95—1421no—
203C24C2×D28411yescharacteristic
204C2×Q8—2842no—
205C2×Q8—1422no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—2844no—
215C22×C4—2842no—
216C22×C4—2842no—
217C22×C4—1422no—
218C22×C4—1421no—
219C22×C4—1421no—
220C22×C4—1428no—
221C22×C4—1424no—
222C22×C4(C2×D71)⋊C412yes—
223C22×C4(C2×D71)⋊C412yes—
224C22×C4C2×C4×D7111yescharacteristic
225C22×C4D4×D7112yes—
226C22×C4C2×C22⋊D7111yescharacteristic
227C22×C4#1176586412yes—
228C2×C8—44no—
229C2×C8—42no—
230C4⋊C4—2842no—
231C4⋊C4—2844no—
232C4⋊C4—2842no—
233C4⋊C4—1422no—
234C22⋊C4—2841no—
235C22⋊C4—2841no—
236C22⋊C4—2842no—
237C42—24no—
238C42—24no—
239C42—14216no—
240C23—22no—
241C23—22no—
242C23C2×(C2×D71)⋊C411yescharacteristic
243C23#1455609811yescharacteristic
244C23#1455611211yescharacteristic
245C23#1455615012yes—
246C23#1455616112yes—
247Q8—5682no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—2844no—
262C2×C4—2848no—
263C2×C4—2848no—
264C2×C4—2844no—
265C2×C4—2841no—
266C2×C4—2841no—
267C2×C4—2842no—
268C2×C4—2842no—
269C2×C4—2842no—
270C2×C4—2842no—
271C2×C4—2842no—
272C2×C4—1428no—
273C2×C4#1455605414yes—
274C2×C4(C22⋊D71)⋊C412yes—
275C2×C4#1455610012yes—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22#1798587611yescharacteristic
284C22#1798597212yes—
285C22#1798599611yescharacteristic
286C22#1798600911yescharacteristic
287C22C2×#1455605412yes—
288C22#1798616312yes—
289C22#1798634112yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—2848no—
295C4—5681no—
296C4—5681no—
297C4—5682no—
298C2—22no—
299C2—22no—
300C2#2258682712yes—
301C2#2258685312yes—
302C2#2258865011yescharacteristic
303C2#2258878111yescharacteristic
304C2#2258879611yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1817C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#1817D71
hypocenterC71#9145
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #17985905C4semidirect
9. #17985897C4semidirect
158. C71#9145semidirect

Maximal subgroups

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

subgroupindex
#225966792
C2×#179858452
C2×#179857372
#225862202
C71×#18172
#914571