Finite Group Databasebeta

← #31894829

Subgroups — #31894829

Order 18176.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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#22597420C211yesmaximal, characteristic
2#22593556C211yesmaximal, characteristic
3#22588333C211yesmaximal, characteristic
4#22587025C211yesmaximal, characteristic
5#22586242C211yesmaximal, characteristic
6#22586241C211yesmaximal, characteristic
7C71×#1549C211yesmaximal, characteristic
8C2×C4×Dic142C2211yescharacteristic
9C8×Dic142—22no—
10#17986123—22no—
11#17986052C2211yescharacteristic
12#17986049—22no—
13#17986023C2211yescharacteristic
14C2×#14556046—21no—
15#17985937C414yes—
16#17985906—22no—
17#17985904—21no—
18#17985864—21no—
19C4×C4⋊Dic71C2211yescharacteristic
20#17985767—21no—
21#17985745—22no—
22C2×C4⋊Q8×C71C2211yescharacteristic
23#431×C71C2211yescharacteristic
24#423×C71C2211yescharacteristic
25#332×C71—22no—
26C22×Dic142D411yescharacteristic
27C2×#11765849—21no—
28C4×Dic142—21no—
29C4×Dic142—21no—
30C4×Dic142—21no—
31C4×Dic142—21no—
32C2×C4⋊Dic71—21no—
33C2×C4⋊Dic71D411yescharacteristic
34C2×Dic71⋊C4C2×C412yes—
35C2×C4×Dic71—21no—
36C2×C4×Dic71D412yes—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)D412yes—
40Dic71⋊C8—24no—
41#14556046—42no—
42C22⋊(C71⋊C8)—21no—
43C4×C71⋊C8—21no—
44C8×Dic71—24no—
45C71⋊#95—24no—
46C71⋊#95—24no—
47C71⋊#95C2×C412yes—
48C22×Q8×C71—21no—
49C2×M16×C71—21no—
50C4⋊Q8×C71—21no—
51C4⋊Q8×C71—21no—
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71—21no—
54C2×C4⋊C4×C71—21no—
55C2×C4⋊C4×C71C2×C412yes—
56C2×C42×C71C2311yescharacteristic
57C4⋊C8×C71—21no—
58C4⋊C8×C71—21no—
59C4⋊C8×C71D412yes—
60C22⋊C8×C71—21no—
61C4×C8×C71—21no—
62C2×Dic142—21no—
63C2×Dic142—21no—
64C2×Dic142—21no—
65C2×Dic142—21no—
66C2×Dic142SD1612yes—
67C2×Dic142Q1612yes—
68C22×Dic71—22no—
69C22×Dic71C22⋊C412yes—
70C22×Dic71C22⋊C412yes—
71#11765849—41no—
72#11765849—41no—
73C2×C71⋊C8—22no—
74C2×C71⋊C8—41no—
75C4⋊Dic71—21no—
76C4⋊Dic71—21no—
77C4⋊Dic71—42no—
78Dic71⋊C4—22no—
79Dic71⋊C4—22no—
80C4×Dic71—22no—
81C4×Dic71—22no—
82C4×Dic71—41no—
83C4×Dic71—41no—
84C2×Q8×C71—44no—
85C2×Q8×C71—41no—
86C2×Q8×C71—41no—
87C22×C4×C71—22no—
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91M16×C71—41no—
92M16×C71—41no—
93C2×C8×C71—22no—
94C2×C8×C71—41no—
95C4⋊C4×C71—22no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C42×C71C2×D411yescharacteristic
100C42×C71C2×D411yescharacteristic
101C42×C71#4111yescharacteristic
102C42×C71#4111yescharacteristic
103C23×C71C2×C22⋊C411yescharacteristic
104Q8×C71—82no—
105Q8×C71—82no—
106C2×C4×C71—22no—
107C2×C4×C71—41no—
108C2×C4×C71—44no—
109C2×C4×C71—42no—
110C2×C4×C71C4×D411yescharacteristic
111C2×C4×C71C4×D411yescharacteristic
112C2×C4×C71#12011yescharacteristic
113C2×C4×C71#12111yescharacteristic
114C2×C4×C71#12111yescharacteristic
115C2×C4×C71#12311yescharacteristic
116C2×C4×C71C2×SD1611yescharacteristic
117C2×C4×C71C2×Q1611yescharacteristic
118C2×C4×C71#13511yescharacteristic
119C2×C4×C71#13511yescharacteristic
120C2×Dic71—22no—
121C2×Dic71—22no—
122C2×Dic71—22no—
123C2×Dic71—42no—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71#10314yes—
128Dic142—22no—
129Dic142—22no—
130Dic142—22no—
131Dic142—22no—
132C8×C71—42no—
133C8×C71—81no—
134C71⋊C8—42no—
135C71⋊C8—81no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71#38611yescharacteristic
139C22×C71C2×#10311yescharacteristic
140C22×C71#41611yescharacteristic
141C4×C71—21no—
142C4×C71—21no—
143C4×C71—21no—
144C4×C71—82no—
145C4×C71#45412yes—
146C4×C71#46511yescharacteristic
147C4×C71#46511yescharacteristic
148C4×C71#46711yescharacteristic
149C4×C71#46711yescharacteristic
150Dic71—24no—
151Dic71—42no—
152Dic71—82no—
153#9396—711nomaximal, Sylow
154C2×C71—21no—
155C2×C71—21no—
156C2×C71#188811yescharacteristic
157C2×C71#189211yescharacteristic
158C2×C71#191511yescharacteristic
159#2982—711no—
160#2568—711no—
161#1801—711no—
162#1549—711no—
163#1549D7111yescharacteristic
164#1303—711no—
165#1303—711no—
166C71#939611yescharacteristic, complemented, Sylow
167C2×C4⋊Q8C2×D7111yescharacteristic
168C2×C4×Q8—711no—
169C8×Q8—1422no—
170C8×Q8—1422no—
171#431—711no—
172#431—1422no—
173#431C2×D7111yescharacteristic
174#423—711no—
175#423C2×D7111yescharacteristic
176C2×#101—1421no—
177#400—714no—
178#391—1422no—
179#391—1421no—
180#382—1421no—
181C4×C4⋊C4—711no—
182#343—1421no—
183#332—22no—
184#332—1422no—
185C22×Q8—21no—
186C22×Q8—711no—
187C2×M16—21no—
188C2×M16—1421no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4×Q8—1421no—
194C4×Q8—1421no—
195C4×Q8—1421no—
196C4×Q8—1421no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—712no—
199C2×C4⋊C4—711no—
200C2×C4⋊C4—1421no—
201C2×C4⋊C4C4×D7112yes—
202C2×C42—712no—
203C2×C42—1421no—
204C2×C42C22×D7111yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—712no—
208C4⋊C8—1424no—
209C4⋊C8—1421no—
210C4⋊C8—1421no—
211C4⋊C8C22⋊D7112yes—
212#101—2842no—
213C22⋊C8—21no—
214C22⋊C8—1421no—
215C4×C8—21no—
216C4×C8—1424no—
217C4×C8—1421no—
218#95—712no—
219#95—1424no—
220#95—1424no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—712no—
225C2×Q8—712no—
226C2×Q8—1421no—
227C2×Q8—1421no—
228C2×Q8—1421no—
229C2×Q8—1421no—
230C22×C4—22no—
231C22×C4—712no—
232C22×C4—712no—
233C22×C4—1422no—
234C22×C4C2×C4×D7111yescharacteristic
235C22×C4D4×D7111yescharacteristic
236C22×C4D4×D7111yescharacteristic
237M16—41no—
238M16—41no—
239M16—2841no—
240M16—2841no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—2841no—
244C2×C8—1422no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—2842no—
250C4⋊C4—1422no—
251C4⋊C4—1422no—
252C4⋊C4—1421no—
253C4⋊C4—1421no—
254C42—2841no—
255C42—2841no—
256C42—1422no—
257C42—1422no—
258C42D4×D7111yescharacteristic
259C42C2×C22⋊D7111yescharacteristic
260C42#1176586411yescharacteristic
261C42#1176586511yescharacteristic
262C23C22⋊C4×D7111yescharacteristic
263Q8—82no—
264Q8—82no—
265Q8—1422no—
266Q8—1422no—
267Q8—1422no—
268Q8—1422no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—2842no—
274C2×C4—2842no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4—714no—
278C2×C4—1422no—
279C2×C4—1422no—
280C2×C4—1422no—
281C2×C4(C22⋊D71)⋊C411yescharacteristic
282C2×C4C4×C22⋊D7111yescharacteristic
283C2×C4#1455609611yescharacteristic
284C2×C4#1455609911yescharacteristic
285C2×C4#1455610111yescharacteristic
286C2×C4#1455611011yescharacteristic
287C2×C4SD16×D7111yescharacteristic
288C2×C4Q16×D7111yescharacteristic
289C2×C4#1455614911yescharacteristic
290C2×C4#1455615111yescharacteristic
291C8—42no—
292C8—81no—
293C8—2842no—
294C8—5681no—
295C22—21no—
296C22—21no—
297C22#1798587511yescharacteristic
298C22#103×D7111yescharacteristic
299C22#1798599411yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—2842no—
305C4—5682no—
306C4—1424no—
307C4#1798617012yes—
308C4#1798622511yescharacteristic
309C4#1798623111yescharacteristic
310C4#1798623711yescharacteristic
311C4#1798623911yescharacteristic
312C2—21no—
313C2—21no—
314C2#2258874011yescharacteristic
315C2#2258876011yescharacteristic
316C2#2258888911yescharacteristic
subgroupG/H
centerC22#17985875
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1549C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1549D71
hypocenterC71#9396
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
166. C71#9396semidirect

Maximal subgroups

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

subgroupindex
#225974202
#225935562
#225883332
#225870252
#225862422
#225862412
C71×#15492
#939671