Finite Group Databasebeta

← #31895412

Subgroups — #31895412

Order 18176.

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

normal subgroups only

This group has 544 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#22597369C211yesmaximal, characteristic
2#22588420C212yesmaximal, complemented
3#22586678C211yesmaximal, characteristic
4#22586246C212yesmaximal, complemented
5C71×#2486C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8#17985963C2212yes—
9#17985963C2212yes—
10C2×C22⋊(C71⋊C8)—21no—
11#17985768—24no—
12#17985725—21no—
13#17985723—21no—
14C2×#123×C71C412yes—
15C2×C4×D4×C71C2211yescharacteristic
16C22×C4⋊C4×C71C2211yescharacteristic
17#398×C71—21no—
18#397×C71—21no—
19#389×C71—21no—
20#381×C71C412yes—
21#380×C71—21no—
22#380×C71C2211yescharacteristic
23C2×#11765849—22no—
24C2×#11765849D414yes—
25C22×C71⋊C8—22no—
26C4⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33C22×D4×C71C2×C411yescharacteristic
34C23×C4×C71C2×C412yes—
35C23×C4×C71D411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37#123×C71—44no—
38C4×D4×C71—22no—
39C4×D4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—22no—
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71—22no—
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71C2×C412yes—
50C2×C22⋊C4×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—22no—
53#95×C71—22no—
54#95×C71—21no—
55#11765849—44no—
56#11765849—44no—
57C2×C71⋊C8—24no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—82no—
60C24×C71C22⋊C411yescharacteristic
61C24×C71C22×C411yescharacteristic
62C2×D4×C71—22no—
63C2×D4×C71—22no—
64C2×D4×C71—41no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—24no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C2×D412yes—
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71#4112yes—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—44no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—41no—
87C22⋊C4×C71—22no—
88C22⋊C4×C71—22no—
89C22⋊C4×C71—44no—
90C22⋊C4×C71—42no—
91C42×C71—21no—
92C42×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71#10012yes—
102C23×C71C2×C22⋊C411yescharacteristic
103C23×C71C4×D412yes—
104D4×C71—44no—
105C2×C4×C71—22no—
106C2×C4×C71—24no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—41no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—81no—
124C2×C4×C71—84no—
125C2×C4×C71#12011yescharacteristic
126C2×C4×C71#12112yes—
127C2×C4×C71#12311yescharacteristic
128C71⋊C8—44no—
129C71⋊C8—82no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—22no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—22no—
136C22×C71—42no—
137C22×C71—42no—
138C22×C71—42no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71—81no—
142C22×C71#38611yescharacteristic
143C22×C71C2×#10011yescharacteristic
144C22×C71#41311yescharacteristic
145C4×C71—22no—
146C4×C71—21no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—82no—
151C4×C71—82no—
152C4×C71#45812yes—
153#9507—711nomaximal, Sylow
154C2×C71—21no—
155C2×C71—21no—
156C2×C71—42no—
157C2×C71—42no—
158C2×C71#189011yescharacteristic
159C2×C71#190011yescharacteristic
160C2×C71#191511yescharacteristic
161#2973—711no—
162#2486D7111yescharacteristic
163#1821—712no—
164#1465—711no—
165#1305—712no—
166C71#950711yescharacteristic, complemented, Sylow
167C2×#123Dic7112yes—
168C2×C4×D4C2×D7111yescharacteristic
169C22×C4⋊C4C2×D7111yescharacteristic
170#435—1424no—
171#432—1421no—
172#407—712no—
173#407—712no—
174C2×C22⋊C8—1421no—
175#398—21no—
176#397—21no—
177#389—21no—
178#381Dic7112yes—
179#380—21no—
180#380C2×D7111yescharacteristic
181#343—1424no—
182#324—1421no—
183#323—1421no—
184C22×D4C2×Dic7111yescharacteristic
185C23×C4C2×Dic7112yes—
186C23×C4C22⋊D7111yescharacteristic
187C23×C4C22×D7111yescharacteristic
188C2×M16—714no—
189C2×M16—1422no—
190C22×C8—1422no—
191#123—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C22⋊D7111yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4C2×Dic7111yescharacteristic
203C2×C22⋊C4C2×Dic7112yes—
204C2×C22⋊C4C22⋊D7111yescharacteristic
205C2×C42C22⋊D7111yescharacteristic
206C4⋊C8—1422no—
207C22⋊C8—1422no—
208C22⋊C8—1422no—
209C22⋊C8—1422no—
210C22⋊C8—1422no—
211C22⋊C8—1422no—
212#97—1422no—
213#95—22no—
214#95—22no—
215#95—21no—
216C24C22⋊Dic7111yescharacteristic
217C24C22×Dic7111yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊Dic7111yescharacteristic
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4D4×D7112yes—
234C22×C4C2×C22⋊D7111yescharacteristic
235C22×C4C2×C22⋊D7111yescharacteristic
236C22×C4#1176586412yes—
237M16—2844no—
238M16—2844no—
239C2×C8—2842no—
240C2×C8—5682no—
241C2×C8—1424no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#1455604512yes—
263C23C2×C22⋊Dic7111yescharacteristic
264C23D4×Dic7112yes—
265D4—44no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—84no—
286C2×C4#1455609611yescharacteristic
287C2×C4#1455610012yes—
288C2×C4#1455610911yescharacteristic
289C8—2844no—
290C8—5682no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1798587911yescharacteristic
304C22C2×#1455604511yescharacteristic
305C22#1798598311yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1798618812yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2258875411yescharacteristic
319C2#2258880411yescharacteristic
320C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2486C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC23×C4×C71C23
hypercenter#2486D71
hypocenterC71#9507
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #22588420C2semidirect
4. #22586246C2semidirect
166. C71#9507semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225884202
#225866782
#225862462
C71×#24862
#950771