Finite Group Databasebeta

← #31912972

Subgroups — #31912972

Order 18176.

1154 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 139 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×#17985796C214yesmaximal, complemented
2#22588268C212yesmaximal, complemented
3C2×#535×C71C211yesmaximal, characteristic
4C2×#14556047C2214yescomplemented
5C2×#14556045—22no—
6#17985963—22no—
7#17985796—28no—
8C22×#41×C71C2211yescharacteristic
9C23×D4×C71C411yescharacteristic
10#535×C71—28no—
11C2×#127×C71C412yes—
12C2×#124×C71C2212yes—
13C2×#121×C71—22no—
14C2×#120×C71—22no—
15C2×#117×C71C411yescharacteristic
16C2×#11765849—24no—
17#14556047—28no—
18#14556045—44no—
19C22⋊(C71⋊C8)—44no—
20C25×C71—21no—
21C2×#41×C71—24no—
22C2×#41×C71—42no—
23C2×#41×C71—42no—
24C22×Q8×C71C2311yescharacteristic
25C22×D4×C71—24no—
26C22×D4×C71—24no—
27C22×D4×C71—21no—
28C22×D4×C71—42no—
29C22×D4×C71—42no—
30C22×D4×C71C2×C411yescharacteristic
31C22×D4×C71C2×C412yes—
32C22×D4×C71D412yes—
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71D412yes—
35#127×C71—24no—
36#127×C71—24no—
37#124×C71—24no—
38#124×C71D418yes—
39#121×C71—216no—
40#120×C71—216no—
41#117×C71—24no—
42C2×C4⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—24no—
45C2×C42×C71C2×C412yes—
46#11765849—44no—
47#11765849—44no—
48C2×C71⋊C8—44no—
49C24×C71—24no—
50C24×C71—22no—
51C24×C71—42no—
52C24×C71—42no—
53C24×C71—84no—
54C24×C71C22⋊C412yes—
55C24×C71C22⋊C411yescharacteristic
56#41×C71—416no—
57C2×Q8×C71—41no—
58C2×Q8×C71—41no—
59C2×Q8×C71C2×D414yes—
60C2×D4×C71—24no—
61C2×D4×C71—24no—
62C2×D4×C71—216no—
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—24no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71—42no—
69C2×D4×C71—42no—
70C2×D4×C71—42no—
71C2×D4×C71—416no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—88no—
75C2×D4×C71C22⋊C414yes—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—22no—
79C22×C4×C71—22no—
80C22×C4×C71—24no—
81C22×C4×C71—24no—
82C22×C4×C71—42no—
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D412yes—
86C4⋊C4×C71—24no—
87C22⋊C4×C71—24no—
88C22⋊C4×C71—216no—
89C42×C71—22no—
90C42×C71C22⋊C414yes—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—24no—
94C23×C71—22no—
95C23×C71—22no—
96C23×C71—22no—
97C23×C71—42no—
98C23×C71—44no—
99C23×C71—48no—
100C23×C71—48no—
101C23×C71—42no—
102C23×C71—42no—
103C23×C71—88no—
104C23×C71—88no—
105C23×C71—88no—
106C23×C71#9912yes—
107C23×C71C2×C22⋊C411yescharacteristic
108Q8×C71—44no—
109D4×C71—48no—
110D4×C71—44no—
111D4×C71—48no—
112D4×C71—44no—
113D4×C71—44no—
114D4×C71—816no—
115C2×C4×C71—28no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—24no—
123C2×C4×C71—48no—
124C2×C4×C71—44no—
125C2×C4×C71#9912yes—
126C2×C4×C71C2×C22⋊C412yes—
127C2×C4×C71#12014yes—
128C71⋊C8—84no—
129C22×C71—22no—
130C22×C71—22no—
131C22×C71—42no—
132C22×C71—42no—
133C22×C71—44no—
134C22×C71—48no—
135C22×C71—42no—
136C22×C71—44no—
137C22×C71—42no—
138C22×C71—44no—
139C22×C71—82no—
140C22×C71—84no—
141C22×C71—816no—
142C22×C71—84no—
143C22×C71#37911yescharacteristic
144C22×C71C2×#9911yescharacteristic
145C22×C71C2×#9911yescharacteristic
146C4×C71—24no—
147C4×C71—22no—
148C4×C71—22no—
149C4×C71—44no—
150#12703—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71—42no—
154C2×C71—82no—
155C2×C71—84no—
156C2×C71#178111yescharacteristic
157C2×C71#213412yes—
158C2×#535D7111yescharacteristic
159C2×#355—714no—
160#1785—712no—
161C71#1270311yescharacteristic, complemented, Sylow
162C22×#41C2×D7111yescharacteristic
163C23×D4Dic7111yescharacteristic
164#535—28no—
165C2×#127Dic7112yes—
166C2×#124C2×D7112yes—
167C2×#121—22no—
168C2×#120—22no—
169C2×#117Dic7111yescharacteristic
170C2×#101—714no—
171C2×#100—1422no—
172#407—1422no—
173#355—1428no—
174C25—21no—
175C2×#41—24no—
176C2×#41—42no—
177C2×#41—42no—
178C22×Q8C22×D7111yescharacteristic
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—21no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4C2×Dic7111yescharacteristic
185C22×D4C2×Dic7112yes—
186C22×D4C22⋊D7112yes—
187C23×C4C2×Dic7111yescharacteristic
188C23×C4C22⋊D7112yes—
189C2×M16—1424no—
190#127—24no—
191#127—24no—
192#124—24no—
193#124C22⋊D7118yes—
194#121—216no—
195#120—216no—
196#117—24no—
197C2×C4⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—24no—
200C2×C42C2×Dic7112yes—
201#101—1428no—
202#100—2844no—
203C22⋊C8—2844no—
204C24—24no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24—84no—
209C24C22⋊Dic7112yes—
210C24C22⋊Dic7111yescharacteristic
211#41—416no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8C2×C22⋊D7114yes—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—216no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—416no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—88no—
230C2×D4C22⋊Dic7114yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4C22⋊Dic7111yescharacteristic
239C22×C4C22×Dic7111yescharacteristic
240C22×C4C2×C22⋊D7112yes—
241M16—2844no—
242M16—2844no—
243C2×C8—2844no—
244C4⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—216no—
247C42—22no—
248C42C22⋊Dic7114yes—
249C23—21no—
250C23—21no—
251C23—24no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—42no—
256C23—44no—
257C23—48no—
258C23—48no—
259C23—42no—
260C23—42no—
261C23—88no—
262C23—88no—
263C23—88no—
264C23#1455604312yes—
265C23C2×C22⋊Dic7111yescharacteristic
266Q8—44no—
267D4—48no—
268D4—44no—
269D4—48no—
270D4—44no—
271D4—44no—
272D4—816no—
273C2×C4—28no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—48no—
282C2×C4—44no—
283C2×C4#1455604312yes—
284C2×C4C2×C22⋊Dic7112yes—
285C2×C4#1455609714yes—
286C8—5684no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—44no—
292C22—48no—
293C22—42no—
294C22—44no—
295C22—42no—
296C22—44no—
297C22—82no—
298C22—84no—
299C22—816no—
300C22—84no—
301C22#1798585511yescharacteristic
302C22C2×#1455604311yescharacteristic
303C22C2×#1455604311yescharacteristic
304C4—24no—
305C4—22no—
306C4—22no—
307C4—44no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—82no—
312C2—84no—
313C2#2258825511yescharacteristic
314C2#2259003212yes—
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
Fitting, prime radicalC2×#535×C71C2
socleC22×C71C2×#99
cosocleC22×Q8×C71C23
hypercenterC2×#535D71
hypocenterC71#12703
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#17985796C2semidirect
2. #22588268C2semidirect
4. C2×#14556047C22semidirect
161. C71#12703semidirect

Maximal subgroups

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

subgroupindex
C2×#179857962
#225882682
C2×#535×C712
#1270371