Finite Group Databasebeta

← #31895322

Subgroups — #31895322

Order 18176.

602 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 119 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#22597381C211yesmaximal, characteristic, complemented
2#22588332C211yesmaximal, characteristic, complemented
3#22586765C212yesmaximal, complemented
4#22586198C212yesmaximal, complemented
5C71×#2535C211yesmaximal, characteristic
6#17986077—24no—
7#17986052C2212yes—
8#17986023C2212yes—
9C2×#14556045—21no—
10#17985963—22no—
11#17985768—21no—
12D4⋊(C71⋊C8)—24no—
13C2×#121×C71—21no—
14C2×C4×D4×C71C2211yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16#397×C71C412yes—
17#392×C71C412yes—
18#386×C71—22no—
19#382×C71C2211yescharacteristic
20#380×C71—21no—
21C2×#11765849—24no—
22C2×#11765849—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556045—42no—
27C22⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8D414yescomplemented
30C22×D4×C71—22no—
31C22×D4×C71C2×C411yescharacteristic
32C22×D4×C71D411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71D412yes—
35#121×C71—44no—
36C4×D4×C71—22no—
37C4×D4×C71—22no—
38C4×D4×C71—22no—
39C4×D4×C71—22no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71C2×C412yes—
47C2×C42×C71C2311yescharacteristic
48#95×C71—22no—
49#95×C71—21no—
50#95×C71C2×C412yes—
51#11765849—42no—
52#11765849—42no—
53#11765849—84no—
54C2×C71⋊C8—24no—
55C2×C71⋊C8—42no—
56C24×C71—21no—
57C24×C71C22⋊C412yes—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—22no—
61C2×D4×C71—21no—
62C2×D4×C71—21no—
63C2×D4×C71—42no—
64C2×D4×C71—44no—
65C2×D4×C71—44no—
66C2×D4×C71—82no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71—41no—
74C22×C4×C71—41no—
75C22×C4×C71—82no—
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C4⋊C4×C71—22no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—42no—
83C22⋊C4×C71—24no—
84C22⋊C4×C71—44no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—82no—
88C42×C71C2×D412yes—
89C42×C71#4112yes—
90C23×C71—22no—
91C23×C71—22no—
92C23×C71—22no—
93C23×C71—22no—
94C23×C71—42no—
95C23×C71—42no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71C2×C22⋊C411yescharacteristic
103D4×C71—24no—
104D4×C71—42no—
105D4×C71—42no—
106D4×C71—42no—
107D4×C71—88no—
108C2×C4×C71—22no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—44no—
112C2×C4×C71—42no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—42no—
119C2×C4×C71—82no—
120C2×C4×C71—82no—
121C2×C4×C71—82no—
122C2×C4×C71C4×D412yes—
123C2×C4×C71#12011yescharacteristic
124C2×C4×C71#12112yes—
125C2×C4×C71#12311yescharacteristic
126C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C71AGL(1,ℤ/8)12yes—
128C2×C4×C71#13711yescharacteristic
129C71⋊C8—44no—
130C71⋊C8—82no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—42no—
134C22×C71—44no—
135C22×C71—42no—
136C22×C71—42no—
137C22×C71—42no—
138C22×C71—42no—
139C22×C71—42no—
140C22×C71—82no—
141C22×C71—81no—
142C22×C71—81no—
143C22×C71—81no—
144C22×C71—81no—
145C22×C71#38611yescharacteristic
146C22×C71#41711yescharacteristic
147C22×C71#41811yescharacteristic
148C4×C71—22no—
149C4×C71—21no—
150C4×C71—21no—
151C4×C71—42no—
152C4×C71—82no—
153C4×C71—81no—
154C4×C71#46812yes—
155C4×C71#46912yes—
156#9489—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—82no—
160C2×C71—82no—
161C2×C71#189111yescharacteristic
162C2×C71#189511yescharacteristic
163C2×C71#191111yescharacteristic
164#2975—711no—
165#2535D7111yescharacteristic
166#1800—711no—
167#1486—712no—
168#1286—712no—
169C71#948911yescharacteristic, complemented, Sylow
170C2×#121—21no—
171C2×C4×D4C2×D7111yescharacteristic
172C2×C4×D4C2×D7111yescharacteristic
173#436—1424no—
174#431—712no—
175#423—712no—
176C2×#100—1421no—
177#407—1422no—
178#397Dic7112yes—
179#392Dic7112yes—
180#386—22no—
181#382C2×D7111yescharacteristic
182#380—21no—
183#343—1421no—
184D4⋊C8—1424no—
185C22×D4—22no—
186C22×D4C2×Dic7111yescharacteristic
187C22×D4C22⋊D7111yescharacteristic
188C23×C4—21no—
189C23×C4C22⋊D7112yes—
190C2×M16—1422no—
191C2×M16—1424no—
192#121—44no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic7111yescharacteristic
199C2×C4⋊C4C22⋊D7111yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C2×Dic7112yes—
204C2×C42C22×D7111yescharacteristic
205C4⋊C8—1422no—
206C4⋊C8—1422no—
207C4⋊C8—1422no—
208#100—2842no—
209C22⋊C8—1424no—
210C22⋊C8—1422no—
211C4×C8—714no—
212#95—22no—
213#95—21no—
214#95C2×Dic7112yes—
215C24—21no—
216C24C22⋊Dic7112yes—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—82no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4C22⋊Dic7112yes—
236C22×C4C22×Dic7111yescharacteristic
237C22×C4C2×C22⋊D7111yescharacteristic
238C22×C4C2×C22⋊D7111yescharacteristic
239M16—2842no—
240M16—2842no—
241M16—5684no—
242C2×C8—2842no—
243C2×C8—1424no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C22⋊C4—24no—
248C22⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—82no—
252C42D4×D7112yes—
253C42#1176586412yes—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic7111yescharacteristic
267D4—24no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4D4×Dic7112yes—
287C2×C4#1455609611yescharacteristic
288C2×C4#1455610012yes—
289C2×C4#1455610911yescharacteristic
290C2×C4#1455615311yescharacteristic
291C2×C4#1455615412yes—
292C2×C4#1455615811yescharacteristic
293C8—2844no—
294C8—5682no—
295C22—21no—
296C22—21no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#1798587911yescharacteristic
310C22#1798600011yescharacteristic
311C22#1798600411yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—82no—
317C4—81no—
318C4#1798624312yes—
319C4#1798625012yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#2258875911yescharacteristic
325C2#2258877911yescharacteristic
326C2#2258887311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#2535C2
socleC22×C71#386
cosocleC2×C42×C71C23
hypercenter#2535D71
hypocenterC71#9489
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #22597381C2semidirect
2. #22588332C2semidirect
3. #22586765C2semidirect
4. #22586198C2semidirect
29. C4×C71⋊C8D4semidirect
169. C71#9489semidirect

Maximal subgroups

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

subgroupindex
#225973812
#225883322
#225867652
#225861982
C71×#25352
#948971