Finite Group Databasebeta

← #31892978

Subgroups — #31892978

Order 18176.

602 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 127 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#22586731C212yesmaximal, complemented
3#22586661C211yesmaximal, characteristic, complemented
4#22586169C212yesmaximal, complemented
5C71×#2328C211yesmaximal, characteristic
6#17986077—24no—
7#17986052C2212yes—
8#17986052C2212yes—
9#17985963—22no—
10D4⋊(C71⋊C8)—24no—
11#17985723—22no—
12C2×#127×C71C411yescharacteristic
13C2×C4×D4×C71C2211yescharacteristic
14C2×C4×D4×C71C2211yescharacteristic
15#390×C71C411yescharacteristic
16#390×C71C411yescharacteristic
17#386×C71—22no—
18#386×C71—22no—
19#382×C71C411yescharacteristic
20#376×C71C2211yescharacteristic
21C2×#11765849—24no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C22⋊(C71⋊C8)—24no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C4×C71⋊C8—22no—
28C4×C71⋊C8D414yescomplemented
29C22×D4×C71—22no—
30C22×D4×C71C2×C411yescharacteristic
31C22×D4×C71C2×C411yescharacteristic
32C23×C4×C71—21no—
33C23×C4×C71D412yes—
34#127×C71—22no—
35#127×C71—22no—
36C4×D4×C71—22no—
37C4×D4×C71—22no—
38C4×D4×C71—22no—
39C4×D4×C71—22no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71D412yes—
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71C2×C411yescharacteristic
48C2×C42×C71C2311yescharacteristic
49#95×C71—22no—
50#95×C71—21no—
51#95×C71—21no—
52#11765849—84no—
53C2×C71⋊C8—24no—
54C2×C71⋊C8—42no—
55C24×C71—21no—
56C24×C71C22⋊C412yes—
57C2×D4×C71—28no—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—21no—
61C2×D4×C71—21no—
62C2×D4×C71—21no—
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—82no—
66C2×D4×C71—82no—
67C2×D4×C71M1612yescomplemented
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—42no—
74C22×C4×C71—82no—
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—
82C22⋊C4×C71—24no—
83C22⋊C4×C71—41no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—82no—
86C22⋊C4×C71—82no—
87C42×C71—21no—
88C42×C71—22no—
89C42×C71M1612yescomplemented
90C42×C71C2×D412yes—
91C42×C71#4112yes—
92C23×C71—22no—
93C23×C71—22no—
94C23×C71—22no—
95C23×C71—22no—
96C23×C71—42no—
97C23×C71—42no—
98C23×C71—42no—
99C23×C71—42no—
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—22no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—44no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—42no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—42no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—82no—
122C2×C4×C71—84no—
123C2×C4×C71C4×D412yes—
124C2×C4×C71#12011yescharacteristic
125C2×C4×C71#12112yes—
126C2×C4×C71#12311yescharacteristic
127C2×C4×C71C2×M1611yescharacteristic
128C2×C4×C71C2×M1611yescharacteristic
129C2×C4×C71#13112yes—
130C71⋊C8—44no—
131C71⋊C8—82no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—44no—
135C22×C71—44no—
136C22×C71—42no—
137C22×C71—42no—
138C22×C71—42no—
139C22×C71—42no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71—81no—
143C22×C71—81no—
144C22×C71#38611yescharacteristic
145C22×C71#40711yescharacteristic
146C22×C71#40811yescharacteristic
147C4×C71—22no—
148C4×C71—21no—
149C4×C71—21no—
150C4×C71—42no—
151C4×C71—42no—
152C4×C71—81no—
153C4×C71—81no—
154C4×C71#43512yes—
155C4×C71#43612yes—
156#9105—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—82no—
160C2×C71—82no—
161C2×C71#187011yescharacteristic
162C2×C71#191011yescharacteristic
163C2×C71#191111yescharacteristic
164#2975—711no—
165#2328D7111yescharacteristic
166#1478—712no—
167#1460—711no—
168#1274—712no—
169C71#910511yescharacteristic, complemented, Sylow
170C2×#127Dic7111yescharacteristic
171C2×C4×D4C2×D7111yescharacteristic
172C2×C4×D4C2×D7111yescharacteristic
173#436—1424no—
174#431—712no—
175#431—712no—
176#407—1422no—
177#390Dic7111yescharacteristic
178#390Dic7111yescharacteristic
179#386—22no—
180#386—22no—
181#382Dic7111yescharacteristic
182#376C2×D7111yescharacteristic
183D4⋊C8—1424no—
184#323—1422no—
185C22×D4—22no—
186C22×D4C2×Dic7111yescharacteristic
187C22×D4C2×Dic7111yescharacteristic
188C23×C4—21no—
189C23×C4C22⋊D7112yes—
190C2×M16—1424no—
191#127—22no—
192#127—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C2×C4⋊C4C2×Dic7111yescharacteristic
198C2×C4⋊C4C2×Dic7111yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊D7112yes—
203C2×C42C2×Dic7111yescharacteristic
204C2×C42C2×Dic7111yescharacteristic
205C2×C42C22×D7111yescharacteristic
206C4⋊C8—1422no—
207C4⋊C8—1422no—
208C22⋊C8—1422no—
209C22⋊C8—1424no—
210C22⋊C8—1422no—
211C4×C8—714no—
212C4×C8—1422no—
213#95—22no—
214#95—21no—
215#95—21no—
216C24—21no—
217C24C22⋊Dic7112yes—
218C2×D4—28no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—82no—
227C2×D4—82no—
228C2×D4#1176584912yescomplemented
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C22⋊Dic7112yes—
238C22×C4C22×Dic7111yescharacteristic
239C22×C4C2×C22⋊D7111yescharacteristic
240C22×C4C2×C22⋊D7111yescharacteristic
241M16—5684no—
242C2×C8—2842no—
243C2×C8—1424no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—21no—
252C42—22no—
253C42#1176584912yescomplemented
254C42D4×D7112yes—
255C42#1176586412yes—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic7111yescharacteristic
267D4—24no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4D4×Dic7112yes—
288C2×C4#1455609611yescharacteristic
289C2×C4#1455610012yes—
290C2×C4#1455610911yescharacteristic
291C2×C4C2×#1176584911yescharacteristic
292C2×C4C2×#1176584911yescharacteristic
293C2×C4#1455613512yes—
294C8—2844no—
295C8—5682no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22#1798587911yescharacteristic
309C22#1798596311yescharacteristic
310C22#1798596611yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4#1798607112yes—
319C4#1798607712yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#2258863811yescharacteristic
325C2#2258886811yescharacteristic
326C2#2258887311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2328C2
socleC22×C71#386
prime radicalC2×#127×C71C4
cosocleC2×C42×C71C23
hypercenter#2328D71
hypocenterC71#9105
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #22597381C2semidirect
2. #22586731C2semidirect
3. #22586661C2semidirect
4. #22586169C2semidirect
28. C4×C71⋊C8D4semidirect
67. C2×D4×C71M16semidirect
89. C42×C71M16semidirect
169. C71#9105semidirect
228. C2×D4#11765849semidirect
253. C42#11765849semidirect

Maximal subgroups

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

subgroupindex
#225973812
#225867312
#225866612
#225861692
C71×#23282
#910571