Finite Group Databasebeta

← #31892451

Subgroups — #31892451

Order 18176.

800 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 147 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×#17985919C211yesmaximal, characteristic
2#22588274C212yesmaximal
3C2×#17985725C211yesmaximal, characteristic
4#22586218C212yesmaximal
5C71×#2316C211yesmaximal, characteristic
6C23×Dic142C2211yescharacteristic
7C22×Dic71⋊C4—21no—
8C2×#14556051—24no—
9C2×C22⋊(C71⋊C8)C2212yes—
10#17985919—22no—
11#17985919C414yescomplemented
12C2×C71⋊#95—21no—
13C2×C71⋊#95C2211yescharacteristic
14#17985725—22no—
15C2×#117×C71—21no—
16C22×C4⋊C4×C71C2211yescharacteristic
17#389×C71—24no—
18#380×C71—21no—
19#380×C71C2212yes—
20C22×Dic142—21no—
21C22×Dic142—22no—
22C22×Dic142—22no—
23C22×Dic142—44no—
24C22×Dic142C2×C412yes—
25C23×Dic71—21no—
26C23×Dic71—21no—
27C22×C71⋊C8—22no—
28C2×Dic71⋊C4—24no—
29C2×Dic71⋊C4—44no—
30#14556051—48no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C71⋊#95—24no—
34C71⋊#95—21no—
35C71⋊#95C2×C412yes—
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39#117×C71—42no—
40#117×C71—42no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71C2×C412yes—
46C2×C4⋊C4×C71D412yes—
47C2×C4⋊C4×C71D414yes—
48C2×C22⋊C4×C71—22no—
49C2×C42×C71—22no—
50#95×C71—22no—
51#95×C71—22no—
52C2×Dic142—28no—
53C2×Dic142—24no—
54C2×Dic142—44no—
55C2×Dic142—48no—
56C2×Dic142—44no—
57C2×Dic142—44no—
58C2×Dic142—42no—
59C22×Dic71—24no—
60C22×Dic71—24no—
61C22×Dic71—22no—
62C22×Dic71—22no—
63C22×Dic71—42no—
64C22×Dic71—42no—
65C22×Dic71—41no—
66C22×Dic71—41no—
67C22×Dic71—41no—
68C22×Dic71—41no—
69C2×C71⋊C8—44no—
70C2×C71⋊C8—42no—
71Dic71⋊C4—48no—
72C24×C71C2×D411yescharacteristic
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—24no—
80C22×C4×C71—42no—
81C22×C4×C71—42no—
82C22×C4×C71C22⋊C412yes—
83C22×C4×C71C22⋊C412yes—
84C22×C4×C71SD1612yes—
85C22×C4×C71Q1612yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D412yes—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71#4112yes—
90C4⋊C4×C71—24no—
91C4⋊C4×C71—24no—
92C4⋊C4×C71—44no—
93C4⋊C4×C71—44no—
94C4⋊C4×C71—82no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—42no—
97C42×C71—44no—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71C2×C22⋊C411yescharacteristic
101C23×C71#12011yescharacteristic
102C23×C71#12311yescharacteristic
103C23×C71C2×SD1611yescharacteristic
104C23×C71C2×Q1611yescharacteristic
105C23×C71AGL(1,ℤ/8)11yescharacteristic
106C23×C71#13711yescharacteristic
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—24no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—44no—
121C2×C4×C71—44no—
122C2×C4×C71—44no—
123C2×C4×C71—81no—
124C2×C4×C71#10314yescomplemented
125C2×C4×C71C4×D412yes—
126C2×C4×C71#12112yes—
127C2×Dic71—24no—
128C2×Dic71—24no—
129C2×Dic71—42no—
130C2×Dic71—44no—
131C2×Dic71—44no—
132C2×Dic71—42no—
133C2×Dic71—44no—
134C2×Dic71—42no—
135C2×Dic71—42no—
136C2×Dic71—44no—
137Dic142—28no—
138Dic142—48no—
139Dic142—44no—
140C71⋊C8—82no—
141C22×C71—22no—
142C22×C71—22no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71#38611yescharacteristic
150C22×C71C2×#10311yescharacteristic
151C22×C71#41711yescharacteristic
152C22×C71#45011yescharacteristic
153C22×C71#45111yescharacteristic
154C22×C71#45612yes—
155C22×C71B(4,𝔽2)12yes—
156C22×C71#48311yescharacteristic
157C22×C71#48411yescharacteristic
158C4×C71—24no—
159C4×C71—22no—
160C4×C71—22no—
161C4×C71—42no—
162C4×C71—84no—
163Dic71—44no—
164Dic71—44no—
165#9015—711nomaximal, Sylow
166C2×C71—21no—
167C2×C71—21no—
168C2×C71—21no—
169C2×C71—21no—
170C2×C71#160412yes—
171C2×C71#160512yes—
172C2×C71#187211yescharacteristic
173C2×C71#188611yescharacteristic
174C2×C71#188911yescharacteristic
175C2×#395—711no—
176#2316D7111yescharacteristic
177#1787—712no—
178C2×#324—711no—
179#1294—712no—
180C71#901511yescharacteristic, complemented, Sylow
181C23×Q8—711no—
182C2×#117—21no—
183C22×C4⋊C4—1421no—
184C22×C4⋊C4C2×D7111yescharacteristic
185C2×#103—1424no—
186C2×C22⋊C8—712no—
187#395—714no—
188#395—1422no—
189#389—24no—
190#380—21no—
191#380C2×D7112yes—
192C2×#95—711no—
193C2×#95—1421no—
194#324—1422no—
195C22×Q8—2844no—
196C22×Q8—712no—
197C22×Q8—1422no—
198C22×Q8—1422no—
199C22×Q8—1421no—
200C23×C4—1421no—
201C23×C4—1421no—
202C23×C4C22⋊D7111yescharacteristic
203C23×C4C22⋊D7111yescharacteristic
204C23×C4C22×D7111yescharacteristic
205C22×C8—1422no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—2844no—
213C2×C4⋊C4—1424no—
214C2×C4⋊C4C4×D7112yes—
215C2×C4⋊C4D28412yes—
216C2×C4⋊C4C22⋊D7114yes—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219#103—2848no—
220C22⋊C8—1422no—
221C22⋊C8—1422no—
222#95—22no—
223#95—22no—
224#95—712no—
225#95—1424no—
226#95—1421no—
227C24C2×C22⋊D7111yescharacteristic
228C2×Q8—2844no—
229C2×Q8—2848no—
230C2×Q8—2844no—
231C2×Q8—2844no—
232C2×Q8—2842no—
233C2×Q8—1428no—
234C2×Q8—1424no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—2842no—
245C22×C4—2842no—
246C22×C4—2841no—
247C22×C4—2841no—
248C22×C4—2841no—
249C22×C4—2841no—
250C22×C4—1424no—
251C22×C4—1424no—
252C22×C4—1422no—
253C22×C4—1422no—
254C22×C4(C2×D71)⋊C412yes—
255C22×C4(C2×D71)⋊C412yes—
256C22×C4#1176585312yes—
257C22×C4#1176585512yes—
258C22×C4C2×C4×D7111yescharacteristic
259C22×C4C2×C22⋊D7112yes—
260C22×C4C2×D28411yescharacteristic
261C22×C4#1176586512yes—
262C2×C8—2844no—
263C2×C8—2842no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—82no—
269C4⋊C4—2848no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—44no—
273C23—22no—
274C23—22no—
275C23C2×(C2×D71)⋊C411yescharacteristic
276C23#1455609711yescharacteristic
277C23#1455611111yescharacteristic
278C23C2×#1176585311yescharacteristic
279C23C2×#1176585511yescharacteristic
280C23#1455615411yescharacteristic
281C23#1455615911yescharacteristic
282Q8—2848no—
283Q8—2844no—
284Q8—1428no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—81no—
302C2×C4—2842no—
303C2×C4—2842no—
304C2×C4—2844no—
305C2×C4—2844no—
306C2×C4—2844no—
307C2×C4—2842no—
308C2×C4—2844no—
309C2×C4—2842no—
310C2×C4—1424no—
311C2×C4—1424no—
312C2×C4#1455605114yescomplemented
313C2×C4C4×C22⋊D7112yes—
314C2×C4#1455610212yes—
315C8—5682no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#1798587811yescharacteristic
325C22C2×#1455605111yescharacteristic
326C22#1798599811yescharacteristic
327C22#1798615011yescharacteristic
328C22#1798615511yescharacteristic
329C22#1798618212yes—
330C22#1798618612yes—
331C22#1798632911yescharacteristic
332C22#1798633411yescharacteristic
333C4—22no—
334C4—22no—
335C4—24no—
336C4—42no—
337C4—84no—
338C4—2844no—
339C4—2844no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2258730912yes—
345C2#2258731412yes—
346C2#2258864711yescharacteristic
347C2#2258873311yescharacteristic
348C2#2258874811yescharacteristic
subgroupG/H
centerC23#14556097
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2316C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2316D71
hypocenterC71#9015
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
11. #17985919C4semidirect
124. C2×C4×C71#103semidirect
180. C71#9015semidirect
312. C2×C4#14556051semidirect

Maximal subgroups

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

subgroupindex
C2×#179859192
#225882742
C2×#179857252
#225862182
C71×#23162
#901571