Finite Group Databasebeta

← #31915749

Subgroups — #31915749

Order 18176.

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

normal subgroups only

This group has 540 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
1C2×#17985789C214yesmaximal
2#22588332C211yesmaximal, characteristic
3C4×#14556043C211yesmaximal, characteristic
4C71×#2486C211yesmaximal, characteristic
5#17986023—21no—
6C2×#14556045C2212yes—
7C2×#14556043C2212yes—
8C4×C22⋊Dic71—21no—
9#17985789—28no—
10#17985768—21no—
11#17985766—21no—
12C2×#123×C71C2212yes—
13C2×C4×D4×C71C2211yescharacteristic
14C22×C4⋊C4×C71C411yescharacteristic, complemented
15#398×C71—21no—
16#397×C71—21no—
17#389×C71—21no—
18#381×C71C412yescomplemented
19#380×C71—21no—
20#380×C71C411yescharacteristic, complemented
21C2×#11765849—22no—
22C2×C22⋊Dic71—22no—
23C2×C4×Dic71—41no—
24C4⋊(C71⋊C8)—41no—
25C4⋊(C71⋊C8)—41no—
26#14556045—24no—
27#14556043—22no—
28#14556043D414yes—
29C71⋊#95—41no—
30C22×D4×C71C2311yescharacteristic
31C23×C4×C71C2×C412yes—
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71D411yescharacteristic
34#123×C71—24no—
35#123×C71—24no—
36C4×D4×C71—41no—
37C4×D4×C71—41no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—41no—
43C2×C4⋊C4×C71—41no—
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71C2×C412yes—
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50#95×C71—22no—
51#95×C71—21no—
52#95×C71—22no—
53C22×Dic71—42no—
54#11765849—42no—
55#11765849—42no—
56C2×C71⋊C8—42no—
57C22⋊Dic71—24no—
58C22⋊Dic71—42no—
59C4×Dic71—82no—
60C24×C71C22×C411yescharacteristic
61C24×C71C2×D411yescharacteristic
62C2×D4×C71—41no—
63C2×D4×C71—41no—
64C2×D4×C71C2×D412yes—
65C2×D4×C71#4112yes—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—24no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—41no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—24no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—44no—
93C42×C71—21no—
94C42×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71C2×C22⋊C411yescharacteristic
104C23×C71C4×D412yes—
105C23×C71#12112yes—
106D4×C71—42no—
107D4×C71—42no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—22no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—81no—
127C2×C4×C71—84no—
128C2×C4×C71#9911yescharacteristic
129C2×C4×C71#9911yescharacteristic
130C2×C4×C71#12411yescharacteristic
131C2×C4×C71#12711yescharacteristic
132C2×Dic71—44no—
133C2×Dic71—82no—
134C2×Dic71—82no—
135C71⋊C8—82no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—42no—
141C22×C71—42no—
142C22×C71—42no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71#39011yescharacteristic
149C22×C71C2×#9911yescharacteristic
150C22×C71#41011yescharacteristic
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—41no—
155C4×C71—41no—
156C4×C71—41no—
157C4×C71—82no—
158C4×C71—82no—
159C4×C71#35212yescomplemented
160Dic71—84no—
161#13220—711nomaximal, Sylow
162C2×C71—21no—
163C2×C71—21no—
164C2×C71—42no—
165C2×C71—42no—
166C2×C71#196411yescharacteristic
167C2×C71C2×#35211yescharacteristic
168C2×C71#212311yescharacteristic
169#2486D7111yescharacteristic
170C2×#352—714no—
171#1800—711no—
172C4×#99—711no—
173C71#1322011yescharacteristic, complemented, Sylow
174C2×#123C2×D7112yes—
175C2×C4×D4C2×D7111yescharacteristic
176C22×C4⋊C4Dic7111yescharacteristic, complemented
177#423—1421no—
178C2×#100—712no—
179C2×#99—712no—
180#398—21no—
181#397—21no—
182#389—21no—
183#381Dic7112yescomplemented
184#380—21no—
185#380Dic7111yescharacteristic, complemented
186C4×C22⋊C4—1421no—
187#352—1428no—
188#343—1421no—
189#342—1421no—
190C22×D4C22×D7111yescharacteristic
191C23×C4C2×Dic7112yes—
192C23×C4C2×Dic7111yescharacteristic
193C23×C4C22⋊D7111yescharacteristic
194C2×M16—1422no—
195#123—24no—
196#123—24no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C22⋊D7111yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1422no—
208C2×C22⋊C4C2×Dic7112yes—
209C2×C22⋊C4C2×Dic7111yescharacteristic
210C2×C22⋊C4C22⋊D7111yescharacteristic
211C2×C42—2841no—
212C2×C42C22⋊D7111yescharacteristic
213C4⋊C8—2841no—
214C4⋊C8—2841no—
215#100—1424no—
216#99—714no—
217#99—1422no—
218#95—22no—
219#95—21no—
220#95—22no—
221#95—2841no—
222C24C22×Dic7111yescharacteristic
223C24C2×C22⋊D7111yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D7112yes—
227C2×D4#1176586412yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—2842no—
240C22×C4C22⋊Dic7111yescharacteristic
241C22×C4C22⋊Dic7111yescharacteristic
242C22×C4C22⋊Dic7111yescharacteristic
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C2×C22⋊D7111yescharacteristic
245M16—2842no—
246M16—2842no—
247C2×C8—2842no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—24no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—44no—
259C22⋊C4—2842no—
260C22⋊C4—1424no—
261C42—21no—
262C42—21no—
263C42—5682no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊Dic7111yescharacteristic
273C23D4×Dic7112yes—
274C23#1455609912yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—84no—
297C2×C4—2844no—
298C2×C4—5682no—
299C2×C4—5682no—
300C2×C4#1455604311yescharacteristic
301C2×C4#1455604311yescharacteristic
302C2×C4#1455611511yescharacteristic
303C2×C4#1455612411yescharacteristic
304C8—5682no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1798590211yescharacteristic
318C22C2×#1455604311yescharacteristic
319C22#1798597311yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—82no—
328C4—5684no—
329C4#1798578912yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2258917711yescharacteristic
335C2C2×#1798578911yescharacteristic
336C2#2258999411yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2486C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenter#2486D71
hypocenterC71#13220
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
14. C22×C4⋊C4×C71C4semidirect
18. #381×C71C4semidirect
20. #380×C71C4semidirect
159. C4×C71#352semidirect
173. C71#13220semidirect
176. C22×C4⋊C4Dic71semidirect
183. #381Dic71semidirect
185. #380Dic71semidirect
329. C4#17985789semidirect

Maximal subgroups

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

subgroupindex
C2×#179857892
#225883322
C4×#145560432
C71×#24862
#1322071