Finite Group Databasebeta

← #31892497

Subgroups — #31892497

Order 18176.

740 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 135 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#22590938C211yesmaximal, characteristic
2#22588430C212yesmaximal, complemented
3C2×#17985723C211yesmaximal, characteristic
4#22586219C212yesmaximal, complemented
5C2×#399×C71C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7#17985892—24no—
8#17985859C2212yes—
9#17985855—21no—
10C4×C22⋊Dic71—21no—
11#17985761—24no—
12#17985723—22no—
13C22×C4⋊C4×C71—21no—
14C22×C4⋊C4×C71C2211yescharacteristic
15C22×C22⋊C4×C71—21no—
16C22×C22⋊C4×C71C2211yescharacteristic
17#399×C71—22no—
18#399×C71C414yescomplemented
19C2×#95×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71D414yes—
22C2×C22⋊Dic71—22no—
23C2×C22⋊Dic71—22no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—22no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C25×C71D411yescharacteristic
31C23×C4×C71—21no—
32C23×C4×C71—21no—
33C23×C4×C71D411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—24no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—44no—
40C2×C4⋊C4×C71C2×C412yes—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—24no—
44C2×C22⋊C4×C71—44no—
45C2×C22⋊C4×C71C2×C412yes—
46C2×C22⋊C4×C71D412yes—
47#95×C71—21no—
48#95×C71C2×C412yes—
49C22×Dic71—24no—
50C2×C71⋊C8—44no—
51C2×C71⋊C8—42no—
52C4⋊Dic71—44no—
53C22⋊Dic71—42no—
54C22⋊Dic71—42no—
55C22⋊Dic71—82no—
56C22⋊Dic71—82no—
57C4×Dic71—82no—
58C24×C71—21no—
59C24×C71—22no—
60C24×C71—21no—
61C24×C71—22no—
62C24×C71—42no—
63C24×C71—42no—
64C24×C71C22⋊C412yes—
65C24×C71C2×D411yescharacteristic
66C22×C4×C71—24no—
67C22×C4×C71—22no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—22no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71C22⋊C412yes—
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D412yes—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71#4112yes—
84C4⋊C4×C71—24no—
85C4⋊C4×C71—48no—
86C4⋊C4×C71—42no—
87C22⋊C4×C71—24no—
88C22⋊C4×C71—24no—
89C22⋊C4×C71—48no—
90C23×C71—22no—
91C23×C71—22no—
92C23×C71—22no—
93C23×C71—22no—
94C23×C71—22no—
95C23×C71—42no—
96C23×C71—42no—
97C23×C71—48no—
98C23×C71—42no—
99C23×C71—42no—
100C23×C71—42no—
101C23×C71—42no—
102C23×C71—81no—
103C23×C71—81no—
104C23×C71—84no—
105C23×C71C2×C22⋊C411yescharacteristic
106C23×C71#12011yescharacteristic
107C23×C71#12311yescharacteristic
108C23×C71#13512yes—
109C23×C71#13712yes—
110C2×C4×C71—24no—
111C2×C4×C71—24no—
112C2×C4×C71—22no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—42no—
116C2×C4×C71—48no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—81no—
125C2×C4×C71—82no—
126C2×C4×C71—82no—
127C2×C4×C71C4×D412yes—
128C2×C4×C71#12112yes—
129C2×Dic71—44no—
130C2×Dic71—44no—
131C2×Dic71—44no—
132C71⋊C8—82no—
133C22×C71—24no—
134C22×C71—24no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—44no—
140C22×C71—42no—
141C22×C71—42no—
142C22×C71—42no—
143C22×C71—82no—
144C22×C71—84no—
145C22×C71—84no—
146C22×C71#38611yescharacteristic
147C22×C71#41611yescharacteristic
148C22×C71#41911yescharacteristic
149C22×C71#45212yes—
150C22×C71#45312yes—
151C22×C71#45812yes—
152C22×C71#48512yes—
153C4×C71—22no—
154C4×C71—22no—
155C4×C71—44no—
156C4×C71—42no—
157C4×C71—82no—
158Dic71—84no—
159#9024—711nomaximal, Sylow
160C2×C71—21no—
161C2×C71—21no—
162C2×C71—22no—
163C2×C71—82no—
164C2×C71#161814yes—
165C2×C71#187311yescharacteristic
166C2×C71#188711yescharacteristic
167C2×C71#189011yescharacteristic
168C2×#399D7111yescharacteristic
169#2312—711no—
170#1824—712no—
171C2×#323—711no—
172#1294—712no—
173C71#902411yescharacteristic, complemented, Sylow
174C22×C4⋊C4—21no—
175C22×C4⋊C4C2×D7111yescharacteristic
176C22×C22⋊C4—21no—
177C22×C22⋊C4C2×D7111yescharacteristic
178C2×C22⋊C8—712no—
179#399—22no—
180#399Dic7114yescomplemented
181#388—1424no—
182#380—712no—
183#379—1421no—
184C4×C22⋊C4—1421no—
185C2×#95C2×D7111yescharacteristic
186#340—1424no—
187#323—1422no—
188C25C22⋊D7111yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C22⋊D7111yescharacteristic
192C23×C4C22×D7111yescharacteristic
193C22×C8—1422no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—714no—
200C2×C4⋊C4C2×Dic7112yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—1422no—
206C2×C22⋊C4—1422no—
207C2×C22⋊C4—1422no—
208C2×C22⋊C4C2×Dic7112yes—
209C2×C22⋊C4C22⋊D7112yes—
210C2×C42—1422no—
211C22⋊C8—1422no—
212C22⋊C8—1422no—
213#95—21no—
214#95—1422no—
215#95—1422no—
216#95C2×Dic7112yes—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24C22⋊Dic7112yes—
224C24C2×C22⋊D7111yescharacteristic
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—1424no—
239C22×C4C22⋊Dic7112yes—
240C22×C4C22×Dic7111yescharacteristic
241C22×C4D4×D7112yes—
242C22×C4C2×C22⋊D7111yescharacteristic
243C22×C4#1176586412yes—
244C2×C8—2844no—
245C2×C8—2842no—
246C4⋊C4—24no—
247C4⋊C4—48no—
248C4⋊C4—42no—
249C4⋊C4—2844no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—48no—
253C22⋊C4—2842no—
254C22⋊C4—2842no—
255C22⋊C4—5682no—
256C22⋊C4—5682no—
257C42—5682no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23C2×C22⋊Dic7111yescharacteristic
274C23#1455609611yescharacteristic
275C23#1455610911yescharacteristic
276C23#1455614812yes—
277C23#1455615912yes—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—2844no—
296C2×C4—2844no—
297C2×C4—2844no—
298C2×C4D4×Dic7112yes—
299C2×C4#1455610012yes—
300C8—5682no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—24no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#1798587911yescharacteristic
315C22#1798599511yescharacteristic
316C22#1798600811yescharacteristic
317C22#1798615912yes—
318C22#1798616612yes—
319C22#1798618812yes—
320C22#1798633712yes—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—82no—
326C4—5684no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#2258738614yes—
332C2#2258865311yescharacteristic
333C2#2258873911yescharacteristic
334C2#2258875411yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#399×C71C2
socleC23×C71#120
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenterC2×#399D71
hypocenterC71#9024
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. #22588430C2semidirect
4. #22586219C2semidirect
18. #399×C71C4semidirect
173. C71#9024semidirect
180. #399Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909382
#225884302
C2×#179857232
#225862192
C2×#399×C712
#902471