Finite Group Databasebeta

← #31895242

Subgroups — #31895242

Order 18176.

556 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 123 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#22597369C211yesmaximal, characteristic
2#22588430C212yesmaximal, complemented
3#22588328C211yesmaximal, characteristic
4#22586207C212yesmaximal, complemented
5C71×#2487C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8C2×#14556043—21no—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)C2212yes—
11#17985859C2212yes—
12#17985766—21no—
13#17985761—24no—
14C2×C4×D4×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#400×C71C412yescomplemented
17#399×C71C412yescomplemented
18#389×C71—21no—
19#386×C71—21no—
20#385×C71—21no—
21#380×C71—21no—
22#380×C71C2211yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8D414yes—
25C2×C4⋊Dic71—22no—
26C2×C22⋊Dic71—22no—
27C4⋊(C71⋊C8)—22no—
28#14556043—42no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33C71⋊#95—22no—
34C22×D4×C71D411yescharacteristic
35C23×C4×C71—21no—
36C23×C4×C71D411yescharacteristic
37C23×C4×C71C2311yescharacteristic
38C4×D4×C71—22no—
39C4×D4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71C2×C412yes—
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—22no—
49C2×C22⋊C4×C71C2×C411yescharacteristic
50C2×C22⋊C4×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—22no—
53#95×C71—22no—
54#95×C71C2×C412yes—
55C22×Dic71—42no—
56#11765849—82no—
57C2×C71⋊C8—24no—
58C2×C71⋊C8—44no—
59C4⋊Dic71—44no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C2×D411yescharacteristic
64C2×D4×C71—22no—
65C2×D4×C71—21no—
66C2×D4×C71—21no—
67C2×D4×C71—22no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71—82no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D412yes—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4112yes—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—24no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—82no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—22no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—41no—
93C22⋊C4×C71—82no—
94C42×C71—21no—
95C42×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×C71#12112yes—
105C23×C71AGL(1,ℤ/8)11yescharacteristic
106C23×C71#13711yescharacteristic
107C23×C71#13712yes—
108D4×C71—42no—
109D4×C71—42no—
110C2×C4×C71—22no—
111C2×C4×C71—24no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—41no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—81no—
129C2×C4×C71—84no—
130C2×C4×C71C4×D412yes—
131C2×C4×C71#12011yescharacteristic
132C2×C4×C71#12311yescharacteristic
133C2×Dic71—82no—
134C2×Dic71—82no—
135C2×Dic71—82no—
136C71⋊C8—44no—
137C22×C71—22no—
138C22×C71—21no—
139C22×C71—22no—
140C22×C71—21no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—42no—
144C22×C71—42no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—81no—
149C22×C71#38611yescharacteristic
150C22×C71#41711yescharacteristic
151C22×C71#41911yescharacteristic
152C22×C71#46812yes—
153C22×C71#47012yes—
154C4×C71—21no—
155C4×C71—22no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71—82no—
160C4×C71—82no—
161C4×C71#45812yes—
162Dic71—162no—
163#9473—711nomaximal, Sylow
164C2×C71—22no—
165C2×C71—21no—
166C2×C71—21no—
167C2×C71—22no—
168C2×C71—81no—
169C2×C71#189011yescharacteristic
170C2×C71#189411yescharacteristic
171C2×C71#191011yescharacteristic
172#2973—711no—
173#2487D7111yescharacteristic
174#1824—712no—
175#1799—711no—
176#1289—712no—
177C71#947311yescharacteristic, complemented, Sylow
178C2×C4×D4C2×D7111yescharacteristic
179C22×C4⋊C4C2×D7111yescharacteristic
180#435—1424no—
181#432—1421no—
182C2×#99—1421no—
183#407—1421no—
184C2×C22⋊C8—712no—
185#400Dic7112yescomplemented
186#399Dic7112yescomplemented
187#389—21no—
188#386—21no—
189#385—21no—
190#380—21no—
191#380—712no—
192#380C2×D7111yescharacteristic
193#342—1421no—
194#340—1424no—
195C22×D4C22⋊D7111yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D7111yescharacteristic
198C23×C4C22×D7111yescharacteristic
199C2×M16—1422no—
200C22×C8—714no—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—1422no—
209C2×C4⋊C4C2×Dic7112yes—
210C2×C4⋊C4C2×Dic7111yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1422no—
214C2×C22⋊C4C2×Dic7111yescharacteristic
215C2×C22⋊C4C22⋊D7111yescharacteristic
216C2×C42C22⋊D7111yescharacteristic
217C4⋊C8—1422no—
218#99—2842no—
219C22⋊C8—1422no—
220C22⋊C8—1422no—
221C22⋊C8—1422no—
222#97—1422no—
223#95—22no—
224#95—22no—
225#95—1422no—
226#95C2×Dic7112yes—
227C24C22⋊Dic7111yescharacteristic
228C24C2×C22⋊D7111yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—2842no—
242C22×C4C22⋊Dic7111yescharacteristic
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22×Dic7111yescharacteristic
246C22×C4D4×D7112yes—
247C22×C4C2×C22⋊D7111yescharacteristic
248C22×C4#1176586412yes—
249M16—5682no—
250C2×C8—2844no—
251C2×C8—1424no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—82no—
258C4⋊C4—2844no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—82no—
264C22⋊C4—2842no—
265C22⋊C4—2842no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×C22⋊Dic7111yescharacteristic
276C23#1455610012yes—
277C23#1455615311yescharacteristic
278C23#1455615811yescharacteristic
279C23#1455615912yes—
280D4—42no—
281D4—42no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—5682no—
303C2×C4—5682no—
304C2×C4—5682no—
305C2×C4D4×Dic7112yes—
306C2×C4#1455609611yescharacteristic
307C2×C4#1455610911yescharacteristic
308C8—2844no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1798587911yescharacteristic
322C22#1798600011yescharacteristic
323C22#1798600811yescharacteristic
324C22#1798624712yes—
325C22#1798625512yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—11362no—
334C4#1798618812yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2258875411yescharacteristic
341C2#2258877411yescharacteristic
342C2#2258886811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2487C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2487D71
hypocenterC71#9473
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #22588430C2semidirect
4. #22586207C2semidirect
16. #400×C71C4semidirect
17. #399×C71C4semidirect
177. C71#9473semidirect
185. #400Dic71semidirect
186. #399Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225884302
#225883282
#225862072
C71×#24872
#947371