Finite Group Databasebeta

← #31895247

Subgroups — #31895247

Order 18176.

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

normal subgroups only

This group has 576 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#22588426C212yesmaximal, complemented
3#22588328C211yesmaximal, characteristic
4#22586207C212yesmaximal, complemented
5C71×#2486C211yesmaximal, 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×#123×C71—21no—
15C2×C4×D4×C71C2211yescharacteristic
16C22×C4⋊C4×C71C2211yescharacteristic
17#398×C71C412yescomplemented
18#397×C71C412yescomplemented
19#389×C71—21no—
20#381×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
38#123×C71—44no—
39C4×D4×C71—22no—
40C4×D4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71C2×C412yes—
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71—22no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71D411yescharacteristic
52C2×C42×C71D411yescharacteristic
53#95×C71—22no—
54#95×C71—22no—
55#95×C71C2×C412yes—
56C22×Dic71—42no—
57#11765849—82no—
58C2×C71⋊C8—24no—
59C2×C71⋊C8—44no—
60C4⋊Dic71—44no—
61C22⋊Dic71—42no—
62C22⋊Dic71—42no—
63C24×C71C22⋊C411yescharacteristic
64C24×C71C2×D411yescharacteristic
65C2×D4×C71—22no—
66C2×D4×C71—22no—
67C2×D4×C71—21no—
68C2×D4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
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⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D412yes—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71#4112yes—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—24no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—44no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C22⋊C4×C71—22no—
93C22⋊C4×C71—22no—
94C22⋊C4×C71—44no—
95C22⋊C4×C71—42no—
96C42×C71—21no—
97C42×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71C2×C22⋊C411yescharacteristic
106C23×C71#12112yes—
107C23×C71AGL(1,ℤ/8)11yescharacteristic
108C23×C71AGL(1,ℤ/8)12yes—
109C23×C71#13711yescharacteristic
110D4×C71—42no—
111D4×C71—42no—
112C2×C4×C71—22no—
113C2×C4×C71—24no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—41no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—81no—
131C2×C4×C71—82no—
132C2×C4×C71—82no—
133C2×C4×C71C4×D412yes—
134C2×C4×C71#12011yescharacteristic
135C2×C4×C71#12311yescharacteristic
136C2×Dic71—82no—
137C2×Dic71—82no—
138C2×Dic71—82no—
139C71⋊C8—44no—
140C22×C71—22no—
141C22×C71—21no—
142C22×C71—22no—
143C22×C71—21no—
144C22×C71—41no—
145C22×C71—42no—
146C22×C71—41no—
147C22×C71—42no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—81no—
152C22×C71#38611yescharacteristic
153C22×C71#41711yescharacteristic
154C22×C71#41811yescharacteristic
155C22×C71#46812yes—
156C22×C71#46912yes—
157C4×C71—21no—
158C4×C71—22no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—82no—
163C4×C71—82no—
164C4×C71#45812yes—
165Dic71—162no—
166#9474—711nomaximal, Sylow
167C2×C71—22no—
168C2×C71—21no—
169C2×C71—21no—
170C2×C71—22no—
171C2×C71—81no—
172C2×C71#189011yescharacteristic
173C2×C71#189511yescharacteristic
174C2×C71#190911yescharacteristic
175#2973—711no—
176#2486D7111yescharacteristic
177#1823—712no—
178#1799—711no—
179#1289—712no—
180C71#947411yescharacteristic, complemented, Sylow
181C2×#123—21no—
182C2×C4×D4C2×D7111yescharacteristic
183C22×C4⋊C4C2×D7111yescharacteristic
184#435—1424no—
185#432—1421no—
186C2×#99—1421no—
187#407—1421no—
188C2×C22⋊C8—712no—
189#398Dic7112yescomplemented
190#397Dic7112yescomplemented
191#389—21no—
192#381—21no—
193#380—21no—
194#380—712no—
195#380C2×D7111yescharacteristic
196#342—1421no—
197#340—1424no—
198C22×D4C22⋊D7111yescharacteristic
199C23×C4—21no—
200C23×C4C22⋊D7111yescharacteristic
201C23×C4C22×D7111yescharacteristic
202C2×M16—1422no—
203C22×C8—714no—
204#123—44no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1422no—
213C2×C4⋊C4C2×Dic7112yes—
214C2×C4⋊C4C2×Dic7111yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1422no—
218C2×C22⋊C4C2×Dic7111yescharacteristic
219C2×C22⋊C4C22⋊D7111yescharacteristic
220C2×C42C22⋊D7111yescharacteristic
221C4⋊C8—1422no—
222#99—2842no—
223C22⋊C8—1422no—
224C22⋊C8—1422no—
225C22⋊C8—1422no—
226#97—1422no—
227#95—22no—
228#95—22no—
229#95—1422no—
230#95C2×Dic7112yes—
231C24C22⋊Dic7111yescharacteristic
232C24C2×C22⋊D7111yescharacteristic
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—2842no—
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22×Dic7111yescharacteristic
252C22×C4D4×D7112yes—
253C22×C4C2×C22⋊D7111yescharacteristic
254C22×C4#1176586412yes—
255M16—5682no—
256C2×C8—2844no—
257C2×C8—1424no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2844no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—2842no—
270C22⋊C4—2842no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic7111yescharacteristic
281C23#1455610012yes—
282C23#1455615311yescharacteristic
283C23#1455615412yes—
284C23#1455615811yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—5682no—
309C2×C4—5682no—
310C2×C4—5682no—
311C2×C4D4×Dic7112yes—
312C2×C4#1455609611yescharacteristic
313C2×C4#1455610911yescharacteristic
314C8—2844no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1798587911yescharacteristic
328C22#1798600011yescharacteristic
329C22#1798600411yescharacteristic
330C22#1798624312yes—
331C22#1798625012yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—11362no—
340C4#1798618812yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2258875411yescharacteristic
347C2#2258877911yescharacteristic
348C2#2258886311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2486C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2486D71
hypocenterC71#9474
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #22588426C2semidirect
4. #22586207C2semidirect
17. #398×C71C4semidirect
18. #397×C71C4semidirect
180. C71#9474semidirect
189. #398Dic71semidirect
190. #397Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225884262
#225883282
#225862072
C71×#24862
#947471