Finite Group Databasebeta

← #31895227

Subgroups — #31895227

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597364C211yesmaximal, characteristic
2#22588426C212yesmaximal, complemented
3#22588328C211yesmaximal, characteristic
4#22586207C212yesmaximal, complemented
5C71×#2486C211yesmaximal, characteristic
6D4×C71⋊C8—24no—
7#17986052—21no—
8C2×#14556043—21no—
9C2×C22⋊(C71⋊C8)—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
23C22×C71⋊C8—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—
32C4×C71⋊C8—22no—
33C71⋊#95—22no—
34C22×D4×C71D411yescharacteristic
35C23×C4×C71—21no—
36C23×C4×C71D411yescharacteristic
37C23×C4×C71C2311yescharacteristic
38#123×C71—42no—
39#123×C71—42no—
40C4×D4×C71—22no—
41C4×D4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71C2×C412yes—
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71—22no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71C2×C411yescharacteristic
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54#95×C71—22no—
55#95×C71—22no—
56#95×C71C2×C412yes—
57C22×Dic71—42no—
58C2×C71⋊C8—24no—
59C2×C71⋊C8—44no—
60C2×C71⋊C8—82no—
61C4⋊Dic71—44no—
62C22⋊Dic71—42no—
63C22⋊Dic71—42no—
64C24×C71C22⋊C411yescharacteristic
65C24×C71C2×D411yescharacteristic
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71D812yes—
69C2×D4×C71SD1612yes—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—42no—
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4112yes—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—24no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—44no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C22⋊C4×C71—22no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—44no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C42×C71—21no—
99C42×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71#10212yes—
107C23×C71C2×C22⋊C411yescharacteristic
108C23×C71#12112yes—
109C23×C71C2×D811yescharacteristic
110C23×C71C2×SD1611yescharacteristic
111C23×C71#13512yes—
112D4×C71—24no—
113D4×C71—24no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—22no—
117C2×C4×C71—24no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—41no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—81no—
132C2×C4×C71—82no—
133C2×C4×C71—82no—
134C2×C4×C71#10212yes—
135C2×C4×C71C4×D412yes—
136C2×C4×C71#12011yescharacteristic
137C2×C4×C71#12311yescharacteristic
138C2×Dic71—82no—
139C2×Dic71—82no—
140C2×Dic71—82no—
141C71⋊C8—44no—
142C22×C71—22no—
143C22×C71—21no—
144C22×C71—22no—
145C22×C71—21no—
146C22×C71—22no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71—42no—
150C22×C71—42no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—81no—
154C22×C71#38611yescharacteristic
155C22×C71C2×#10211yescharacteristic
156C22×C71#41611yescharacteristic
157C22×C71#46512yes—
158C22×C71#46612yes—
159C4×C71—21no—
160C4×C71—22no—
161C4×C71—22no—
162C4×C71—41no—
163C4×C71—41no—
164C4×C71—82no—
165C4×C71—82no—
166C4×C71#45812yes—
167Dic71—162no—
168#9470—711nomaximal, Sylow
169C2×C71—22no—
170C2×C71—21no—
171C2×C71—21no—
172C2×C71—22no—
173C2×C71—42no—
174C2×C71#189011yescharacteristic
175C2×C71#189311yescharacteristic
176C2×C71#190911yescharacteristic
177#2972—711no—
178#2486D7111yescharacteristic
179#1823—712no—
180#1799—711no—
181#1289—712no—
182C71#947011yescharacteristic, complemented, Sylow
183C2×#123—21no—
184C2×C4×D4C2×D7111yescharacteristic
185C22×C4⋊C4C2×D7111yescharacteristic
186C8×D4—1424no—
187#431—1421no—
188C2×#99—1421no—
189C2×C22⋊C8—712no—
190C2×C22⋊C8—1421no—
191#398Dic7112yescomplemented
192#397Dic7112yescomplemented
193#389—21no—
194#381—21no—
195#380—21no—
196#380—712no—
197#380C2×D7111yescharacteristic
198#342—1421no—
199#340—1424no—
200C22×D4C22⋊D7111yescharacteristic
201C23×C4—21no—
202C23×C4C22⋊D7111yescharacteristic
203C23×C4C22×D7111yescharacteristic
204C22×C8—714no—
205C22×C8—1422no—
206#123—42no—
207#123—42no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1422no—
216C2×C4⋊C4C2×Dic7112yes—
217C2×C4⋊C4C2×Dic7111yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—1422no—
221C2×C22⋊C4C2×Dic7111yescharacteristic
222C2×C22⋊C4C22⋊D7111yescharacteristic
223C2×C42C22⋊D7111yescharacteristic
224C4⋊C8—1422no—
225#99—2842no—
226C22⋊C8—1422no—
227C22⋊C8—1422no—
228C22⋊C8—1422no—
229C4×C8—1422no—
230#95—22no—
231#95—22no—
232#95—1422no—
233#95C2×Dic7112yes—
234C24C22⋊Dic7111yescharacteristic
235C24C2×C22⋊D7111yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1176585012yes—
239C2×D4#1176585312yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—2842no—
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22⋊Dic7111yescharacteristic
253C22×C4C22⋊Dic7111yescharacteristic
254C22×C4C22×Dic7111yescharacteristic
255C22×C4D4×D7112yes—
256C22×C4C2×C22⋊D7111yescharacteristic
257C22×C4#1176586412yes—
258C2×C8—2844no—
259C2×C8—5682no—
260C2×C8—1424no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—2844no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—44no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—2842no—
274C22⋊C4—2842no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1455604912yes—
284C23C2×C22⋊Dic7111yescharacteristic
285C23#1455610012yes—
286C23C2×#1176585011yescharacteristic
287C23C2×#1176585311yescharacteristic
288C23#1455614812yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—5682no—
312C2×C4—5682no—
313C2×C4—5682no—
314C2×C4#1455604912yes—
315C2×C4D4×Dic7112yes—
316C2×C4#1455609611yescharacteristic
317C2×C4#1455610911yescharacteristic
318C8—2844no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1798587911yescharacteristic
332C22C2×#1455604911yescharacteristic
333C22#1798599511yescharacteristic
334C22#1798622612yes—
335C22#1798623312yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—82no—
342C4—82no—
343C4—11362no—
344C4#1798618812yes—
345C2—21no—
346C2—22no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2258875411yescharacteristic
351C2#2258876911yescharacteristic
352C2#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#9470
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
182. C71#9470semidirect
191. #398Dic71semidirect
192. #397Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973642
#225884262
#225883282
#225862072
C71×#24862
#947071