Finite Group Databasebeta

← #31891003

Subgroups — #31891003

Order 18176.

530 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588328C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586219C212yesmaximal, complemented
4#22586207C212yesmaximal, complemented
5C71×#2349C211yesmaximal, characteristic
6C2×#14556043—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985859C2212yes—
9#17985766—21no—
10#17985725—21no—
11#17985723—21no—
12C2×C4×D4×C71—21no—
13C2×C4×D4×C71C2211yescharacteristic
14#389×C71—21no—
15#385×C71—21no—
16#385×C71—21no—
17#382×C71C411yescharacteristic, complemented
18#380×C71C411yescharacteristic, complemented
19#380×C71C2211yescharacteristic
20C4×C4⋊C4×C71C411yescharacteristic, complemented
21C2×#95×C71C411yescharacteristic, complemented
22C2×#95×C71C2211yescharacteristic
23C22×C71⋊C8—22no—
24C2×C4⋊Dic71—22no—
25C2×C22⋊Dic71—22no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C71⋊#95—22no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71—21no—
32C23×C4×C71—21no—
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C4×D4×C71—44no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C42×C71—21no—
48C2×C42×C71C2×C411yescharacteristic
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71C2×C411yescharacteristic
53#95×C71C2×C411yescharacteristic
54#95×C71C2×C411yescharacteristic
55#95×C71C2×C411yescharacteristic
56C22×Dic71—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—82no—
59C4⋊Dic71—82no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C2×D411yescharacteristic
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71SD1612yes—
67C2×D4×C71Q1612yes—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71—81no—
79C22×C4×C71—81no—
80C22×C4×C71—81no—
81C22×C4×C71—81no—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4112yes—
88C22×C4×C71#4112yes—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—42no—
92C4⋊C4×C71—81no—
93C4⋊C4×C71—81no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—42no—
98C42×C71—21no—
99C42×C71—21no—
100C42×C71—42no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71#10312yes—
108C23×C71C2×C22⋊C411yescharacteristic
109C23×C71#12312yes—
110C23×C71C2×SD1611yescharacteristic
111C23×C71C2×Q1611yescharacteristic
112C23×C71AGL(1,ℤ/8)12yes—
113D4×C71—24no—
114D4×C71—24no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—22no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—81no—
138C2×C4×C71—81no—
139C2×C4×C71—84no—
140C2×C4×C71—84no—
141C2×C4×C71#10312yes—
142C2×C4×C71#11711yescharacteristic
143C2×C4×C71#11711yescharacteristic
144C2×C4×C71#12312yes—
145C2×Dic71—82no—
146C2×Dic71—82no—
147C2×Dic71—82no—
148C71⋊C8—82no—
149C22×C71—22no—
150C22×C71—21no—
151C22×C71—22no—
152C22×C71—21no—
153C22×C71—22no—
154C22×C71—42no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—42no—
158C22×C71—42no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71#38111yescharacteristic
162C22×C71C2×#10311yescharacteristic
163C22×C71#41811yescharacteristic
164C22×C71#48112yes—
165C22×C71#48412yes—
166C4×C71—22no—
167C4×C71—22no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—42no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71—81no—
176Dic71—162no—
177#8761—711nomaximal, Sylow
178C2×C71—22no—
179C2×C71—21no—
180C2×C71—21no—
181C2×C71—22no—
182C2×C71—42no—
183C2×C71#182411yescharacteristic
184C2×C71#182711yescharacteristic
185C2×C71#182811yescharacteristic
186#2349D7111yescharacteristic
187#1799—711no—
188#1459—711no—
189#1294—712no—
190#1289—712no—
191C71#876111yescharacteristic, complemented, Sylow
192C2×C4×D4—21no—
193C2×C4×D4C2×D7111yescharacteristic
194C2×#99—1421no—
195C2×C22⋊C8—712no—
196#389—21no—
197#385—21no—
198#385—21no—
199#382Dic7111yescharacteristic, complemented
200#380—712no—
201#380Dic7111yescharacteristic, complemented
202#380C2×D7111yescharacteristic
203C4×C4⋊C4Dic7111yescharacteristic, complemented
204C2×#95Dic7111yescharacteristic, complemented
205C2×#95C2×D7111yescharacteristic
206#342—1421no—
207#324—1421no—
208#323—1421no—
209C22×D4C22⋊D7111yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C2×Dic7111yescharacteristic
213C23×C4C22×D7111yescharacteristic
214C22×C8—1422no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—1422no—
223C2×C4⋊C4C22⋊D7111yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1422no—
227C2×C22⋊C4C22⋊D7111yescharacteristic
228C2×C22⋊C4C22⋊D7111yescharacteristic
229C2×C42—21no—
230C2×C42C2×Dic7111yescharacteristic
231#99—2842no—
232C22⋊C8—1422no—
233C22⋊C8—1422no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—1422no—
238#95C2×Dic7111yescharacteristic
239#95C2×Dic7111yescharacteristic
240#95C2×Dic7111yescharacteristic
241#95C2×Dic7111yescharacteristic
242C24C22⋊Dic7111yescharacteristic
243C24C2×C22⋊D7111yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4#1176585212yes—
247C2×D4#1176585512yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—2842no—
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22⋊Dic7111yescharacteristic
265C22×C4C22⋊Dic7111yescharacteristic
266C22×C4C22×Dic7111yescharacteristic
267C22×C4C2×C22⋊D7111yescharacteristic
268C22×C4#1176586512yes—
269C22×C4#1176586512yes—
270C2×C8—2842no—
271C2×C8—5682no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—5682no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—2842no—
283C22⋊C4—2842no—
284C42—21no—
285C42—21no—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#1455605212yes—
294C23C2×C22⋊Dic7111yescharacteristic
295C23#1455611112yes—
296C23C2×#1176585211yescharacteristic
297C23C2×#1176585511yescharacteristic
298C23#1455615412yes—
299D4—24no—
300D4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—5682no—
328C2×C4—5682no—
329C2×C4—5682no—
330C2×C4#1455605212yes—
331C2×C4#1455608711yescharacteristic
332C2×C4#1455608711yescharacteristic
333C2×C4#1455611112yes—
334C8—5682no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22#1798586211yescharacteristic
348C22C2×#1455605211yescharacteristic
349C22#1798600411yescharacteristic
350C22#1798631712yes—
351C22#1798633412yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—11362no—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2#2258843011yescharacteristic
369C2#2258844411yescharacteristic
370C2#2258844811yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2349C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2349D71
hypocenterC71#8761
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22586219C2semidirect
4. #22586207C2semidirect
17. #382×C71C4semidirect
18. #380×C71C4semidirect
20. C4×C4⋊C4×C71C4semidirect
21. C2×#95×C71C4semidirect
191. C71#8761semidirect
199. #382Dic71semidirect
201. #380Dic71semidirect
203. C4×C4⋊C4Dic71semidirect
204. C2×#95Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866582
#225862192
#225862072
C71×#23492
#876171