Finite Group Databasebeta

← #31908174

Subgroups — #31908174

Order 18176.

496 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588414C211yesmaximal, characteristic
2#22586667C211yesmaximal, characteristic
3#22586364C212yesmaximal, complemented
4#22586361C212yesmaximal, complemented
5C71×#2626C211yesmaximal, characteristic
6#17985963—21no—
7#17985855—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic, complemented
13C22×C22⋊C4×C71C2211yescharacteristic
14#399×C71C411yescharacteristic, complemented
15#399×C71C2211yescharacteristic
16#396×C71—21no—
17#396×C71—21no—
18#393×C71C411yescharacteristic, complemented
19#393×C71C2211yescharacteristic
20#385×C71—21no—
21#380×C71C411yescharacteristic, complemented
22#379×C71—21no—
23C2×#11765849—41no—
24C2×C22⋊Dic71—22no—
25C2×C22⋊Dic71—41no—
26C2×C22⋊Dic71—41no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—41no—
29C25×C71D411yescharacteristic
30C22×Q8×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C22⋊Q8×C71—41no—
34C22⋊Q8×C71—41no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—41no—
47C2×C22⋊C4×C71—41no—
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C22⋊C4×C71D411yescharacteristic
51C2×C22⋊C4×C71C2311yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71C2×C411yescharacteristic
55#95×C71C2×C411yescharacteristic
56C22×Dic71—42no—
57#11765849—82no—
58C2×C71⋊C8—42no—
59C22⋊Dic71—42no—
60C22⋊Dic71—42no—
61C22⋊Dic71—82no—
62C22⋊Dic71—82no—
63C24×C71—21no—
64C24×C71—21no—
65C24×C71—22no—
66C24×C71—41no—
67C24×C71—81no—
68C24×C71—81no—
69C24×C71C22⋊C411yescharacteristic
70C24×C71C22⋊C411yescharacteristic
71C24×C71C2×D411yescharacteristic
72C2×Q8×C71—41no—
73C2×Q8×C71—81no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—81no—
83C22×C4×C71—81no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—41no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—81no—
91C4⋊C4×C71—81no—
92C4⋊C4×C71—81no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—81no—
101C22⋊C4×C71—81no—
102C22⋊C4×C71—81no—
103C22⋊C4×C71—81no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—42no—
120C23×C71—42no—
121C23×C71—81no—
122C23×C71—81no—
123C23×C71—84no—
124C23×C71—84no—
125C23×C71#9911yescharacteristic
126C23×C71#9911yescharacteristic
127C23×C71C2×C22⋊C411yescharacteristic
128C23×C71#13712yes—
129Q8×C71—82no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—42no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—81no—
152C2×C4×C71—81no—
153C2×C4×C71—81no—
154C2×Dic71—82no—
155C2×Dic71—82no—
156C2×Dic71—82no—
157C71⋊C8—82no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—22no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—42no—
168C22×C71—42no—
169C22×C71—42no—
170C22×C71—42no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71—81no—
177C22×C71—81no—
178C22×C71—84no—
179C22×C71—84no—
180C22×C71C2×#9911yescharacteristic
181C22×C71#41911yescharacteristic
182C22×C71#42111yescharacteristic
183C4×C71—41no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—81no—
188C4×C71—81no—
189C4×C71—81no—
190C4×C71—81no—
191Dic71—162no—
192#11811—711nomaximal, Sylow
193C2×C71—21no—
194C2×C71—21no—
195C2×C71—42no—
196C2×C71—41no—
197C2×C71—41no—
198C2×C71—81no—
199C2×C71#151111yescharacteristic
200C2×C71#212111yescharacteristic
201C2×C71#212711yescharacteristic
202#2626D7111yescharacteristic
203#1819—711no—
204#1462—711no—
205#1352—712no—
206#1351—712no—
207C71#1181111yescharacteristic, complemented, Sylow
208C2×C22⋊Q8Dic7111yescharacteristic, complemented
209C22×C22⋊C4C2×D7111yescharacteristic
210#407—1421no—
211#399Dic7111yescharacteristic, complemented
212#399C2×D7111yescharacteristic
213#396—21no—
214#396—21no—
215#393Dic7111yescharacteristic, complemented
216#393C2×D7111yescharacteristic
217#385—21no—
218#380Dic7111yescharacteristic, complemented
219#379—21no—
220#379—1421no—
221#342—712no—
222#342—1421no—
223#323—712no—
224#323—1421no—
225C25C22⋊D7111yescharacteristic
226C22×Q8C2×Dic7111yescharacteristic
227C23×C4C2×Dic7111yescharacteristic
228C23×C4C22⋊D7111yescharacteristic
229C2×M16—2841no—
230C22⋊Q8—41no—
231C22⋊Q8—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic7111yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—2841no—
246C2×C22⋊C4—2841no—
247C2×C22⋊C4—1422no—
248C2×C22⋊C4C2×Dic7111yescharacteristic
249C2×C22⋊C4C22⋊D7111yescharacteristic
250C2×C22⋊C4C22⋊D7111yescharacteristic
251C2×C22⋊C4C22×D7111yescharacteristic
252C22⋊C8—2841no—
253C22⋊C8—1422no—
254#95—21no—
255#95—21no—
256#95C2×Dic7111yescharacteristic
257#95C2×Dic7111yescharacteristic
258C24—21no—
259C24—21no—
260C24—22no—
261C24—41no—
262C24—81no—
263C24—81no—
264C24C22⋊Dic7111yescharacteristic
265C24C22⋊Dic7111yescharacteristic
266C24C2×C22⋊D7111yescharacteristic
267C2×Q8—41no—
268C2×Q8—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—2842no—
280C22×C4C22⋊Dic7111yescharacteristic
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22×Dic7111yescharacteristic
283C22×C4C2×C22⋊D7111yescharacteristic
284M16—5682no—
285C2×C8—2842no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—2842no—
303C22⋊C4—2842no—
304C22⋊C4—5682no—
305C22⋊C4—5682no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—84no—
326C23—84no—
327C23#1455604311yescharacteristic
328C23#1455604311yescharacteristic
329C23C2×C22⋊Dic7111yescharacteristic
330C23#1455615912yes—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—5682no—
357C2×C4—5682no—
358C2×C4—5682no—
359C8—5682no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×#1455604311yescharacteristic
383C22#1798600811yescharacteristic
384C22#1798601611yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—11362no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2#2258687311yescharacteristic
401C2#2258998611yescharacteristic
402C2#2259000911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2626C2
socleC22×C71C2×#99
prime radicalC25×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2626D71
hypocenterC71#11811
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #22586364C2semidirect
4. #22586361C2semidirect
12. C2×C22⋊Q8×C71C4semidirect
14. #399×C71C4semidirect
18. #393×C71C4semidirect
21. #380×C71C4semidirect
207. C71#11811semidirect
208. C2×C22⋊Q8Dic71semidirect
211. #399Dic71semidirect
215. #393Dic71semidirect
218. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884142
#225866672
#225863642
#225863612
C71×#26262
#1181171