Finite Group Databasebeta

← #31908228

Subgroups — #31908228

Order 18176.

446 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 91 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#22588328C211yesmaximal, characteristic
2#22586678C211yesmaximal, characteristic
3#22586361C212yesmaximal, complemented
4#22586340C212yesmaximal, complemented
5C71×#2398C211yesmaximal, characteristic
6C2×#14556043—21no—
7#17985963—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×C4×Q8×C71C411yescharacteristic, complemented
13C2×C4×D4×C71C2211yescharacteristic
14#400×C71—21no—
15#397×C71C411yescharacteristic, complemented
16#397×C71C2211yescharacteristic
17#394×C71—21no—
18#393×C71C411yescharacteristic, complemented
19#393×C71C2211yescharacteristic
20C4×C22⋊C4×C71—21no—
21C4×C22⋊C4×C71—21no—
22#376×C71C411yescharacteristic, complemented
23C2×#11765849—41no—
24C2×C4⋊Dic71—41no—
25C2×C22⋊Dic71—22no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—41no—
29C71⋊#95—41no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×D4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C4×Q8×C71—41no—
35C4×Q8×C71—41no—
36C4×D4×C71—41no—
37C4×D4×C71—41no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71C2311yescharacteristic
47C2×C42×C71—21no—
48C2×C42×C71—21no—
49C2×C42×C71C2×C411yescharacteristic
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71C2×C411yescharacteristic
56#95×C71C2×C411yescharacteristic
57C22×Dic71—42no—
58#11765849—82no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—82no—
61C22⋊Dic71—42no—
62C22⋊Dic71—42no—
63C24×C71C22⋊C411yescharacteristic
64C24×C71C2×D411yescharacteristic
65C2×Q8×C71—42no—
66C2×Q8×C71—41no—
67C2×D4×C71—21no—
68C2×D4×C71—21no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—42no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C42×C71—21no—
97C42×C71—21no—
98C42×C71—42no—
99C42×C71—41no—
100C42×C71—41no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71C2×C22⋊C411yescharacteristic
111C23×C71AGL(1,ℤ/8)11yescharacteristic
112C23×C71#13711yescharacteristic
113Q8×C71—42no—
114Q8×C71—42no—
115D4×C71—42no—
116D4×C71—42no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
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—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—81no—
142C2×C4×C71—81no—
143C2×C4×C71#9911yescharacteristic
144C2×C4×C71#9911yescharacteristic
145C2×Dic71—82no—
146C2×Dic71—82no—
147C2×Dic71—82no—
148C71⋊C8—82no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—42no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71C2×#9911yescharacteristic
164C22×C71#41711yescharacteristic
165C22×C71#42111yescharacteristic
166C4×C71—21no—
167C4×C71—42no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—81no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71#35412yes—
176Dic71—162no—
177#11823—711nomaximal, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—41no—
181C2×C71—41no—
182C2×C71—81no—
183C2×C71#151311yescharacteristic
184C2×C71#212011yescharacteristic
185C2×C71C2×#35411yescharacteristic
186#2398D7111yescharacteristic
187#1799—711no—
188#1465—711no—
189#1351—712no—
190#1344—712no—
191C71#1182311yescharacteristic, complemented, Sylow
192C2×C4×Q8Dic7111yescharacteristic, complemented
193C2×C4×D4C2×D7111yescharacteristic
194C2×#99—1421no—
195#407—1421no—
196#400—21no—
197#397Dic7111yescharacteristic, complemented
198#397C2×D7111yescharacteristic
199#394—21no—
200#393Dic7111yescharacteristic, complemented
201#393C2×D7111yescharacteristic
202#380—1421no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205#376Dic7111yescharacteristic, complemented
206#342—712no—
207#324—1421no—
208#323—712no—
209C22×Q8C2×Dic7111yescharacteristic
210C22×D4C22⋊D7111yescharacteristic
211C23×C4C22⋊D7111yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C2×M16—2841no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—2841no—
220C2×C4⋊C4C2×Dic7111yescharacteristic
221C2×C4⋊C4C22⋊D7111yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1422no—
227C2×C22⋊C4C2×Dic7111yescharacteristic
228C2×C22⋊C4C22×D7111yescharacteristic
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C2×Dic7111yescharacteristic
232#99—2842no—
233C22⋊C8—2841no—
234C22⋊C8—1422no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—2841no—
241#95C2×Dic7111yescharacteristic
242#95C2×Dic7111yescharacteristic
243C24C22⋊Dic7111yescharacteristic
244C24C2×C22⋊D7111yescharacteristic
245C2×Q8—42no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—2842no—
262C22×C4C22⋊Dic7111yescharacteristic
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22⋊Dic7111yescharacteristic
265C22×C4C22×Dic7111yescharacteristic
266C22×C4C2×C22⋊D7111yescharacteristic
267M16—5682no—
268C2×C8—2842no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—5682no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—2842no—
281C22⋊C4—2842no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23C2×C22⋊Dic7111yescharacteristic
297C23#1455615311yescharacteristic
298C23#1455615811yescharacteristic
299Q8—42no—
300Q8—42no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—5682no—
330C2×C4—5682no—
331C2×C4—5682no—
332C2×C4#1455604311yescharacteristic
333C2×C4#1455604311yescharacteristic
334C8—5682no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#1455604311yescharacteristic
350C22#1798600011yescharacteristic
351C22#1798601611yescharacteristic
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—11362no—
362C4#1798579412yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#2258688311yescharacteristic
369C2#2258998111yescharacteristic
370C2C2×#1798579411yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2398C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2398D71
hypocenterC71#11823
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #22586361C2semidirect
4. #22586340C2semidirect
12. C2×C4×Q8×C71C4semidirect
15. #397×C71C4semidirect
18. #393×C71C4semidirect
22. #376×C71C4semidirect
191. C71#11823semidirect
192. C2×C4×Q8Dic71semidirect
197. #397Dic71semidirect
200. #393Dic71semidirect
205. #376Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866782
#225863612
#225863402
C71×#23982
#1182371