Finite Group Databasebeta

← #31907008

Subgroups — #31907008

Order 18176.

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

normal subgroups only

This group has 496 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#22586670C211yesmaximal, characteristic, complemented
2C2×#17985725C211yesmaximal, characteristic, complemented
3#22586286C212yesmaximal, complemented
4#22586272C212yesmaximal, complemented
5C71×#2501C211yesmaximal, characteristic
6#17985963—21no—
7C2×C22⋊(C71⋊C8)—21no—
8#17985725—21no—
9#17985725—21no—
10#17985725C2212yes—
11#17985725C2212yes—
12C2×#123×C71—21no—
13C2×#123×C71—21no—
14C2×#121×C71C411yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#399×C71—21no—
17#399×C71—21no—
18#390×C71C411yescharacteristic
19#390×C71C2211yescharacteristic
20#384×C71C411yescharacteristic
21#384×C71C2211yescharacteristic
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—41no—
24C22×C71⋊C8—41no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28C22⋊(C71⋊C8)—41no—
29C22×D4×C71—21no—
30C22×D4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34#123×C71—44no—
35#123×C71—44no—
36#121×C71—41no—
37#121×C71—41no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—41no—
43C2×C4⋊C4×C71—41no—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71C2311yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C42×C71C2×C411yescharacteristic
56C2×C42×C71C2×C411yescharacteristic
57#95×C71—21no—
58#11765849—82no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C24×C71—21no—
63C24×C71C22⋊C411yescharacteristic
64C2×D4×C71—22no—
65C2×D4×C71—41no—
66C2×D4×C71—41no—
67C2×D4×C71—42no—
68C2×D4×C71—42no—
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—22no—
80C22×C4×C71—21no—
81C22×C4×C71—41no—
82C22×C4×C71—42no—
83C22×C4×C71—42no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—42no—
97C4⋊C4×C71—42no—
98C4⋊C4×C71—42no—
99C22⋊C4×C71—24no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—42no—
103C22⋊C4×C71—42no—
104C22⋊C4×C71—42no—
105C42×C71—22no—
106C42×C71—41no—
107C23×C71—22no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71#9911yescharacteristic
119C23×C71#9911yescharacteristic
120C23×C71C2×C22⋊C411yescharacteristic
121D4×C71—82no—
122D4×C71—82no—
123D4×C71—82no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—22no—
127C2×C4×C71—22no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—42no—
140C2×C4×C71—42no—
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—81no—
151C2×C4×C71—81no—
152C2×C4×C71#13112yes—
153C71⋊C8—82no—
154C71⋊C8—82no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—42no—
163C22×C71—42no—
164C22×C71—81no—
165C22×C71—81no—
166C22×C71—81no—
167C22×C71—81no—
168C22×C71—81no—
169C22×C71—81no—
170C22×C71—81no—
171C22×C71#35312yes—
172C22×C71#40811yescharacteristic
173C22×C71C2×#9911yescharacteristic
174C22×C71#41311yescharacteristic
175C4×C71—22no—
176C4×C71—42no—
177C4×C71—42no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—81no—
184#11549—711nomaximal, Sylow
185C2×C71—21no—
186C2×C71—21no—
187C2×C71—22no—
188C2×C71—41no—
189C2×C71—161no—
190C2×C71#146811yescharacteristic
191C2×C71C2×#35311yescharacteristic
192C2×C71#213311yescharacteristic
193#2501D7111yescharacteristic
194#1463—711no—
195C2×#324—711no—
196#1325—712no—
197#1318—712no—
198C71#1154911yescharacteristic, complemented, Sylow
199C2×#123—21no—
200C2×#123—21no—
201C2×#121Dic7111yescharacteristic
202C22×C4⋊C4C2×D7111yescharacteristic
203#407—1421no—
204C2×C22⋊C8—1421no—
205#399—21no—
206#399—21no—
207#390Dic7111yescharacteristic
208#390C2×D7111yescharacteristic
209#384Dic7111yescharacteristic
210#384C2×D7111yescharacteristic
211C4×C22⋊C4Dic7111yescharacteristic
212#324—712no—
213#324—712no—
214#324—1421no—
215#324—1421no—
216C22×D4—21no—
217C22×D4C2×Dic7111yescharacteristic
218C23×C4C2×Dic7111yescharacteristic
219C23×C4C22⋊D7111yescharacteristic
220C23×C4C22⋊D7111yescharacteristic
221C2×M16—2841no—
222C22×C8—2841no—
223#123—44no—
224#123—44no—
225#121—41no—
226#121—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C2×Dic7111yescharacteristic
234C2×C4⋊C4C2×Dic7111yescharacteristic
235C2×C4⋊C4C22⋊D7111yescharacteristic
236C2×C4⋊C4C22⋊D7111yescharacteristic
237C2×C4⋊C4C22×D7111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42C2×Dic7111yescharacteristic
245C2×C42C2×Dic7111yescharacteristic
246C22⋊C8—2841no—
247C22⋊C8—2841no—
248C22⋊C8—1422no—
249C22⋊C8—1422no—
250#95—21no—
251C24—21no—
252C24C22⋊Dic7111yescharacteristic
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C22⋊Dic7111yescharacteristic
274C22×C4C22⋊Dic7111yescharacteristic
275C22×C4C22⋊Dic7111yescharacteristic
276C22×C4C22×Dic7111yescharacteristic
277C22×C4C2×C22⋊D7111yescharacteristic
278C22×C4C2×C22⋊D7111yescharacteristic
279M16—5682no—
280C2×C8—2842no—
281C2×C8—2842no—
282C2×C8—5682no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—22no—
299C42—41no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23#1455604311yescharacteristic
312C23#1455604311yescharacteristic
313C23C2×C22⋊Dic7111yescharacteristic
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4#1455613512yes—
346C8—5682no—
347C8—5682no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#1798579212yes—
365C22#1798596611yescharacteristic
366C22C2×#1455604311yescharacteristic
367C22#1798598311yescharacteristic
368C4—22no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2#2258668811yescharacteristic
383C2C2×#1798579211yescharacteristic
384C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2501C2
socleC22×C71C2×#99
prime radicalC2×#121×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2501D71
hypocenterC71#11549
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
1. #22586670C2semidirect
2. C2×#17985725C2semidirect
3. #22586286C2semidirect
4. #22586272C2semidirect
198. C71#11549semidirect

Maximal subgroups

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

subgroupindex
#225866702
C2×#179857252
#225862862
#225862722
C71×#25012
#1154971