Finite Group Databasebeta

← #31908096

Subgroups — #31908096

Order 18176.

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

normal subgroups only

This group has 444 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
1C4×#14556045C211yesmaximal, characteristic
2#22586678C211yesmaximal, characteristic
3#22586370C212yesmaximal, complemented
4#22586367C212yesmaximal, complemented
5C71×#2564C211yesmaximal, characteristic
6C2×#14556045—21no—
7#17985963—21no—
8C4×#11765849—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C22⋊Q8×C71C411yescharacteristic
13C2×C4×D4×C71C2211yescharacteristic
14#400×C71C2211yescharacteristic
15#399×C71C411yescharacteristic
16#395×C71C2211yescharacteristic
17#393×C71C411yescharacteristic
18#387×C71—21no—
19#386×C71—21no—
20#385×C71—21no—
21#380×C71—21no—
22C2×#95×C71C411yescharacteristic
23C2×#11765849—22no—
24C2×#11765849—41no—
25#14556045—24no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28#14556039—41no—
29C4×C71⋊C8—41no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×D4×C71D411yescharacteristic
32C23×C4×C71—21no—
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C22⋊Q8×C71—41no—
36C22⋊Q8×C71—41no—
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
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
57#11765849—24no—
58#11765849—42no—
59#11765849—82no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C24×C71C22⋊C411yescharacteristic
63C24×C71C22⋊C411yescharacteristic
64C2×Q8×C71—41no—
65C2×Q8×C71—81no—
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—41no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
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—42no—
78C22×C4×C71—42no—
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×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—81no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—81no—
98C42×C71—21no—
99C42×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71#10412yes—
110C23×C71C2×C22⋊C411yescharacteristic
111Q8×C71—82no—
112D4×C71—44no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—22no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—41no—
120C2×C4×C71—42no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
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—81no—
136C2×C4×C71—81no—
137C2×C4×C71—81no—
138C2×C4×C71—84no—
139C2×C4×C71#9911yescharacteristic
140C2×C4×C71#9911yescharacteristic
141C2×C4×C71#13712yes—
142C71⋊C8—44no—
143C71⋊C8—82no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—22no—
147C22×C71—42no—
148C22×C71—42no—
149C22×C71—42no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—81no—
156C22×C71C2×#9911yescharacteristic
157C22×C71#41911yescharacteristic
158C22×C71C2×#10411yescharacteristic
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—22no—
162C4×C71—42no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—81no—
167C4×C71—81no—
168C4×C71—81no—
169#11793—711nomaximal, Sylow
170C2×C71—21no—
171C2×C71—21no—
172C2×C71—42no—
173C2×C71—81no—
174C2×C71#150811yescharacteristic
175C2×C71#212111yescharacteristic
176C2×C71#213211yescharacteristic
177#2564D7111yescharacteristic
178C4×#100—711no—
179#1465—711no—
180#1354—712no—
181#1353—712no—
182C71#1179311yescharacteristic, complemented, Sylow
183C2×C22⋊Q8Dic7111yescharacteristic
184C2×C4×D4C2×D7111yescharacteristic
185C2×#100—1421no—
186#407—1421no—
187C4×M16—1421no—
188#400C2×D7111yescharacteristic
189#399Dic7111yescharacteristic
190#395C2×D7111yescharacteristic
191#393Dic7111yescharacteristic
192#387—21no—
193#386—21no—
194#385—21no—
195#380—21no—
196C2×#95Dic7111yescharacteristic
197#343—712no—
198#324—712no—
199#323—1421no—
200C22×Q8C2×Dic7111yescharacteristic
201C22×D4C22⋊D7111yescharacteristic
202C23×C4—21no—
203C23×C4C2×Dic7111yescharacteristic
204C23×C4C22⋊D7111yescharacteristic
205C2×M16—2841no—
206C2×M16—1422no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic7111yescharacteristic
215C2×C4⋊C4C22×D7111yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×Dic7111yescharacteristic
220C2×C22⋊C4C22⋊D7111yescharacteristic
221C2×C42C22⋊D7111yescharacteristic
222#100—1424no—
223C22⋊C8—2841no—
224C22⋊C8—1422no—
225#97—2841no—
226C4×C8—2841no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic7111yescharacteristic
233#95C2×Dic7111yescharacteristic
234C24C22⋊Dic7111yescharacteristic
235C24C22⋊Dic7111yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22×Dic7111yescharacteristic
257C22×C4C2×C22⋊D7111yescharacteristic
258C22×C4C2×C22⋊D7111yescharacteristic
259M16—2842no—
260M16—5682no—
261M16—1424no—
262C2×C8—2842no—
263C2×C8—2842no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1455605512yes—
287C23C2×C22⋊Dic7111yescharacteristic
288Q8—82no—
289D4—44no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—84no—
316C2×C4#1455604311yescharacteristic
317C2×C4#1455604311yescharacteristic
318C2×C4#1455615912yes—
319C8—2844no—
320C8—5682no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22C2×#1455604311yescharacteristic
334C22#1798600811yescharacteristic
335C22C2×#1455605511yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2258686011yescharacteristic
351C2#2258998611yescharacteristic
352C2#2259002511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2564C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C4⋊C4×C71C23
hypercenter#2564D71
hypocenterC71#11793
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #22586370C2semidirect
4. #22586367C2semidirect
182. C71#11793semidirect

Maximal subgroups

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

subgroupindex
C4×#145560452
#225866782
#225863702
#225863672
C71×#25642
#1179371