Finite Group Databasebeta

← #31907958

Subgroups — #31907958

Order 18176.

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

normal subgroups only

This group has 428 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#22588079C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3#22586370C212yesmaximal, complemented
4#22586346C212yesmaximal, complemented
5C71×#2599C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17986056—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C4×Q8×C71C411yescharacteristic
13C2×#117×C71C2211yescharacteristic
14#398×C71C2211yescharacteristic
15#397×C71C411yescharacteristic
16#395×C71C2211yescharacteristic
17#393×C71C411yescharacteristic
18#388×C71—21no—
19#387×C71—21no—
20#387×C71—21no—
21#385×C71—21no—
22#383×C71C411yescharacteristic
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C22×C71⋊C8—41no—
28C4⋊(C71⋊C8)—41no—
29C22⋊(C71⋊C8)—22no—
30#14556039—41no—
31C22×Q8×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C4×Q8×C71—41no—
34C4×Q8×C71—41no—
35#117×C71—41no—
36#117×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71C2311yescharacteristic
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C42×C71—21no—
48C2×C42×C71C2×C411yescharacteristic
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×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58#11765849—24no—
59#11765849—44no—
60#11765849—42no—
61C2×C71⋊C8—44no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C24×C71C22⋊C411yescharacteristic
65C2×Q8×C71—42no—
66C2×Q8×C71—41no—
67C22×C4×C71—22no—
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—21no—
75C22×C4×C71—21no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—42no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—21no—
98C42×C71—42no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71#10412yes—
104C23×C71C2×C22⋊C411yescharacteristic
105Q8×C71—42no—
106Q8×C71—42no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
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—21no—
120C2×C4×C71—41no—
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#9911yescharacteristic
136C2×C4×C71#9911yescharacteristic
137C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C71#13711yescharacteristic
139C71⋊C8—44no—
140C71⋊C8—82no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—22no—
144C22×C71—42no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71C2×#9911yescharacteristic
149C22×C71#41711yescharacteristic
150C22×C71C2×#10411yescharacteristic
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—41no—
156C4×C71—42no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71—41no—
160C4×C71—81no—
161C4×C71—81no—
162C4×C71—81no—
163#11763—711nomaximal, Sylow
164C2×C71—21no—
165C2×C71—21no—
166C2×C71—42no—
167C2×C71#150411yescharacteristic
168C2×C71#212011yescharacteristic
169C2×C71#213511yescharacteristic
170#2599D7111yescharacteristic
171#1742—711no—
172#1464—711no—
173#1354—712no—
174#1346—712no—
175C71#1176311yescharacteristic, complemented, Sylow
176C22×M16—1421no—
177C2×C4×Q8Dic7111yescharacteristic
178C2×#117C2×D7111yescharacteristic
179#432—1421no—
180#398C2×D7111yescharacteristic
181#397Dic7111yescharacteristic
182#395C2×D7111yescharacteristic
183#393Dic7111yescharacteristic
184#388—21no—
185#387—21no—
186#387—21no—
187#385—21no—
188#383Dic7111yescharacteristic
189#343—712no—
190#343—1421no—
191#324—712no—
192#323—1421no—
193C22×Q8C2×Dic7111yescharacteristic
194C23×C4C22⋊D7111yescharacteristic
195C2×M16—2841no—
196C2×M16—2842no—
197C2×M16—1424no—
198C2×M16—1422no—
199C22×C8—2841no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×Dic7111yescharacteristic
208C2×C4⋊C4C22⋊D7111yescharacteristic
209C2×C4⋊C4C22×D7111yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4C2×Dic7111yescharacteristic
213C2×C22⋊C4C22⋊D7111yescharacteristic
214C2×C42—21no—
215C2×C42C2×Dic7111yescharacteristic
216C2×C42C22⋊D7111yescharacteristic
217C4⋊C8—2841no—
218C22⋊C8—1422no—
219#97—2841no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C2×Dic7111yescharacteristic
227#95C2×Dic7111yescharacteristic
228C24C22⋊Dic7111yescharacteristic
229C2×Q8—42no—
230C2×Q8—41no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4C22⋊Dic7111yescharacteristic
241C22×C4C22⋊Dic7111yescharacteristic
242C22×C4C22⋊Dic7111yescharacteristic
243C22×C4C22×Dic7111yescharacteristic
244C22×C4C2×C22⋊D7111yescharacteristic
245C22×C4C2×C22⋊D7111yescharacteristic
246M16—2844no—
247M16—2842no—
248M16—1424no—
249C2×C8—2844no—
250C2×C8—2842no—
251C2×C8—2842no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1455605512yes—
274C23C2×C22⋊Dic7111yescharacteristic
275Q8—42no—
276Q8—42no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4#1455604311yescharacteristic
306C2×C4#1455604311yescharacteristic
307C2×C4#1455615311yescharacteristic
308C2×C4#1455615811yescharacteristic
309C8—2844no—
310C8—5682no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1455604311yescharacteristic
319C22#1798600011yescharacteristic
320C22C2×#1455605511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2258684411yescharacteristic
337C2#2258998111yescharacteristic
338C2#2259003511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2599C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2599D71
hypocenterC71#11763
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
3. #22586370C2semidirect
4. #22586346C2semidirect
175. C71#11763semidirect

Maximal subgroups

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

subgroupindex
#225880792
#225866742
#225863702
#225863462
C71×#25992
#1176371