Finite Group Databasebeta

← #31892547

Subgroups — #31892547

Order 18176.

482 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 119 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#22596978C211yesmaximal, characteristic
2#22586769C212yesmaximal, complemented
3#22586654C211yesmaximal, characteristic
4#22586243C212yesmaximal, complemented
5C71×#2396C211yesmaximal, characteristic
6#17986060—24no—
7#17986052C2212yes—
8#17986023C2212yes—
9#17985963—21no—
10C4×#11765849—21no—
11Q8⋊(C71⋊C8)—24no—
12#17985725—21no—
13#17985723—21no—
14C2×C4×Q8×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#400×C71C411yescharacteristic
17#399×C71C411yescharacteristic
18#398×C71—21no—
19#396×C71—21no—
20#395×C71C411yescharacteristic
21#393×C71C411yescharacteristic
22C4×C4⋊C4×C71—21no—
23C4×C4⋊C4×C71C2211yescharacteristic
24C4×C22⋊C4×C71—21no—
25C2×#11765849—22no—
26C4⋊(C71⋊C8)—22no—
27C4⋊(C71⋊C8)—22no—
28C4⋊(C71⋊C8)D414yes—
29C22⋊(C71⋊C8)—24no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32C4×C71⋊C8—22no—
33C4×C71⋊C8—22no—
34C22×Q8×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C4×Q8×C71—22no—
37C4×Q8×C71—22no—
38#117×C71—22no—
39#117×C71—22no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71C2×C411yescharacteristic
50C2×C22⋊C4×C71D411yescharacteristic
51C2×C42×C71—21no—
52C2×C42×C71—21no—
53C2×C42×C71D411yescharacteristic
54C2×C42×C71C2311yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71C2×C411yescharacteristic
60#95×C71C2×C411yescharacteristic
61#11765849—42no—
62#11765849—42no—
63C2×C71⋊C8—24no—
64C2×C71⋊C8—42no—
65C24×C71C22⋊C411yescharacteristic
66C2×Q8×C71—22no—
67C2×Q8×C71—42no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
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—22no—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—22no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—22no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—41no—
96C42×C71—41no—
97C42×C71—42no—
98C42×C71C2×D412yes—
99C42×C71#4112yes—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71C2×C22⋊C411yescharacteristic
106Q8×C71—24no—
107Q8×C71—42no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
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×C71C4×D412yes—
132C2×C4×C71#12011yescharacteristic
133C2×C4×C71#12112yes—
134C2×C4×C71#12311yescharacteristic
135C2×C4×C71#13512yes—
136C2×C4×C71#13712yes—
137C71⋊C8—44no—
138C71⋊C8—82no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—42no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—81no—
146C22×C71#38611yescharacteristic
147C22×C71#41611yescharacteristic
148C22×C71#41911yescharacteristic
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—42no—
152C4×C71—41no—
153C4×C71—41no—
154C4×C71—41no—
155C4×C71—42no—
156C4×C71—81no—
157C4×C71—81no—
158C4×C71#45212yes—
159C4×C71#48512yes—
160#9034—711nomaximal, Sylow
161C2×C71—21no—
162C2×C71—21no—
163C2×C71—81no—
164C2×C71#187311yescharacteristic
165C2×C71#188411yescharacteristic
166C2×C71#189111yescharacteristic
167#2921—711no—
168#2396D7111yescharacteristic
169#1487—712no—
170#1458—711no—
171#1303—712no—
172C71#903411yescharacteristic, complemented, Sylow
173C2×C4×Q8C2×D7111yescharacteristic
174C2×#117C2×D7111yescharacteristic
175#433—1424no—
176#431—712no—
177#423—712no—
178#407—1421no—
179C4×M16—1421no—
180#400Dic7111yescharacteristic
181#399Dic7111yescharacteristic
182#398—21no—
183#396—21no—
184#395Dic7111yescharacteristic
185#393Dic7111yescharacteristic
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2×D7111yescharacteristic
188C4×C22⋊C4—21no—
189Q8⋊C8—1424no—
190#324—1421no—
191#323—1421no—
192C22×Q8C2×Dic7111yescharacteristic
193C23×C4C22⋊D7111yescharacteristic
194C2×M16—1422no—
195C4×Q8—22no—
196C4×Q8—22no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C2×Dic7111yescharacteristic
204C2×C4⋊C4C2×Dic7111yescharacteristic
205C2×C4⋊C4C22⋊D7111yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4C2×Dic7111yescharacteristic
209C2×C22⋊C4C22⋊D7111yescharacteristic
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22⋊D7111yescharacteristic
213C2×C42C22×D7111yescharacteristic
214C4⋊C8—714no—
215C4⋊C8—1422no—
216C4⋊C8—1422no—
217C22⋊C8—1422no—
218C22⋊C8—1424no—
219#97—1422no—
220C4×C8—1422no—
221C4×C8—1422no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C2×Dic7111yescharacteristic
227#95C2×Dic7111yescharacteristic
228C24C22⋊Dic7111yescharacteristic
229C2×Q8—22no—
230C2×Q8—42no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C22⋊Dic7111yescharacteristic
240C22×C4C22⋊Dic7111yescharacteristic
241C22×C4C22⋊Dic7111yescharacteristic
242C22×C4C22×Dic7111yescharacteristic
243C22×C4C2×C22⋊D7111yescharacteristic
244C22×C4C2×C22⋊D7111yescharacteristic
245M16—2842no—
246M16—2842no—
247C2×C8—2842no—
248C2×C8—1424no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—21no—
261C42—21no—
262C42—41no—
263C42—41no—
264C42—42no—
265C42D4×D7112yes—
266C42#1176586412yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊Dic7111yescharacteristic
273Q8—24no—
274Q8—42no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
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×C4D4×Dic7112yes—
299C2×C4#1455609611yescharacteristic
300C2×C4#1455610012yes—
301C2×C4#1455610911yescharacteristic
302C2×C4#1455614812yes—
303C2×C4#1455615912yes—
304C8—2844no—
305C8—5682no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1798587911yescharacteristic
314C22#1798599511yescharacteristic
315C22#1798600811yescharacteristic
316C4—22no—
317C4—22no—
318C4—42no—
319C4—41no—
320C4—42no—
321C4—41no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4#1798615912yes—
326C4#1798633712yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2258865311yescharacteristic
331C2#2258872411yescharacteristic
332C2#2258875911yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2396C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2396D71
hypocenterC71#9034
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #22586769C2semidirect
4. #22586243C2semidirect
172. C71#9034semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225867692
#225866542
#225862432
C71×#23962
#903471