Finite Group Databasebeta

← #31907732

Subgroups — #31907732

Order 18176.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#22588073C211yesmaximal, characteristic
2#22586654C211yesmaximal, characteristic
3#22586364C212yesmaximal
4#22586361C212yesmaximal
5C71×#2396C211yesmaximal, characteristic
6C2×#14556087—21no—
7#17986052—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×C4×Q8×C71C411yescharacteristic, complemented
13C2×#117×C71C2211yescharacteristic
14#400×C71C411yescharacteristic, complemented
15#399×C71C2211yescharacteristic
16#398×C71—21no—
17#396×C71—21no—
18#395×C71C411yescharacteristic, complemented
19#393×C71C2211yescharacteristic
20C4×C4⋊C4×C71—21no—
21C4×C4⋊C4×C71C411yescharacteristic, complemented
22C4×C22⋊C4×C71—21no—
23#14556087—24no—
24#14556087—42no—
25C2×C4⋊Dic71—41no—
26C2×C22⋊Dic71—22no—
27C2×C4×Dic71—41no—
28C4⋊(C71⋊C8)—41no—
29C22⋊(C71⋊C8)—22no—
30C4×C71⋊C8—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×C71—21no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71C2311yescharacteristic
48C2×C42×C71—21no—
49C2×C42×C71—21no—
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58C22×Dic71—42no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—44no—
61C22⋊Dic71—24no—
62C22⋊Dic71—42no—
63C4×Dic71—44no—
64C24×C71C2×D411yescharacteristic
65C2×Q8×C71—42no—
66C2×Q8×C71—41no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71SD1612yes—
79C22×C4×C71Q1612yes—
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—41no—
98C42×C71—41no—
99C42×C71—42no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71C2×C22⋊C411yescharacteristic
105C23×C71C2×SD1611yescharacteristic
106C23×C71C2×Q1611yescharacteristic
107Q8×C71—42no—
108Q8×C71—42no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
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×C71#10312yes—
138C2×C4×C71#10312yes—
139C2×C4×C71#10412yes—
140C2×Dic71—44no—
141C2×Dic71—82no—
142C2×Dic71—82no—
143C71⋊C8—82no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—81no—
153C22×C71C2×#9911yescharacteristic
154C22×C71C2×#10311yescharacteristic
155C22×C71C2×#10411yescharacteristic
156C4×C71—22no—
157C4×C71—42no—
158C4×C71—41no—
159C4×C71—42no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71—81no—
164C4×C71—81no—
165C4×C71#32812yes—
166C4×C71#32812yes—
167Dic71—84no—
168#11715—711nomaximal, Sylow
169C2×C71—21no—
170C2×C71—21no—
171C2×C71—41no—
172C2×C71—41no—
173C2×C71C2×#32811yescharacteristic
174C2×C71#212011yescharacteristic
175C2×C71#212611yescharacteristic
176#2396D7111yescharacteristic
177#1740—711no—
178#1458—711no—
179#1352—712no—
180#1351—712no—
181C71#1171511yescharacteristic, complemented, Sylow
182C2×C4×Q8Dic7111yescharacteristic, complemented
183C2×#117—1421no—
184C2×#117C2×D7111yescharacteristic
185#431—1421no—
186#400Dic7111yescharacteristic, complemented
187#399C2×D7111yescharacteristic
188#398—21no—
189#396—21no—
190#395Dic7111yescharacteristic, complemented
191#393C2×D7111yescharacteristic
192C4×C4⋊C4—21no—
193C4×C4⋊C4Dic7111yescharacteristic, complemented
194C4×C22⋊C4—21no—
195#342—712no—
196#342—1421no—
197#324—1421no—
198#323—712no—
199C22×Q8C2×Dic7111yescharacteristic
200C23×C4C22⋊D7111yescharacteristic
201C4×Q8—41no—
202C4×Q8—41no—
203#117—41no—
204#117—41no—
205#117—2842no—
206#117—1424no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—2841no—
212C2×C4⋊C4C2×Dic7111yescharacteristic
213C2×C4⋊C4C2×Dic7111yescharacteristic
214C2×C4⋊C4C22⋊D7111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1422no—
218C2×C22⋊C4C22⋊D7111yescharacteristic
219C2×C22⋊C4C22×D7111yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—2841no—
223C2×C42C2×Dic7111yescharacteristic
224C2×C42C22⋊D7111yescharacteristic
225C4⋊C8—2841no—
226C22⋊C8—1422no—
227C4×C8—2841no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic7111yescharacteristic
233#95C2×Dic7111yescharacteristic
234C24C2×C22⋊D7111yescharacteristic
235C2×Q8—42no—
236C2×Q8—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—2842no—
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22⋊Dic7111yescharacteristic
247C22×C4C22⋊Dic7111yescharacteristic
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4#1176585212yes—
250C22×C4#1176585512yes—
251C22×C4C22×Dic7111yescharacteristic
252C22×C4C2×C22⋊D7111yescharacteristic
253C2×C8—2842no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—2844no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—2842no—
267C22⋊C4—1424no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—41no—
273C42—41no—
274C42—42no—
275C42—2844no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic7111yescharacteristic
281C23C2×#1176585211yescharacteristic
282C23C2×#1176585511yescharacteristic
283Q8—42no—
284Q8—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
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—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—2844no—
312C2×C4—5682no—
313C2×C4—5682no—
314C2×C4#1455604311yescharacteristic
315C2×C4#1455604311yescharacteristic
316C2×C4#1455605212yes—
317C2×C4#1455605212yes—
318C2×C4#1455605512yes—
319C8—5682no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1455604311yescharacteristic
330C22C2×#1455605211yescharacteristic
331C22C2×#1455605511yescharacteristic
332C4—22no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—5684no—
342C4#1798573612yes—
343C4#1798573612yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1798573611yescharacteristic
349C2#2258998111yescharacteristic
350C2#2259000511yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2396C2
socleC22×C71C2×#99
prime radicalC24×C71C2×D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2396D71
hypocenterC71#11715
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C71C4semidirect
14. #400×C71C4semidirect
18. #395×C71C4semidirect
21. C4×C4⋊C4×C71C4semidirect
181. C71#11715semidirect
182. C2×C4×Q8Dic71semidirect
186. #400Dic71semidirect
190. #395Dic71semidirect
193. C4×C4⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225880732
#225866542
#225863642
#225863612
C71×#23962
#1171571