Finite Group Databasebeta

← #31916114

Subgroups — #31916114

Order 18176.

616 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 111 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#22588332C211yesmaximal, characteristic
2C4×#14556043C211yesmaximal, characteristic
3#22586458C214yesmaximal
4C71×#2409C211yesmaximal, characteristic
5#17986023—21no—
6C2×#14556045C2212yes—
7C2×#14556043C2212yes—
8C4×C22⋊Dic71—21no—
9#17985768—21no—
10#17985766—21no—
11C23×Q8×C71C411yescharacteristic, complemented
12C2×C22⋊Q8×C71—21no—
13C2×C4×D4×C71C2211yescharacteristic
14#395×C71—21no—
15#393×C71—21no—
16#391×C71—21no—
17#386×C71C2212yes—
18#385×C71C412yescomplemented
19#380×C71C411yescharacteristic, complemented
20C2×#11765849—22no—
21C2×C22⋊Dic71—22no—
22C2×C4×Dic71—41no—
23C4⋊(C71⋊C8)—41no—
24C4⋊(C71⋊C8)—41no—
25#14556045—24no—
26#14556043—22no—
27#14556043Q814yescomplemented
28C71⋊#95—41no—
29C22×Q8×C71—21no—
30C22×Q8×C71—21no—
31C22×Q8×C71—41no—
32C22×Q8×C71—41no—
33C22×Q8×C71—44no—
34C22×D4×C71C2311yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C412yes—
37C23×C4×C71D411yescharacteristic
38C22⋊Q8×C71—44no—
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71C2×C412yes—
46C2×C22⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49#95×C71—21no—
50#95×C71—22no—
51#95×C71—22no—
52C22×Dic71—42no—
53#11765849—42no—
54#11765849—42no—
55C2×C71⋊C8—42no—
56C22⋊Dic71—24no—
57C22⋊Dic71—42no—
58C4×Dic71—82no—
59C24×C71C22×C411yescharacteristic
60C24×C71C2×D411yescharacteristic
61C2×Q8×C71—24no—
62C2×Q8×C71—42no—
63C2×Q8×C71—44no—
64C2×Q8×C71—42no—
65C2×Q8×C71—44no—
66C2×Q8×C71—48no—
67C2×Q8×C71—44no—
68C2×Q8×C71—41no—
69C2×Q8×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71C2×Q812yes—
73C2×D4×C71#4112yes—
74C22×C4×C71—22no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—42no—
81C22×C4×C71—44no—
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
89C4⋊C4×C71—41no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—44no—
92C22⋊C4×C71—41no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—44no—
95C22⋊C4×C71—42no—
96C42×C71—21no—
97C42×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71C2×C22⋊C411yescharacteristic
107C23×C71C4×Q812yes—
108C23×C71C22⋊Q812yes—
109Q8×C71—44no—
110Q8×C71—48no—
111Q8×C71—44no—
112D4×C71—42no—
113D4×C71—42no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—22no—
117C2×C4×C71—24no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—44no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—44no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—81no—
135C2×C4×C71#9911yescharacteristic
136C2×C4×C71#9911yescharacteristic
137C2×C4×C71#12411yescharacteristic
138C2×C4×C71C4⋊Q811yescharacteristic
139C2×Dic71—44no—
140C2×Dic71—82no—
141C2×Dic71—82no—
142C71⋊C8—82no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—22no—
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—81no—
155C22×C71#39111yescharacteristic
156C22×C71C2×#9911yescharacteristic
157C22×C71#41011yescharacteristic
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—44no—
162C4×C71—41no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71—82no—
166C4×C71AΣL(1,𝔽16)12yes—
167Dic71—84no—
168#13296—711nomaximal, Sylow
169C2×C71—21no—
170C2×C71—21no—
171C2×C71—42no—
172C2×C71—42no—
173C2×C71#199611yescharacteristic
174C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
175C2×C71#212311yescharacteristic
176#2409D7111yescharacteristic
177#1800—711no—
178C4×#99—711no—
179#1390—714no—
180C71#1329611yescharacteristic, complemented, Sylow
181C23×Q8Dic7111yescharacteristic, complemented
182C2×C22⋊Q8—21no—
183C2×C4×D4C2×D7111yescharacteristic
184#423—1421no—
185C2×#100—712no—
186C2×#99—712no—
187#395—21no—
188#393—21no—
189#391—21no—
190#386C2×D7112yes—
191#385Dic7112yescomplemented
192#380Dic7111yescharacteristic, complemented
193C4×C22⋊C4—1421no—
194#343—1421no—
195#342—1421no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—41no—
199C22×Q8—41no—
200C22×Q8—44no—
201C22×D4C22×D7111yescharacteristic
202C23×C4C2×Dic7111yescharacteristic
203C23×C4C2×Dic7112yes—
204C23×C4C22⋊D7111yescharacteristic
205C2×M16—1422no—
206C22⋊Q8—44no—
207C4×D4—41no—
208C4×D4—41no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D7111yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1422no—
214C2×C22⋊C4C2×Dic7112yes—
215C2×C22⋊C4C2×Dic7111yescharacteristic
216C2×C22⋊C4C22⋊D7111yescharacteristic
217C2×C42—2841no—
218C2×C42C22⋊D7111yescharacteristic
219C4⋊C8—2841no—
220C4⋊C8—2841no—
221#100—1424no—
222#99—714no—
223#99—1422no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—2841no—
228C24C22×Dic7111yescharacteristic
229C24C2×C22⋊D7111yescharacteristic
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—48no—
236C2×Q8—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4Q8×D7112yes—
242C2×D4#1176586612yes—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—2842no—
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4C2×C22⋊D7111yescharacteristic
259M16—2842no—
260M16—2842no—
261C2×C8—2842no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—2842no—
270C22⋊C4—1424no—
271C42—21no—
272C42—21no—
273C42—5682no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C2×C22⋊Dic7111yescharacteristic
283C23Q8×Dic7112yes—
284C23#1455610412yes—
285Q8—44no—
286Q8—48no—
287Q8—44no—
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—2844no—
312C2×C4—5682no—
313C2×C4—5682no—
314C2×C4#1455604311yescharacteristic
315C2×C4#1455604311yescharacteristic
316C2×C4#1455611711yescharacteristic
317C2×C4#1455612611yescharacteristic
318C8—5682no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1798590611yescharacteristic
332C22C2×#1455604311yescharacteristic
333C22#1798597311yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—44no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—82no—
342C4—5684no—
343C4#1798578612yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2#2258932611yescharacteristic
349C2C2×#1798578611yescharacteristic
350C2#2258999411yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2409C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenter#2409D71
hypocenterC71#13296
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
11. C23×Q8×C71C4semidirect
18. #385×C71C4semidirect
19. #380×C71C4semidirect
27. #14556043Q8semidirect
180. C71#13296semidirect
181. C23×Q8Dic71semidirect
191. #385Dic71semidirect
192. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883322
C4×#145560432
#225864582
C71×#24092
#1329671