Finite Group Databasebeta

← #31913694

Subgroups — #31913694

Order 18176.

656 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 107 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#22588268C211yesmaximal, characteristic
2#22588258C211yesmaximal, characteristic
3#22586458C214yesmaximal, complemented
4C71×#2297C211yesmaximal, characteristic
5C2×#14556047—21no—
6C2×#14556045C2212yes—
7C2×#14556043—21no—
8C2×#14556043C2212yes—
9#17985963—21no—
10#17985862—21no—
11C22×#41×C71C2211yescharacteristic
12C2×#117×C71C411yescharacteristic, complemented
13C22×C42×C71C411yescharacteristic, complemented
14#391×C71—21no—
15#390×C71—21no—
16#386×C71C412yescomplemented
17#386×C71C2212yes—
18C4×C22⋊C4×C71—21no—
19C4×C22⋊C4×C71—21no—
20C2×#11765849—22no—
21C2×C22⋊Dic71—22no—
22#14556047—42no—
23#14556045—24no—
24#14556043—24no—
25#14556043—42no—
26C22⋊(C71⋊C8)—42no—
27C71⋊#95—42no—
28C2×#41×C71—24no—
29C2×#41×C71—42no—
30C2×#41×C71—42no—
31C22×Q8×C71D411yescharacteristic
32C22×D4×C71C2×C411yescharacteristic
33C22×D4×C71D411yescharacteristic
34C22×D4×C71C2311yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C412yes—
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39#117×C71—42no—
40C2×C4⋊C4×C71—21no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71C2×C412yes—
44C2×C42×C71—21no—
45C2×C42×C71—21no—
46C2×C42×C71—21no—
47C2×C42×C71—42no—
48#95×C71—22no—
49#95×C71—22no—
50C22×Dic71—42no—
51#11765849—42no—
52#11765849—42no—
53C2×C71⋊C8—42no—
54C22⋊Dic71—42no—
55C22⋊Dic71—42no—
56C24×C71C22⋊C411yescharacteristic
57C24×C71C22×C411yescharacteristic
58C24×C71C2×D411yescharacteristic
59#41×C71—416no—
60C2×Q8×C71—22no—
61C2×Q8×C71—22no—
62C2×Q8×C71—41no—
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71#4114yes—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—22no—
79C22×C4×C71—22no—
80C22×C4×C71—22no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—44no—
84C22×C4×C71—42no—
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—42no—
90C22⋊C4×C71—44no—
91C22⋊C4×C71—44no—
92C22⋊C4×C71—42no—
93C42×C71—42no—
94C42×C71—42no—
95C42×C71—42no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—42no—
104C23×C71—42no—
105C23×C71—42no—
106C23×C71C2×C22⋊C411yescharacteristic
107C23×C71#11712yes—
108C23×C71#12312yes—
109Q8×C71—44no—
110D4×C71—44no—
111D4×C71—44no—
112D4×C71—44no—
113C2×C4×C71—28no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—24no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—44no—
125C2×C4×C71—44no—
126C2×C4×C71—42no—
127C2×C4×C71—44no—
128C2×C4×C71—42no—
129C2×C4×C71—44no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71#9912yes—
134C2×C4×C71#9912yes—
135C2×C4×C71#12312yes—
136C2×Dic71—82no—
137C2×Dic71—84no—
138C71⋊C8—82no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71—22no—
142C22×C71—42no—
143C22×C71—42no—
144C22×C71—42no—
145C22×C71—42no—
146C22×C71—42no—
147C22×C71—42no—
148C22×C71—41no—
149C22×C71—42no—
150C22×C71—81no—
151C22×C71—81no—
152C22×C71#38111yescharacteristic
153C22×C71C2×#9911yescharacteristic
154C22×C71C2×#9911yescharacteristic
155C4×C71—22no—
156C4×C71—22no—
157C4×C71—22no—
158C4×C71—41no—
159C4×C71—44no—
160C4×C71—82no—
161Dic71—162no—
162#12853—711nomaximal, Sylow
163C2×C71—21no—
164C2×C71—21no—
165C2×C71—42no—
166C2×C71—42no—
167C2×C71—81no—
168C2×C71#181911yescharacteristic
169C2×C71#212012yes—
170#2297D7111yescharacteristic
171#1785—711no—
172#1782—711no—
173#1390—714no—
174C71#1285311yescharacteristic, complemented, Sylow
175C22×#41C2×D7111yescharacteristic
176C2×#117Dic7111yescharacteristic, complemented
177C22×C42Dic7111yescharacteristic, complemented
178C2×#101—1421no—
179C2×#100—712no—
180C2×#99—712no—
181C2×#99—1421no—
182#407—1421no—
183#391—21no—
184#390—21no—
185#386Dic7112yescomplemented
186#386C2×D7112yes—
187#381—1421no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—42no—
193C22×Q8C22⋊D7111yescharacteristic
194C22×D4C2×Dic7111yescharacteristic
195C22×D4C22⋊D7111yescharacteristic
196C22×D4C22×D7111yescharacteristic
197C23×C4C2×Dic7111yescharacteristic
198C23×C4C2×Dic7112yes—
199C23×C4C22⋊D7111yescharacteristic
200C23×C4C22⋊D7111yescharacteristic
201C2×M16—1422no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1422no—
207C2×C22⋊C4C2×Dic7112yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—42no—
212#101—2842no—
213#100—1424no—
214#99—2842no—
215#99—1424no—
216C22⋊C8—2842no—
217#95—22no—
218#95—22no—
219#95—2842no—
220C24C22⋊Dic7111yescharacteristic
221C24C22×Dic7111yescharacteristic
222C24C2×C22⋊D7111yescharacteristic
223#41—416no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4#1176586514yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—2842no—
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22⋊Dic7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254M16—2842no—
255M16—2842no—
256C2×C8—2842no—
257C4⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—2842no—
262C22⋊C4—2842no—
263C42—42no—
264C42—42no—
265C42—42no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23C2×C22⋊Dic7111yescharacteristic
277C23#1455608712yes—
278C23#1455611112yes—
279Q8—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—5682no—
304C2×C4—5684no—
305C2×C4#1455604312yes—
306C2×C4#1455604312yes—
307C2×C4#1455611112yes—
308C8—5682no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22#1798586211yescharacteristic
323C22C2×#1455604311yescharacteristic
324C22C2×#1455604311yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—44no—
329C4—41no—
330C4—82no—
331C4—11362no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2—81no—
337C2#2258841411yescharacteristic
338C2#2258998112yes—
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2297C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C22
cosocleC22×D4×C71C23
hypercenter#2297D71
hypocenterC71#12853
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. #22586458C2semidirect
12. C2×#117×C71C4semidirect
13. C22×C42×C71C4semidirect
16. #386×C71C4semidirect
174. C71#12853semidirect
176. C2×#117Dic71semidirect
177. C22×C42Dic71semidirect
185. #386Dic71semidirect

Maximal subgroups

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

subgroupindex
#225882682
#225882582
#225864582
C71×#22972
#1285371