Finite Group Databasebeta

← #31896976

Subgroups — #31896976

Order 18176.

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

normal subgroups only

This group has 510 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
1C2×#17986240C211yesmaximal, characteristic
2#22589324C212yesmaximal
3#22588116C211yesmaximal, characteristic
4#22586911C212yesmaximal
5C71×#1807C211yesmaximal, characteristic
6C22×Dic284C411yescharacteristic, complemented
7C2×#14556107C2212yes—
8#17986240—21no—
9#17986240—21no—
10#17986240—21no—
11#17986240—21no—
12C2×#14556062C411yescharacteristic, complemented
13C2×#14556053C412yescomplemented
14C2×C22⋊(C71⋊C8)C2211yescharacteristic
15C2×#14556038—21no—
16C4×C22⋊Dic71C2211yescharacteristic
17#17985765—24no—
18#17985753—21no—
19#17985735—24no—
20C23×C8×C71C2211yescharacteristic
21C2×#106×C71—21no—
22#380×C71C2212yes—
23#340×C71—21no—
24C22×Dic142C2×C412yes—
25C2×Dic284—21no—
26C2×Dic284—21no—
27C2×Dic284—42no—
28C22×C71⋊C8—21no—
29#14556107—22no—
30#14556107—22no—
31#14556107—22no—
32#14556107—22no—
33C2×C4⋊Dic71C2×C412yes—
34C2×Dic71⋊C4—22no—
35C2×C22⋊Dic71—22no—
36C2×C22⋊Dic71D412yes—
37C2×C4×Dic71—21no—
38#14556062—21no—
39#14556062—21no—
40#14556053—22no—
41#14556053—22no—
42C22⋊(C71⋊C8)—21no—
43C22⋊(C71⋊C8)D412yes—
44#14556038—42no—
45C71⋊#95—21no—
46C23×C4×C71C2311yescharacteristic
47C22×C8×C71—42no—
48C22×C8×C71C2×C411yescharacteristic
49C22×C8×C71C2×C411yescharacteristic
50C22×C8×C71D411yescharacteristic
51C22×C8×C71D411yescharacteristic
52C22×C8×C71D411yescharacteristic
53C22×C8×C71D411yescharacteristic
54C2×C4⋊C4×C71—22no—
55C2×C22⋊C4×C71—22no—
56#106×C71—42no—
57#95×C71—22no—
58C2×Dic142—22no—
59C2×Dic142—22no—
60C2×Dic142—42no—
61C2×Dic142—42no—
62C22×Dic71—22no—
63C22×Dic71—22no—
64C22×Dic71—22no—
65Dic284—42no—
66Dic284—42no—
67C2×C71⋊C8—22no—
68C2×C71⋊C8—41no—
69C2×C71⋊C8—41no—
70C4⋊Dic71—22no—
71C4⋊Dic71—22no—
72Dic71⋊C4—42no—
73Dic71⋊C4—42no—
74C22⋊Dic71—24no—
75C22⋊Dic71—42no—
76C22⋊Dic71—42no—
77C4×Dic71—41no—
78C4×Dic71—41no—
79C24×C71C2×D411yescharacteristic
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71—42no—
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4111yescharacteristic
88C22×C4×C71#4111yescharacteristic
89C2×C8×C71—22no—
90C2×C8×C71—22no—
91C2×C8×C71—22no—
92C2×C8×C71—41no—
93C2×C8×C71—41no—
94C2×C8×C71—41no—
95C2×C8×C71C22⋊C411yescharacteristic
96C2×C8×C71C22⋊C411yescharacteristic
97C2×C8×C71C22⋊C411yescharacteristic
98C2×C8×C71C22⋊C411yescharacteristic
99C4⋊C4×C71—42no—
100C4⋊C4×C71—42no—
101C22⋊C4×C71—44no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71C4×D411yescharacteristic
105C23×C71#12111yescharacteristic
106C23×C71#12711yescharacteristic
107C23×C71#13714yes—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—82no—
118C2×C4×C71—82no—
119C2×C4×C71—82no—
120C2×C4×C71C2×C22⋊C411yescharacteristic
121C2×C4×C71C4×D411yescharacteristic
122C2×C4×C71#12111yescharacteristic
123C2×C4×C71#12411yescharacteristic
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—42no—
128C2×Dic71—42no—
129C2×Dic71—82no—
130C2×Dic71—82no—
131Dic142—42no—
132Dic142—42no—
133Dic142—82no—
134C8×C71—21no—
135C8×C71—21no—
136C8×C71—21no—
137C8×C71#9912yes—
138C71⋊C8—42no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—44no—
146C22×C71#39011yescharacteristic
147C22×C71#44112yes—
148C22×C71#47012yes—
149C22×C71#49712yes—
150C4×C71—21no—
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—162no—
154C4×C71C2×#9911yescharacteristic
155C4×C71#41011yescharacteristic
156Dic71—82no—
157Dic71—82no—
158Dic71—82no—
159#9784—711nomaximal, Sylow
160C2×C71—22no—
161C2×C71—22no—
162C2×C71—22no—
163C2×C71#196411yescharacteristic
164C2×C71#197111yescharacteristic
165C2×C71#197611yescharacteristic
166C2×#467—711no—
167#1996—712no—
168#1807D7111yescharacteristic
169#1752—711no—
170#1520—712no—
171C71#978411yescharacteristic, complemented, Sylow
172C22×Q16—711no—
173C23×C8C2×D7111yescharacteristic
174C2×C22⋊Q8—712no—
175#467—1421no—
176#467—1421no—
177#467—1421no—
178#467—1421no—
179C2×#107—711no—
180C2×#106—21no—
181C2×#103—712no—
182C2×C22⋊C8—711no—
183C2×#97—1421no—
184#380C2×D7112yes—
185C4×C22⋊C4—711no—
186#342—1424no—
187#340—21no—
188#336—1421no—
189#328—1424no—
190C22×Q8—712no—
191C23×C4C22×D7111yescharacteristic
192C2×Q16—2842no—
193C2×Q16—1421no—
194C2×Q16—1421no—
195C22×C8—42no—
196C22×C8—1421no—
197C22×C8C4×D7111yescharacteristic
198C22×C8C4×D7111yescharacteristic
199C22×C8D28411yescharacteristic
200C22×C8D28411yescharacteristic
201C22×C8C22⋊D7111yescharacteristic
202C22×C8C22⋊D7111yescharacteristic
203C22⋊Q8—1422no—
204C22⋊Q8—1422no—
205C22⋊Q8—1422no—
206C22⋊Q8—1422no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—712no—
209C2×C4⋊C4—1422no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—712no—
212C2×C22⋊C4—1422no—
213C2×C42—1421no—
214#107—1421no—
215#107—1421no—
216#106—42no—
217#103—1422no—
218#103—1422no—
219C22⋊C8—712no—
220C22⋊C8—1421no—
221#97—2842no—
222#95—22no—
223#95—1421no—
224C24D4×D7111yescharacteristic
225C2×Q8—2842no—
226C2×Q8—2842no—
227C2×Q8—1422no—
228C2×Q8—1422no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—1422no—
233C22×C4—1422no—
234C22×C4—1422no—
235C22×C4C2×C4×D7111yescharacteristic
236C22×C4D4×D7111yescharacteristic
237C22×C4C2×C22⋊D7111yescharacteristic
238C22×C4C2×D28411yescharacteristic
239C22×C4#1176586611yescharacteristic
240C22×C4#1176586611yescharacteristic
241Q16—2842no—
242Q16—2842no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—2841no—
250C2×C8—2841no—
251C2×C8—1422no—
252C2×C8(C2×D71)⋊C411yescharacteristic
253C2×C8(C2×D71)⋊C411yescharacteristic
254C2×C8(C2×D71)⋊C411yescharacteristic
255C2×C8(C2×D71)⋊C411yescharacteristic
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—2842no—
259C4⋊C4—2842no—
260C4⋊C4—1422no—
261C4⋊C4—1422no—
262C22⋊C4—44no—
263C22⋊C4—2842no—
264C22⋊C4—2842no—
265C22⋊C4—1424no—
266C42—2841no—
267C42—2841no—
268C23—41no—
269C23—41no—
270C23#1455609011yescharacteristic
271C23#1455610311yescharacteristic
272C23#1455612411yescharacteristic
273C23#1455615714yes—
274Q8—2842no—
275Q8—2842no—
276Q8—5682no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—2842no—
290C2×C4—2842no—
291C2×C4—2842no—
292C2×C4—2842no—
293C2×C4—2842no—
294C2×C4—5682no—
295C2×C4—5682no—
296C2×C4C2×(C2×D71)⋊C411yescharacteristic
297C2×C4#1455609011yescharacteristic
298C2×C4#1455610311yescharacteristic
299C2×C4#1455611711yescharacteristic
300C8—21no—
301C8—21no—
302C8—21no—
303C8—2842no—
304C8#1455604212yes—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—44no—
312C22#1798590011yescharacteristic
313C22#1798610512yes—
314C22#1798625812yes—
315C22#1798639012yes—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—162no—
320C4—5682no—
321C4—5682no—
322C4—5682no—
323C4C2×#1455604211yescharacteristic
324C4#1798597211yescharacteristic
325C2—22no—
326C2—22no—
327C2—22no—
328C2#2258917511yescharacteristic
329C2#2258920911yescharacteristic
330C2#2258923611yescharacteristic
subgroupG/H
centerC22#17985900
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1807C2
socleC22×C71#390
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#1807D71
hypocenterC71#9784
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
6. C22×Dic284C4semidirect
12. C2×#14556062C4semidirect
13. C2×#14556053C4semidirect
171. C71#9784semidirect

Maximal subgroups

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

subgroupindex
C2×#179862402
#225893242
#225881162
#225869112
C71×#18072
#978471