Finite Group Databasebeta

← #31897175

Subgroups — #31897175

Order 18176.

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

normal subgroups only

This group has 484 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
1C2×#17986240C211yesmaximal, characteristic
2#22588116C211yesmaximal, characteristic
3#22586861C212yesmaximal
4#22586705C212yesmaximal
5C71×#1810C211yesmaximal, 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#17985753—21no—
18#17985737—24no—
19#17985724—24no—
20C23×C8×C71C2211yescharacteristic
21C2×#108×C71—21no—
22#407×C71C2212yes—
23#341×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—21no—
36C2×C22⋊Dic71—22no—
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⋊#95D412yes—
46C23×C4×C71C2311yescharacteristic
47C2×M16×C71—22no—
48C22×C8×C71—42no—
49C22×C8×C71C2×C411yescharacteristic
50C22×C8×C71C2×C411yescharacteristic
51C22×C8×C71D411yescharacteristic
52C22×C8×C71D411yescharacteristic
53C22×C8×C71D411yescharacteristic
54C22×C8×C71D411yescharacteristic
55#108×C71—42no—
56C22⋊C8×C71—22no—
57C22⋊C8×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—42no—
75C22⋊Dic71—42no—
76C22⋊Dic71—42no—
77C4×Dic71—41no—
78C4×Dic71—41no—
79C24×C71#4111yescharacteristic
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4111yescharacteristic
88M16×C71—42no—
89M16×C71—42no—
90C2×C8×C71—22no—
91C2×C8×C71—22no—
92C2×C8×C71—22no—
93C2×C8×C71—41no—
94C2×C8×C71—41no—
95C2×C8×C71—41no—
96C2×C8×C71—42no—
97C2×C8×C71C22⋊C411yescharacteristic
98C2×C8×C71C22⋊C411yescharacteristic
99C2×C8×C71C22⋊C411yescharacteristic
100C2×C8×C71C22⋊C411yescharacteristic
101C23×C71—22no—
102C23×C71—22no—
103C23×C71—41no—
104C23×C71C4×D411yescharacteristic
105C23×C71#12111yescharacteristic
106C23×C71#12411yescharacteristic
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71C2×C22⋊C411yescharacteristic
117C2×C4×C71C4×D411yescharacteristic
118C2×C4×C71#12111yescharacteristic
119C2×C4×C71#12711yescharacteristic
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124C2×Dic71—44no—
125C2×Dic71—42no—
126C2×Dic71—82no—
127Dic142—42no—
128Dic142—42no—
129Dic142—82no—
130C8×C71—21no—
131C8×C71—21no—
132C8×C71—21no—
133C8×C71—82no—
134C8×C71#9912yes—
135C71⋊C8—42no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—44no—
144C22×C71#39011yescharacteristic
145C22×C71#44412yes—
146C22×C71#47312yes—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—21no—
150C4×C71C2×#9911yescharacteristic
151C4×C71#41011yescharacteristic
152Dic71—82no—
153Dic71—82no—
154Dic71—82no—
155#9817—711nomaximal, Sylow
156C2×C71—22no—
157C2×C71—22no—
158C2×C71—41no—
159C2×C71#196411yescharacteristic
160C2×C71#197511yescharacteristic
161C2×C71#197911yescharacteristic
162C2×#467—711no—
163#1810D7111yescharacteristic
164#1752—711no—
165#1508—712no—
166#1472—712no—
167C71#981711yescharacteristic, complemented, Sylow
168C22×Q16—711no—
169C23×C8C2×D7111yescharacteristic
170C2×C22⋊Q8—712no—
171#467—1421no—
172#467—1421no—
173#467—1421no—
174#467—1421no—
175C2×#108—21no—
176C2×#107—711no—
177C2×#103—712no—
178#407C2×D7112yes—
179C2×C22⋊C8—711no—
180C2×#97—1421no—
181C4×C22⋊C4—711no—
182#341—21no—
183#336—1421no—
184#328—1424no—
185#324—1424no—
186C22×Q8—712no—
187C23×C4C22×D7111yescharacteristic
188C2×Q16—2842no—
189C2×Q16—1421no—
190C2×Q16—1421no—
191C2×M16—22no—
192C22×C8—42no—
193C22×C8—1421no—
194C22×C8C4×D7111yescharacteristic
195C22×C8C4×D7111yescharacteristic
196C22×C8D28411yescharacteristic
197C22×C8D28411yescharacteristic
198C22×C8C22⋊D7111yescharacteristic
199C22×C8C22⋊D7111yescharacteristic
200C22⋊Q8—1422no—
201C22⋊Q8—1422no—
202C22⋊Q8—1422no—
203C22⋊Q8—1422no—
204C2×C4⋊C4—712no—
205C2×C4⋊C4—1422no—
206C2×C22⋊C4—1422no—
207C2×C22⋊C4—1421no—
208C2×C42—1421no—
209#108—42no—
210#107—1421no—
211#107—1421no—
212#103—1422no—
213#103—1422no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—712no—
217C22⋊C8—1421no—
218#97—2842no—
219#95—712no—
220C24#1176586611yescharacteristic
221C2×Q8—2842no—
222C2×Q8—2842no—
223C2×Q8—1422no—
224C2×Q8—1422no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—1422no—
228C22×C4—1422no—
229C22×C4—1422no—
230C22×C4C2×C4×D7111yescharacteristic
231C22×C4D4×D7111yescharacteristic
232C22×C4D4×D7111yescharacteristic
233C22×C4C2×C22⋊D7111yescharacteristic
234C22×C4C2×D28411yescharacteristic
235C22×C4#1176586611yescharacteristic
236Q16—2842no—
237Q16—2842no—
238M16—42no—
239M16—42no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—2841no—
248C2×C8—2841no—
249C2×C8—1422no—
250C2×C8(C2×D71)⋊C411yescharacteristic
251C2×C8(C2×D71)⋊C411yescharacteristic
252C2×C8(C2×D71)⋊C411yescharacteristic
253C2×C8(C2×D71)⋊C411yescharacteristic
254C4⋊C4—2842no—
255C4⋊C4—2842no—
256C4⋊C4—1422no—
257C4⋊C4—1422no—
258C22⋊C4—2842no—
259C22⋊C4—2842no—
260C22⋊C4—2842no—
261C42—2841no—
262C42—2841no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23#1455609011yescharacteristic
267C23#1455610311yescharacteristic
268C23#1455611711yescharacteristic
269Q8—2842no—
270Q8—2842no—
271Q8—5682no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—2842no—
282C2×C4—2842no—
283C2×C4—2844no—
284C2×C4—2842no—
285C2×C4—2842no—
286C2×C4—2842no—
287C2×C4—5682no—
288C2×C4C2×(C2×D71)⋊C411yescharacteristic
289C2×C4#1455609011yescharacteristic
290C2×C4#1455610311yescharacteristic
291C2×C4#1455612411yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—82no—
296C8—2842no—
297C8#1455604212yes—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—44no—
306C22#1798590011yescharacteristic
307C22#1798612012yes—
308C22#1798627512yes—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—5682no—
313C4—5682no—
314C4—5682no—
315C4C2×#1455604211yescharacteristic
316C4#1798597211yescharacteristic
317C2—22no—
318C2—22no—
319C2—41no—
320C2#2258917511yescharacteristic
321C2#2258923111yescharacteristic
322C2#2258925311yescharacteristic
subgroupG/H
centerC22#17985900
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1810C2
socleC22×C71#390
prime radicalC24×C71#41
cosocleC23×C4×C71C23
hypercenter#1810D71
hypocenterC71#9817
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
6. C22×Dic284C4semidirect
12. C2×#14556062C4semidirect
13. C2×#14556053C4semidirect
167. C71#9817semidirect

Maximal subgroups

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

subgroupindex
C2×#179862402
#225881162
#225868612
#225867052
C71×#18102
#981771