Finite Group Databasebeta

← #31896259

Subgroups — #31896259

Order 18176.

558 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C4×#14556116C211yesmaximal, characteristic
2#22588457C211yesmaximal, characteristic, complemented
3C2×#17985731C212yesmaximal, complemented
4C2×#17985728C212yesmaximal
5C71×#1744C211yesmaximal, characteristic, complemented
6C2×#14556116C411yescharacteristic
7C2×C4×Dic142C2211yescharacteristic
8C2×C4×D284C2211yescharacteristic
9C2×C4⋊(C71⋊C8)C2212yes—
10#17985905C411yescharacteristic
11#17985901C411yescharacteristic
12#17985890—22no—
13#17985865C411yescharacteristic
14C4×(C2×D71)⋊C4—22no—
15#17985754—22no—
16#17985731—24no—
17#17985728—24no—
18C2×C4×C8×C71C2212yes—
19C43×C71C2211yescharacteristic
20#336×C71—22no—
21C22×Dic142C2×C411yescharacteristic
22C22×D284C2×C411yescharacteristic
23C22×C4×D71—21no—
24C22×C71⋊C8—22no—
25#14556116—22no—
26#14556116—22no—
27C4×Dic142—22no—
28C4×Dic142—22no—
29C4×D284—22no—
30C4×D284—22no—
31C2×C4⋊Dic71C2×C411yescharacteristic
32C2×C4⋊Dic71C2×C411yescharacteristic
33C2×Dic71⋊C4—21no—
34C2×(C2×D71)⋊C4—22no—
35C2×(C2×D71)⋊C4—22no—
36C2×(C2×D71)⋊C4—21no—
37C2×C4×Dic71—21no—
38C4⋊(C71⋊C8)—22no—
39C4⋊(C71⋊C8)—22no—
40C71⋊#95—22no—
41C71⋊#95—22no—
42C22×C8×C71—22no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D412yes—
46C2×C42×C71D412yes—
47C2×C42×C71C2311yescharacteristic
48C4×C8×C71—22no—
49C4×C8×C71D414yes—
50C23×D71—21no—
51C2×Dic142—22no—
52C2×Dic142—22no—
53C2×Dic142—21no—
54C2×Dic142M1612yes—
55C2×D284—22no—
56C2×D284—22no—
57C2×D284—21no—
58C2×D284M1612yes—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C22×Dic71—21no—
62C2×C4×D71—41no—
63C2×C4×D71—41no—
64C2×C4×D71—44no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—44no—
67C4⋊Dic71—22no—
68C4⋊Dic71—22no—
69Dic71⋊C4—42no—
70(C2×D71)⋊C4—28no—
71(C2×D71)⋊C4—44no—
72(C2×D71)⋊C4—42no—
73C4×Dic71—41no—
74C4×Dic71—41no—
75C22×C4×C71C22⋊C412yes—
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C2×C8×C71—24no—
81C2×C8×C71—44no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—22no—
85C42×C71—24no—
86C42×C71—42no—
87C42×C71C2×D412yes—
88C42×C71#4112yes—
89C23×C71C2×C22⋊C411yescharacteristic
90C22×D71—22no—
91C22×D71—41no—
92C22×D71—41no—
93C22×D71—42no—
94C22×D71—42no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71C4×D412yes—
103C2×C4×C71#12112yes—
104C2×C4×C71#12411yescharacteristic
105C2×C4×C71#12711yescharacteristic
106C2×C4×C71C2×M1611yescharacteristic
107C2×C4×C71C2×M1611yescharacteristic
108C2×C4×C71#13112yes—
109C2×C4×C71#13514yes—
110C2×Dic71—22no—
111C2×Dic71—41no—
112C2×Dic71—41no—
113C2×Dic71—41no—
114C2×Dic71—41no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117D284—24no—
118D284—42no—
119C4×D71—82no—
120C4×D71—82no—
121Dic142—24no—
122Dic142—42no—
123C8×C71—44no—
124C71⋊C8—82no—
125C22×C71#39011yescharacteristic
126C22×C71#40711yescharacteristic
127C22×C71#40811yescharacteristic
128C22×C71#41612yes—
129C22×C71#42112yes—
130C2×D71—42no—
131C2×D71—44no—
132C2×D71—81no—
133C2×D71—81no—
134C2×D71—82no—
135C4×C71—22no—
136C4×C71—42no—
137C4×C71—42no—
138C4×C71#43512yes—
139C4×C71#43512yes—
140C4×C71#48812yes—
141C4×C71#49512yes—
142Dic71—42no—
143Dic71—81no—
144Dic71—81no—
145#9645—711nomaximal, Sylow
146C2×C71#148212yes—
147C2×C71#148812yes—
148C2×C71#195511yescharacteristic
149C2×C71#195811yescharacteristic
150C2×C71#196311yescharacteristic
151D71—82no—
152C4×#124—711no—
153#1831—711no—
154#1744D7111yescharacteristic, complemented
155C2×Q8⋊C8—712no—
156C2×D4⋊C8—712no—
157C71#964511yescharacteristic, complemented, Sylow
158C2×#124—711no—
159C2×C4×Q8—711no—
160C2×C4×D4—711no—
161C2×C4⋊C8—712no—
162C2×C4×C8C2×D7112yes—
163#391—711no—
164#390—711no—
165#388—1422no—
166#382—711no—
167C4×C22⋊C4—1422no—
168C43C2×D7111yescharacteristic
169#336—22no—
170#336—1422no—
171Q8⋊C8—1424no—
172D4⋊C8—1424no—
173C22×Q8—711no—
174C22×D4—711no—
175C23×C4—1421no—
176C22×C8—22no—
177C22×C8—1422no—
178#124—1422no—
179#124—1422no—
180C4×Q8—1422no—
181C4×Q8—1422no—
182C4×D4—1422no—
183C4×D4—1422no—
184C2×C4⋊C4—711no—
185C2×C4⋊C4—711no—
186C2×C4⋊C4—1421no—
187C2×C22⋊C4—1422no—
188C2×C22⋊C4—1421no—
189C2×C22⋊C4—1422no—
190C2×C42—1421no—
191C2×C42C4×D7111yescharacteristic
192C2×C42C4×D7111yescharacteristic
193C2×C42D28412yes—
194C2×C42C22⋊D7112yes—
195C2×C42C22×D7111yescharacteristic
196C4⋊C8—1422no—
197C4⋊C8—1422no—
198C4×C8—22no—
199C4×C8D28414yes—
200#95—1422no—
201#95—1422no—
202C24—1421no—
203C2×Q8—712no—
204C2×Q8—1422no—
205C2×Q8—1422no—
206C2×Q8—1421no—
207C2×D4—712no—
208C2×D4—1422no—
209C2×D4—1421no—
210C2×D4—1422no—
211C22×C4—2841no—
212C22×C4—2841no—
213C22×C4—2844no—
214C22×C4—1421no—
215C22×C4—1421no—
216C22×C4—1421no—
217C22×C4(C2×D71)⋊C412yes—
218C22×C4(C2×D71)⋊C412yes—
219C22×C4C2×C4×D7111yescharacteristic
220C22×C4C2×C22⋊D7111yescharacteristic
221C22×C4C2×D28411yescharacteristic
222C2×C8—24no—
223C2×C8—44no—
224C2×C8—2842no—
225C2×C8—2844no—
226C4⋊C4—2842no—
227C4⋊C4—1422no—
228C4⋊C4—1422no—
229C22⋊C4—2844no—
230C22⋊C4—2842no—
231C22⋊C4—1428no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—24no—
236C42—42no—
237C42—2841no—
238C42—2841no—
239C42C2×D28412yes—
240C42#1176586512yes—
241C23—2841no—
242C23—2841no—
243C23—2842no—
244C23—2842no—
245C23—1422no—
246C23C2×(C2×D71)⋊C411yescharacteristic
247Q8—2842no—
248Q8—1424no—
249D4—2842no—
250D4—1424no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—2841no—
259C2×C4—2841no—
260C2×C4—2841no—
261C2×C4—2841no—
262C2×C4—2842no—
263C2×C4—2842no—
264C2×C4—5682no—
265C2×C4—5682no—
266C2×C4—1422no—
267C2×C4C4×D28412yes—
268C2×C4#1455610212yes—
269C2×C4#1455611611yescharacteristic
270C2×C4#1455612511yescharacteristic
271C2×C4M16×D7111yescharacteristic
272C2×C4M16×D7111yescharacteristic
273C2×C4#1455613612yes—
274C2×C4#1455615014yes—
275C8—44no—
276C8—5682no—
277C22—2844no—
278C22—2842no—
279C22—5682no—
280C22—5681no—
281C22—5681no—
282C22#1798590111yescharacteristic
283C22#1798596211yescharacteristic
284C22#1798596711yescharacteristic
285C22#1798599612yes—
286C22#1798601712yes—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—2842no—
291C4—5681no—
292C4—5681no—
293C4#1798607212yes—
294C4#1798607212yes—
295C4#1798635412yes—
296C4#1798638312yes—
297C2—5682no—
298C2#2258674912yes—
299C2#2258677512yes—
300C2#2258913811yescharacteristic
301C2#2258915111yescharacteristic
302C2#2258917211yescharacteristic
subgroupG/H
centerC22×C4C2×D284
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1744C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1744D71
hypocenterC71#9645
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1744

Complements

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

NH ≅ G/Ntype
2. #22588457C2semidirect
3. C2×#17985731C2semidirect
5. C71×#1744C2semidirect
154. #1744D71semidirect
157. C71#9645semidirect

Maximal subgroups

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

subgroupindex
C4×#145561162
#225884572
C2×#179857312
C2×#179857282
C71×#17442
#964571