Finite Group Databasebeta

← #31915778

Subgroups — #31915778

Order 18176.

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

normal subgroups only

This group has 472 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#22593325C211yesmaximal, characteristic
2C2×#17985797C214yesmaximal
3#22588336C211yesmaximal, characteristic
4C4×#101×C71C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556126C2212yes—
7#17986023—21no—
8C2×#14556047C2212yes—
9#17985929—21no—
10#17985929—21no—
11#17985904C412yes—
12#17985897—21no—
13#17985897—21no—
14C4×C4⋊Dic71C411yescharacteristic
15#17985797—28no—
16#17985768—21no—
17C2×C4×Q8×C71C2211yescharacteristic
18C2×#101×C71C2212yes—
19C4×M16×C71—21no—
20#343×C71—21no—
21C22×Dic142C2×C412yes—
22C2×#11765849—22no—
23#14556128—41no—
24#14556128—41no—
25#14556126—24no—
26#14556126—24no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—21no—
30C2×Dic71⋊C4—22no—
31C2×Dic71⋊C4—22no—
32C2×C4×Dic71C2×C412yes—
33C4⋊(C71⋊C8)—41no—
34C4⋊(C71⋊C8)—41no—
35#14556047—24no—
36C71⋊#95—21no—
37C71⋊#95—22no—
38C71⋊#95—22no—
39C22×Q8×C71C2311yescharacteristic
40C2×M16×C71—22no—
41C4×Q8×C71—41no—
42C4×Q8×C71—41no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49#101×C71—22no—
50#101×C71D414yes—
51#97×C71—41no—
52C4×C8×C71—41no—
53C2×Dic142—24no—
54C2×Dic142—42no—
55C2×Dic142—42no—
56C22×Dic71—22no—
57C22×Dic71—22no—
58#11765849—42no—
59#11765849—42no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—24no—
62C4⋊Dic71—42no—
63C4⋊Dic71—41no—
64C4⋊Dic71—41no—
65Dic71⋊C4—44no—
66Dic71⋊C4—42no—
67Dic71⋊C4—42no—
68C4×Dic71—24no—
69C2×Q8×C71—41no—
70C2×Q8×C71—41no—
71C2×Q8×C71C2×D412yes—
72C2×Q8×C71#4112yes—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80M16×C71—24no—
81M16×C71—42no—
82C2×C8×C71—42no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C42×C71—21no—
87C42×C71—21no—
88C42×C71—21no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—21no—
92C23×C71C2×C22⋊C411yescharacteristic
93Q8×C71—42no—
94Q8×C71—42no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71#9911yescharacteristic
109C2×C4×C71#9911yescharacteristic
110C2×C4×C71C4×D412yes—
111C2×C4×C71#12112yes—
112C2×C4×C71#12411yescharacteristic
113C2×C4×C71#12711yescharacteristic
114C2×Dic71—24no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—42no—
120Dic142—84no—
121C8×C71—44no—
122C71⋊C8—82no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71#39011yescharacteristic
126C22×C71C2×#9911yescharacteristic
127C22×C71#41011yescharacteristic
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—21no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71#35612yescomplemented
137Dic71—44no—
138Dic71—82no—
139#13227—711nomaximal, Sylow
140C2×C71—21no—
141C2×C71—21no—
142C2×C71#196411yescharacteristic
143C2×C71C2×#35611yescharacteristic
144C2×C71#213511yescharacteristic
145#2546—711no—
146C2×#356—714no—
147#1801—711no—
148C4×#101D7111yescharacteristic
149C71#1322711yescharacteristic, complemented, Sylow
150C2×C4⋊Q8—712no—
151C2×C4⋊Q8—711no—
152C2×C4×Q8C2×D7111yescharacteristic
153#423—1421no—
154C2×#101—712no—
155C2×#101C2×D7112yes—
156C4×M16—21no—
157#398—1421no—
158#398—1421no—
159#391—712no—
160#389—1421no—
161#389—1421no—
162C4×C4⋊C4—711no—
163#356—1428no—
164#343—21no—
165#343—1421no—
166C22×Q8—712no—
167C22×Q8C22×D7111yescharacteristic
168C2×M16—22no—
169C2×M16—1422no—
170C4⋊Q8—2841no—
171C4⋊Q8—2841no—
172C4⋊Q8—1424no—
173C4⋊Q8—1424no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—1421no—
177C2×C4⋊C4—1421no—
178C2×C4⋊C4—1422no—
179C2×C4⋊C4—1422no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4C4×D7111yescharacteristic
182C2×C4⋊C4D28411yescharacteristic
183C2×C4⋊C4C22⋊D7111yescharacteristic
184C2×C42—712no—
185C2×C42C4×D7111yescharacteristic
186C2×C42D28411yescharacteristic
187C2×C42C22⋊D7111yescharacteristic
188C4⋊C8—2841no—
189C4⋊C8—2841no—
190#101—22no—
191#101—1424no—
192#101D28414yes—
193#97—41no—
194C4×C8—41no—
195#95—1421no—
196#95—1422no—
197#95—1422no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—2842no—
201C2×Q8—2842no—
202C2×Q8—1424no—
203C2×Q8C2×D28412yes—
204C2×Q8#1176586512yes—
205C22×C4—1422no—
206C22×C4—1422no—
207C22×C4(C2×D71)⋊C411yescharacteristic
208C22×C4(C2×D71)⋊C411yescharacteristic
209C22×C4(C2×D71)⋊C411yescharacteristic
210C22×C4(C2×D71)⋊C411yescharacteristic
211C22×C4C2×C4×D7111yescharacteristic
212C22×C4C2×C22⋊D7111yescharacteristic
213C22×C4C2×D28411yescharacteristic
214M16—24no—
215M16—42no—
216M16—2842no—
217M16—2842no—
218C2×C8—42no—
219C2×C8—2842no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—2842no—
224C4⋊C4—2841no—
225C4⋊C4—2841no—
226C4⋊C4—2844no—
227C4⋊C4—2842no—
228C4⋊C4—2842no—
229C4⋊C4—1424no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1424no—
237C23C2×(C2×D71)⋊C411yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—5684no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—2842no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—2842no—
259C2×C4—1424no—
260C2×C4#1455604211yescharacteristic
261C2×C4#1455604211yescharacteristic
262C2×C4C4×D28412yes—
263C2×C4#1455610212yes—
264C2×C4#1455611611yescharacteristic
265C2×C4#1455612511yescharacteristic
266C8—44no—
267C8—5682no—
268C22—21no—
269C22—21no—
270C22#1798590111yescharacteristic
271C22C2×#1455604211yescharacteristic
272C22#1798597211yescharacteristic
273C4—21no—
274C4—22no—
275C4—21no—
276C4—22no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—2844no—
282C4—5682no—
283C4#1798579712yescomplemented
284C2—21no—
285C2—21no—
286C2#2258917611yescharacteristic
287C2C2×#1798579711yescharacteristic
288C2#2259003411yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC4×#101×C71C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC22×Q8×C71C23
hypercenterC4×#101D71
hypocenterC71#13227
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
136. C4×C71#356semidirect
149. C71#13227semidirect
283. C4#17985797semidirect

Maximal subgroups

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

subgroupindex
#225933252
C2×#179857972
#225883362
C4×#101×C712
#1322771