Finite Group Databasebeta

← #31907117

Subgroups — #31907117

Order 18176.

358 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 95 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#22586684C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586284C214yesmaximal
4C71×#2568C211yesmaximal, characteristic
5#17986056—21no—
6#17986052—21no—
7#17985725—21no—
8#17985725—21no—
9#17985725C2212yes—
10#17985725C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4×Q8×C71C2211yescharacteristic
13#400×C71—21no—
14#400×C71—21no—
15#391×C71—21no—
16#391×C71—21no—
17#391×C71C412yes—
18#382×C71C2212yes—
19C4×C4⋊C4×C71C411yescharacteristic
20C4⋊(C71⋊C8)—41no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24#14556039—41no—
25C4×C71⋊C8—41no—
26C22×Q8×C71—21no—
27C22×Q8×C71C2×C411yescharacteristic
28C4⋊Q8×C71—41no—
29C4⋊Q8×C71—41no—
30C4×Q8×C71—41no—
31C4×Q8×C71—41no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71C2×C412yes—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C4⋊C4×C71C2311yescharacteristic
38C2×C42×C71C2×C412yes—
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71D411yescharacteristic
41C2×C42×C71D411yescharacteristic
42#95×C71—21no—
43#95×C71—22no—
44#95×C71—22no—
45#95×C71—22no—
46#95×C71—22no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C2×Q8×C71—22no—
50C2×Q8×C71—22no—
51C2×Q8×C71—41no—
52C2×Q8×C71—44no—
53C2×Q8×C71—41no—
54C2×Q8×C71—41no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22×C411yescharacteristic
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71C2×D411yescharacteristic
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C42×C71—21no—
71C42×C71—21no—
72C42×C71—21no—
73C42×C71—21no—
74C42×C71—21no—
75C42×C71—21no—
76C42×C71—24no—
77C23×C71C2×C22⋊C411yescharacteristic
78Q8×C71—44no—
79Q8×C71—82no—
80Q8×C71—82no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—44no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71#9911yescharacteristic
101C2×C4×C71#9911yescharacteristic
102C2×C4×C71#10112yes—
103C2×C4×C71#13112yes—
104C71⋊C8—82no—
105C71⋊C8—82no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71#40811yescharacteristic
109C22×C71C2×#9911yescharacteristic
110C22×C71C2×#10111yescharacteristic
111C4×C71—21no—
112C4×C71—22no—
113C4×C71—22no—
114C4×C71—41no—
115C4×C71—41no—
116C4×C71—41no—
117C4×C71—41no—
118C4×C71—82no—
119C4×C71—82no—
120C4×C71#35512yes—
121#11573—711nomaximal, Sylow
122C2×C71—21no—
123C2×C71—21no—
124C2×C71#147211yescharacteristic
125C2×C71#212911yescharacteristic
126C2×C71C2×#35511yescharacteristic
127#2568D7111yescharacteristic
128#1467—711no—
129#1461—711no—
130#1324—714no—
131C71#1157311yescharacteristic, complemented, Sylow
132C2×C4⋊Q8Dic7111yescharacteristic
133C2×C4×Q8C2×D7111yescharacteristic
134#432—1421no—
135#431—1421no—
136#400—21no—
137#400—21no—
138#391—21no—
139#391—21no—
140#391Dic7112yes—
141#382C2×D7112yes—
142C4×C4⋊C4Dic7111yescharacteristic
143#324—712no—
144#324—712no—
145#324—1421no—
146#324—1421no—
147C22×Q8—21no—
148C22×Q8C2×Dic7111yescharacteristic
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4C2×Dic7112yes—
156C2×C4⋊C4C22⋊D7111yescharacteristic
157C2×C4⋊C4C22⋊D7111yescharacteristic
158C2×C4⋊C4C22×D7111yescharacteristic
159C2×C42C2×Dic7112yes—
160C2×C42C2×Dic7111yescharacteristic
161C2×C42C22⋊D7111yescharacteristic
162C2×C42C22⋊D7111yescharacteristic
163C4⋊C8—2841no—
164C4⋊C8—2841no—
165C22⋊C8—1422no—
166C22⋊C8—1422no—
167#97—2841no—
168C4×C8—2841no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—44no—
178C2×Q8—41no—
179C2×Q8—41no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C22⋊Dic7111yescharacteristic
183C22×C4C22⋊Dic7111yescharacteristic
184C22×C4C22⋊Dic7111yescharacteristic
185C22×C4C22⋊Dic7111yescharacteristic
186C22×C4C22×Dic7111yescharacteristic
187C22×C4C2×C22⋊D7111yescharacteristic
188C22×C4C2×C22⋊D7111yescharacteristic
189C2×C8—2842no—
190C2×C8—2842no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—24no—
204C23C2×C22⋊Dic7111yescharacteristic
205Q8—44no—
206Q8—82no—
207Q8—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4#1455604311yescharacteristic
228C2×C4#1455604311yescharacteristic
229C2×C4#1455604712yes—
230C2×C4#1455613512yes—
231C8—5682no—
232C8—5682no—
233C22—21no—
234C22—21no—
235C22#1798596611yescharacteristic
236C22C2×#1455604311yescharacteristic
237C22C2×#1455604711yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—82no—
247C4#1798579612yes—
248C2—21no—
249C2—21no—
250C2#2258670411yescharacteristic
251C2#2259001511yescharacteristic
252C2C2×#1798579611yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2568C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2568D71
hypocenterC71#11573
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
131. C71#11573semidirect

Maximal subgroups

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

subgroupindex
#225866842
#225866642
#225862842
C71×#25682
#1157371