Finite Group Databasebeta

← #31891053

Subgroups — #31891053

Order 18176.

426 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 115 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#22588332C211yesmaximal, characteristic
2#22586661C211yesmaximal, characteristic
3#22586243C212yesmaximal, complemented
4#22586198C212yesmaximal, complemented
5C71×#2357C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2212yes—
8C2×#14556045—21no—
9#17985768—21no—
10#17985723—22no—
11C2×C4×D4×C71C2211yescharacteristic
12#390×C71—21no—
13#388×C71—22no—
14#382×C71C2211yescharacteristic
15#381×C71C412yes—
16C4×C4⋊C4×C71—21no—
17C4×C4⋊C4×C71C2211yescharacteristic
18C4×C22⋊C4×C71C412yes—
19C2×#11765849—22no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)—22no—
22C4⋊(C71⋊C8)—22no—
23#14556045—42no—
24C22⋊(C71⋊C8)—22no—
25C4×C71⋊C8—22no—
26C22×D4×C71D411yescharacteristic
27C23×C4×C71C2×C412yes—
28C4×D4×C71—22no—
29C4×D4×C71—22no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71D411yescharacteristic
33C2×C22⋊C4×C71—22no—
34C2×C22⋊C4×C71—22no—
35C2×C22⋊C4×C71D412yes—
36C2×C42×C71—21no—
37C2×C42×C71—22no—
38C2×C42×C71C2311yescharacteristic
39#95×C71—22no—
40#95×C71—21no—
41#95×C71C2×C411yescharacteristic
42#95×C71C2×C412yes—
43#95×C71C2×C411yescharacteristic
44#11765849—42no—
45#11765849—42no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C24×C71C22⋊C412yes—
49C2×D4×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71—21no—
52C2×D4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—22no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71—21no—
58C22×C4×C71—82no—
59C22×C4×C71C22⋊C412yes—
60C22×C4×C71C22×C411yescharacteristic
61C22×C4×C71C2×D411yescharacteristic
62C22×C4×C71C2×D411yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C22⋊C4×C71—24no—
68C22⋊C4×C71—44no—
69C22⋊C4×C71—82no—
70C42×C71—42no—
71C42×C71—82no—
72C42×C71#4112yes—
73C42×C71#4112yes—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71C2×C22⋊C411yescharacteristic
80D4×C71—42no—
81D4×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—42no—
95C2×C4×C71—82no—
96C2×C4×C71—82no—
97C2×C4×C71#11711yescharacteristic
98C2×C4×C71#11711yescharacteristic
99C2×C4×C71#12312yes—
100C2×C4×C71#12312yes—
101C2×C4×C71#13512yes—
102C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C71#13711yescharacteristic
104C71⋊C8—82no—
105C71⋊C8—82no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71—42no—
111C22×C71—82no—
112C22×C71—82no—
113C22×C71#38111yescharacteristic
114C22×C71#41611yescharacteristic
115C22×C71#41711yescharacteristic
116C4×C71—22no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—42no—
120C4×C71—42no—
121C4×C71—81no—
122C4×C71—82no—
123C4×C71#48212yes—
124C4×C71#48512yes—
125#8771—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71—82no—
129C2×C71#182511yescharacteristic
130C2×C71#182711yescharacteristic
131C2×C71#182911yescharacteristic
132#2357D7111yescharacteristic
133#1800—711no—
134#1460—711no—
135#1303—712no—
136#1286—712no—
137C71#877111yescharacteristic, complemented, Sylow
138C2×C4×D4C2×D7111yescharacteristic
139#431—712no—
140#423—712no—
141C2×#100—1421no—
142#390—21no—
143#388—22no—
144#382C2×D7111yescharacteristic
145#381Dic7112yes—
146C4×C4⋊C4—21no—
147C4×C4⋊C4C2×D7111yescharacteristic
148C4×C22⋊C4Dic7112yes—
149#343—1421no—
150#323—1422no—
151C22×D4C22⋊D7111yescharacteristic
152C23×C4C2×Dic7112yes—
153C2×M16—1422no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C22⋊D7111yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22⋊D7112yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C22×D7111yescharacteristic
165C4⋊C8—1422no—
166C4⋊C8—1422no—
167C4⋊C8—1422no—
168#100—2842no—
169C22⋊C8—1422no—
170C4×C8—1422no—
171#95—22no—
172#95—21no—
173#95C2×Dic7111yescharacteristic
174#95C2×Dic7112yes—
175#95C2×Dic7111yescharacteristic
176C24C22⋊Dic7112yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—82no—
187C22×C4C22⋊Dic7112yes—
188C22×C4C22×Dic7111yescharacteristic
189C22×C4C2×C22⋊D7111yescharacteristic
190C22×C4C2×C22⋊D7111yescharacteristic
191M16—2842no—
192M16—2842no—
193C2×C8—2842no—
194C2×C8—2842no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C22⋊C4—24no—
200C22⋊C4—44no—
201C22⋊C4—82no—
202C42—42no—
203C42—82no—
204C42#1176586512yes—
205C42#1176586512yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×C22⋊Dic7111yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4#1455608711yescharacteristic
230C2×C4#1455608711yescharacteristic
231C2×C4#1455611112yes—
232C2×C4#1455611112yes—
233C2×C4#1455614812yes—
234C2×C4#1455615311yescharacteristic
235C2×C4#1455615811yescharacteristic
236C8—5682no—
237C8—5682no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1798586211yescharacteristic
246C22#1798599511yescharacteristic
247C22#1798600011yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4#1798632312yes—
256C4#1798634012yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2258843511yescharacteristic
261C2#2258844411yescharacteristic
262C2#2258845111yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2357C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2357D71
hypocenterC71#8771
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. #22586243C2semidirect
4. #22586198C2semidirect
137. C71#8771semidirect

Maximal subgroups

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

subgroupindex
#225883322
#225866612
#225862432
#225861982
C71×#23572
#877171