Finite Group Databasebeta

← #31908155

Subgroups — #31908155

Order 18176.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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#22594106C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3#22586345C214yesmaximal, complemented
4C71×#1822C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556107—21no—
7#17986056—21no—
8#17985929—21no—
9#17985929C2212yes—
10#17985924—21no—
11#17985924C412yes—
12#17985905—21no—
13#17985865C411yescharacteristic
14#17985725C2212yes—
15#17985723—21no—
16C2×#117×C71C2211yescharacteristic
17#407×C71—21no—
18#343×C71—21no—
19#343×C71C2212yes—
20C22×Dic142—21no—
21#14556128—41no—
22#14556128—41no—
23#14556107—44no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×C22⋊Dic71—22no—
29C4⋊(C71⋊C8)—41no—
30C22⋊(C71⋊C8)—22no—
31#14556039—41no—
32C71⋊#95—21no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95C2×C412yes—
36C23×C4×C71D411yescharacteristic
37C2×M16×C71—22no—
38#117×C71—41no—
39#117×C71—41no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C22⋊C4×C71C2×C411yescharacteristic
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C22⋊C8×C71—41no—
47C22⋊C8×C71—41no—
48C2×Dic142—42no—
49C2×Dic142—42no—
50C2×Dic142—41no—
51C2×Dic142—41no—
52C22×Dic71—22no—
53C22×Dic71—22no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—42no—
56C4⋊Dic71—42no—
57C4⋊Dic71—42no—
58Dic71⋊C4—44no—
59C22⋊Dic71—44no—
60C24×C71C22⋊C411yescharacteristic
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—22no—
64C22×C4×C71—41no—
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71M16×C71—42no—
72M16×C71—42no—
73C2×C8×C71—42no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C22⋊C4×C71—41no—
77C22⋊C4×C71—41no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C23×C71—21no—
83C23×C71—21no—
84C23×C71—21no—
85C23×C71—41no—
86C23×C71—41no—
87C23×C71C2×C22⋊C411yescharacteristic
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71#9911yescharacteristic
103C2×C4×C71#9911yescharacteristic
104C2×C4×C71AGL(1,ℤ/8)12yes—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111Dic142—82no—
112Dic142—82no—
113C8×C71—82no—
114C71⋊C8—82no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71—42no—
118C22×C71—41no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71—81no—
122C22×C71C2×#9911yescharacteristic
123C22×C71#41811yescharacteristic
124C22×C71#42111yescharacteristic
125C4×C71—22no—
126C4×C71—21no—
127C4×C71—21no—
128C4×C71—41no—
129C4×C71—41no—
130C4×C71—41no—
131C4×C71—41no—
132Dic71—82no—
133Dic71—82no—
134#11807—711nomaximal, Sylow
135C2×C71—21no—
136C2×C71—21no—
137C2×C71—81no—
138C2×C71#151011yescharacteristic
139C2×C71#212311yescharacteristic
140C2×C71#213511yescharacteristic
141#2621—711no—
142#1822D7111yescharacteristic
143#1464—711no—
144#1346—714no—
145C71#1180711yescharacteristic, complemented, Sylow
146C2×C4⋊Q8—711no—
147C2×C22⋊Q8—1421no—
148C2×#117C2×D7111yescharacteristic
149#432—1421no—
150#407—21no—
151#398—712no—
152#398—1421no—
153#397—712no—
154#397—1421no—
155#391—1421no—
156#382—711no—
157#343—21no—
158#343C2×D7112yes—
159#324—712no—
160#323—1421no—
161C22×Q8—1421no—
162C23×C4C22⋊D7111yescharacteristic
163C2×M16—22no—
164C4⋊Q8—2841no—
165C4⋊Q8—2841no—
166C22⋊Q8—2844no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—1421no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4—1422no—
173C2×C4⋊C4C22⋊D7111yescharacteristic
174C2×C4⋊C4C22×D7111yescharacteristic
175C2×C22⋊C4—1422no—
176C2×C22⋊C4C4×D7111yescharacteristic
177C2×C22⋊C4D28411yescharacteristic
178C2×C42C4×D7111yescharacteristic
179C2×C42D28411yescharacteristic
180C4⋊C8—2841no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—1422no—
184#97—2841no—
185#95—712no—
186#95—1421no—
187#95—1422no—
188#95—1422no—
189C24(C2×D71)⋊C411yescharacteristic
190C2×Q8—2842no—
191C2×Q8—2842no—
192C2×Q8—2841no—
193C2×Q8—2841no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4(C2×D71)⋊C411yescharacteristic
201C22×C4(C2×D71)⋊C411yescharacteristic
202C22×C4(C2×D71)⋊C411yescharacteristic
203C22×C4C2×C4×D7111yescharacteristic
204C22×C4C2×C22⋊D7111yescharacteristic
205C22×C4C2×D28411yescharacteristic
206M16—42no—
207M16—42no—
208C2×C8—42no—
209C2×C8—2842no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—2842no—
213C4⋊C4—2842no—
214C4⋊C4—2842no—
215C4⋊C4—2844no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—2844no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×(C2×D71)⋊C411yescharacteristic
229Q8—5682no—
230Q8—5682no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—2842no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4#1455604211yescharacteristic
252C2×C4#1455604211yescharacteristic
253C2×C4#1455615312yes—
254C8—82no—
255C8—5682no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×#1455604211yescharacteristic
264C22#1798600311yescharacteristic
265C22#1798601711yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—5682no—
274C4—5682no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2#2258686811yescharacteristic
279C2#2258999511yescharacteristic
280C2#2259003411yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1822C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1822D71
hypocenterC71#11807
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #22586345C2semidirect
145. C71#11807semidirect

Maximal subgroups

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

subgroupindex
#225941062
#225866742
#225863452
C71×#18222
#1180771