Finite Group Databasebeta

← #31890853

Subgroups — #31890853

Order 18176.

554 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 111 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#382×D71C211yesmaximal, characteristic
2#22588103C212yesmaximal
3#22586222C212yesmaximal, complemented
4#22586221C211yesmaximal, characteristic, complemented
5C71×#1295C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D71C412yes—
7C2×C42×D71C2211yescharacteristic
8C2×(C2×D71)⋊C8—22no—
9C2×#14556039C2212yes—
10#17985865C2211yescharacteristic
11#17985864C412yes—
12#17985863—22no—
13#95×D71—22no—
14#17985753—22no—
15C2×#97×C71C2212yes—
16#382×C71C2211yescharacteristic
17C22×C4×D71—22no—
18C22×C4×D71C2×C411yescharacteristic
19C22×C4×D71D412yes—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71C2×C412yes—
22C2×Dic71⋊C4—22no—
23C4⋊C4×D71—44no—
24C2×C4×Dic71C2×C411yescharacteristic
25C2×C4×Dic71D412yes—
26C42×D71—21no—
27C42×D71—22no—
28C42×D71—21no—
29(C2×D71)⋊C8—28no—
30#14556039—22no—
31#14556039—22no—
32C71⋊#95—24no—
33C71⋊#95—22no—
34C22×C8×C71—22no—
35C2×C4⋊C4×C71C2×C412yes—
36C2×C42×C71C2311yescharacteristic
37#97×C71—22no—
38#97×C71—22no—
39#95×C71—22no—
40C23×D71C22⋊C411yescharacteristic
41C22×Dic71—22no—
42C22×Dic71C22⋊C412yes—
43C22×Dic71C22⋊C411yescharacteristic
44C2×C4×D71—22no—
45C2×C4×D71—21no—
46C2×C4×D71—21no—
47C2×C4×D71—22no—
48C2×C4×D71—24no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—42no—
52C2×C4×D71—44no—
53C2×C4×D71—44no—
54C2×C4×D71—44no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—42no—
59Dic71⋊C4—44no—
60C4×Dic71—21no—
61C4×Dic71—21no—
62C4×Dic71—22no—
63C4×Dic71—22no—
64C22×C4×C71—22no—
65C22×C4×C71C22×C411yescharacteristic
66C22×C4×C71C2×D412yes—
67C2×C8×C71—24no—
68C2×C8×C71—42no—
69C2×C8×C71—42no—
70C4⋊C4×C71—42no—
71C42×C71#4111yescharacteristic
72C42×C71#4112yes—
73C42×C71#4111yescharacteristic
74C23×C71C2×C22⋊C411yescharacteristic
75C22×D71—21no—
76C22×D71—21no—
77C22×D71—22no—
78C22×D71#10112yes—
79C22×D71#10414yes—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71#11711yescharacteristic
88C2×C4×C71#11711yescharacteristic
89C2×C4×C71#12312yes—
90C2×C4×C71#12312yes—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—42no—
97C2×Dic71—42no—
98C2×Dic71—42no—
99C2×Dic71#10012yes—
100C4×D71—24no—
101C4×D71—24no—
102C4×D71—44no—
103C4×D71—44no—
104C4×D71—88no—
105C8×C71—44no—
106C71⋊C8—44no—
107C22×C71#38111yescharacteristic
108C22×C71C2×#10011yescharacteristic
109C22×C71C2×#10111yescharacteristic
110C22×C71C2×#10412yes—
111C22×C71#42112yes—
112C2×D71—22no—
113C2×D71—24no—
114C2×D71—24no—
115C2×D71—22no—
116C2×D71#33014yes—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—82no—
122Dic71—22no—
123Dic71—42no—
124Dic71—82no—
125#8719—711nomaximal, Sylow
126C2×C71C2×#33011yescharacteristic
127C2×C71#152712yes—
128C2×C71#153411yescharacteristic
129C2×C71#182111yescharacteristic
130C2×C71#182612yes—
131D71—24no—
132C2×#382—711no—
133#1749—712no—
134#1295—712no—
135#1295—711no—
136#1295D7111yescharacteristic, complemented
137C71#871911yescharacteristic, complemented, Sylow
138C22×C4⋊C4—712no—
139C22×C42—711no—
140C2×C22⋊C8—1422no—
141C2×#97—712no—
142C2×#97C2×D7112yes—
143#382—712no—
144#382—711no—
145#382—1422no—
146#382C2×D7111yescharacteristic
147C2×#95—1422no—
148#336—1422no—
149C23×C4—712no—
150C23×C4—711no—
151C23×C4—1422no—
152C22×C8—22no—
153C22×C8—1422no—
154C2×C4⋊C4—2844no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—1422no—
157C2×C4⋊C4C4×D7112yes—
158C2×C42—711no—
159C2×C42—712no—
160C2×C42—1421no—
161C2×C42—1422no—
162C2×C42—1421no—
163C2×C42C22×D7111yescharacteristic
164C22⋊C8—1428no—
165#97—22no—
166#97—22no—
167#97—1422no—
168#97—1422no—
169#95—22no—
170#95—1424no—
171#95—1422no—
172C24—711no—
173C22×C4—22no—
174C22×C4—2842no—
175C22×C4—2844no—
176C22×C4—2844no—
177C22×C4—2844no—
178C22×C4—711no—
179C22×C4—712no—
180C22×C4—1424no—
181C22×C4—1422no—
182C22×C4—1422no—
183C22×C4—1422no—
184C22×C4—1422no—
185C22×C4—1422no—
186C22×C4—1421no—
187C22×C4—1421no—
188C22×C4C2×C4×D7111yescharacteristic
189C22×C4D4×D7112yes—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—2842no—
194C2×C8—2842no—
195C2×C8—1424no—
196C4⋊C4—42no—
197C4⋊C4—2842no—
198C4⋊C4—2844no—
199C42—1421no—
200C42—1421no—
201C42—1422no—
202C42—1422no—
203C42#1176586411yescharacteristic
204C42#1176586512yes—
205C42#1176586611yescharacteristic
206C23—712no—
207C23—714no—
208C23—1421no—
209C23—1421no—
210C23—1422no—
211C23C22⋊C4×D7111yescharacteristic
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—2844no—
220C2×C4—2844no—
221C2×C4—2842no—
222C2×C4—2842no—
223C2×C4—2842no—
224C2×C4—5688no—
225C2×C4—712no—
226C2×C4—1424no—
227C2×C4—1424no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1421no—
232C2×C4—1421no—
233C2×C4#1455608511yescharacteristic
234C2×C4#1455608611yescharacteristic
235C2×C4#1455611012yes—
236C2×C4#1455611312yes—
237C8—44no—
238C8—2844no—
239C22—714no—
240C22—1422no—
241C22—1424no—
242C22—1424no—
243C22—1422no—
244C22#1798586111yescharacteristic
245C22#100×D7111yescharacteristic
246C22#101×D7111yescharacteristic
247C22#104×D7112yes—
248C22#1798601412yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—2842no—
255C4—5682no—
256C4—1422no—
257C2—1424no—
258C2#330×D7111yescharacteristic
259C2#2258693612yes—
260C2#2258696511yescharacteristic
261C2#2258841911yescharacteristic
262C2#2258843812yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1295D71
hypocenterC71#8719
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #22586222C2semidirect
4. #22586221C2semidirect
5. C71×#1295C2semidirect
136. #1295D71semidirect
137. C71#8719semidirect

Maximal subgroups

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

subgroupindex
#382×D712
#225881032
#225862222
#225862212
C71×#12952
#871971