Finite Group Databasebeta

← #31906644

Subgroups — #31906644

Order 18176.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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#22596901C211yesmaximal, characteristic
2#349×D71C212yesmaximal
3#22586293C211yesmaximal, characteristic, complemented
4#22586292C212yesmaximal, complemented
5C71×#1328C211yesmaximal, characteristic, complemented
6C22×#11765848C411yescharacteristic
7C2×M16×D71C2211yescharacteristic
8#117×D71C411yescharacteristic
9M32×D71—22no—
10#17986027C2211yescharacteristic
11#17986026C411yescharacteristic
12#17986025C411yescharacteristic
13#17986024—22no—
14#17986021—22no—
15#17985781C2212yes—
16#17985780—22no—
17#424×C71C2211yescharacteristic
18#349×C71C2212yes—
19C22×C4×D71C2×C411yescharacteristic
20C2×#11765849C2×C411yescharacteristic
21C2×#11765848—21no—
22C2×#11765848—21no—
23C2×#11765848—42no—
24C2×#11765848D412yes—
25M16×D71—42no—
26C22×C71⋊C8C2×C411yescharacteristic
27C2×C8×D71D412yes—
28#14556087C2×C411yescharacteristic
29#14556086—21no—
30#14556085—21no—
31#14556084C812yes—
32C4⋊C4×D71—21no—
33C22⋊C4×D71C812yes—
34C42×D71—21no—
35#14556068—22no—
36#14556067—42no—
37C16×D71—42no—
38C4⋊(C71⋊C8)—22no—
39Dic71⋊C8—22no—
40Dic71⋊C8—22no—
41C2×M16×C71C2311yescharacteristic
42C22×C8×C71C2×C411yescharacteristic
43#117×C71C2×C411yescharacteristic
44M32×C71—22no—
45C4⋊C8×C71—22no—
46C23×D71C2×C811yescharacteristic
47C22×Dic71C2×C811yescharacteristic
48C2×C4×D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71C22⋊C411yescharacteristic
52C2×C4×D71C22⋊C411yescharacteristic
53C2×C4×D71M1611yescharacteristic
54C2×C4×D71M1611yescharacteristic
55#11765849—41no—
56#11765848—42no—
57#11765848—42no—
58#11765848—42no—
59C2×C71⋊C8—21no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8C22⋊C412yes—
62C8×D71—22no—
63C8×D71—22no—
64C4⋊Dic71—21no—
65Dic71⋊C4—22no—
66C22⋊Dic71C2×C812yes—
67(C2×D71)⋊C4—22no—
68C4×Dic71—21no—
69C4×Dic71—22no—
70C71⋊C16—42no—
71C22×C4×C71C22×C411yescharacteristic
72M16×C71—41no—
73C2×C8×C71—21no—
74C2×C8×C71—41no—
75C2×C8×C71C2×D412yes—
76C4⋊C4×C71—21no—
77C22⋊C4×C71C2×C812yes—
78C42×C71—21no—
79C16×C71—42no—
80C23×C71C22×C811yescharacteristic
81C22×D71—22no—
82C22×D71—81no—
83C22×D71C22⋊C811yescharacteristic
84C22×D71C22⋊C811yescharacteristic
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71C2×C22⋊C411yescharacteristic
89C2×C4×C71C2×M1611yescharacteristic
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71C22⋊C811yescharacteristic
93C2×Dic71C22⋊C811yescharacteristic
94C4×D71—22no—
95C4×D71—21no—
96C4×D71—21no—
97C4×D71—22no—
98C4×D71—82no—
99C4×D71#9912yes—
100C4×D71#10112yes—
101C8×C71—22no—
102C8×C71—41no—
103C71⋊C8—22no—
104C71⋊C8—41no—
105C22×C71—21no—
106C22×C71—41no—
107C22×C71C2×C22⋊C811yescharacteristic
108C2×D71—21no—
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—82no—
112C2×D71#32412yes—
113C4×C71—21no—
114C4×C71—42no—
115C4×C71C2×#9911yescharacteristic
116C4×C71C2×#10111yescharacteristic
117Dic71—21no—
118Dic71—42no—
119Dic71#32412yes—
120#11458—711nomaximal, Sylow
121C2×C71—21no—
122C2×C71—41no—
123C2×C71C2×#32411yescharacteristic
124D71—22no—
125D71—41no—
126#2905—711no—
127C2×#349—712no—
128#1328—711no—
129#1328—712no—
130#1328D7111yescharacteristic, complemented
131C71#1145811yescharacteristic, complemented, Sylow
132C22×M16—711no—
133C22×M16—711no—
134C2×#117—711no—
135C2×M32—1422no—
136#424—711no—
137#424—711no—
138#424—711no—
139#424—1422no—
140#424C2×D7111yescharacteristic
141#423—1422no—
142#349—712no—
143#349—1422no—
144#349C2×D7112yes—
145C23×C4—711no—
146C2×M16—2842no—
147C2×M16—2842no—
148C2×M16—711no—
149C2×M16—712no—
150C2×M16—1421no—
151C2×M16—1421no—
152C2×M16C22×D7111yescharacteristic
153C22×C8—711no—
154C22×C8—712no—
155C22×C8C4×D7111yescharacteristic
156#117—711no—
157#117—712no—
158#117—1421no—
159#117—1421no—
160#117C4×D7111yescharacteristic
161C2×C4⋊C4—1421no—
162C2×C22⋊C4—712no—
163C2×C42—1421no—
164M32—22no—
165M32—2842no—
166M32—1422no—
167C2×C16—2842no—
168C4⋊C8—22no—
169C4⋊C8—1422no—
170C4⋊C8—1422no—
171C4⋊C8—1422no—
172C24—711no—
173C22×C4—711no—
174C22×C4—711no—
175C22×C4—711no—
176C22×C4—711no—
177C22×C4—711no—
178C22×C4—1422no—
179C22×C4—1422no—
180C22×C4—1422no—
181C22×C4C2×C4×D7111yescharacteristic
182M16—41no—
183M16—2842no—
184M16—2842no—
185M16—2842no—
186M16—2841no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—2841no—
190C2×C8—712no—
191C2×C8—1422no—
192C2×C8—1422no—
193C2×C8—1421no—
194C2×C8D4×D7112yes—
195C4⋊C4—21no—
196C4⋊C4—1422no—
197C4⋊C4—1421no—
198C22⋊C4—712no—
199C22⋊C4—1422no—
200C22⋊C4C8×D7112yes—
201C42—21no—
202C42—1422no—
203C42—1421no—
204C16—42no—
205C16—2842no—
206C23—5681no—
207C23—711no—
208C23—711no—
209C23—1422no—
210C23C2×C8×D7111yescharacteristic
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—5682no—
215C2×C4—711no—
216C2×C4—712no—
217C2×C4—712no—
218C2×C4—711no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1422no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4C22⋊C4×D7111yescharacteristic
226C2×C4M16×D7111yescharacteristic
227C8—22no—
228C8—41no—
229C8—2841no—
230C8—1422no—
231C22—21no—
232C22—41no—
233C22—2841no—
234C22—2841no—
235C22—5682no—
236C22—712no—
237C22—1421no—
238C22C22⋊C8×D7111yescharacteristic
239C4—21no—
240C4—42no—
241C4—2842no—
242C4—712no—
243C4—1421no—
244C4#99×D7111yescharacteristic
245C4#101×D7111yescharacteristic
246C2—21no—
247C2—41no—
248C2—2841no—
249C2—1422no—
250C2#324×D7111yescharacteristic
subgroupG/H
centerC4#99×D71
commutatorC23×C71C22×C8
FrattiniC2×M16C22×D71
FittingC71×#1328C2
socleC2×C71C2×#324
prime radicalC23×D71C2×C8
cosocleC2×M16×C71C23
hypercenter#1328D71
hypocenterC71#11458
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
3. #22586293C2semidirect
4. #22586292C2semidirect
5. C71×#1328C2semidirect
130. #1328D71semidirect
131. C71#11458semidirect

Maximal subgroups

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

subgroupindex
#225969012
#349×D712
#225862932
#225862922
C71×#13282
#1145871