Finite Group Databasebeta

← #31908299

Subgroups — #31908299

Order 18176.

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

normal subgroups only

This group has 324 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#22590152C211yesmaximal, characteristic, complemented
2#22588489C211yesmaximal, characteristic, complemented
3#22586331C212yesmaximal, complemented
4#22586325C212yesmaximal, complemented
5C71×#3184C211yesmaximal, characteristic
6#17985820—21no—
7#17985781C2212yes—
8#17985777—21no—
9#17985770—21no—
10#17985759—21no—
11C71⋊#338C2211yescharacteristic
12#17985756C2211yescharacteristic
13#532×C71—21no—
14#498×C71C411yescharacteristic, complemented
15#493×C71C411yescharacteristic, complemented
16#486×C71—21no—
17#469×C71—21no—
18#469×C71C2212yes—
19#442×C71—21no—
20#442×C71C412yescomplemented
21C4×M16×C71C2211yescharacteristic
22C2×#11765849—21no—
23#14556087—21no—
24#14556068—22no—
25C2×#41×C71—41no—
26C2×D8×C71—21no—
27C2×D8×C71C2×C412yes—
28C2×M16×C71D411yescharacteristic
29C2×M16×C71C2311yescharacteristic
30C4⋊Q8×C71—21no—
31#127×C71—21no—
32#124×C71—41no—
33#121×C71—22no—
34#121×C71—41no—
35C4×D4×C71—22no—
36C4×D4×C71—41no—
37C2×C42×C71D411yescharacteristic
38#106×C71—21no—
39#106×C71C2×C412yes—
40#102×C71—22no—
41#102×C71—22no—
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71C2×C411yescharacteristic
45C4×C8×C71D411yescharacteristic
46#11765849—41no—
47#11765849—41no—
48C2×C71⋊C8—41no—
49C4⋊Dic71—41no—
50C22⋊Dic71—41no—
51C4×Dic71—41no—
52C71⋊C16—42no—
53#41×C71—84no—
54C2×Q8×C71—41no—
55C2×D4×C71—22no—
56C2×D4×C71—41no—
57C2×D4×C71—41no—
58C2×D4×C71—42no—
59C22×C4×C71—21no—
60C22×C4×C71—42no—
61C22×C4×C71C2×D411yescharacteristic
62D8×C71—42no—
63D8×C71—42no—
64M16×C71—41no—
65M16×C71—41no—
66C2×C8×C71C22⋊C411yescharacteristic
67C2×C8×C71C22⋊C411yescharacteristic
68C2×C8×C71C22×C411yescharacteristic
69C2×C8×C71C2×D411yescharacteristic
70C4⋊C4×C71—22no—
71C4⋊C4×C71—41no—
72C22⋊C4×C71—42no—
73C22⋊C4×C71—42no—
74C42×C71—21no—
75C42×C71C22⋊C411yescharacteristic
76C42×C71C22⋊C411yescharacteristic
77C23×C71—42no—
78C23×C71AGL(1,ℤ/8)11yescharacteristic
79Q8×C71—82no—
80D4×C71—42no—
81D4×C71—42no—
82D4×C71—82no—
83D4×C71—82no—
84D4×C71—84no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—84no—
92C2×C4×C71C2×C22⋊C411yescharacteristic
93C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
94C2×Dic71—81no—
95C8×C71—21no—
96C8×C71—21no—
97C8×C71—21no—
98C8×C71—21no—
99C71⋊C8—81no—
100C22×C71—21no—
101C22×C71—41no—
102C22×C71—82no—
103C22×C71—82no—
104C22×C71—82no—
105C22×C71#41811yescharacteristic
106C4×C71—22no—
107C4×C71—21no—
108C4×C71—41no—
109C4×C71—82no—
110C4×C71#41011yescharacteristic
111C4×C71#42111yescharacteristic
112Dic71—161no—
113#11837—711nomaximal, Sylow
114C2×C71—21no—
115C2×C71—41no—
116C2×C71—162no—
117C2×C71#151411yescharacteristic
118#3184D7111yescharacteristic
119#2165—711no—
120#1841—711no—
121#1341—712no—
122#1339—712no—
123C71#1183711yescharacteristic, complemented, Sylow
124#532—21no—
125#498Dic7111yescharacteristic, complemented
126#493Dic7111yescharacteristic, complemented
127#486—21no—
128#469—21no—
129#469C2×D7112yes—
130#442—21no—
131#442Dic7112yescomplemented
132C4×M16C2×D7111yescharacteristic
133#364—1421no—
134#349—712no—
135#347—1421no—
136#344—1421no—
137#339—1421no—
138#338—711no—
139H(3,ℤ/4)—711no—
140C2×#41—41no—
141C2×D8—21no—
142C2×D8C2×Dic7112yes—
143C2×M16—1421no—
144C2×M16C22⋊D7111yescharacteristic
145C2×M16C22×D7111yescharacteristic
146C4⋊Q8—21no—
147#127—21no—
148#124—41no—
149#121—22no—
150#121—41no—
151C4×D4—22no—
152C4×D4—41no—
153#117—1421no—
154C2×C42C22⋊D7111yescharacteristic
155M32—1422no—
156#106—21no—
157#106C2×Dic7112yes—
158#102—22no—
159#102—22no—
160#97C2×Dic7111yescharacteristic
161#97C22⋊D7111yescharacteristic
162C4×C8C2×Dic7111yescharacteristic
163C4×C8C22⋊D7111yescharacteristic
164#41—84no—
165C2×Q8—41no—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4C2×C22⋊D7111yescharacteristic
173D8—42no—
174D8—42no—
175M16—41no—
176M16—41no—
177M16—2841no—
178M16—2841no—
179C2×C8—2841no—
180C2×C8C22⋊Dic7111yescharacteristic
181C2×C8C22⋊Dic7111yescharacteristic
182C2×C8C22×Dic7111yescharacteristic
183C2×C8C2×C22⋊D7111yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—2841no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—2841no—
190C42—21no—
191C42—2841no—
192C42C22⋊Dic7111yescharacteristic
193C42C22⋊Dic7111yescharacteristic
194C16—2842no—
195C23—42no—
196C23#1455615411yescharacteristic
197Q8—82no—
198D4—42no—
199D4—42no—
200D4—82no—
201D4—82no—
202D4—84no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4—5681no—
211C2×C4C2×C22⋊Dic7111yescharacteristic
212C2×C4#1455615411yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—5681no—
218C22—21no—
219C22—41no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22#1798600411yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—82no—
228C4—11361no—
229C4#1798597311yescharacteristic
230C4#1798601611yescharacteristic
231C2—21no—
232C2—41no—
233C2—162no—
234C2#2258688711yescharacteristic
subgroupG/H
centerC2#22586887
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
Fitting, prime radicalC71×#3184C2
socleC2×C71#1514
cosocleC2×M16×C71C23
hypercenter#3184D71
hypocenterC71#11837
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. #22590152C2semidirect
2. #22588489C2semidirect
3. #22586331C2semidirect
4. #22586325C2semidirect
14. #498×C71C4semidirect
15. #493×C71C4semidirect
20. #442×C71C4semidirect
123. C71#11837semidirect
125. #498Dic71semidirect
126. #493Dic71semidirect
131. #442Dic71semidirect

Maximal subgroups

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

subgroupindex
#225901522
#225884892
#225863312
#225863252
C71×#31842
#1183771