Finite Group Databasebeta

← #31911682

Subgroups — #31911682

Order 18176.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#22590173C211yesmaximal, characteristic, complemented
2#22587702C211yesmaximal, characteristic, complemented
3#22587692C211yesmaximal, characteristic
4#22586412C211yesmaximal, characteristic, complemented
5#22586389C211yesmaximal, characteristic, complemented
6#22586386C211yesmaximal, characteristic, complemented
7C71×#2248C211yesmaximal, characteristic, complemented
8C2×#14556067—21no—
9#17986276—21no—
10#17986191—21no—
11C8×D284C2211yescharacteristic
12C4⋊(C71⋊C16)C2211yescharacteristic
13(C2×D71)⋊C16—21no—
14#17985752C2211yescharacteristic
15#17985752C2211yescharacteristic
16#17985745—21no—
17#17985743—21no—
18C2×SD32×C71—21no—
19C2×SD32×C71—21no—
20#494×C71C2211yescharacteristic
21#493×C71C2211yescharacteristic
22C4×C16×C71C2211yescharacteristic
23C2×#11765850—41no—
24C2×C8×D71—21no—
25C4×D284D411yescharacteristic
26#14556067—42no—
27C2×C71⋊C16—21no—
28#14556061—41no—
29C4⋊(C71⋊C8)—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)D411yescharacteristic
32#14556049—41no—
33#14556048—41no—
34(C2×D71)⋊C8—21no—
35C2×Q16×C71—21no—
36C2×Q16×C71—21no—
37C2×D8×C71—21no—
38C2×D8×C71—21no—
39C4⋊Q8×C71—21no—
40#127×C71—21no—
41SD32×C71—42no—
42SD32×C71—42no—
43C2×C16×C71D411yescharacteristic
44C2×C16×C71D411yescharacteristic
45C4×C8×C71C2311yescharacteristic
46C2×D284D811yescharacteristic
47C2×C4×D71—21no—
48#11765850—82no—
49C2×C71⋊C8—21no—
50C2×C71⋊C8—41no—
51C2×C71⋊C8—41no—
52C8×D71—41no—
53C8×D71—41no—
54C4⋊Dic71D811yescharacteristic
55(C2×D71)⋊C4—21no—
56C71⋊C16—41no—
57C2×Q8×C71—41no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61Q16×C71—41no—
62Q16×C71—41no—
63Q16×C71—81no—
64D8×C71—41no—
65D8×C71—41no—
66D8×C71—81no—
67C2×C8×C71C2×D411yescharacteristic
68C2×C8×C71#4111yescharacteristic
69C4⋊C4×C71—41no—
70C4⋊C4×C71—41no—
71C42×C71C2×D411yescharacteristic
72C16×C71—21no—
73C16×C71D812yescomplemented
74C23×C71—81no—
75C22×D71—21no—
76Q8×C71—81no—
77Q8×C71—81no—
78D4×C71—81no—
79D4×C71—81no—
80D4×C71—81no—
81D4×C71—81no—
82D4×C71—161no—
83C2×C4×C71—81no—
84C2×C4×C71#12111yescharacteristic
85C2×C4×C71C2×D811yescharacteristic
86C2×C4×C71#13511yescharacteristic
87C2×Dic71—21no—
88D284—21no—
89D284—21no—
90C4×D71—41no—
91C4×D71—41no—
92C8×C71—21no—
93C8×C71C2×D811yescharacteristic
94C8×C71AGL(1,ℤ/8)11yescharacteristic
95C71⋊C8—41no—
96C71⋊C8—81no—
97C71⋊C8—81no—
98C22×C71—161no—
99C22×C71—161no—
100C22×C71—161no—
101C22×C71#46611yescharacteristic
102C2×D71—41no—
103C2×D71—41no—
104C2×D71—41no—
105C4×C71—21no—
106C4×C71—161no—
107C4×C71#45911yescharacteristic
108C4×C71#47211yescharacteristic
109C4×C71#50911yescharacteristic
110C4×C71#51011yescharacteristic
111Dic71—41no—
112#12435—711nomaximal, Sylow
113C2×C71—321no—
114C2×C71#166711yescharacteristic
115C2×C71#222011yescharacteristic
116C2×C71#222311yescharacteristic
117D71—81no—
118#2248D7111yescharacteristic, complemented
119#2169—711no—
120#1669—711no—
121#1667—711no—
122#1371—711no—
123#1362—711no—
124#1361—711no—
125C71#1243511yescharacteristic, complemented, Sylow
126C2×SD32—21no—
127C2×SD32—21no—
128C2×M32—1421no—
129#494C2×D7111yescharacteristic
130#493C2×D7111yescharacteristic
131#474—1421no—
132#459—1421no—
133C8×D4—711no—
134C4⋊C16—711no—
135C22⋊C16—1421no—
136C4×C16C2×D7111yescharacteristic
137#335—711no—
138#335—711no—
139#332—1421no—
140#331—1421no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—2841no—
146C22×C8—1421no—
147C4⋊Q8—21no—
148#127—21no—
149C4×D4—711no—
150SD32—42no—
151SD32—42no—
152M32—2842no—
153C2×C16—1421no—
154C2×C16C22⋊D7111yescharacteristic
155C2×C16C22⋊D7111yescharacteristic
156#107—2841no—
157C4⋊C8—711no—
158C4⋊C8—1421no—
159C4⋊C8—1421no—
160#102—2841no—
161#102—2841no—
162C22⋊C8—1421no—
163C4×C8C22×D7111yescharacteristic
164C2×Q8—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—711no—
169C22×C4—1421no—
170Q16—41no—
171Q16—41no—
172Q16—81no—
173D8—41no—
174D8—41no—
175D8—81no—
176D8—5682no—
177C2×C8—2841no—
178C2×C8—2841no—
179C2×C8—2841no—
180C2×C8—2841no—
181C2×C8—1421no—
182C2×C8C2×C22⋊D7111yescharacteristic
183C2×C8#1176586411yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—711no—
187C22⋊C4—1421no—
188C42D4×D7111yescharacteristic
189C16—21no—
190C16—2841no—
191C16#1176585012yescomplemented
192C23—81no—
193C23—1421no—
194Q8—81no—
195Q8—81no—
196D4—81no—
197D4—81no—
198D4—81no—
199D4—81no—
200D4—161no—
201D4—1421no—
202D4—1421no—
203C2×C4—81no—
204C2×C4—2841no—
205C2×C4—2841no—
206C2×C4—1421no—
207C2×C4#1455609911yescharacteristic
208C2×C4D8×D7111yescharacteristic
209C2×C4#1455614711yescharacteristic
210C8—21no—
211C8—2841no—
212C8—5681no—
213C8—5681no—
214C8C2×#1176585011yescharacteristic
215C8#1455615411yescharacteristic
216C22—161no—
217C22—161no—
218C22—161no—
219C22—2841no—
220C22—2841no—
221C22—2841no—
222C22#1798623211yescharacteristic
223C4—21no—
224C4—161no—
225C4—2841no—
226C4#1798619111yescharacteristic
227C4#1798626611yescharacteristic
228C4#1798643611yescharacteristic
229C4#1798643811yescharacteristic
230C2—321no—
231C2—5681no—
232C2#2258769211yescharacteristic
233C2#2259045011yescharacteristic
234C2#2259046711yescharacteristic
subgroupG/H
centerC22#17986232
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2248C2
socleC22×C71#466
prime radical#22587692C2
hypercenter#2248D71
hypocenterC71#12435
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #22590173C2semidirect
2. #22587702C2semidirect
4. #22586412C2semidirect
5. #22586389C2semidirect
6. #22586386C2semidirect
7. C71×#2248C2semidirect
73. C16×C71D8semidirect
118. #2248D71semidirect
125. C71#12435semidirect
191. C16#11765850semidirect

Maximal subgroups

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

subgroupindex
#225901732
#225877022
#225876922
#225864122
#225863892
#225863862
C71×#22482
#1243571