Finite Group Databasebeta

← #31911588

Subgroups — #31911588

Order 18176.

240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 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#22590493C211yesmaximal, characteristic, complemented
2#22590377C211yesmaximal, characteristic
3#22587929C211yesmaximal, characteristic
4#22587146C211yesmaximal, characteristic, complemented
5#22586403C211yesmaximal, characteristic, complemented
6#22586399C211yesmaximal, characteristic, complemented
7C71×#1361C211yesmaximal, characteristic
8C2×#14556072—21no—
9#17986404C2211yescharacteristic
10#17986371—21no—
11#17986345—21no—
12#17986123—21no—
13C4×Dic284C2211yescharacteristic
14#17985825—21no—
15C4⋊(C71⋊C16)C2211yescharacteristic
16#17985804—21no—
17#17985800—21no—
18#17985729—21no—
19C8×C71⋊C8C2211yescharacteristic
20#493×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284D411yescharacteristic
24C2×#11765853—42no—
25#14556128—21no—
26C4×Dic142—21no—
27#14556072—42no—
28C2×C71⋊C16—21no—
29#14556062—21no—
30#14556062—21no—
31#14556062D411yescharacteristic
32Dic71⋊C8—41no—
33#14556053—21no—
34#14556049—41no—
35#14556049—41no—
36C4×C71⋊C8D411yescharacteristic
37C4×C71⋊C8D411yescharacteristic
38C8×Dic71—41no—
39C2×D8×C71—21no—
40C2×D8×C71—21no—
41#127×C71—21no—
42C2×C16×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—21no—
46C2×Dic142—41no—
47Dic284—21no—
48Dic284SD1612yescomplemented
49#11765853—84no—
50C2×C71⋊C8—21no—
51C2×C71⋊C8D811yescharacteristic
52C2×C71⋊C8D811yescharacteristic
53C4⋊Dic71—21no—
54C4⋊Dic71—41no—
55C4⋊Dic71—41no—
56Dic71⋊C4—41no—
57C4×Dic71—41no—
58C71⋊C16—41no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61C2×D4×C71—41no—
62D8×C71—41no—
63D8×C71—41no—
64D8×C71—81no—
65C2×C8×C71—41no—
66C2×C8×C71C2×D411yescharacteristic
67C2×C8×C71#4111yescharacteristic
68C42×C71C2×D411yescharacteristic
69C16×C71—41no—
70C23×C71—81no—
71D4×C71—82no—
72D4×C71—81no—
73D4×C71—81no—
74D4×C71—81no—
75D4×C71—81no—
76C2×C4×C71#12111yescharacteristic
77C2×C4×C71C2×D811yescharacteristic
78C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
79C2×Dic71—41no—
80C2×Dic71—41no—
81C2×Dic71—81no—
82Dic142—22no—
83Dic142—41no—
84Dic142—82no—
85C8×C71—21no—
86C8×C71—81no—
87C8×C71C2×SD1611yescharacteristic
88C8×C71#13511yescharacteristic
89C71⋊C8—22no—
90C71⋊C8—21no—
91C71⋊C8—21no—
92C22×C71—161no—
93C22×C71—161no—
94C22×C71—161no—
95C22×C71#45911yescharacteristic
96C4×C71—21no—
97C4×C71#46511yescharacteristic
98C4×C71#47111yescharacteristic
99C4×C71#50811yescharacteristic
100C4×C71#51011yescharacteristic
101Dic71—42no—
102Dic71—81no—
103Dic71—161no—
104#12419—711nomaximal, Sylow
105C2×C71—321no—
106C2×C71#166511yescharacteristic
107C2×C71#221111yescharacteristic
108C2×C71#221311yescharacteristic
109#2227—711no—
110#2207—711no—
111#1706—711no—
112C8×Q16—711no—
113#1367—711no—
114#1365—711no—
115#1361D7111yescharacteristic
116C71#1241911yescharacteristic, complemented, Sylow
117C2×SD32—1421no—
118#500—711no—
119#493C2×D7111yescharacteristic
120#492—1421no—
121#486—1421no—
122C8×Q8—1421no—
123C4×Q16—711no—
124#366—1421no—
125C4⋊C16—711no—
126C4⋊C16C2×D7111yescharacteristic
127#358—1421no—
128#357—1421no—
129#335C2×D7111yescharacteristic
130Q8⋊C8—1421no—
131C82—711no—
132C2×Q16—711no—
133C2×SD16—2842no—
134C2×D8—21no—
135C2×D8—21no—
136C4⋊Q8—1421no—
137#127—21no—
138C4×Q8—1421no—
139SD32—2842no—
140C2×C16—21no—
141C2×C16—1421no—
142#107—711no—
143#107—1421no—
144#107—1421no—
145C4⋊C8—21no—
146C4⋊C8—2841no—
147#103—1421no—
148#102—2841no—
149#102—2841no—
150C4×C8—2841no—
151C4×C8—711no—
152C4×C8—711no—
153C4×C8C22×D7111yescharacteristic
154C2×Q8—2841no—
155C2×Q8—1421no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159Q16—712no—
160Q16—1421no—
161SD16—5684no—
162D8—41no—
163D8—41no—
164D8—81no—
165C2×C8—41no—
166C2×C8—711no—
167C2×C8—711no—
168C2×C8—1421no—
169C2×C8D4×D7111yescharacteristic
170C2×C8#1176586511yescharacteristic
171C4⋊C4—2841no—
172C4⋊C4—2841no—
173C4⋊C4—2841no—
174C4⋊C4—1421no—
175C42—2841no—
176C42D4×D7111yescharacteristic
177C16—41no—
178C16—2841no—
179C23—81no—
180Q8—2841no—
181Q8—5682no—
182Q8—1422no—
183D4—82no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188C2×C4—2841no—
189C2×C4—2841no—
190C2×C4—5681no—
191C2×C4#1455610111yescharacteristic
192C2×C4D8×D7111yescharacteristic
193C2×C4#1455615511yescharacteristic
194C8—21no—
195C8—81no—
196C8—1422no—
197C8—1421no—
198C8—1421no—
199C8SD16×D7111yescharacteristic
200C8#1455614711yescharacteristic
201C22—161no—
202C22—161no—
203C22—161no—
204C22#1798619311yescharacteristic
205C4—21no—
206C4—2842no—
207C4—5681no—
208C4—11361no—
209C4#1798622711yescharacteristic
210C4#1798626111yescharacteristic
211C4#1798642811yescharacteristic
212C4#1798643811yescharacteristic
213C2—321no—
214C2#2258768011yescharacteristic
215C2#2259040111yescharacteristic
216C2#2259041511yescharacteristic
subgroupG/H
centerC22#17986193
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1361C2
socleC22×C71#459
prime radical#493×C71C22
hypercenter#1361D71
hypocenterC71#12419
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
1. #22590493C2semidirect
4. #22587146C2semidirect
5. #22586403C2semidirect
6. #22586399C2semidirect
48. Dic284SD16semidirect
116. C71#12419semidirect

Maximal subgroups

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

subgroupindex
#225904932
#225903772
#225879292
#225871462
#225864032
#225863992
C71×#13612
#1241971