Finite Group Databasebeta

← #31911578

Subgroups — #31911578

Order 18176.

240 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 65 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#22590479C211yesmaximal, characteristic, complemented
2#22590381C211yesmaximal, characteristic
3#22587933C211yesmaximal, characteristic
4#22587146C211yesmaximal, characteristic, complemented
5#22586403C211yesmaximal, characteristic, complemented
6#22586385C211yesmaximal, characteristic
7C71×#1364C211yesmaximal, characteristic, complemented
8C2×#14556073—21no—
9#17986376C2211yescharacteristic
10#17986373—21no—
11#17986347—21no—
12#17986123—21no—
13C4×Dic284C2211yescharacteristic
14#17985826—21no—
15C4⋊(C71⋊C16)C2211yescharacteristic
16#17985802—21no—
17#17985801—21no—
18#17985729—21no—
19C8×C71⋊C8C2211yescharacteristic
20#500×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284D411yescharacteristic
24C2×#11765852—41no—
25C2×#11765852—41no—
26C2×D568—21no—
27C2×D568—21no—
28#14556125—21no—
29C4×Dic142—21no—
30#14556073—42no—
31C2×C71⋊C16—21no—
32#14556062D411yescharacteristic
33Dic71⋊C8—41no—
34#14556053—21no—
35#14556048—42no—
36C4×C71⋊C8D411yescharacteristic
37C4×C71⋊C8D411yescharacteristic
38C8×Dic71—41no—
39C4⋊Q8×C71—21no—
40C2×C16×C71—21no—
41#107×C71—21no—
42#107×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—21no—
46C2×D284—41no—
47C2×D284—41no—
48C2×D284—41no—
49Dic284—21no—
50Dic284SD1612yescomplemented
51#11765852—82no—
52#11765852—82no—
53D568—41no—
54D568—41no—
55D568—81no—
56C2×C71⋊C8—21no—
57C2×C71⋊C8D811yescharacteristic
58C2×C71⋊C8D811yescharacteristic
59C4⋊Dic71—21no—
60Dic71⋊C4—41no—
61C4×Dic71—41no—
62C71⋊C16—41no—
63C2×Q8×C71—41no—
64C2×C8×C71—41no—
65C2×C8×C71C2×D411yescharacteristic
66C2×C8×C71#4111yescharacteristic
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C42×C71C2×D411yescharacteristic
70C16×C71—41no—
71C22×D71—81no—
72Q8×C71—81no—
73Q8×C71—81no—
74C2×C4×C71—81no—
75C2×C4×C71#12111yescharacteristic
76C2×C4×C71C2×D811yescharacteristic
77C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
78C2×Dic71—41no—
79C2×Dic71—41no—
80D284—82no—
81D284—81no—
82D284—81no—
83D284—81no—
84D284—81no—
85Dic142—22no—
86Dic142—41no—
87C8×C71—21no—
88C8×C71—81no—
89C8×C71C2×SD1611yescharacteristic
90C8×C71#13511yescharacteristic
91C71⋊C8—21no—
92C71⋊C8—41no—
93C71⋊C8SD3212yescomplemented
94C22×C71#45911yescharacteristic
95C2×D71—161no—
96C2×D71—161no—
97C2×D71—161no—
98C4×C71—21no—
99C4×C71—161no—
100C4×C71#46511yescharacteristic
101C4×C71#47111yescharacteristic
102C4×C71C2×SD3211yescharacteristic
103C4×C71#50911yescharacteristic
104Dic71—42no—
105Dic71—81no—
106#12417—711nomaximal, Sylow
107C2×C71#166511yescharacteristic
108C2×C71#220711yescharacteristic
109C2×C71#221211yescharacteristic
110D71—321no—
111#2225—711no—
112#2207—711no—
113#1706—711no—
114C8×Q16—711no—
115#1367—711no—
116#1364D7111yescharacteristic, complemented
117#1361—711no—
118C71#1241711yescharacteristic, complemented, Sylow
119C2×SD32—1421no—
120#500C2×D7111yescharacteristic
121#493—711no—
122#492—1421no—
123#486—1421no—
124C8×Q8—1421no—
125C4×Q16—711no—
126#367—1421no—
127C4⋊C16—711no—
128C4⋊C16C2×D7111yescharacteristic
129#358—1421no—
130#357—1421no—
131#335C2×D7111yescharacteristic
132Q8⋊C8—1421no—
133C82—711no—
134C2×Q16—711no—
135C2×SD16—2841no—
136C2×SD16—2841no—
137C2×D8—1421no—
138C2×D8—1421no—
139C4⋊Q8—21no—
140#127—1421no—
141C4×Q8—1421no—
142SD32—2842no—
143C2×C16—21no—
144C2×C16—1421no—
145#107—21no—
146#107—21no—
147#107—711no—
148C4⋊C8—21no—
149C4⋊C8—2841no—
150#103—1421no—
151#102—2842no—
152C4×C8—2841no—
153C4×C8—711no—
154C4×C8—711no—
155C4×C8C22×D7111yescharacteristic
156C2×Q8—41no—
157C2×Q8—1421no—
158C2×D4—2841no—
159C2×D4—2841no—
160C2×D4—2841no—
161Q16—712no—
162Q16—1421no—
163SD16—5682no—
164SD16—5682no—
165D8—2841no—
166D8—2841no—
167D8—5681no—
168C2×C8—41no—
169C2×C8—711no—
170C2×C8—711no—
171C2×C8—1421no—
172C2×C8D4×D7111yescharacteristic
173C2×C8#1176586511yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2841no—
177C4⋊C4—1421no—
178C42—2841no—
179C42D4×D7111yescharacteristic
180C16—41no—
181C16—2841no—
182C23—5681no—
183Q8—81no—
184Q8—81no—
185Q8—2841no—
186Q8—1422no—
187D4—5682no—
188D4—5681no—
189D4—5681no—
190D4—5681no—
191D4—5681no—
192C2×C4—81no—
193C2×C4—2841no—
194C2×C4—2841no—
195C2×C4#1455610111yescharacteristic
196C2×C4D8×D7111yescharacteristic
197C2×C4#1455615511yescharacteristic
198C8—21no—
199C8—81no—
200C8—2841no—
201C8—712no—
202C8—1421no—
203C8SD16×D7111yescharacteristic
204C8#1455615111yescharacteristic
205C22—11361no—
206C22—11361no—
207C22—11361no—
208C22#1798619311yescharacteristic
209C4—21no—
210C4—161no—
211C4—2842no—
212C4—5681no—
213C4#1798623111yescharacteristic
214C4#1798626511yescharacteristic
215C4SD32×D7111yescharacteristic
216C4#1798643311yescharacteristic
217C2—22721no—
218C2#2258768411yescharacteristic
219C2#2259037911yescharacteristic
220C2#2259041211yescharacteristic
subgroupG/H
centerC22#17986193
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1364C2
socleC22×C71#459
prime radical#17986376C22
hypercenter#1364D71
hypocenterC71#12417
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
1. #22590479C2semidirect
4. #22587146C2semidirect
5. #22586403C2semidirect
7. C71×#1364C2semidirect
50. Dic284SD16semidirect
93. C71⋊C8SD32semidirect
116. #1364D71semidirect
118. C71#12417semidirect

Maximal subgroups

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

subgroupindex
#225904792
#225903812
#225879332
#225871462
#225864032
#225863852
C71×#13642
#1241771