Finite Group Databasebeta

← #31911592

Subgroups — #31911592

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#22590489C211yesmaximal, characteristic, complemented
2#22590381C211yesmaximal, characteristic
3#22587933C211yesmaximal, characteristic
4#22587146C211yesmaximal, characteristic, complemented
5#22586403C211yesmaximal, characteristic, complemented
6#22586385C211yesmaximal, characteristic
7C71×#1365C211yesmaximal, characteristic, complemented
8C2×#14556073—21no—
9#17986376C2211yescharacteristic
10#17986373—21no—
11#17986347—21no—
12#17986123—21no—
13C4×Dic284C2211yescharacteristic
14#17985824—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—42no—
25C2×D568—21no—
26C2×D568—21no—
27#14556125—21no—
28C4×Dic142—21no—
29#14556073—42no—
30C2×C71⋊C16—21no—
31#14556062D411yescharacteristic
32Dic71⋊C8—41no—
33#14556053—21no—
34#14556048—41no—
35#14556048—41no—
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—84no—
52D568—41no—
53D568—41no—
54D568—81no—
55C2×C71⋊C8—21no—
56C2×C71⋊C8D811yescharacteristic
57C2×C71⋊C8D811yescharacteristic
58C4⋊Dic71—21no—
59Dic71⋊C4—41no—
60C4×Dic71—41no—
61C71⋊C16—41no—
62C2×Q8×C71—41no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71#4111yescharacteristic
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C42×C71C2×D411yescharacteristic
69C16×C71—41no—
70C22×D71—81no—
71Q8×C71—82no—
72C2×C4×C71—81no—
73C2×C4×C71#12111yescharacteristic
74C2×C4×C71C2×D811yescharacteristic
75C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
76C2×Dic71—41no—
77C2×Dic71—41no—
78D284—82no—
79D284—81no—
80D284—81no—
81D284—81no—
82D284—81no—
83Dic142—22no—
84Dic142—41no—
85C8×C71—21no—
86C8×C71—81no—
87C8×C71C2×SD1611yescharacteristic
88C8×C71#13511yescharacteristic
89C71⋊C8—22no—
90C71⋊C8—21no—
91C71⋊C8—21no—
92C22×C71#45911yescharacteristic
93C2×D71—161no—
94C2×D71—161no—
95C2×D71—161no—
96C4×C71—21no—
97C4×C71—161no—
98C4×C71#46511yescharacteristic
99C4×C71#47111yescharacteristic
100C4×C71#50811yescharacteristic
101C4×C71#51011yescharacteristic
102Dic71—42no—
103Dic71—81no—
104#12419—711nomaximal, Sylow
105C2×C71#166511yescharacteristic
106C2×C71#221111yescharacteristic
107C2×C71#221311yescharacteristic
108D71—321no—
109#2227—711no—
110#2207—711no—
111#1706—711no—
112C8×Q16—711no—
113#1367—711no—
114#1365D7111yescharacteristic, complemented
115#1361—711no—
116C71#1241911yescharacteristic, complemented, Sylow
117C2×SD32—1421no—
118#500C2×D7111yescharacteristic
119#493—711no—
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—1421no—
135C2×D8—1421no—
136C4⋊Q8—21no—
137#127—1421no—
138C4×Q8—1421no—
139SD32—2842no—
140C2×C16—21no—
141C2×C16—1421no—
142#107—21no—
143#107—21no—
144#107—711no—
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—41no—
155C2×Q8—1421no—
156C2×D4—2841no—
157C2×D4—2841no—
158C2×D4—2841no—
159Q16—712no—
160Q16—1421no—
161SD16—5684no—
162D8—2841no—
163D8—2841no—
164D8—5681no—
165C2×C8—41no—
166C2×C8—711no—
167C2×C8—711no—
168C2×C8—1421no—
169C2×C8D4×D7111yescharacteristic
170C2×C8#1176586511yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—2841no—
174C4⋊C4—1421no—
175C42—2841no—
176C42D4×D7111yescharacteristic
177C16—41no—
178C16—2841no—
179C23—5681no—
180Q8—82no—
181Q8—2841no—
182Q8—1422no—
183D4—5682no—
184D4—5681no—
185D4—5681no—
186D4—5681no—
187D4—5681no—
188C2×C4—81no—
189C2×C4—2841no—
190C2×C4—2841no—
191C2×C4#1455610111yescharacteristic
192C2×C4D8×D7111yescharacteristic
193C2×C4#1455615511yescharacteristic
194C8—21no—
195C8—81no—
196C8—1422no—
197C8—1421no—
198C8—1421no—
199C8SD16×D7111yescharacteristic
200C8#1455615111yescharacteristic
201C22—11361no—
202C22—11361no—
203C22—11361no—
204C22#1798619311yescharacteristic
205C4—21no—
206C4—161no—
207C4—2842no—
208C4—5681no—
209C4#1798623111yescharacteristic
210C4#1798626511yescharacteristic
211C4#1798643011yescharacteristic
212C4#1798644111yescharacteristic
213C2—22721no—
214C2#2258768411yescharacteristic
215C2#2259040511yescharacteristic
216C2#2259041511yescharacteristic
subgroupG/H
centerC22#17986193
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1365C2
socleC22×C71#459
prime radical#17986376C22
hypercenter#1365D71
hypocenterC71#12419
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
1. #22590489C2semidirect
4. #22587146C2semidirect
5. #22586403C2semidirect
7. C71×#1365C2semidirect
50. Dic284SD16semidirect
114. #1365D71semidirect
116. C71#12419semidirect

Maximal subgroups

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

subgroupindex
#225904892
#225903812
#225879332
#225871462
#225864032
#225863852
C71×#13652
#1241971