Finite Group Databasebeta

← #31911595

Subgroups — #31911595

Order 18176.

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

normal subgroups only

This group has 234 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#22590488C211yesmaximal, characteristic
2#22590382C211yesmaximal, characteristic
3#22587934C211yesmaximal, characteristic, complemented
4#22587134C211yesmaximal, characteristic
5#22586403C211yesmaximal, characteristic, complemented
6#22586399C211yesmaximal, characteristic, complemented
7C71×#1362C211yesmaximal, characteristic, complemented
8C2×#14556071—21no—
9#17986404C2211yescharacteristic
10#17986395—21no—
11#17986351—21no—
12C4×D568C2211yescharacteristic
13#17986064—21no—
14#17985827—21no—
15C4⋊(C71⋊C16)C2211yescharacteristic
16#17985803—21no—
17#17985801—21no—
18#17985726—21no—
19C8×C71⋊C8C2211yescharacteristic
20#494×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×D568D411yescharacteristic
24C2×C8×D71—41no—
25#14556128—21no—
26C4×D284—21no—
27#14556071—42no—
28C2×C71⋊C16—21no—
29#14556062—21no—
30#14556062—21no—
31#14556062D411yescharacteristic
32#14556059—41no—
33#14556059—41no—
34#14556051—42no—
35#14556050—21no—
36(C2×D71)⋊C8—41no—
37C4×C71⋊C8D411yescharacteristic
38C4×C71⋊C8D411yescharacteristic
39C2×Q16×C71—21no—
40C2×Q16×C71—21no—
41C4⋊Q8×C71—21no—
42C2×C16×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46C2×D284—21no—
47C2×C4×D71—41no—
48D568—21no—
49D568SD1612yes—
50C2×C71⋊C8—21no—
51C2×C71⋊C8D811yescharacteristic
52C2×C71⋊C8D811yescharacteristic
53C8×D71—82no—
54C4⋊Dic71—21no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57(C2×D71)⋊C4—41no—
58C71⋊C16—41no—
59C2×Q8×C71—41no—
60Q16×C71—41no—
61Q16×C71—41no—
62Q16×C71—81no—
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—41no—
71Q8×C71—81no—
72Q8×C71—81no—
73C2×C4×C71—81no—
74C2×C4×C71#12111yescharacteristic
75C2×C4×C71C2×D811yescharacteristic
76C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
77C2×Dic71—41no—
78C2×Dic71—81no—
79D284—22no—
80D284—41no—
81C4×D71—81no—
82C4×D71—81no—
83Dic142—82no—
84C8×C71—21no—
85C8×C71—81no—
86C8×C71C2×SD1611yescharacteristic
87C8×C71#13511yescharacteristic
88C71⋊C8—21no—
89C71⋊C8—41no—
90C71⋊C8SD3212yescomplemented
91C22×C71#45911yescharacteristic
92C2×D71—42no—
93C2×D71—81no—
94C2×D71—81no—
95C4×C71—21no—
96C4×C71—161no—
97C4×C71#46511yescharacteristic
98C4×C71#47111yescharacteristic
99C4×C71C2×SD3211yescharacteristic
100C4×C71#51011yescharacteristic
101Dic71—81no—
102Dic71—161no—
103#12420—711nomaximal, Sylow
104C2×C71#166511yescharacteristic
105C2×C71#220811yescharacteristic
106C2×C71#221211yescharacteristic
107D71—82no—
108#2226—711no—
109#2208—711no—
110#1707—711no—
111C8×D8—711no—
112#1367—711no—
113#1365—711no—
114#1362D7111yescharacteristic, complemented
115C71#1242011yescharacteristic, complemented, Sylow
116C2×SD32—1421no—
117#500—711no—
118#498—1421no—
119#494C2×D7111yescharacteristic
120#487—1421no—
121C4×D8—711no—
122C8×D4—1421no—
123#367—1421no—
124C4⋊C16—711no—
125C4⋊C16C2×D7111yescharacteristic
126#358—1421no—
127#357—1421no—
128#335C2×D7111yescharacteristic
129D4⋊C8—1421no—
130C82—711no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×D8—711no—
134C22×C8—2841no—
135C4⋊Q8—21no—
136C4⋊Q8—1421no—
137C4×D4—1421no—
138SD32—2842no—
139C2×C16—21no—
140C2×C16—1421no—
141#107—711no—
142#107—1421no—
143#107—1421no—
144#106—2841no—
145#106—2841no—
146C4⋊C8—21no—
147#103—2842no—
148#102—1421no—
149C22⋊C8—2841no—
150C4×C8—711no—
151C4×C8—711no—
152C4×C8C22×D7111yescharacteristic
153C2×Q8—41no—
154C2×Q8—2841no—
155C2×D4—1421no—
156C22×C4—2841no—
157Q16—41no—
158Q16—41no—
159Q16—81no—
160D8—712no—
161D8—1421no—
162C2×C8—41no—
163C2×C8—5682no—
164C2×C8—711no—
165C2×C8—711no—
166C2×C8—1421no—
167C2×C8D4×D7111yescharacteristic
168C2×C8#1176586511yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—2841no—
172C4⋊C4—2841no—
173C4⋊C4—1421no—
174C22⋊C4—2841no—
175C42D4×D7111yescharacteristic
176C16—41no—
177C16—2841no—
178C23—2841no—
179Q8—81no—
180Q8—81no—
181Q8—5682no—
182D4—2841no—
183D4—1422no—
184C2×C4—81no—
185C2×C4—2841no—
186C2×C4—5681no—
187C2×C4—5681no—
188C2×C4—5681no—
189C2×C4#1455610111yescharacteristic
190C2×C4D8×D7111yescharacteristic
191C2×C4#1455615511yescharacteristic
192C8—21no—
193C8—81no—
194C8—2841no—
195C8—712no—
196C8—1421no—
197C8SD16×D7111yescharacteristic
198C8#1455614711yescharacteristic
199C22—2842no—
200C22—5681no—
201C22—5681no—
202C22#1798619311yescharacteristic
203C4—21no—
204C4—161no—
205C4—5681no—
206C4—11361no—
207C4#1798622711yescharacteristic
208C4#1798626111yescharacteristic
209C4SD32×D7111yescharacteristic
210C4#1798644111yescharacteristic
211C2—5682no—
212C2#2258768011yescharacteristic
213C2#2259038411yescharacteristic
214C2#2259041211yescharacteristic
subgroupG/H
centerC22#17986193
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1362C2
socleC22×C71#459
prime radicalC2×D568D4
hypercenter#1362D71
hypocenterC71#12420
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
3. #22587934C2semidirect
5. #22586403C2semidirect
6. #22586399C2semidirect
7. C71×#1362C2semidirect
90. C71⋊C8SD32semidirect
114. #1362D71semidirect
115. C71#12420semidirect

Maximal subgroups

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

subgroupindex
#225904882
#225903822
#225879342
#225871342
#225864032
#225863992
C71×#13622
#1242071