Finite Group Databasebeta

← #31911594

Subgroups — #31911594

Order 18176.

234 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 69 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#22590487C211yesmaximal, characteristic
2#22587934C211yesmaximal, characteristic, complemented
3D8×C71⋊C8C211yesmaximal, characteristic
4#22586403C211yesmaximal, characteristic, complemented
5#22586397C211yesmaximal, characteristic, complemented
6#22586389C211yesmaximal, characteristic, complemented
7C71×#2208C211yesmaximal, characteristic
8#17986404C2211yescharacteristic
9#17986395—21no—
10#17986351—21no—
11D4×C71⋊C8—21no—
12#17985826—21no—
13C4⋊(C71⋊C16)C2211yescharacteristic
14#17985800—21no—
15#17985752C2211yescharacteristic
16D4⋊(C71⋊C8)—21no—
17C8×C71⋊C8C2211yescharacteristic
18C2×SD32×C71—21no—
19#494×C71C2211yescharacteristic
20C4×D8×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#358×C71—21no—
23C22×C71⋊C8—41no—
24#14556128—21no—
25C2×C71⋊C16—21no—
26#14556062—21no—
27#14556062—21no—
28#14556059—41no—
29#14556059—41no—
30C4⋊(C71⋊C8)—21no—
31#14556051—42no—
32C22⋊(C71⋊C8)—41no—
33C4×C71⋊C8D411yescharacteristic
34C4×C71⋊C8D411yescharacteristic
35C2×Q16×C71—21no—
36C2×Q16×C71—21no—
37C2×D8×C71D411yescharacteristic
38C4⋊Q8×C71—21no—
39C4×D4×C71—21no—
40SD32×C71—42no—
41C2×C16×C71—21no—
42#107×C71D411yescharacteristic
43#102×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46C2×C71⋊C8—21no—
47C2×C71⋊C8—41no—
48C2×C71⋊C8—82no—
49C2×C71⋊C8D811yescharacteristic
50C2×C71⋊C8D811yescharacteristic
51C4⋊Dic71—41no—
52C4⋊Dic71—41no—
53C71⋊C16—41no—
54C2×Q8×C71—41no—
55C2×D4×C71—21no—
56C22×C4×C71—41no—
57Q16×C71—41no—
58Q16×C71—41no—
59Q16×C71—81no—
60D8×C71—21no—
61D8×C71SD1612yes—
62C2×C8×C71C2×D411yescharacteristic
63C2×C8×C71#4111yescharacteristic
64C4⋊C4×C71—21no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C22⋊C4×C71—41no—
68C42×C71C2×D411yescharacteristic
69C16×C71—41no—
70C23×C71—41no—
71Q8×C71—81no—
72Q8×C71—81no—
73D4×C71—22no—
74D4×C71—41no—
75C2×C4×C71—41no—
76C2×C4×C71—81no—
77C2×C4×C71—81no—
78C2×C4×C71—81no—
79C2×C4×C71#12111yescharacteristic
80C2×C4×C71C2×D811yescharacteristic
81C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
82C2×Dic71—81no—
83Dic142—82no—
84C8×C71—21no—
85C8×C71C2×SD1611yescharacteristic
86C8×C71#13511yescharacteristic
87C71⋊C8—21no—
88C71⋊C8—41no—
89C71⋊C8—81no—
90C71⋊C8SD3212yescomplemented
91C22×C71—42no—
92C22×C71—81no—
93C22×C71—81no—
94C22×C71#45911yescharacteristic
95C4×C71—21no—
96C4×C71—81no—
97C4×C71—161no—
98C4×C71#46511yescharacteristic
99C4×C71#47111yescharacteristic
100C4×C71C2×SD3211yescharacteristic
101C4×C71#51011yescharacteristic
102Dic71—161no—
103#12420—711nomaximal, Sylow
104C2×C71—82no—
105C2×C71#166511yescharacteristic
106C2×C71#220811yescharacteristic
107C2×C71#221211yescharacteristic
108#2226—711no—
109#2208D7111yescharacteristic
110#1707—711no—
111C8×D8—711no—
112#1367—711no—
113#1365—711no—
114#1362—711no—
115C71#1242011yescharacteristic, complemented, Sylow
116C2×SD32—21no—
117#500—711no—
118#498—1421no—
119#494C2×D7111yescharacteristic
120#487—1421no—
121C4×D8C2×D7111yescharacteristic
122C8×D4—1421no—
123#367—1421no—
124C4⋊C16—711no—
125C4⋊C16C2×D7111yescharacteristic
126#358—21no—
127#357—1421no—
128#335—711no—
129D4⋊C8—1421no—
130C82—711no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×D8C22⋊D7111yescharacteristic
134C22×C8—2841no—
135C4⋊Q8—21no—
136C4⋊Q8—1421no—
137C4×D4—21no—
138SD32—42no—
139C2×C16—21no—
140C2×C16—1421no—
141#107—1421no—
142#107—1421no—
143#107C22⋊D7111yescharacteristic
144#106—2841no—
145#106—2841no—
146C4⋊C8—1421no—
147#103—2842no—
148#102—21no—
149C22⋊C8—2841no—
150C4×C8—711no—
151C4×C8—711no—
152C4×C8C22×D7111yescharacteristic
153C2×Q8—41no—
154C2×Q8—2841no—
155C2×D4—21no—
156C22×C4—41no—
157Q16—41no—
158Q16—41no—
159Q16—81no—
160D8—21no—
161D8#1176585312yes—
162C2×C8—2841no—
163C2×C8—5682no—
164C2×C8—711no—
165C2×C8—711no—
166C2×C8—1421no—
167C2×C8C2×C22⋊D7111yescharacteristic
168C2×C8#1176586411yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—2841no—
173C4⋊C4—2841no—
174C22⋊C4—41no—
175C42D4×D7111yescharacteristic
176C16—41no—
177C16—2841no—
178C23—41no—
179Q8—81no—
180Q8—81no—
181Q8—5682no—
182D4—22no—
183D4—41no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—5681no—
189C2×C4#1455610011yescharacteristic
190C2×C4D8×D7111yescharacteristic
191C2×C4#1455615211yescharacteristic
192C8—21no—
193C8—2841no—
194C8—5681no—
195C8—712no—
196C8—1421no—
197C8C2×#1176585311yescharacteristic
198C8#1455614811yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22#1798619211yescharacteristic
203C4—21no—
204C4—81no—
205C4—161no—
206C4—11361no—
207C4#1798622611yescharacteristic
208C4#1798626011yescharacteristic
209C4SD32×D7111yescharacteristic
210C4#1798643811yescharacteristic
211C2—82no—
212C2#2258767911yescharacteristic
213C2#2259038311yescharacteristic
214C2#2259041111yescharacteristic
subgroupG/H
centerC22#17986192
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2208C2
socleC22×C71#459
prime radicalC2×D8×C71D4
hypercenter#2208D71
hypocenterC71#12420
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
2. #22587934C2semidirect
4. #22586403C2semidirect
5. #22586397C2semidirect
6. #22586389C2semidirect
90. C71⋊C8SD32semidirect
115. C71#12420semidirect

Maximal subgroups

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

subgroupindex
#225904872
#225879342
D8×C71⋊C82
#225864032
#225863972
#225863892
C71×#22082
#1242071