Finite Group Databasebeta

← #31911613

Subgroups — #31911613

Order 18176.

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

normal subgroups only

This group has 234 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#22590474C211yesmaximal, characteristic
2#22590386C211yesmaximal, characteristic
3#22587938C211yesmaximal, characteristic, complemented
4#22587134C211yesmaximal, characteristic
5#22586403C211yesmaximal, characteristic, complemented
6#22586388C211yesmaximal, characteristic, complemented
7C71×#1364C211yesmaximal, characteristic, complemented
8C2×#14556073—21no—
9#17986395—21no—
10#17986379C2211yescharacteristic
11#17986349—21no—
12C4×D568C2211yescharacteristic
13#17986064—21no—
14#17985824—21no—
15C4⋊(C71⋊C16)C2211yescharacteristic
16#17985804—21no—
17#17985799—21no—
18#17985726—21no—
19C8×C71⋊C8C2211yescharacteristic
20#500×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284—21no—
24C2×Dic284—21no—
25C2×D568D411yescharacteristic
26C2×C8×D71—41no—
27#14556128—21no—
28C4×D284—21no—
29#14556073—42no—
30C2×C71⋊C16—21no—
31#14556062D411yescharacteristic
32#14556059—42no—
33#14556052—41no—
34#14556052—41no—
35#14556050—21no—
36(C2×D71)⋊C8—41no—
37C4×C71⋊C8D411yescharacteristic
38C4×C71⋊C8D411yescharacteristic
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—41no—
46C2×D284—21no—
47C2×C4×D71—41no—
48Dic284—41no—
49Dic284—41no—
50Dic284—81no—
51D568—21no—
52D568SD1612yes—
53C2×C71⋊C8—21no—
54C2×C71⋊C8D811yescharacteristic
55C2×C71⋊C8D811yescharacteristic
56C8×D71—82no—
57C4⋊Dic71—21no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60(C2×D71)⋊C4—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—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—81no—
84Dic142—81no—
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—42no—
94C2×D71—81no—
95C2×D71—81no—
96C4×C71—21no—
97C4×C71—161no—
98C4×C71#46511yescharacteristic
99C4×C71#47111yescharacteristic
100C4×C71#50811yescharacteristic
101C4×C71#50911yescharacteristic
102Dic71—81no—
103Dic71—161no—
104#12422—711nomaximal, Sylow
105C2×C71#166511yescharacteristic
106C2×C71#221011yescharacteristic
107C2×C71#221411yescharacteristic
108D71—82no—
109#2224—711no—
110#2208—711no—
111#1707—711no—
112C8×D8—711no—
113#1367—711no—
114#1364D7111yescharacteristic, complemented
115#1362—711no—
116C71#1242211yescharacteristic, complemented, Sylow
117C2×SD32—1421no—
118#500C2×D7111yescharacteristic
119#498—1421no—
120#494—711no—
121#487—1421no—
122C4×D8—711no—
123C8×D4—1421no—
124#366—1421no—
125C4⋊C16—711no—
126C4⋊C16C2×D7111yescharacteristic
127#358—1421no—
128#357—1421no—
129#335C2×D7111yescharacteristic
130D4⋊C8—1421no—
131C82—711no—
132C2×Q16—1421no—
133C2×Q16—1421no—
134C2×D8—711no—
135C22×C8—2841no—
136C4⋊Q8—21no—
137C4⋊Q8—1421no—
138C4×D4—1421no—
139SD32—2842no—
140C2×C16—21no—
141C2×C16—1421no—
142#107—21no—
143#107—21no—
144#107—711no—
145#106—2842no—
146C4⋊C8—21no—
147#103—2841no—
148#103—2841no—
149#102—1421no—
150C22⋊C8—2841no—
151C4×C8—711no—
152C4×C8—711no—
153C4×C8C22×D7111yescharacteristic
154C2×Q8—41no—
155C2×Q8—2841no—
156C2×D4—1421no—
157C22×C4—2841no—
158Q16—2841no—
159Q16—2841no—
160Q16—5681no—
161D8—712no—
162D8—1421no—
163C2×C8—41no—
164C2×C8—5682no—
165C2×C8—711no—
166C2×C8—711no—
167C2×C8—1421no—
168C2×C8D4×D7111yescharacteristic
169C2×C8#1176586511yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—2841no—
173C4⋊C4—2841no—
174C4⋊C4—1421no—
175C22⋊C4—2841no—
176C42D4×D7111yescharacteristic
177C16—41no—
178C16—2841no—
179C23—2841no—
180Q8—81no—
181Q8—81no—
182Q8—5681no—
183Q8—5681no—
184D4—2841no—
185D4—1422no—
186C2×C4—81no—
187C2×C4—2841no—
188C2×C4—5681no—
189C2×C4—5681no—
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#1455615111yescharacteristic
201C22—2842no—
202C22—5681no—
203C22—5681no—
204C22#1798619311yescharacteristic
205C4—21no—
206C4—161no—
207C4—5681no—
208C4—11361no—
209C4#1798623111yescharacteristic
210C4#1798626511yescharacteristic
211C4#1798643011yescharacteristic
212C4#1798643311yescharacteristic
213C2—5682no—
214C2#2258768411yescharacteristic
215C2#2259039411yescharacteristic
216C2#2259042011yescharacteristic
subgroupG/H
centerC22#17986193
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1364C2
socleC22×C71#459
prime radicalC2×D568D4
hypercenter#1364D71
hypocenterC71#12422
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
3. #22587938C2semidirect
5. #22586403C2semidirect
6. #22586388C2semidirect
7. C71×#1364C2semidirect
114. #1364D71semidirect
116. C71#12422semidirect

Maximal subgroups

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

subgroupindex
#225904742
#225903862
#225879382
#225871342
#225864032
#225863882
C71×#13642
#1242271