Finite Group Databasebeta

← #31911722

Subgroups — #31911722

Order 18176.

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

normal subgroups only

This group has 232 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#22590597C211yesmaximal, characteristic
2#22590242C211yesmaximal, characteristic, complemented
3#22587707C211yesmaximal, characteristic, complemented
4#22587701C211yesmaximal, characteristic
5#22586413C211yesmaximal, characteristic, complemented
6#22586387C211yesmaximal, characteristic
7C71×#1362C211yesmaximal, characteristic, complemented
8C2×#14556071—21no—
9C2×#14556071—21no—
10#17986376C2211yescharacteristic
11#17986281—21no—
12#17986200—21no—
13C8×Dic142C2211yescharacteristic
14Dic71⋊C16—21no—
15C4×C71⋊C16C2211yescharacteristic
16C16×Dic71—21no—
17#17985752C2211yescharacteristic
18#17985745—21no—
19#17985742—21no—
20#494×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×#11765854—41no—
24C2×D568—21no—
25C2×D568—21no—
26#14556125—21no—
27C4×Dic142D411yescharacteristic
28#14556071—42no—
29#14556071—42no—
30C2×C71⋊C16D411yescharacteristic
31C2×C71⋊C16D411yescharacteristic
32#14556059—41no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)D411yescharacteristic
35Dic71⋊C8—21no—
36#14556051—41no—
37#14556050—41no—
38C8×Dic71—21no—
39C2×Q16×C71—21no—
40C2×Q16×C71—21no—
41C4⋊Q8×C71—21no—
42C2×C16×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142D811yescharacteristic
46C2×D284—41no—
47C2×D284—41no—
48C2×D284—41no—
49#11765854—82no—
50D568—41no—
51D568—41no—
52D568—81no—
53C2×C71⋊C8—21no—
54C2×C71⋊C8—41no—
55C4⋊Dic71D811yescharacteristic
56Dic71⋊C4—21no—
57C4×Dic71—21no—
58C71⋊C16—21no—
59C71⋊C16SD1612yescomplemented
60C2×Q8×C71—41no—
61Q16×C71—41no—
62Q16×C71—41no—
63Q16×C71—81no—
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×C71#13511yescharacteristic
78C2×Dic71—21no—
79C2×Dic71—21no—
80D284—81no—
81D284—81no—
82D284—81no—
83D284—81no—
84D284—161no—
85Dic142—21no—
86Dic142SD3212yescomplemented
87C8×C71—21no—
88C8×C71—81no—
89C8×C71C2×SD1611yescharacteristic
90C8×C71AGL(1,ℤ/8)11yescharacteristic
91C71⋊C8—41no—
92C71⋊C8—81no—
93C22×C71#46611yescharacteristic
94C2×D71—161no—
95C2×D71—161no—
96C2×D71—161no—
97C4×C71—21no—
98C4×C71—161no—
99C4×C71#46011yescharacteristic
100C4×C71#47211yescharacteristic
101C4×C71C2×SD3211yescharacteristic
102C4×C71#50811yescharacteristic
103Dic71—22no—
104Dic71—41no—
105#12441—711nomaximal, Sylow
106C2×C71#166811yescharacteristic
107C2×C71#221711yescharacteristic
108C2×C71#222311yescharacteristic
109D71—321no—
110#2248—711no—
111Q8×C16—711no—
112#1670—711no—
113#1668—711no—
114#1372—711no—
115#1362D7111yescharacteristic, complemented
116#1361—711no—
117C71#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1421no—
119C2×SD32—1421no—
120#494C2×D7111yescharacteristic
121#493—711no—
122#475—1421no—
123#460—1421no—
124C8×Q8—711no—
125C4⋊C16—1421no—
126C4⋊C16C2×D7111yescharacteristic
127C4×C16—711no—
128C4×C16—1421no—
129#335—711no—
130#335C2×D7111yescharacteristic
131#332—1421no—
132#331—1421no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2841no—
136C2×D8—1421no—
137C2×D8—1421no—
138C4⋊Q8—21no—
139#127—1421no—
140C4×Q8—711no—
141SD32—2842no—
142SD32—2842no—
143C2×C16—21no—
144C2×C16—711no—
145C2×C16—711no—
146#106—2841no—
147C4⋊C8—21no—
148C4⋊C8—711no—
149C4⋊C8—1421no—
150C4⋊C8—1421no—
151#103—2841no—
152#102—2841no—
153C4×C8—1421no—
154C4×C8C22×D7111yescharacteristic
155C2×Q8—41no—
156C2×Q8—711no—
157C2×D4—2841no—
158C2×D4—2841no—
159C2×D4—2841no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—5682no—
164D8—2841no—
165D8—2841no—
166D8—5681no—
167C2×C8—41no—
168C2×C8—2841no—
169C2×C8—1421no—
170C2×C8D4×D7111yescharacteristic
171C2×C8#1176586511yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—711no—
175C4⋊C4—1421no—
176C42—1421no—
177C42D4×D7111yescharacteristic
178C16—41no—
179C16—712no—
180C16—1421no—
181C23—5681no—
182Q8—81no—
183Q8—81no—
184Q8—712no—
185Q8—1421no—
186D4—5681no—
187D4—5681no—
188D4—5681no—
189D4—5681no—
190D4—11361no—
191C2×C4—81no—
192C2×C4—1421no—
193C2×C4—1421no—
194C2×C4#1455610111yescharacteristic
195C2×C4D8×D7111yescharacteristic
196C2×C4#1455614911yescharacteristic
197C8—21no—
198C8—81no—
199C8—2841no—
200C8—5681no—
201C8SD16×D7111yescharacteristic
202C8#1455615211yescharacteristic
203C22—11361no—
204C22—11361no—
205C22—11361no—
206C22#1798623411yescharacteristic
207C4—21no—
208C4—161no—
209C4—2841no—
210C4—1422no—
211C4#1798619811yescharacteristic
212C4#1798626811yescharacteristic
213C4SD32×D7111yescharacteristic
214C4#1798643211yescharacteristic
215C2—22721no—
216C2#2258769911yescharacteristic
217C2#2259043911yescharacteristic
218C2#2259047311yescharacteristic
subgroupG/H
centerC22#17986234
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1362C2
socleC22×C71#466
prime radical#17986376C22
hypercenter#1362D71
hypocenterC71#12441
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
2. #22590242C2semidirect
3. #22587707C2semidirect
5. #22586413C2semidirect
7. C71×#1362C2semidirect
59. C71⋊C16SD16semidirect
86. Dic142SD32semidirect
115. #1362D71semidirect
117. C71#12441semidirect

Maximal subgroups

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

subgroupindex
#225905972
#225902422
#225877072
#225877012
#225864132
#225863872
C71×#13622
#1244171