Finite Group Databasebeta

← #31911734

Subgroups — #31911734

Order 18176.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#22590595C211yesmaximal, characteristic
2#22590249C211yesmaximal, characteristic, complemented
3#22587710C211yesmaximal, characteristic, complemented
4#22586411C211yesmaximal, characteristic, complemented
5#22586388C211yesmaximal, characteristic, complemented
6#22586386C211yesmaximal, characteristic
7C71×#1668C211yesmaximal, characteristic
8C2×#14556072—21no—
9C2×#14556072—21no—
10#17986379C2211yescharacteristic
11#17986284—21no—
12C4⋊(C71⋊C16)—21no—
13C4⋊(C71⋊C16)C2211yescharacteristic
14#17985775—21no—
15C4×C71⋊C16C2211yescharacteristic
16#17985752C2211yescharacteristic
17#17985746—21no—
18#493×C71C2211yescharacteristic
19#460×C71—21no—
20C8×Q8×C71C2211yescharacteristic
21#335×C71C2211yescharacteristic
22#331×C71—21no—
23C2×Dic284—21no—
24C2×Dic284—21no—
25#14556128—21no—
26#14556072—42no—
27#14556072—42no—
28C2×C71⋊C16—21no—
29C2×C71⋊C16D411yescharacteristic
30C2×C71⋊C16D411yescharacteristic
31#14556059—41no—
32C4⋊(C71⋊C8)—21no—
33#14556052—41no—
34C2×SD16×C71—41no—
35C2×D8×C71—21no—
36C2×D8×C71—21no—
37#127×C71—21no—
38C4×Q8×C71D411yescharacteristic
39C4⋊C8×C71—21no—
40C4⋊C8×C71—21no—
41C4⋊C8×C71D411yescharacteristic
42#102×C71—41no—
43C4×C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46Dic284—41no—
47Dic284—41no—
48Dic284—81no—
49C2×C71⋊C8—41no—
50C4⋊Dic71—41no—
51C4⋊Dic71—41no—
52C71⋊C16—21no—
53C71⋊C16—41no—
54C71⋊C16SD1612yescomplemented
55C2×Q8×C71D811yescharacteristic
56C2×D4×C71—41no—
57C2×D4×C71—41no—
58C2×D4×C71—41no—
59SD16×C71—82no—
60D8×C71—41no—
61D8×C71—41no—
62D8×C71—81no—
63C2×C8×C71—21no—
64C2×C8×C71—41no—
65C2×C8×C71C2×D411yescharacteristic
66C2×C8×C71#4111yescharacteristic
67C4⋊C4×C71—21no—
68C4⋊C4×C71D811yescharacteristic
69C42×C71—21no—
70C42×C71C2×D411yescharacteristic
71C23×C71—81no—
72Q8×C71—21no—
73Q8×C71—21no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—81no—
77D4×C71—81no—
78D4×C71—161no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71#12111yescharacteristic
82C2×C4×C71C2×D811yescharacteristic
83C2×C4×C71#13511yescharacteristic
84C2×Dic71—81no—
85Dic142—81no—
86Dic142—81no—
87C8×C71—21no—
88C8×C71—41no—
89C8×C71—81no—
90C8×C71C2×SD1611yescharacteristic
91C8×C71AGL(1,ℤ/8)11yescharacteristic
92C71⋊C8—81no—
93C22×C71—161no—
94C22×C71—161no—
95C22×C71—161no—
96C22×C71#46611yescharacteristic
97C4×C71—21no—
98C4×C71—41no—
99C4×C71—41no—
100C4×C71#46011yescharacteristic
101C4×C71#47211yescharacteristic
102C4×C71#50911yescharacteristic
103C4×C71#51011yescharacteristic
104Dic71—161no—
105#12444—711nomaximal, Sylow
106C2×C71—321no—
107C2×C71#166811yescharacteristic
108C2×C71#222011yescharacteristic
109C2×C71#222311yescharacteristic
110#2248—711no—
111#2183—711no—
112#1670—711no—
113#1668D7111yescharacteristic
114#1371—711no—
115#1362—711no—
116#1361—711no—
117C71#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1421no—
119C2×SD32—1421no—
120#494—711no—
121#493C2×D7111yescharacteristic
122#475—1421no—
123#460—21no—
124C8×Q8C2×D7111yescharacteristic
125C4⋊C16—711no—
126C4⋊C16—1421no—
127#346—1421no—
128C4×C16—711no—
129#335—711no—
130#335C2×D7111yescharacteristic
131#332—1421no—
132#331—21no—
133C2×Q16—1421no—
134C2×Q16—1421no—
135C2×SD16—41no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1421no—
139#127—21no—
140C4×Q8C22⋊D7111yescharacteristic
141SD32—2842no—
142SD32—2842no—
143C2×C16—711no—
144C2×C16—711no—
145C2×C16—1421no—
146#106—2841no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—1421no—
150C4⋊C8C22⋊D7111yescharacteristic
151#103—2841no—
152#102—41no—
153C4×C8—21no—
154C4×C8C22×D7111yescharacteristic
155C2×Q8—2841no—
156C2×Q8#1176585011yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—2841no—
161Q16—2841no—
162Q16—5681no—
163SD16—82no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—2841no—
170C2×C8D4×D7111yescharacteristic
171C2×C8#1176586411yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—2841no—
174C4⋊C4—2841no—
175C4⋊C4#1176585011yescharacteristic
176C42—21no—
177C42C2×C22⋊D7111yescharacteristic
178C16—2841no—
179C16—712no—
180C16—1421no—
181C23—81no—
182Q8—21no—
183Q8—21no—
184Q8—5681no—
185Q8—5681no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—5681no—
194C2×C4#1455610011yescharacteristic
195C2×C4C2×#1176585011yescharacteristic
196C2×C4#1455614811yescharacteristic
197C8—21no—
198C8—41no—
199C8—81no—
200C8—5681no—
201C8SD16×D7111yescharacteristic
202C8#1455615511yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1798623311yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—11361no—
211C4#1798619711yescharacteristic
212C4#1798626711yescharacteristic
213C4#1798643511yescharacteristic
214C4#1798643911yescharacteristic
215C2—321no—
216C2#2258769811yescharacteristic
217C2#2259045111yescharacteristic
218C2#2259046811yescharacteristic
subgroupG/H
centerC22#17986233
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1668C2
socleC22×C71#466
prime radical#493×C71C22
hypercenter#1668D71
hypocenterC71#12444
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. #22590249C2semidirect
3. #22587710C2semidirect
4. #22586411C2semidirect
5. #22586388C2semidirect
54. C71⋊C16SD16semidirect
117. C71#12444semidirect

Maximal subgroups

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

subgroupindex
#225905952
#225902492
#225877102
#225864112
#225863882
#225863862
C71×#16682
#1244471