Finite Group Databasebeta

← #31918294

Subgroups — #31918294

Order 18176.

204 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#17985838C211yesmaximal, characteristic, complemented
2C2×C32×D71C211yesmaximal, characteristic, complemented
3#22590432C211yesmaximal, characteristic, complemented
4#22590427C211yesmaximal, characteristic
5#22586520C211yesmaximal, characteristic, complemented
6#22586517C211yesmaximal, characteristic, complemented
7C71×#1424C211yesmaximal, characteristic, complemented
8C2×Dic568C2211yescharacteristic
9C2×D1136C2211yescharacteristic
10C2×C16×D71C2211yescharacteristic
11#17986241—21no—
12#17986236—21no—
13#17985838—22no—
14C2×C71⋊C32C2211yescharacteristic
15C32×D71—22no—
16#17985802—21no—
17#17985799—21no—
18C2×C32×C71C2211yescharacteristic
19#366×C71C2211yescharacteristic
20#366×C71C2211yescharacteristic
21C2×Dic284—21no—
22C2×D568—21no—
23C2×C8×D71D411yescharacteristic
24#14556108—41no—
25#14556103—41no—
26Dic568—21no—
27Dic568—21no—
28D1136—21no—
29D1136—21no—
30C2×C71⋊C16D411yescharacteristic
31C16×D71—21no—
32C16×D71—21no—
33#14556051—41no—
34#14556048—41no—
35C71⋊C32—21no—
36C2×C16×C71C2311yescharacteristic
37#107×C71—21no—
38#107×C71—21no—
39C32×C71D412yescomplemented
40C2×Dic142—41no—
41C2×D284—41no—
42C2×D284—81no—
43C2×C4×D71D811yescharacteristic
44Dic284—41no—
45Dic284—41no—
46D568—41no—
47D568—41no—
48C2×C71⋊C8D811yescharacteristic
49C8×D71—21no—
50C8×D71—21no—
51C4⋊Dic71—81no—
52(C2×D71)⋊C4—81no—
53(C2×D71)⋊C4—81no—
54C71⋊C16—21no—
55C2×C8×C71C2×D411yescharacteristic
56C4⋊C4×C71—41no—
57C4⋊C4×C71—41no—
58C16×C71C2×D411yescharacteristic
59C16×C71#4111yescharacteristic
60C22×D71—81no—
61C22×D71D1611yescharacteristic
62C2×C4×C71—81no—
63C2×C4×C71—81no—
64C2×C4×C71C2×D811yescharacteristic
65C2×Dic71—81no—
66C2×Dic71D1611yescharacteristic
67D284—81no—
68D284—81no—
69D284—162no—
70C4×D71—21no—
71C4×D71—21no—
72Dic142—81no—
73Dic142—81no—
74C8×C71#12111yescharacteristic
75C8×C71#13511yescharacteristic
76C71⋊C8—21no—
77C22×C71C2×D1611yescharacteristic
78C2×D71—21no—
79C2×D71—21no—
80C2×D71—161no—
81C2×D71—161no—
82C2×D71—161no—
83C2×D71SD6412yescomplemented
84C4×C71—161no—
85C4×C71—161no—
86C4×C71#46611yescharacteristic
87C4×C71#50811yescharacteristic
88Dic71—21no—
89Dic71—161no—
90#13696—711nomaximal, Sylow
91C2×C71#221511yescharacteristic
92C2×C71C2×SD6411yescharacteristic
93C2×C71#226211yescharacteristic
94D71—22no—
95D71—321no—
96C2×SD64—711no—
97C22×C32—711no—
98#2216—711no—
99#2215—711no—
100#1424D7111yescharacteristic, complemented
101#1416—711no—
102#1415—711no—
103C71#1369611yescharacteristic, complemented, Sylow
104C2×Q32—711no—
105C2×D16—711no—
106C22×C16—711no—
107#467—1421no—
108#466—1421no—
109SD64—1422no—
110C2×C32—711no—
111C2×C32—1422no—
112C2×C32C2×D7111yescharacteristic
113#366C2×D7111yescharacteristic
114#366C2×D7111yescharacteristic
115#358—1421no—
116#357—1421no—
117C2×Q16—1421no—
118C2×D8—1421no—
119C22×C8—711no—
120C22⋊Q8—2841no—
121#121—2841no—
122Q32—1421no—
123Q32—1421no—
124D16—1421no—
125D16—1421no—
126C2×C16—711no—
127C2×C16—1421no—
128C2×C16—1421no—
129C2×C16C22×D7111yescharacteristic
130#107—21no—
131#107—21no—
132#103—2841no—
133#102—2841no—
134C32—1421no—
135C32D28412yescomplemented
136C2×Q8—2841no—
137C2×D4—2841no—
138C2×D4—5681no—
139C22×C4—711no—
140Q16—2841no—
141Q16—2841no—
142D8—2841no—
143D8—2841no—
144C2×C8—711no—
145C2×C8—1421no—
146C2×C8—1421no—
147C2×C8D4×D7111yescharacteristic
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—5681no—
151C22⋊C4—5681no—
152C22⋊C4—5681no—
153C16—1421no—
154C16C2×D28411yescharacteristic
155C16#1176586611yescharacteristic
156C23—5681no—
157C23—711no—
158Q8—5681no—
159Q8—5681no—
160D4—5681no—
161D4—5681no—
162D4—11362no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—5681no—
166C2×C4—711no—
167C2×C4—1421no—
168C2×C4—1421no—
169C2×C4D8×D7111yescharacteristic
170C8—1421no—
171C8#1455610311yescharacteristic
172C8#1455615111yescharacteristic
173C22—712no—
174C22—11361no—
175C22—11361no—
176C22—11361no—
177C22—1421no—
178C22—1421no—
179C22D16×D7111yescharacteristic
180C4—161no—
181C4—161no—
182C4—11361no—
183C4—1421no—
184C4#1798623611yescharacteristic
185C4#1798643211yescharacteristic
186C2—1422no—
187C2—22721no—
188C2#2259042711yescharacteristic
189C2SD64×D7111yescharacteristic
190C2#2259064811yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1424C2
socleC22×C71C2×D16
prime radical#22590427C2
hypercenter#1424D71
hypocenterC71#13696
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1424

Complements

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

NH ≅ G/Ntype
1. C2×#17985838C2semidirect
2. C2×C32×D71C2semidirect
3. #22590432C2semidirect
5. #22586520C2semidirect
6. #22586517C2semidirect
7. C71×#1424C2semidirect
39. C32×C71D4semidirect
83. C2×D71SD64semidirect
100. #1424D71semidirect
103. C71#13696semidirect
135. C32D284semidirect

Maximal subgroups

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

subgroupindex
C2×#179858382
C2×C32×D712
#225904322
#225904272
#225865202
#225865172
C71×#14242
#1369671