Finite Group Databasebeta

← #31918287

Subgroups — #31918287

Order 18176.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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×#17985839C211yesmaximal, characteristic, complemented
2C2×C32×D71C211yesmaximal, characteristic
3#22590432C211yesmaximal, characteristic
4#22590428C211yesmaximal, characteristic
5#22586539C211yesmaximal, characteristic, complemented
6#22586522C211yesmaximal, characteristic, complemented
7C71×#1416C211yesmaximal, characteristic, complemented
8C2×Dic568C2211yescharacteristic
9C2×C16×D71C2211yescharacteristic
10#17986241—21no—
11#17986237—21no—
12#17985839—22no—
13C2×C71⋊C32C2211yescharacteristic
14C32×D71—22no—
15#17985825C2211yescharacteristic
16#17985803—21no—
17#17985802—21no—
18C2×Q32×C71C2211yescharacteristic
19C2×C32×C71C2211yescharacteristic
20#366×C71C2211yescharacteristic
21C2×Dic284—21no—
22C2×C8×D71D411yescharacteristic
23#14556108—41no—
24#14556104—41no—
25Dic568—21no—
26Dic568—21no—
27C2×C71⋊C16D411yescharacteristic
28C16×D71—21no—
29C16×D71—21no—
30#14556062—21no—
31#14556052—41no—
32#14556051—41no—
33C71⋊C32D412yes—
34C2×Q16×C71—21no—
35Q32×C71—21no—
36Q32×C71—21no—
37C2×C16×C71C2311yescharacteristic
38#107×C71—21no—
39C32×C71—21no—
40C2×Dic142—41no—
41C2×C4×D71D811yescharacteristic
42Dic284—41no—
43Dic284—41no—
44C2×C71⋊C8D811yescharacteristic
45C8×D71—21no—
46C8×D71—21no—
47C4⋊Dic71—41no—
48C4⋊Dic71—81no—
49Dic71⋊C4—81no—
50(C2×D71)⋊C4—81no—
51(C2×D71)⋊C4—81no—
52C71⋊C16—21no—
53C2×Q8×C71—41no—
54Q16×C71—41no—
55Q16×C71—41no—
56C2×C8×C71C2×D411yescharacteristic
57C4⋊C4×C71—41no—
58C16×C71C2×D411yescharacteristic
59C16×C71#4111yescharacteristic
60C22×D71D1611yescharacteristic
61Q8×C71—81no—
62Q8×C71—81no—
63C2×C4×C71—81no—
64C2×C4×C71—81no—
65C2×C4×C71C2×D811yescharacteristic
66C2×Dic71—81no—
67C2×Dic71—81no—
68C2×Dic71D1611yescharacteristic
69C4×D71—21no—
70C4×D71—21no—
71Dic142—81no—
72Dic142—81no—
73C8×C71#12111yescharacteristic
74C8×C71#13511yescharacteristic
75C71⋊C8—21no—
76C22×C71C2×D1611yescharacteristic
77C2×D71—21no—
78C2×D71—21no—
79C2×D71Q6412yescomplemented
80C4×C71—161no—
81C4×C71—161no—
82C4×C71#46611yescharacteristic
83C4×C71#50811yescharacteristic
84Dic71—21no—
85Dic71—161no—
86Dic71—161no—
87#13695—711nomaximal, Sylow
88C2×C71#221511yescharacteristic
89C2×C71C2×Q6411yescharacteristic
90C2×C71#226211yescharacteristic
91D71—22no—
92C2×Q64—711no—
93C22×C32—711no—
94#2216—711no—
95#2216—711no—
96#1423—711no—
97#1416—711no—
98#1416D7111yescharacteristic, complemented
99C71#1369511yescharacteristic, complemented, Sylow
100C2×Q32—711no—
101C2×Q32C2×D7111yescharacteristic
102C22×C16—711no—
103#467—1421no—
104#467—1421no—
105Q64—1422no—
106C2×C32—711no—
107C2×C32—1422no—
108C2×C32C2×D7111yescharacteristic
109#366—711no—
110#366C2×D7111yescharacteristic
111#358—1421no—
112#358—1421no—
113C2×Q16—21no—
114C2×Q16—1421no—
115C22×C8—711no—
116C22⋊Q8—2841no—
117C22⋊Q8—2841no—
118Q32—21no—
119Q32—21no—
120Q32—1421no—
121Q32—1421no—
122C2×C16—711no—
123C2×C16—1421no—
124C2×C16—1421no—
125C2×C16C22×D7111yescharacteristic
126#107—21no—
127#107—1421no—
128#103—2841no—
129#103—2841no—
130C32—21no—
131C32—712no—
132C2×Q8—41no—
133C2×Q8—2841no—
134C22×C4—711no—
135Q16—41no—
136Q16—41no—
137Q16—2841no—
138Q16—2841no—
139C2×C8—711no—
140C2×C8—1421no—
141C2×C8—1421no—
142C2×C8D4×D7111yescharacteristic
143C4⋊C4—41no—
144C4⋊C4—2841no—
145C4⋊C4—5681no—
146C4⋊C4—5681no—
147C22⋊C4—5681no—
148C22⋊C4—5681no—
149C16—1421no—
150C16D4×D7111yescharacteristic
151C16#1176586511yescharacteristic
152C23—711no—
153Q8—81no—
154Q8—81no—
155Q8—5681no—
156Q8—5681no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—5681no—
160C2×C4—5681no—
161C2×C4—711no—
162C2×C4—1421no—
163C2×C4—1421no—
164C2×C4D8×D7111yescharacteristic
165C8—1421no—
166C8#1455610111yescharacteristic
167C8#1455614911yescharacteristic
168C22—712no—
169C22—1421no—
170C22—1421no—
171C22D16×D7111yescharacteristic
172C4—161no—
173C4—161no—
174C4—11361no—
175C4—11361no—
176C4—1421no—
177C4#1798623411yescharacteristic
178C4#1798643011yescharacteristic
179C2—1422no—
180C2#2259042511yescharacteristic
181C2Q64×D7111yescharacteristic
182C2#2259064811yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1416C2
socleC22×C71C2×D16
prime radicalC22×D71D16
hypercenter#1416D71
hypocenterC71#13695
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1416

Complements

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

NH ≅ G/Ntype
1. C2×#17985839C2semidirect
5. #22586539C2semidirect
6. #22586522C2semidirect
7. C71×#1416C2semidirect
79. C2×D71Q64semidirect
98. #1416D71semidirect
99. C71#13695semidirect

Maximal subgroups

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

subgroupindex
C2×#179858392
C2×C32×D712
#225904322
#225904282
#225865392
#225865222
C71×#14162
#1369571