Finite Group Databasebeta

← #31918111

Subgroups — #31918111

Order 18176.

284 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 61 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×Q32×D71C211yesmaximal, characteristic
2C2×#17985839C211yesmaximal, characteristic, complemented
3#22590428C211yesmaximal, characteristic
4#22586522C211yesmaximal, characteristic, complemented
5#22586521C211yesmaximal, characteristic, complemented
6(C2×D71)⋊C32C211yesmaximal, characteristic
7C2×Q64×C71C211yesmaximal, characteristic, complemented
8C2×Q16×D71—21no—
9C2×Dic568C2211yescharacteristic
10C2×#14556074—21no—
11Q32×D71—22no—
12Q32×D71—22no—
13C2×C16×D71C2211yescharacteristic
14#17986237—21no—
15#17985839—22no—
16C2×C71⋊C32C2211yescharacteristic
17#17985825C2211yescharacteristic
18#17985803—21no—
19C2×Q32×C71C2211yescharacteristic
20C2×Q32×C71C2211yescharacteristic
21Q64×C71—22no—
22C2×C32×C71C2211yescharacteristic
23C2×Q8×D71—41no—
24C2×Dic284—21no—
25C2×#11765855—41no—
26Q16×D71—24no—
27Q16×D71—42no—
28C2×C8×D71D411yescharacteristic
29#14556104—41no—
30Dic568—21no—
31Dic568D412yes—
32#14556074—24no—
33C2×C71⋊C16D411yescharacteristic
34C16×D71—21no—
35C16×D71—21no—
36#14556062—21no—
37#14556052—41no—
38C71⋊C32—21no—
39C2×Q16×C71—21no—
40C2×Q16×C71—21no—
41Q32×C71—21no—
42Q32×C71—21no—
43Q32×C71—21no—
44Q32×C71D412yes—
45C2×C16×C71C2311yescharacteristic
46C32×C71—21no—
47C2×Dic142—41no—
48C2×Dic142—81no—
49Q8×D71—44no—
50Q8×D71—82no—
51C2×C4×D71—81no—
52C2×C4×D71D811yescharacteristic
53Dic284—22no—
54Dic284—41no—
55#11765855—44no—
56C2×C71⋊C8D811yescharacteristic
57C8×D71—21no—
58C8×D71—21no—
59C4⋊Dic71—41no—
60Dic71⋊C4—81no—
61(C2×D71)⋊C4—81no—
62C71⋊C16—21no—
63C2×Q8×C71—41no—
64C2×Q8×C71—41no—
65Q16×C71—22no—
66Q16×C71—41no—
67Q16×C71—41no—
68Q16×C71—41no—
69C2×C8×C71C2×D411yescharacteristic
70C16×C71C2×D411yescharacteristic
71C16×C71C2×D411yescharacteristic
72C22×D71D1611yescharacteristic
73Q8×C71—42no—
74Q8×C71—81no—
75Q8×C71—81no—
76Q8×C71—81no—
77C2×C4×C71—81no—
78C2×C4×C71—81no—
79C2×C4×C71C2×D811yescharacteristic
80C2×Dic71—81no—
81C2×Dic71—81no—
82C2×Dic71D1611yescharacteristic
83C4×D71—21no—
84C4×D71—21no—
85C4×D71—84no—
86Dic142—42no—
87Dic142—84no—
88Dic142—81no—
89C8×C71#12011yescharacteristic
90C8×C71AGL(1,ℤ/8)11yescharacteristic
91C71⋊C8—21no—
92C22×C71C2×D1611yescharacteristic
93C2×D71—21no—
94C2×D71—21no—
95C2×D71Q6412yescomplemented
96C4×C71—82no—
97C4×C71—161no—
98C4×C71#44711yescharacteristic
99C4×C71#50911yescharacteristic
100Dic71—21no—
101Dic71—82no—
102Dic71—161no—
103#13665—711nomaximal, Sylow
104C2×C71#218411yescharacteristic
105C2×C71C2×Q6411yescharacteristic
106C2×C71#226411yescharacteristic
107D71—22no—
108C22×Q32—711no—
109C2×Q64—711no—
110C2×Q64D7111yescharacteristic, complemented
111#2216—711no—
112#1416—711no—
113#1416—711no—
114C22⋊C32—711no—
115C71#1366511yescharacteristic, complemented, Sylow
116C22×Q16—1421no—
117C2×Q32—711no—
118C2×Q32—1421no—
119C2×Q32—1422no—
120C2×Q32—1422no—
121C2×Q32C2×D7111yescharacteristic
122C2×Q32C2×D7111yescharacteristic
123C22×C16—711no—
124#467—1421no—
125Q64—22no—
126Q64—1422no—
127C2×C32—711no—
128C2×C32C2×D7111yescharacteristic
129#366—711no—
130#358—1421no—
131C22×Q8—2841no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—2841no—
135C2×Q16—2842no—
136C2×Q16—1424no—
137C2×Q16—1421no—
138C22×C8—711no—
139C22⋊Q8—2841no—
140Q32—21no—
141Q32—21no—
142Q32—21no—
143Q32—712no—
144Q32—1424no—
145Q32—1421no—
146Q32C22⋊D7112yes—
147C2×C16—711no—
148C2×C16—1421no—
149C2×C16—1421no—
150C2×C16C22×D7111yescharacteristic
151#107—1421no—
152#103—2841no—
153C32—21no—
154C32—1421no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—2844no—
158C2×Q8—2841no—
159C2×Q8—5681no—
160C2×Q8—5682no—
161C22×C4—5681no—
162C22×C4—711no—
163Q16—22no—
164Q16—41no—
165Q16—41no—
166Q16—41no—
167Q16—2844no—
168Q16—2841no—
169Q16—1422no—
170C2×C8—711no—
171C2×C8—1421no—
172C2×C8—1421no—
173C2×C8D4×D7111yescharacteristic
174C4⋊C4—2841no—
175C4⋊C4—5681no—
176C22⋊C4—5681no—
177C16—1421no—
178C16D4×D7111yescharacteristic
179C16C2×C22⋊D7111yescharacteristic
180C23—711no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184Q8—81no—
185Q8—2842no—
186Q8—5684no—
187Q8—5681no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—5684no—
191C2×C4—5681no—
192C2×C4—5681no—
193C2×C4—711no—
194C2×C4—1421no—
195C2×C4—1421no—
196C2×C4D8×D7111yescharacteristic
197C8—1421no—
198C8#1455609611yescharacteristic
199C8#1455615211yescharacteristic
200C22—712no—
201C22—1421no—
202C22—1421no—
203C22D16×D7111yescharacteristic
204C4—82no—
205C4—161no—
206C4—5682no—
207C4—11361no—
208C4—1421no—
209C4#1798613111yescharacteristic
210C4#1798643311yescharacteristic
211C2—1422no—
212C2#2259025011yescharacteristic
213C2Q64×D7111yescharacteristic
214C2#2259065911yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC2×Q64×C71C2
socleC22×C71C2×D16
prime radicalC22×D71D16
hypercenterC2×Q64D71
hypocenterC71#13665
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×Q64

Complements

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

NH ≅ G/Ntype
2. C2×#17985839C2semidirect
4. #22586522C2semidirect
5. #22586521C2semidirect
7. C2×Q64×C71C2semidirect
95. C2×D71Q64semidirect
110. C2×Q64D71semidirect
115. C71#13665semidirect

Maximal subgroups

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

subgroupindex
C2×Q32×D712
C2×#179858392
#225904282
#225865222
#225865212
(C2×D71)⋊C322
C2×Q64×C712
#1366571