Finite Group Databasebeta

← #31918296

Subgroups — #31918296

Order 18176.

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

normal subgroups only

This group has 204 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×#17985836C211yesmaximal, characteristic, complemented
2C2×C32×D71C211yesmaximal, characteristic, complemented
3#22590428C211yesmaximal, characteristic, complemented
4#22590427C211yesmaximal, characteristic
5#22586541C211yesmaximal, characteristic, complemented
6#22586519C211yesmaximal, characteristic, complemented
7C71×#1416C211yesmaximal, characteristic, complemented
8C2×D1136C2211yescharacteristic
9C2×C16×D71C2211yescharacteristic
10#17986237—21no—
11#17986236—21no—
12#17985836—22no—
13C2×C71⋊C32C2211yescharacteristic
14C32×D71—22no—
15#17985825C2211yescharacteristic
16#17985803—21no—
17#17985799—21no—
18C2×Q32×C71C2211yescharacteristic
19C2×C32×C71C2211yescharacteristic
20#366×C71C2211yescharacteristic
21C2×D568—21no—
22C2×C8×D71D411yescharacteristic
23#14556104—41no—
24#14556103—41no—
25D1136—21no—
26D1136—21no—
27C2×C71⋊C16D411yescharacteristic
28C16×D71—21no—
29C16×D71—21no—
30#14556062—21no—
31#14556052—41no—
32#14556048—41no—
33C71⋊C32D412yescomplemented
34C2×Q16×C71—21no—
35Q32×C71—21no—
36Q32×C71—21no—
37C2×C16×C71C2311yescharacteristic
38#107×C71—21no—
39C32×C71—21no—
40C2×D284—41no—
41C2×D284—81no—
42C2×C4×D71D811yescharacteristic
43D568—41no—
44D568—41no—
45C2×C71⋊C8D811yescharacteristic
46C8×D71—21no—
47C8×D71—21no—
48C4⋊Dic71—41no—
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×D71—81no—
61C22×D71D1611yescharacteristic
62Q8×C71—81no—
63Q8×C71—81no—
64C2×C4×C71—81no—
65C2×C4×C71—81no—
66C2×C4×C71C2×D811yescharacteristic
67C2×Dic71—81no—
68C2×Dic71D1611yescharacteristic
69D284—81no—
70D284—81no—
71D284—162no—
72C4×D71—21no—
73C4×D71—21no—
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#1424—711no—
101#1416D7111yescharacteristic, complemented
102#1415—711no—
103C71#1369611yescharacteristic, complemented, Sylow
104C2×Q32C2×D7111yescharacteristic
105C2×D16—711no—
106C22×C16—711no—
107#467—1421no—
108#466—1421no—
109SD64—1422no—
110C2×C32—711no—
111C2×C32—1422no—
112C2×C32C2×D7111yescharacteristic
113#366—711no—
114#366C2×D7111yescharacteristic
115#358—1421no—
116#357—1421no—
117C2×Q16—21no—
118C2×D8—1421no—
119C22×C8—711no—
120C22⋊Q8—2841no—
121#121—2841no—
122Q32—21no—
123Q32—21no—
124D16—1421no—
125D16—1421no—
126C2×C16—711no—
127C2×C16—1421no—
128C2×C16—1421no—
129C2×C16C22×D7111yescharacteristic
130#107—21no—
131#107—1421no—
132#103—2841no—
133#102—2841no—
134C32—21no—
135C32—712no—
136C2×Q8—41no—
137C2×D4—2841no—
138C2×D4—5681no—
139C22×C4—711no—
140Q16—41no—
141Q16—41no—
142D8—2841no—
143D8—2841no—
144C2×C8—711no—
145C2×C8—1421no—
146C2×C8—1421no—
147C2×C8D4×D7111yescharacteristic
148C4⋊C4—41no—
149C4⋊C4—2841no—
150C4⋊C4—5681no—
151C22⋊C4—5681no—
152C22⋊C4—5681no—
153C16—1421no—
154C16D4×D7111yescharacteristic
155C16#1176586511yescharacteristic
156C23—5681no—
157C23—711no—
158Q8—81no—
159Q8—81no—
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#1455610111yescharacteristic
172C8#1455614911yescharacteristic
173C22—712no—
174C22—11361no—
175C22—11361no—
176C22—11361no—
177C22—1421no—
178C22—1421no—
179C22D16×D7111yescharacteristic
180C4—161no—
181C4—161no—
182C4—11361no—
183C4—1421no—
184C4#1798623411yescharacteristic
185C4#1798643011yescharacteristic
186C2—1422no—
187C2—22721no—
188C2#2259042511yescharacteristic
189C2SD64×D7111yescharacteristic
190C2#2259065011yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1416C2
socleC22×C71C2×D16
prime radical#22590427C2
hypercenter#1416D71
hypocenterC71#13696
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×#17985836C2semidirect
2. C2×C32×D71C2semidirect
3. #22590428C2semidirect
5. #22586541C2semidirect
6. #22586519C2semidirect
7. C71×#1416C2semidirect
33. C71⋊C32D4semidirect
83. C2×D71SD64semidirect
101. #1416D71semidirect
103. C71#13696semidirect

Maximal subgroups

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

subgroupindex
C2×#179858362
C2×C32×D712
#225904282
#225904272
#225865412
#225865192
C71×#14162
#1369671