Finite Group Databasebeta

← #31918282

Subgroups — #31918282

Order 18176.

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

normal subgroups only

This group has 214 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×#17985834C211yesmaximal, characteristic, complemented
2C2×C32×D71C211yesmaximal, characteristic, complemented
3#22590427C211yesmaximal, characteristic, complemented
4#22590423C211yesmaximal, characteristic, complemented
5#22586539C211yesmaximal, characteristic, complemented
6#22586519C211yesmaximal, characteristic, complemented
7C71×#1415C211yesmaximal, characteristic, complemented
8C2×D1136C2211yescharacteristic
9C2×C16×D71C2211yescharacteristic
10#17986236—21no—
11#17986232—21no—
12#17985834—22no—
13C2×C71⋊C32C2211yescharacteristic
14C32×D71—22no—
15#17985825C2211yescharacteristic
16#17985800—21no—
17#17985799—21no—
18C2×D16×C71C2211yescharacteristic
19C2×C32×C71C2211yescharacteristic
20#366×C71C2211yescharacteristic
21C2×D568—21no—
22C2×C8×D71D411yescharacteristic
23#14556103—41no—
24#14556099—41no—
25D1136—21no—
26D1136—21no—
27C2×C71⋊C16D411yescharacteristic
28C16×D71—21no—
29C16×D71—21no—
30#14556062—21no—
31#14556049—41no—
32#14556048—41no—
33C71⋊C32D412yescomplemented
34C2×D8×C71—21no—
35D16×C71—21no—
36D16×C71—21no—
37C2×C16×C71C2311yescharacteristic
38#107×C71—21no—
39C32×C71—21no—
40C2×D284—41no—
41C2×D284—81no—
42C2×C22⋊D71—81no—
43C2×C4×D71D811yescharacteristic
44D568—41no—
45D568—41no—
46C2×C71⋊C8D811yescharacteristic
47C8×D71—21no—
48C8×D71—21no—
49C4⋊Dic71—41no—
50C22⋊Dic71—81no—
51(C2×D71)⋊C4—81no—
52C71⋊C16—21no—
53C2×D4×C71—41no—
54D8×C71—41no—
55D8×C71—41no—
56C2×C8×C71C2×D411yescharacteristic
57C4⋊C4×C71—41no—
58C16×C71C2×D411yescharacteristic
59C16×C71#4111yescharacteristic
60C23×C71—81no—
61C22×D71—81no—
62C22×D71D1611yescharacteristic
63D4×C71—81no—
64D4×C71—81no—
65C2×C4×C71—81no—
66C2×C4×C71C2×D811yescharacteristic
67C22⋊D71—162no—
68C2×Dic71—81no—
69C2×Dic71D1611yescharacteristic
70D284—81no—
71D284—81no—
72D284—162no—
73C4×D71—21no—
74C4×D71—21no—
75C8×C71#12111yescharacteristic
76C8×C71#13511yescharacteristic
77C71⋊C8—21no—
78C22×C71—161no—
79C22×C71—161no—
80C22×C71—161no—
81C22×C71C2×D1611yescharacteristic
82C2×D71—21no—
83C2×D71—21no—
84C2×D71—161no—
85C2×D71—161no—
86C2×D71—161no—
87C2×D71D3212yescomplemented
88C4×C71—161no—
89C4×C71#46611yescharacteristic
90C4×C71#50811yescharacteristic
91Dic71—21no—
92Dic71—161no—
93#13694—711nomaximal, Sylow
94C2×C71—321no—
95C2×C71#221511yescharacteristic
96C2×C71C2×D3211yescharacteristic
97C2×C71#226211yescharacteristic
98D71—22no—
99D71—321no—
100C2×D32—711no—
101C22×C32—711no—
102#2215—711no—
103#2215—711no—
104#1423—711no—
105#1415—711no—
106#1415D7111yescharacteristic, complemented
107C71#1369411yescharacteristic, complemented, Sylow
108C2×D16—711no—
109C2×D16C2×D7111yescharacteristic
110C22×C16—711no—
111#466—1421no—
112#466—1421no—
113D32—1422no—
114C2×C32—711no—
115C2×C32—1422no—
116C2×C32C2×D7111yescharacteristic
117#366—711no—
118#366C2×D7111yescharacteristic
119#357—1421no—
120#357—1421no—
121C2×D8—21no—
122C2×D8—1421no—
123C22×C8—711no—
124#121—2841no—
125#121—2841no—
126D16—21no—
127D16—21no—
128D16—1421no—
129D16—1421no—
130C2×C16—711no—
131C2×C16—1421no—
132C2×C16—1421no—
133C2×C16C22×D7111yescharacteristic
134#107—21no—
135#107—1421no—
136#102—2841no—
137#102—2841no—
138C32—21no—
139C32—712no—
140C2×D4—41no—
141C2×D4—2841no—
142C2×D4—5681no—
143C2×D4—5681no—
144C22×C4—711no—
145D8—41no—
146D8—41no—
147D8—2841no—
148D8—2841no—
149C2×C8—711no—
150C2×C8—1421no—
151C2×C8—1421no—
152C2×C8D4×D7111yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—2841no—
155C22⋊C4—5681no—
156C22⋊C4—5681no—
157C16—1421no—
158C16D4×D7111yescharacteristic
159C16#1176586511yescharacteristic
160C23—81no—
161C23—5681no—
162C23—711no—
163D4—81no—
164D4—81no—
165D4—5681no—
166D4—5681no—
167D4—11362no—
168D4—11362no—
169C2×C4—81no—
170C2×C4—5681no—
171C2×C4—711no—
172C2×C4—1421no—
173C2×C4—1421no—
174C2×C4D8×D7111yescharacteristic
175C8—1421no—
176C8#1455610111yescharacteristic
177C8#1455614911yescharacteristic
178C22—161no—
179C22—161no—
180C22—161no—
181C22—712no—
182C22—11361no—
183C22—11361no—
184C22—11361no—
185C22—1421no—
186C22—1421no—
187C22D16×D7111yescharacteristic
188C4—161no—
189C4—11361no—
190C4—1421no—
191C4#1798623411yescharacteristic
192C4#1798643011yescharacteristic
193C2—321no—
194C2—1422no—
195C2—22721no—
196C2#2259042511yescharacteristic
197C2D32×D7111yescharacteristic
198C2#2259064811yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1415C2
socleC22×C71C2×D16
hypercenter#1415D71
hypocenterC71#13694
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1415

Complements

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

NH ≅ G/Ntype
1. C2×#17985834C2semidirect
2. C2×C32×D71C2semidirect
3. #22590427C2semidirect
4. #22590423C2semidirect
5. #22586539C2semidirect
6. #22586519C2semidirect
7. C71×#1415C2semidirect
33. C71⋊C32D4semidirect
87. C2×D71D32semidirect
106. #1415D71semidirect
107. C71#13694semidirect

Maximal subgroups

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

subgroupindex
C2×#179858342
C2×C32×D712
#225904272
#225904232
#225865392
#225865192
C71×#14152
#1369471