Finite Group Databasebeta

← #31918231

Subgroups — #31918231

Order 18176.

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

normal subgroups only

This group has 232 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×D2272C211yesmaximal, characteristic, complemented
2C2×#17985834C211yesmaximal, characteristic, complemented
3#22590588C211yesmaximal, characteristic, complemented
4#22590186C211yesmaximal, characteristic, complemented
5Dic71⋊C32C211yesmaximal, characteristic, complemented
6#22586517C211yesmaximal, characteristic, complemented
7C71×#1415C211yesmaximal, characteristic, complemented
8C2×D1136C2211yescharacteristic
9C2×D1136C2211yescharacteristic
10C2×#14556069—21no—
11#17986374—21no—
12#17986081—21no—
13D2272—22no—
14#17985834—22no—
15C2×C71⋊C32C2211yescharacteristic
16#17985799—21no—
17C16×Dic71C2211yescharacteristic
18C2×D16×C71C2211yescharacteristic
19C2×C32×C71C2211yescharacteristic
20#366×C71C2211yescharacteristic
21C2×D568—21no—
22C2×D568—21no—
23C2×#11765850—41no—
24#14556124—41no—
25#14556090—41no—
26D1136—21no—
27D1136—21no—
28D1136—21no—
29D1136D412yescomplemented
30#14556069—42no—
31C2×C71⋊C16D411yescharacteristic
32#14556048—41no—
33C8×Dic71D411yescharacteristic
34C71⋊C32—21no—
35C2×D8×C71—21no—
36D16×C71—21no—
37D16×C71—21no—
38C2×C16×C71C2311yescharacteristic
39#107×C71—21no—
40C32×C71—21no—
41C2×D284—41no—
42C2×D284—41no—
43C2×C22⋊D71—81no—
44C2×C22⋊D71—81no—
45C2×C4×D71—81no—
46D568—22no—
47D568—41no—
48D568—41no—
49D568—41no—
50#11765850—82no—
51C2×C71⋊C8D811yescharacteristic
52(C2×D71)⋊C4—81no—
53C4×Dic71D811yescharacteristic
54C71⋊C16—21no—
55C2×D4×C71—41no—
56D8×C71—41no—
57D8×C71—41no—
58C2×C8×C71C2×D411yescharacteristic
59C4⋊C4×C71—41no—
60C16×C71C2×D411yescharacteristic
61C16×C71#4111yescharacteristic
62C23×C71—81no—
63C22×D71—81no—
64C22×D71—81no—
65D4×C71—81no—
66D4×C71—81no—
67C2×C4×C71—81no—
68C2×C4×C71C2×D811yescharacteristic
69C22⋊D71—162no—
70C22⋊D71—162no—
71C2×Dic71D1611yescharacteristic
72C2×Dic71D1611yescharacteristic
73D284—42no—
74D284—81no—
75D284—81no—
76D284—81no—
77C4×D71—162no—
78C8×C71#12111yescharacteristic
79C8×C71AGL(1,ℤ/8)11yescharacteristic
80C71⋊C8—21no—
81C22×C71—161no—
82C22×C71—161no—
83C22×C71—161no—
84C22×C71C2×D1611yescharacteristic
85C2×D71—82no—
86C2×D71—161no—
87C2×D71—161no—
88C2×D71—161no—
89C2×D71—161no—
90C2×D71—161no—
91C4×C71—161no—
92C4×C71#45911yescharacteristic
93C4×C71#50911yescharacteristic
94Dic71—21no—
95Dic71D3212yescomplemented
96#13685—711nomaximal, Sylow
97C2×C71—321no—
98C2×C71#220611yescharacteristic
99C2×C71C2×D3211yescharacteristic
100C2×C71#226311yescharacteristic
101D71—162no—
102D71—321no—
103C2×D32—711no—
104C2×D32—711no—
105#2246—711no—
106C4×D16—711no—
107C4⋊C32—711no—
108#1415—711no—
109#1415D7111yescharacteristic, complemented
110C71#1368511yescharacteristic, complemented, Sylow
111C2×D16—711no—
112C2×D16—711no—
113C2×D16—1421no—
114C2×D16C2×D7111yescharacteristic
115#493—1421no—
116C4×D8—1421no—
117D32—1422no—
118D32—1422no—
119C2×C32—711no—
120C2×C32C2×D7111yescharacteristic
121#366C2×D7111yescharacteristic
122#357—1421no—
123C4×C16—711no—
124C2×D8—21no—
125C2×D8—2841no—
126C2×D8—1421no—
127C2×D8—1421no—
128#127—2841no—
129C4×D4—2841no—
130D16—21no—
131D16—21no—
132D16—2842no—
133D16—712no—
134D16—1421no—
135D16—1421no—
136D16—1421no—
137C2×C16—711no—
138C2×C16C22×D7111yescharacteristic
139#107—21no—
140#102—2841no—
141C4×C8—711no—
142C32—21no—
143C32—1421no—
144C2×D4—41no—
145C2×D4—2841no—
146C2×D4—2841no—
147C2×D4—5681no—
148C2×D4—5681no—
149C22×C4—5681no—
150D8—41no—
151D8—41no—
152D8—2841no—
153D8—2841no—
154D8—2841no—
155D8—5682no—
156D8—1422no—
157C2×C8—711no—
158C2×C8D4×D7111yescharacteristic
159C4⋊C4—41no—
160C22⋊C4—5681no—
161C42—711no—
162C16—1421no—
163C16D4×D7111yescharacteristic
164C16#1176586511yescharacteristic
165C23—81no—
166C23—5681no—
167C23—5681no—
168D4—81no—
169D4—81no—
170D4—2842no—
171D4—5681no—
172D4—5681no—
173D4—5681no—
174D4—11362no—
175D4—11362no—
176C2×C4—81no—
177C2×C4—711no—
178C2×C4—711no—
179C2×C4—11362no—
180C2×C4D8×D7111yescharacteristic
181C8—1421no—
182C8#1455610111yescharacteristic
183C8#1455615511yescharacteristic
184C22—161no—
185C22—161no—
186C22—161no—
187C22—5682no—
188C22—11361no—
189C22—11361no—
190C22—11361no—
191C22—11361no—
192C22—11361no—
193C22D16×D7111yescharacteristic
194C4—161no—
195C4—712no—
196C4—1421no—
197C4#1798619311yescharacteristic
198C4#1798643611yescharacteristic
199C2—321no—
200C2—11362no—
201C2—22721no—
202C2#2259037411yescharacteristic
203C2D32×D7111yescharacteristic
204C2#2259065411yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1415C2
socleC22×C71C2×D16
hypercenter#1415D71
hypocenterC71#13685
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×D2272C2semidirect
2. C2×#17985834C2semidirect
3. #22590588C2semidirect
4. #22590186C2semidirect
5. Dic71⋊C32C2semidirect
6. #22586517C2semidirect
7. C71×#1415C2semidirect
29. D1136D4semidirect
95. Dic71D32semidirect
109. #1415D71semidirect
110. C71#13685semidirect

Maximal subgroups

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

subgroupindex
C2×D22722
C2×#179858342
#225905882
#225901862
Dic71⋊C322
#225865172
C71×#14152
#1368571