Finite Group Databasebeta

← #14528121

Subgroups — #14528121

Order 2176.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#11763559C211yesmaximal, characteristic
2#11763218C211yesmaximal, characteristic, complemented
3#11763178C211yesmaximal, characteristic, complemented
4#11763168C211yesmaximal, characteristic
5#11762550C211yesmaximal, characteristic, complemented
6#11762550C211yesmaximal, characteristic, complemented
7C17×#329C211yesmaximal, characteristic
8C17×#124C2211yescharacteristic
9#10587630—21no—
10#10587621—21no—
11#10587565—21no—
12#10587565—21no—
13#10587564—21no—
14#10587564—21no—
15#10587556C2211yescharacteristic
16#10587556C2211yescharacteristic
17#10587555C2211yescharacteristic
18C17×#97C2211yescharacteristic
19C17×#97C2211yescharacteristic
20#10587506—21no—
21#10587503—21no—
22#10587489—21no—
23#10587488—21no—
24#10587487—21no—
25#10587487—21no—
26#10587474C2211yescharacteristic
27C2×Q8×C17D411yescharacteristic
28C2×D4×C17D411yescharacteristic
29C22×Dic17—21no—
30C2×Dic34—21no—
31C2×Dic34—21no—
32C2×C8×C17—21no—
33C2×C8×C17—21no—
34C22⋊C4×C17—21no—
35C22⋊C4×C17—21no—
36C42×C17C2311yescharacteristic
37C22⋊Dic17—21no—
38C22⋊Dic17—21no—
39C22⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17D411yescharacteristic
43C4⋊Dic17D411yescharacteristic
44C4⋊Dic17D411yescharacteristic
45C4⋊Dic17D411yescharacteristic
46#63225—21no—
47#63225—21no—
48#63225—21no—
49#63225—21no—
50#63225—21no—
51#63225—21no—
52C4×Dic17—21no—
53C2×#3622—21no—
54C23×C17—21no—
55Q8×C17—41no—
56D4×C17—41no—
57C2×C4×C17—21no—
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C2×Dic17—21no—
62C2×Dic17—21no—
63C2×Dic17—21no—
64C2×Dic17—21no—
65C2×Dic17—41no—
66C2×Dic17—81no—
67Dic34—42no—
68Dic34—81no—
69C8×C17—22no—
70C8×C17—41no—
71#3622—41no—
72#1664—171nomaximal, Sylow
73C22×C17—41no—
74C22×C17—41no—
75C22×C17—41no—
76C22×C17#12011yescharacteristic
77C4×C17—21no—
78C4×C17—21no—
79C4×C17—21no—
80C4×C17—41no—
81Dic17—41no—
82Dic17—41no—
83Dic17—41no—
84Dic17—41no—
85Dic17—81no—
86#563—171no—
87#501—171no—
88#491—171no—
89#489—171no—
90#330—171no—
91#330—171no—
92#329D1711yescharacteristic
93C2×C17—81no—
94C2×C17#45411yescharacteristic
95C2×C17#45511yescharacteristic
96C2×C17#45611yescharacteristic
97C4⋊Q8—171no—
98C4⋊Q8—341no—
99#126—341no—
100#126—341no—
101#125—171no—
102#125—171no—
103#124C2×D1711yescharacteristic
104#123—341no—
105C22⋊Q8—341no—
106C22⋊Q8—341no—
107#107—341no—
108#106—341no—
109#103—341no—
110#103—341no—
111#103—341no—
112#102—341no—
113#97—171no—
114#97C2×D1711yescharacteristic
115#97C2×D1711yescharacteristic
116C17#166411yescharacteristic, complemented, Sylow
117C2×Q8—341no—
118C2×Q8—341no—
119C2×Q8D6811yescharacteristic
120C2×D4D6811yescharacteristic
121C22×C4—341no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—341no—
125C4⋊C4—171no—
126C4⋊C4—171no—
127C4⋊C4—171no—
128C4⋊C4—171no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4—341no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4—341no—
136C4⋊C4—341no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—341no—
140C22⋊C4—341no—
141C22⋊C4—341no—
142C42—341no—
143C42C22×D1711yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—682no—
147Q8—1361no—
148D4—41no—
149C2×C4—21no—
150C2×C4—341no—
151C2×C4—341no—
152C2×C4—341no—
153C2×C4—341no—
154C2×C4—681no—
155C2×C4—1361no—
156C2×C4C2×D6811yescharacteristic
157C2×C4D4×D1711yescharacteristic
158C2×C4D4×D1711yescharacteristic
159C8—22no—
160C8—41no—
161C8—681no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1058756811yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—681no—
171C4—681no—
172C4—681no—
173C4—681no—
174C4—1361no—
175C2—81no—
176C2#1176298311yescharacteristic
177C2#1176298911yescharacteristic
178C2#1176299311yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#329C2
socleC22×C17#120
prime radicalC2×D4×C17D4
hypercenter#329D17
hypocenterC17#1664
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #11763218C2semidirect
3. #11763178C2semidirect
5. #11762550C2semidirect
6. #11762550C2semidirect
116. C17#1664semidirect

Maximal subgroups

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

subgroupindex
#117635592
#117632182
#117631782
#117631682
#117625502
C17×#3292
#166417