Finite Group Databasebeta

← #14544217

Subgroups — #14544217

Order 2176.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#11763828C211yesmaximal, characteristic
2#11763753C211yesmaximal, characteristic
3#11763663C212yesmaximal
4#11763662C212yesmaximal
5#11763363C211yesmaximal, characteristic
6C2×C4⋊#595—21no—
7C4×Q8×C17C411yescharacteristic, complemented
8#10587632C2211yescharacteristic
9#10587630C411yescharacteristic, complemented
10#10587583C2211yescharacteristic
11#10587582—21no—
12#10587575—21no—
13#10587574C411yescharacteristic, complemented
14#10587558—21no—
15C42×D17C2211yescharacteristic
16C4×Dic34—21no—
17C4×Dic34C411yescharacteristic, complemented
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587543C2212yes—
21C4⋊#3624—21no—
22C4×#3624—21no—
23C22×#595—22no—
24C2×Q8×C17C2×C411yescharacteristic
25C2×C4×D17D411yescharacteristic
26C2×C4×D17D411yescharacteristic
27C2×C4×D17C2311yescharacteristic
28C2×Dic34—21no—
29C2×Dic34C2×C411yescharacteristic
30C2×#3624—22no—
31C4⋊#595—42no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17C2×C411yescharacteristic
34C42×C17—21no—
35C42×C17C2×C411yescharacteristic
36#63227—21no—
37#63227—21no—
38C4⋊Dic17—21no—
39C4⋊Dic17C2×C411yescharacteristic
40#63225—21no—
41#63225—21no—
42#63225—21no—
43#63225—21no—
44#63225—21no—
45C4×Dic17—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C22×D17C2×D411yescharacteristic
50C2×#595—42no—
51C2×#595—42no—
52C2×#595—42no—
53Q8×C17—41no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C4×D17—41no—
65C4×D17—41no—
66C4×D17SD1612yes—
67C4×D17Q1612yes—
68Dic34—42no—
69Dic34—41no—
70#3624—42no—
71#1518—171nomaximal, Sylow
72C22×C17C2×C22⋊C411yescharacteristic
73C2×D17—21no—
74C2×D17—21no—
75C2×D17C2×SD1611yescharacteristic
76C2×D17C2×Q1611yescharacteristic
77#595—82no—
78C4×C17—21no—
79C4×C17—21no—
80C4×C17—41no—
81C4×C17—41no—
82C4×C17#10312yes—
83Dic17—21no—
84Dic17—21no—
85Dic17—41no—
86Dic17—41no—
87Dic17#10312yes—
88#533—171no—
89#431—171no—
90#382—171no—
91#328—172no—
92#328—172no—
93C2×C17#41011yescharacteristic
94C2×C17C2×#10311yescharacteristic
95C2×C17#42111yescharacteristic
96D17—21no—
97D17—21no—
98C4⋊Q8—171no—
99C4⋊Q8—171no—
100#125—341no—
101C22⋊Q8—171no—
102C22⋊Q8—171no—
103C22⋊Q8—341no—
104C4×Q8—171no—
105C4×Q8—341no—
106C4×Q8#59511yescharacteristic, complemented
107#117—341no—
108C2×C4⋊C4—341no—
109C2×C42—171no—
110C4⋊C8—341no—
111C22⋊C8—172no—
112C4×C8—341no—
113#95—172no—
114#95—341no—
115C17#151811yescharacteristic, complemented, Sylow
116C2×Q8—171no—
117C2×Q8—341no—
118C2×Q8C2×#59511yescharacteristic
119C22×C4—171no—
120C22×C4—171no—
121C22×C4—171no—
122C22×C4—342no—
123C2×C8—342no—
124C4⋊C4—21no—
125C4⋊C4—171no—
126C4⋊C4—341no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4—682no—
133C4⋊C4C2×#59511yescharacteristic
134C22⋊C4—341no—
135C22⋊C4—341no—
136C42—21no—
137C42—171no—
138C42—171no—
139C42—171no—
140C42—341no—
141C42C2×#59511yescharacteristic
142C23—171no—
143Q8—41no—
144Q8—681no—
145Q8—682no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—171no—
149C2×C4—172no—
150C2×C4—172no—
151C2×C4—171no—
152C2×C4—171no—
153C2×C4—341no—
154C2×C4—341no—
155C2×C4—681no—
156C2×C4—681no—
157C2×C4—682no—
158C2×C4—682no—
159C2×C4—682no—
160C2×C4C22⋊#59511yescharacteristic
161C2×C4C22⋊#59511yescharacteristic
162C2×C4C22×#59511yescharacteristic
163C8—682no—
164C22—171no—
165C22—171no—
166C22—341no—
167C22—341no—
168C22C2×C22⋊#59511yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—172no—
174C4—341no—
175C4—341no—
176C4—681no—
177C4—681no—
178C4—1362no—
179C4#1058754912yes—
180C2—341no—
181C2—341no—
182C2#1176378811yescharacteristic
183C2C2×#1058754911yescharacteristic
184C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4×Q8#595
hypocenterC17#1518
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
7. C4×Q8×C17C4semidirect
9. #10587630C4semidirect
13. #10587574C4semidirect
17. C4×Dic34C4semidirect
106. C4×Q8#595semidirect
115. C17#1518semidirect

Maximal subgroups

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

subgroupindex
#117638282
#117637532
#117636632
#117636622
#117633632
#151817