Finite Group Databasebeta

← #14527255

Subgroups — #14527255

Order 2176.

202 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 65 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#11763363C211yesmaximal, characteristic
2#11762865C211yesmaximal, characteristic
3#11762860C211yesmaximal, characteristic
4#11762541C211yesmaximal, characteristic, complemented
5Q8⋊#3622C211yesmaximal, characteristic, complemented
6#11762539C211yesmaximal, characteristic, complemented
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C4×Q8×C17C2211yescharacteristic
9#10587632C411yescharacteristic
10#10587630C411yescharacteristic
11#10587583C411yescharacteristic
12#10587582—21no—
13#10587575—21no—
14#10587574C411yescharacteristic
15#10587558—21no—
16C42×D17C2211yescharacteristic
17C4×Dic34—21no—
18C4×Dic34C2211yescharacteristic
19C17×C4⋊C8C2211yescharacteristic
20C4×C8×C17C2211yescharacteristic
21#10587491—21no—
22#10587491—21no—
23#10587486—21no—
24#10587485—21no—
25C4⋊#3622C2211yescharacteristic
26C4×#3622C2211yescharacteristic
27C2×Q8×C17C2×C411yescharacteristic
28C2×C4×D17C2×C411yescharacteristic
29C2×C4×D17D411yescharacteristic
30C2×C4×D17D411yescharacteristic
31C2×Dic34—21no—
32C2×Dic34C2×C411yescharacteristic
33C2×C8×C17—21no—
34C2×C8×C17—21no—
35C4⋊C4×C17—21no—
36C4⋊C4×C17C2×C411yescharacteristic
37C42×C17—21no—
38C42×C17C2311yescharacteristic
39#63227—21no—
40#63227—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17C2×C411yescharacteristic
43#63225—21no—
44#63225—21no—
45#63225—21no—
46#63225—21no—
47#63225—21no—
48C4×Dic17—21no—
49C4×Dic17C2×C411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17D411yescharacteristic
52C2×#3622—21no—
53C2×#3622—21no—
54C22×D17C22⋊C411yescharacteristic
55Q8×C17—21no—
56Q8×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—21no—
59C2×C4×C17C22×C411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C2×C4×C17C2×D411yescharacteristic
62C2×Dic17—21no—
63C2×Dic17—21no—
64C2×Dic17C22⋊C411yescharacteristic
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C4×D17—21no—
68C4×D17—21no—
69C4×D17—21no—
70C4×D17—21no—
71C4×D17—41no—
72Dic34—21no—
73Dic34—21no—
74Dic34—42no—
75C8×C17—41no—
76C8×C17—41no—
77#3622—41no—
78#3622—41no—
79#1497—171nomaximal, Sylow
80C22×C17C2×C22⋊C411yescharacteristic
81C2×D17—21no—
82C2×D17—21no—
83C2×D17—21no—
84C4×C17—21no—
85C4×C17—41no—
86C4×C17—41no—
87C4×C17#13111yescharacteristic
88C4×C17#13111yescharacteristic
89C4×C17#13711yescharacteristic
90C4×C17#13711yescharacteristic
91Dic17—21no—
92Dic17—21no—
93Dic17—21no—
94Dic17—41no—
95Dic17—41no—
96#533—171no—
97#432—171no—
98#431—171no—
99Q8⋊C8—171no—
100Q8⋊C8—171no—
101Q8⋊C8—171no—
102Q8⋊C8D1711yescharacteristic, complemented
103C2×C17#40811yescharacteristic
104C2×C17#41911yescharacteristic
105C2×C17#42111yescharacteristic
106D17—41no—
107C4⋊Q8—171no—
108C4⋊Q8—171no—
109#125—341no—
110C22⋊Q8—171no—
111C22⋊Q8—171no—
112C22⋊Q8—341no—
113C4×Q8—171no—
114C4×Q8—341no—
115C4×Q8C2×D1711yescharacteristic
116#117—341no—
117C2×C42—171no—
118C4⋊C8—171no—
119C4⋊C8—341no—
120C4⋊C8C2×D1711yescharacteristic
121C22⋊C8—341no—
122C22⋊C8—341no—
123#97—341no—
124C4×C8—171no—
125C4×C8C2×D1711yescharacteristic
126C17#149711yescharacteristic, complemented, Sylow
127C2×Q8—171no—
128C2×Q8—341no—
129C2×Q8C4×D1711yescharacteristic
130C22×C4—171no—
131C22×C4—171no—
132C22×C4—171no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—341no—
136C2×C8—341no—
137C4⋊C4—21no—
138C4⋊C4—171no—
139C4⋊C4—341no—
140C4⋊C4—341no—
141C4⋊C4—341no—
142C4⋊C4—341no—
143C4⋊C4—341no—
144C4⋊C4—341no—
145C4⋊C4C4×D1711yescharacteristic
146C22⋊C4—341no—
147C22⋊C4—341no—
148C42—21no—
149C42—171no—
150C42—171no—
151C42—171no—
152C42—341no—
153C42C22×D1711yescharacteristic
154C23—171no—
155Q8—21no—
156Q8—21no—
157Q8—341no—
158Q8—341no—
159Q8—682no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—171no—
163C2×C4—171no—
164C2×C4—171no—
165C2×C4—341no—
166C2×C4—341no—
167C2×C4—341no—
168C2×C4—341no—
169C2×C4—341no—
170C2×C4—341no—
171C2×C4—681no—
172C2×C4C2×C4×D1711yescharacteristic
173C2×C4D4×D1711yescharacteristic
174C2×C4D4×D1711yescharacteristic
175C8—41no—
176C8—41no—
177C8—681no—
178C8—681no—
179C22—341no—
180C22—341no—
181C22—341no—
182C22C22⋊C4×D1711yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—341no—
187C4—341no—
188C4—341no—
189C4—681no—
190C4—681no—
191C4#1058758711yescharacteristic
192C4#1058759311yescharacteristic
193C4#1058760111yescharacteristic
194C4#1058760411yescharacteristic
195C2—681no—
196C2#1176277411yescharacteristic
197C2#1176281611yescharacteristic
198C2#1176282411yescharacteristic
subgroupG/H
centerC2×C4D4×D17
commutatorC2×C4×C17C22×C4
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C22⋊C4
cosocleC42×C17C23
hypercenterQ8⋊C8D17
hypocenterC17#1497
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #11762541C2semidirect
5. Q8⋊#3622C2semidirect
6. #11762539C2semidirect
7. C17×Q8⋊C8C2semidirect
102. Q8⋊C8D17semidirect
126. C17#1497semidirect

Maximal subgroups

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

subgroupindex
#117633632
#117628652
#117628602
#117625412
Q8⋊#36222
#117625392
C17×Q8⋊C82
#149717