Finite Group Databasebeta

← #17982021

Subgroups — #17982021

Order 4352.

342 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 99 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#14545982C212yesmaximal
2#14545818C212yesmaximal
3#14545817C212yesmaximal
4C2×#11763858C211yesmaximal, characteristic
5#11763912—22no—
6C22⋊#63241C2214yes—
7C4×#63241—22no—
8#11763858—22no—
9#11763858—22no—
10C2×#10587544C411yescharacteristic
11C2×#10587544C2211yescharacteristic
12C2×C4⋊#3624C2211yescharacteristic
13C2×C4×#3624C411yescharacteristic
14C2×C4×#3624C411yescharacteristic
15C2×C4×#3624C2211yescharacteristic
16C2×#10587573C411yescharacteristic
17C22×Dic34C811yescharacteristic
18C22×#3624C2×C411yescharacteristic
19C22×#3624C2×C411yescharacteristic
20C22×#3624C2×C411yescharacteristic
21C22×#3624C2311yescharacteristic
22C2×#63241—24no—
23C2×C4⋊C4×C17C811yescharacteristic
24C2×#63225C811yescharacteristic
25C2×#63225C811yescharacteristic
26C2×C4×Dic17C2×C411yescharacteristic
27C2×C4×Dic17C2×C411yescharacteristic
28C2×C4×Dic17C2×C411yescharacteristic
29#10587573—22no—
30#10587573—22no—
31#10587544—22no—
32#10587544—22no—
33C4⋊#3624—22no—
34C4×#3624—21no—
35C4×#3624—21no—
36C4×#3624—21no—
37C4×#3624—21no—
38C4×#3624D412yes—
39C4×#3624D412yes—
40C22×C4×C17C2×C811yescharacteristic
41C22×C4×C17C2×C811yescharacteristic
42C22×C4×C17C2×C811yescharacteristic
43C22×Dic17C2×C811yescharacteristic
44C22×Dic17C2×C811yescharacteristic
45C22×Dic17C2×C811yescharacteristic
46C22×Dic17C22×C411yescharacteristic
47C2×Dic34—22no—
48C2×Dic34—22no—
49C2×Dic34C1614yescomplemented
50C2×#3624—22no—
51C2×#3624—21no—
52C2×#3624—21no—
53C2×#3624—41no—
54C2×#3624—41no—
55C2×#3624—41no—
56C2×#3624C22⋊C411yescharacteristic
57C2×#3624C22⋊C411yescharacteristic
58C2×#3624C2×D412yes—
59#63241—44no—
60C4⋊C4×C17—22no—
61#63225—22no—
62#63225—22no—
63C4×Dic17—21no—
64C4×Dic17—21no—
65C4×Dic17—21no—
66C4×Dic17—21no—
67C4×Dic17C22⋊C412yes—
68C4×Dic17M1612yes—
69#11493—171nomaximal, Sylow
70C23×C17C22×C811yescharacteristic
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—41no—
75C2×C4×C17—41no—
76C2×C4×C17C22⋊C812yes—
77C2×C4×C17C2×C1612yes—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—21no—
83C2×Dic17—41no—
84C2×Dic17C22⋊C812yes—
85C2×Dic17C2×C1612yes—
86C2×Dic17C2×C1612yes—
87C2×Dic17M3212yes—
88C2×Dic17C2×C22⋊C411yescharacteristic
89C2×Dic17C2×M1611yescharacteristic
90Dic34—28no—
91#3624—22no—
92#3624—21no—
93#3624—41no—
94#3624—41no—
95#3624#10112yes—
96C2×C8×Q8—171no—
97#2162—172no—
98#1322—172no—
99#1322—172no—
100C22×C17—21no—
101C22×C17—21no—
102C22×C17C2×C22⋊C811yescharacteristic
103C22×C17C22×C1611yescharacteristic
104C22×C17C2×M3211yescharacteristic
105C4×C17—22no—
106C4×C17—22no—
107C4×C17—41no—
108Dic17—22no—
109Dic17—21no—
110Dic17—41no—
111Dic17C22⋊C1614yes—
112Dic17#41011yescharacteristic
113Dic17C2×#10111yescharacteristic
114C2×C4×Q8—171no—
115C8×Q8—342no—
116C8×Q8—342no—
117C2×C4⋊C8—171no—
118C2×C4⋊C8—171no—
119C2×C4⋊C8—171no—
120C2×C4×C8—171no—
121C2×C4×C8—171no—
122C2×C4×C8—171no—
123C4⋊C16—342no—
124C22⋊C16—174no—
125C4×C16—342no—
126C2×C17—21no—
127C2×C17—21no—
128C2×C17#146111yescharacteristic
129C2×C17C2×C22⋊C1611yescharacteristic
130C2×C17#211511yescharacteristic
131C22×Q8—171no—
132C22×C8—171no—
133C22×C8—171no—
134C22×C8—171no—
135C22×C8—171no—
136C4×Q8—342no—
137C4×Q8—342no—
138C2×C4⋊C4—171no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4#363311yescharacteristic
141C2×C42—171no—
142C2×C42—171no—
143C2×C42—171no—
144C2×C16—344no—
145C4⋊C8—342no—
146C4⋊C8—342no—
147C4⋊C8—342no—
148C4×C8—172no—
149C4×C8—172no—
150C4×C8—341no—
151C4×C8—341no—
152C4×C8—341no—
153C4×C8—341no—
154C17#1149311yescharacteristic, complemented, Sylow
155C2×Q8—174no—
156C2×Q8—342no—
157C2×Q8—342no—
158C22×C4—171no—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4C2×#363311yescharacteristic
163C22×C4C2×#363311yescharacteristic
164C22×C4C2×#363311yescharacteristic
165C2×C8—171no—
166C2×C8—171no—
167C2×C8—172no—
168C2×C8—342no—
169C2×C8—341no—
170C2×C8—341no—
171C2×C8—681no—
172C2×C8—681no—
173C2×C8—681no—
174C4⋊C4—22no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C42—172no—
178C42—172no—
179C42—341no—
180C42—341no—
181C42—341no—
182C42—341no—
183C16—684no—
184C23C22×#363311yescharacteristic
185Q8—348no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—171no—
192C2×C4—172no—
193C2×C4—172no—
194C2×C4—172no—
195C2×C4—171no—
196C2×C4—172no—
197C2×C4—341no—
198C2×C4—341no—
199C2×C4—341no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—681no—
203C2×C4#1058766512yes—
204C2×C4C22⋊#363312yes—
205C8—172no—
206C8—342no—
207C8—341no—
208C8—681no—
209C8—681no—
210C22—21no—
211C22—21no—
212C22C2×C22⋊#363311yescharacteristic
213C22C2×#1058766511yescharacteristic
214C22#1176392811yescharacteristic
215C4—22no—
216C4—22no—
217C4—41no—
218C4—174no—
219C4—171no—
220C4—171no—
221C4—342no—
222C4—341no—
223C4—681no—
224C2—21no—
225C2—21no—
226C2#1454589911yescharacteristic
227C2#1454594911yescharacteristic
228C2#1454596011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3633
commutatorC22×C17C22×C16
FrattiniC23C22×#3633
FittingC2×C4⋊C4×C17C8
socleC22×C17C2×C22⋊C8
prime radicalC23×C17C22×C8
cosocleC22×#3624C23
hypercenterC2×C4⋊C4#3633
hypocenterC17#11493
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
49. C2×Dic34C16semidirect
154. C17#11493semidirect

Maximal subgroups

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

subgroupindex
#145459822
#145458182
#145458172
C2×#117638582
#1149317