Finite Group Databasebeta

← #28403601

Subgroups — #28403601

Order 14464.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#20299321C211yesmaximal, characteristic
2#20298990C212yesmaximal
3#20298977C211yesmaximal, characteristic, complemented
4#20298820C212yesmaximal
5#20298808C211yesmaximal, characteristic, complemented
6#20298805C212yesmaximal, complemented
7#20298800C211yesmaximal, characteristic
8#20298729C211yesmaximal, characteristic, complemented
9#20298703C212yesmaximal, complemented
10C4×#14268634C211yesmaximal, characteristic
11#557×C113C211yesmaximal, characteristic, complemented
12C2×#14268637C2212yes—
13C2×#14268637C2212yes—
14C2×#14268637C2212yes—
15C2×#14268634C2211yescharacteristic
16C2×#14268634C2212yes—
17#16813622C2211yescharacteristic
18#16813617—22no—
19#16813610C2212yes—
20#16813598—21no—
21C4×Dic226C2212yes—
22C4×D452C2211yescharacteristic
23#16813580—21no—
24#16813553C2211yescharacteristic
25C4⋊#11744751C2211yescharacteristic
26C4⋊#11744751C2212yes—
27#16813546C2212yes—
28#16813546C2211yescharacteristic
29#16813545C2212yes—
30#16813542C2211yescharacteristic
31#16813533C2212yes—
32C4×#11744751C2211yescharacteristic
33C4⋊Q8×C113C2211yescharacteristic
34C4⋊Q8×C113C2212yes—
35#125×C113—21no—
36#125×C113—22no—
37C4×Q8×C113—21no—
38C4×Q8×C113C2211yescharacteristic
39C4×Q8×C113C2212yes—
40C4×Q8×C113C2211yescharacteristic
41C2×Dic226C2312yes—
42C2×D452C2311yescharacteristic
43C2×C4×D113—21no—
44#14268637—24no—
45#14268637—42no—
46#14268637—42no—
47#14268634—22no—
48#14268634—42no—
49C2×#11744751C2312yes—
50C2×#11744751C2311yescharacteristic
51C2×#11744751C2311yescharacteristic
52C4⋊Dic113—21no—
53C4⋊Dic113—21no—
54C4⋊Dic113C2311yescharacteristic
55#14268626—22no—
56#14268624—22no—
57#14268624—21no—
58#14268624—22no—
59C4×Dic113—22no—
60C2×Q8×C113D411yescharacteristic
61C2×Q8×C113C2311yescharacteristic
62C2×Q8×C113C2312yes—
63C4⋊C4×C113—22no—
64C4⋊C4×C113—21no—
65C4⋊C4×C113—22no—
66C4⋊C4×C113—21no—
67C4⋊C4×C113—22no—
68C4⋊C4×C113—21no—
69C4⋊C4×C113D411yescharacteristic
70C4⋊C4×C113D412yes—
71C4⋊C4×C113C2311yescharacteristic
72C42×C113—22no—
73C42×C113—21no—
74C42×C113C2311yescharacteristic
75C42×C113C2312yes—
76C22×D113—21no—
77Q8×C113—22no—
78Q8×C113—21no—
79Q8×C113—22no—
80Q8×C113—22no—
81Q8×C113—21no—
82C2×C4×C113—21no—
83C2×C4×C113—21no—
84C2×C4×C113—22no—
85C2×C4×C113C2×D412yes—
86C2×C4×C113C2×D412yes—
87C2×C4×C113C2×D411yescharacteristic
88C2×C4×C113C2×D411yescharacteristic
89C2×C4×C113C2411yescharacteristic
90C2×Dic113—21no—
91C2×Dic113—22no—
92D452—21no—
93D452—21no—
94C4×D113—42no—
95Dic226—22no—
96Dic226—22no—
97#11744751—22no—
98#11744751—21no—
99#11744751#4112yes—
100C22×C113C22×D411yescharacteristic
101C2×D113—41no—
102C2×D113—41no—
103C2×D113—41no—
104C4×C113—22no—
105C4×C113—22no—
106C4×C113—21no—
107C4×C113—41no—
108C4×C113—42no—
109C4×C113—41no—
110C4×C113#13712yes—
111C4×C113C2×#4111yescharacteristic
112C4×C113H(5,𝔽2)11yescharacteristic
113Dic113—42no—
114Dic113—41no—
115C2×C113#55011yescharacteristic
116C2×C113C2×#13711yescharacteristic
117C2×C113#57811yescharacteristic
118D113—81no—
119#3398—1131nomaximal, Sylow
120C113#339811yescharacteristic, complemented, Sylow
121#557D11311yescharacteristic, complemented
122#556—1131no—
123#497—1132no—
124#494—1131no—
125#464—1132no—
126#462—1132no—
127#462—1131no—
128#461—1131no—
129#446—1131no—
130#441—1132no—
131C4×SD16—1131no—
132C2×Q16—1132no—
133C2×Q16—1132no—
134C2×Q16—1132no—
135C2×SD16—1132no—
136C2×SD16—1131no—
137C4⋊Q8—1131no—
138C4⋊Q8C2×D11311yescharacteristic
139C4⋊Q8C2×D11312yes—
140#126—2262no—
141#125—21no—
142#125—22no—
143#124—1132no—
144C22⋊Q8—2261no—
145C4×Q8—21no—
146C4×Q8—1132no—
147C4×Q8C2×D11311yescharacteristic
148C4×Q8C2×D11312yes—
149C4×Q8C2×D11311yescharacteristic
150C4×D4—1131no—
151#117—2261no—
152#106—1131no—
153C4⋊C8—1131no—
154C4⋊C8—1132no—
155#103—1131no—
156#103—1132no—
157#103—1132no—
158#102—1131no—
159#97—1132no—
160C4×C8—1131no—
161C2×Q8—1132no—
162C2×Q8C22⋊D11311yescharacteristic
163C2×Q8C22×D11311yescharacteristic
164C2×Q8C22×D11312yes—
165C2×D4—1131no—
166C22×C4—2261no—
167Q16—4522no—
168Q16—4522no—
169Q16—2264no—
170SD16—4522no—
171SD16—2262no—
172C2×C8—1131no—
173C2×C8—1131no—
174C2×C8—1132no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—1131no—
182C4⋊C4—2261no—
183C4⋊C4—2262no—
184C4⋊C4—2261no—
185C4⋊C4C22⋊D11311yescharacteristic
186C4⋊C4C22⋊D11312yes—
187C4⋊C4C22×D11311yescharacteristic
188C22⋊C4—2262no—
189C22⋊C4—2261no—
190C22⋊C4—2262no—
191C42—22no—
192C42—21no—
193C42—2262no—
194C42C22×D11311yescharacteristic
195C42C22×D11312yes—
196C23—2261no—
197Q8—22no—
198Q8—21no—
199Q8—22no—
200Q8—22no—
201Q8—21no—
202Q8—2262no—
203Q8—2262no—
204D4—2261no—
205D4—2261no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—4522no—
210C2×C4—2261no—
211C2×C4—2262no—
212C2×C4C2×C22⋊D11312yes—
213C2×C4C2×C22⋊D11312yes—
214C2×C4C2×C22⋊D11311yescharacteristic
215C2×C4C2×C22⋊D11311yescharacteristic
216C2×C4C23×D11311yescharacteristic
217C8—1132no—
218C8—2262no—
219C8—2261no—
220C22—4521no—
221C22—4521no—
222C22—4521no—
223C22C22×C22⋊D11311yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—41no—
230C4—4522no—
231C4—4521no—
232C4#1681365312yes—
233C4#41×D11311yescharacteristic
234C4#1681366811yescharacteristic
235C2—9041no—
236C2#2029926711yescharacteristic
237C2C2×#1681365311yescharacteristic
238C2#2029941311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#557×C113C2
socleC22×C113C22×D4
prime radicalC2×D452C23
hypercenter#557D113
hypocenterC113#3398
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
3. #20298977C2semidirect
5. #20298808C2semidirect
6. #20298805C2semidirect
8. #20298729C2semidirect
9. #20298703C2semidirect
11. #557×C113C2semidirect
120. C113#3398semidirect
121. #557D113semidirect

Maximal subgroups

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

subgroupindex
#202993212
#202989902
#202989772
#202988202
#202988082
#202988052
#202988002
#202987292
#202987032
C4×#142686342
#557×C1132
#3398113