Finite Group Databasebeta

← #23774346

Subgroups — #23774346

Order 10112.

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

normal subgroups only

This group has 384 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#19099115C211yesmaximal, characteristic
2C2×#14579159C211yesmaximal, characteristic
3#19098941C212yesmaximal, complemented
4#19098912C212yesmaximal, complemented
5#19098772C212yesmaximal, complemented
6#19098771C212yesmaximal, complemented
7#19098721C211yesmaximal, characteristic
8#19098719C211yesmaximal, characteristic
9#19098718C211yesmaximal, characteristic
10#19098613C211yesmaximal, characteristic
11#451×C79C211yesmaximal, characteristic
12C22×Dic158C2211yescharacteristic
13C2×Dic316C2212yes—
14C2×#11929614C2212yes—
15C2×#11929614C2212yes—
16#14579160C2211yescharacteristic
17#14579159—21no—
18#14579159—21no—
19#14579159C2212yes—
20#14579159C2211yescharacteristic
21#14579159C2212yes—
22#14579151—21no—
23#14579140C2211yescharacteristic
24#14579138—21no—
25Q8×Dic79C2211yescharacteristic
26#14579126—21no—
27#14579126C2211yescharacteristic
28#14579117—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×C4×Dic79C2211yescharacteristic
32Dic79⋊C8C2212yes—
33#14579086C2212yes—
34#14579085C2212yes—
35#14579084C2212yes—
36C22⋊(C79⋊C8)C2211yescharacteristic
37C8×Dic79C2212yes—
38C22×Q8×C79C2211yescharacteristic
39C2×Q16×C79C2212yes—
40C22⋊Q8×C79C2211yescharacteristic
41#103×C79C2212yes—
42C22⋊C8×C79C2211yescharacteristic
43C2×Dic158—21no—
44C2×Dic158—21no—
45C2×Dic158—42no—
46C2×Dic158C2312yes—
47C2×Dic158C2311yescharacteristic
48C22×Dic79—21no—
49C22×Dic79D411yescharacteristic
50C22×Dic79D411yescharacteristic
51Dic316—42no—
52#11929614—42no—
53#11929614—42no—
54C2×C79⋊C8C2312yes—
55C4⋊Dic79—21no—
56C4⋊Dic79C2311yescharacteristic
57Dic79⋊C4—22no—
58Dic79⋊C4—22no—
59Dic79⋊C4—22no—
60Dic79⋊C4—22no—
61Dic79⋊C4—21no—
62Dic79⋊C4—21no—
63Dic79⋊C4—21no—
64Dic79⋊C4—21no—
65Dic79⋊C4—21no—
66C22⋊Dic79—21no—
67C4×Dic79—21no—
68C4×Dic79—21no—
69C4×Dic79D411yescharacteristic
70C4×Dic79D411yescharacteristic
71C4×Dic79C2311yescharacteristic
72C4×Dic79C2311yescharacteristic
73C2×Q8×C79—21no—
74C2×Q8×C79—42no—
75C2×Q8×C79C2312yes—
76C2×Q8×C79C2311yescharacteristic
77C22×C4×C79—21no—
78C22×C4×C79C2311yescharacteristic
79Q16×C79—42no—
80C2×C8×C79C2312yes—
81C4⋊C4×C79—21no—
82C4⋊C4×C79C2311yescharacteristic
83C22⋊C4×C79—21no—
84C23×C79C2×D411yescharacteristic
85Q8×C79—22no—
86Q8×C79—22no—
87Q8×C79—22no—
88Q8×C79—42no—
89C2×C4×C79—22no—
90C2×C4×C79—21no—
91C2×C4×C79—21no—
92C2×C4×C79—22no—
93C2×C4×C79—42no—
94C2×C4×C79C2×D411yescharacteristic
95C2×C4×C79C2411yescharacteristic
96C2×Dic79—22no—
97C2×Dic79—21no—
98C2×Dic79—21no—
99C2×Dic79—22no—
100C2×Dic79—42no—
101C2×Dic79Q1614yescomplemented
102C2×Dic79C2×D411yescharacteristic
103C2×Dic79C2×D411yescharacteristic
104C2×Dic79C2×D411yescharacteristic
105C2×Dic79C2×D411yescharacteristic
106Dic158—22no—
107Dic158—22no—
108Dic158—22no—
109Dic158—42no—
110Dic158—42no—
111C8×C79—22no—
112C79⋊C8—22no—
113C22×C79—21no—
114C22×C79—21no—
115C22×C79C2×Q1612yes—
116C22×C79C22×D411yescharacteristic
117C4×C79—21no—
118C4×C79—42no—
119C4×C79—41no—
120C4×C79—41no—
121C4×C79H(5,𝔽2)12yes—
122Dic79—21no—
123Dic79—42no—
124Dic79—41no—
125Dic79—41no—
126Dic79C2×Q1612yes—
127Dic79C2×Q1612yes—
128Dic79#13712yes—
129C2×C79—22no—
130C2×C79#53511yescharacteristic
131C2×C79C22×Q1611yescharacteristic
132C2×C79C2×#13711yescharacteristic
133#3209—791nomaximal, Sylow
134C79#320911yescharacteristic, complemented, Sylow
135#533—791no—
136C2×C4⋊Q8—791no—
137#494—792no—
138#487—792no—
139#462—792no—
140#462—792no—
141#451—791no—
142#451—791no—
143#451—791no—
144#451D7911yescharacteristic
145#431—791no—
146C22×Q8—791no—
147C22×Q8C2×D7911yescharacteristic
148C2×Q16—792no—
149C2×Q16—792no—
150C2×Q16—792no—
151C2×Q16C2×D7912yes—
152C4⋊Q8—791no—
153C4⋊Q8—792no—
154C4⋊Q8—792no—
155C4⋊Q8—791no—
156C4⋊Q8—1581no—
157C4⋊Q8—1581no—
158#125—1581no—
159C22⋊Q8—791no—
160C22⋊Q8—1581no—
161C22⋊Q8C2×D7911yescharacteristic
162C4×Q8—791no—
163C4×Q8—791no—
164C4×Q8—1581no—
165#117—1581no—
166C2×C4⋊C4—1581no—
167C2×C4⋊C4—1581no—
168C2×C42—791no—
169C4⋊C8—792no—
170#103—792no—
171#103—792no—
172#103—792no—
173#103C2×D7912yes—
174C22⋊C8—791no—
175C22⋊C8C2×D7911yescharacteristic
176C4×C8—792no—
177C2×Q8—21no—
178C2×Q8—42no—
179C2×Q8—3162no—
180C2×Q8—792no—
181C2×Q8—791no—
182C2×Q8—1581no—
183C2×Q8—1581no—
184C2×Q8C22×D7912yes—
185C2×Q8C22×D7911yescharacteristic
186C22×C4—21no—
187C22×C4—791no—
188C22×C4—791no—
189C22×C4—1581no—
190C22×C4C22×D7911yescharacteristic
191Q16—42no—
192Q16—3162no—
193Q16—3162no—
194Q16—3162no—
195C2×C8—792no—
196C2×C8C22×D7912yes—
197C4⋊C4—21no—
198C4⋊C4—791no—
199C4⋊C4—1582no—
200C4⋊C4—1582no—
201C4⋊C4—1582no—
202C4⋊C4—1582no—
203C4⋊C4—1581no—
204C4⋊C4—1581no—
205C4⋊C4—1581no—
206C4⋊C4—1581no—
207C4⋊C4—1581no—
208C4⋊C4—1581no—
209C4⋊C4C22×D7911yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—1581no—
212C42—791no—
213C42—791no—
214C42—791no—
215C42—791no—
216C42—1581no—
217C42—1581no—
218C23D4×D7911yescharacteristic
219Q8—22no—
220Q8—22no—
221Q8—22no—
222Q8—42no—
223Q8—3162no—
224Q8—3162no—
225Q8—1582no—
226Q8—1582no—
227Q8—1582no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—3162no—
234C2×C4—794no—
235C2×C4—791no—
236C2×C4—791no—
237C2×C4—791no—
238C2×C4—791no—
239C2×C4—1581no—
240C2×C4—1581no—
241C2×C4—1582no—
242C2×C4—1582no—
243C2×C4D4×D7911yescharacteristic
244C2×C4C23×D7911yescharacteristic
245C8—22no—
246C8—1582no—
247C22—21no—
248C22—21no—
249C22Q16×D7912yes—
250C22C2×D4×D7911yescharacteristic
251C4—21no—
252C4—41no—
253C4—42no—
254C4—41no—
255C4—3161no—
256C4—3162no—
257C4—3161no—
258C4—792no—
259C4—792no—
260C4—792no—
261C4—1581no—
262C4#1457920612yes—
263C2—22no—
264C2#1909912811yescharacteristic
265C2C2×Q16×D7911yescharacteristic
266C2#137×D7911yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#451×C79C2
socleC22×C79C22×D4
prime radicalC23×C79C2×D4
hypercenter#451D79
hypocenterC79#3209
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #19098941C2semidirect
4. #19098912C2semidirect
5. #19098772C2semidirect
6. #19098771C2semidirect
101. C2×Dic79Q16semidirect
134. C79#3209semidirect