Finite Group Databasebeta

← #38769052

Subgroups — #38769052

Order 25472.

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

normal subgroups only

This group has 316 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
1C2×#20263913C211yesmaximal, characteristic
2C2×#20263889C211yesmaximal, characteristic, complemented
3C2×#20263886C211yesmaximal, characteristic, complemented
4C2×(C2×D199)⋊C8C211yesmaximal, characteristic
5#27163875C211yesmaximal, characteristic
6#27163780C211yesmaximal, characteristic, complemented
7#340×C199C211yesmaximal, characteristic, complemented
8C22×C4×D199C2211yescharacteristic
9C22×C199⋊C8C2211yescharacteristic
10#20263913—22no—
11#20263913—22no—
12C2×C4⋊Dic199C2211yescharacteristic
13C2×C4⋊Dic199C2211yescharacteristic
14C2×(C2×D199)⋊C4—21no—
15C2×(C2×D199)⋊C4—21no—
16C2×C4×Dic199—21no—
17#20263889—22no—
18#20263886—22no—
19(C2×D199)⋊C8C414yescomplemented
20C199⋊#95—21no—
21C22×C8×C199C2211yescharacteristic
22C2×C4⋊C4×C199C2211yescharacteristic
23C2×C4⋊C4×C199C2211yescharacteristic
24C23×D199D411yescharacteristic
25C22×Dic199—21no—
26C22×Dic199—21no—
27C22×Dic199D411yescharacteristic
28C2×C4×D199—21no—
29C2×C4×D199—21no—
30C2×C4×D199—21no—
31C2×C4×D199—21no—
32C2×C4×D199C2×C412yes—
33C2×C4×D199Q812yes—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36C2×C199⋊C8C2×C412yes—
37C4⋊Dic199—21no—
38C4⋊Dic199—21no—
39C4⋊Dic199—21no—
40C4⋊Dic199D412yes—
41(C2×D199)⋊C4—24no—
42(C2×D199)⋊C4—42no—
43C4×Dic199—24no—
44C22×C4×C199—21no—
45C22×C4×C199—21no—
46C22×C4×C199C2311yescharacteristic
47C2×C8×C199—21no—
48C2×C8×C199—21no—
49C2×C8×C199C2×C412yes—
50C4⋊C4×C199—21no—
51C4⋊C4×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199D412yes—
54C22×D199—21no—
55C22×D199—21no—
56C22×D199—21no—
57C22×D199—21no—
58C22×D199—21no—
59C22×D199—21no—
60C22×D199C4⋊C412yes—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—22no—
64C2×Dic199—41no—
65C2×Dic199—41no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C2×Dic199C4⋊C412yes—
70C4×D199—22no—
71C4×D199—22no—
72C4×D199—22no—
73C4×D199—22no—
74C199⋊C8—22no—
75C23×C199C2×D411yescharacteristic
76C2×C4×C199—22no—
77C2×C4×C199—41no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199—41no—
81C2×C4×C199—41no—
82C2×C4×C199C22×C411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×D411yescharacteristic
85C2×C4×C199C2×Q811yescharacteristic
86C2×C4×C199#4111yescharacteristic
87C2×C4×C199#4111yescharacteristic
88C8×C199—22no—
89C22×C199C2×C4⋊C411yescharacteristic
90C22×C199#12011yescharacteristic
91C22×C199#12311yescharacteristic
92C22×C199#13511yescharacteristic
93C22×C199#13511yescharacteristic
94C22×C199AGL(1,ℤ/8)11yescharacteristic
95C22×C199#13711yescharacteristic
96C2×D199—22no—
97C2×D199—22no—
98C2×D199—22no—
99C2×D199—41no—
100C2×D199—41no—
101C2×D199—41no—
102C2×D199—41no—
103C4×C199—42no—
104C4×C199—81no—
105C4×C199C4×D411yescharacteristic
106C4×C199C4×D411yescharacteristic
107C4×C199C22⋊Q811yescharacteristic
108C4×C199C22⋊Q811yescharacteristic
109Dic199—22no—
110Dic199—42no—
111Dic199—81no—
112C2×C199#38511yescharacteristic
113C2×C199#42711yescharacteristic
114C2×C199#42811yescharacteristic
115C2×C199#44911yescharacteristic
116C2×C199#45211yescharacteristic
117C2×C199#48211yescharacteristic
118C2×C199#48511yescharacteristic
119D199—42no—
120C199#185411yescharacteristic, complemented, Sylow
121#1854—1991nomaximal, Sylow
122C2×#117—1991no—
123C2×#107—1991no—
124C2×#106—1991no—
125C2×C22⋊C8—1991no—
126#380—1991no—
127#340—1991no—
128#340D19911yescharacteristic, complemented
129C23×C4—1991no—
130C22×C8—1991no—
131C22×C8C2×D19911yescharacteristic
132#117—3982no—
133#117—3982no—
134C2×C4⋊C4—1991no—
135C2×C4⋊C4—1991no—
136C2×C4⋊C4C2×D19911yescharacteristic
137C2×C4⋊C4C2×D19911yescharacteristic
138C2×C22⋊C4—3981no—
139C2×C22⋊C4—3981no—
140C2×C42—3981no—
141#107—3982no—
142#106—3982no—
143C22⋊C8—1994no—
144#95—3981no—
145C24—1991no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3981no—
149C22×C4—3981no—
150C22×C4—3981no—
151C22×C4—3981no—
152C22×C4—3981no—
153C22×C4—3981no—
154C22×C4—1991no—
155C22×C4—1992no—
156C22×C4—1992no—
157C22×C4C22×D19911yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—3981no—
161C2×C8—3981no—
162C2×C8—1992no—
163C2×C8C4×D19912yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—3981no—
168C4⋊C4—3981no—
169C4⋊C4—3981no—
170C4⋊C4—1992no—
171C4⋊C4D79612yes—
172C22⋊C4—7962no—
173C22⋊C4—3984no—
174C42—3984no—
175C23—3981no—
176C23—3981no—
177C23—3981no—
178C23—3981no—
179C23—3981no—
180C23—3981no—
181C23—1992no—
182C23D4×D19911yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—7961no—
190C2×C4—7961no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—7961no—
194C2×C4—3982no—
195C2×C4—3981no—
196C2×C4—3981no—
197C2×C4—3982no—
198C2×C4—3982no—
199C2×C4—3982no—
200C2×C4—3982no—
201C2×C4—1992no—
202C2×C4C2×C4×D19911yescharacteristic
203C2×C4D4×D19911yescharacteristic
204C2×C4C2×D79611yescharacteristic
205C2×C4Q8×D19911yescharacteristic
206C2×C4#1569188511yescharacteristic
207C2×C4#1569188611yescharacteristic
208C8—22no—
209C8—3982no—
210C22—7961no—
211C22—7961no—
212C22—7961no—
213C22—7961no—
214C22—3982no—
215C22—3982no—
216C22—3982no—
217C22C4⋊C4×D19911yescharacteristic
218C22#2026392511yescharacteristic
219C22#2026393711yescharacteristic
220C22#2026397411yescharacteristic
221C22#2026397811yescharacteristic
222C22#2026398211yescharacteristic
223C22#2026398411yescharacteristic
224C4—42no—
225C4—81no—
226C4—7962no—
227C4—15921no—
228C4—3982no—
229C4(C22⋊D199)⋊C411yescharacteristic
230C4C4×D79611yescharacteristic
231C4#2026393311yescharacteristic
232C4#2026393511yescharacteristic
233C2—7962no—
234C2#2716389211yescharacteristic
235C2#2716405511yescharacteristic
236C2#2716405711yescharacteristic
237C2#2716416611yescharacteristic
238C2#2716418111yescharacteristic
239C2#2716434011yescharacteristic
240C2#2716435811yescharacteristic
subgroupG/H
centerC23D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#340×C199C2
socleC23×C199C2×D4
prime radicalC23×D199D4
cosocleC22×C4×C199C23
hypercenter#340D199
hypocenterC199#1854
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20263889C2semidirect
3. C2×#20263886C2semidirect
6. #27163780C2semidirect
7. #340×C199C2semidirect
19. (C2×D199)⋊C8C4semidirect
120. C199#1854semidirect
128. #340D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639132
C2×#202638892
C2×#202638862
C2×(C2×D199)⋊C82
#271638752
#271637802
#340×C1992
#1854199