Finite Group Databasebeta

← #38768825

Subgroups — #38768825

Order 25472.

536 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 121 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#27164035C212yesmaximal, complemented
2#27164027C212yesmaximal, complemented
3#27164002C211yesmaximal, characteristic, complemented
4#27163992C211yesmaximal, characteristic, complemented
5C2×2−1+4×C199C211yesmaximal, characteristic
6C2×#15691870C2212yes—
7#20263914C2212yes—
8#20263882—24no—
9#20263879—24no—
10#20263874—21no—
11#20263872—21no—
12#20263870—22no—
132−1+4×C199—24no—
142−1+4×C199—24no—
15C2×#41×C199—21no—
16C2×#41×C199—22no—
17C2×#41×C199C412yescomplemented
18C2×#41×C199C2211yescharacteristic
19C2×#41×C199C2211yescharacteristic
20C22×Q8×C199—21no—
21C22×Q8×C199C412yescomplemented
22C22×Q8×C199C2211yescharacteristic
23#15691870—22no—
24#15691870—22no—
25C2×C199⋊C8—22no—
26C4⋊Dic199—22no—
27C22⋊Dic199—22no—
28C4×Dic199—22no—
29#41×C199—216no—
30#41×C199—28no—
31#41×C199—24no—
32#41×C199—24no—
33#41×C199—22no—
34#41×C199—22no—
35#41×C199—22no—
36#41×C199D414yes—
37C2×Q8×C199—28no—
38C2×Q8×C199—24no—
39C2×Q8×C199—24no—
40C2×Q8×C199—22no—
41C2×Q8×C199—21no—
42C2×Q8×C199—22no—
43C2×Q8×C199C2×C411yescharacteristic
44C2×Q8×C199C2×C412yes—
45C2×Q8×C199D414yes—
46C2×Q8×C199D411yescharacteristic
47C2×D4×C199—22no—
48C2×D4×C199—21no—
49C2×D4×C199C2×C411yescharacteristic
50C2×D4×C199D412yes—
51C2×D4×C199D411yescharacteristic
52C22×C4×C199—22no—
53C22×C4×C199—21no—
54C22×C4×C199—22no—
55C22×C4×C199—21no—
56C22×C4×C199C2×C412yes—
57C22×C4×C199C2311yescharacteristic
58C2×Dic199—42no—
59C199⋊C8—42no—
60C23×C199—21no—
61C23×C199C22⋊C412yes—
62C23×C199C2×D411yescharacteristic
63Q8×C199—28no—
64Q8×C199—21no—
65Q8×C199—22no—
66Q8×C199—24no—
67Q8×C199—22no—
68Q8×C199C22⋊C412yes—
69Q8×C199C22⋊C414yes—
70D4×C199—28no—
71D4×C199—24no—
72D4×C199—24no—
73D4×C199—21no—
74D4×C199—22no—
75D4×C199C22⋊C412yes—
76C2×C4×C199—28no—
77C2×C4×C199—28no—
78C2×C4×C199—21no—
79C2×C4×C199—22no—
80C2×C4×C199—24no—
81C2×C4×C199—22no—
82C2×C4×C199—22no—
83C2×C4×C199—24no—
84C2×C4×C199C22⋊C412yes—
85C2×C4×C199C22×C411yescharacteristic
86C2×C4×C199C2×D412yes—
87C2×C4×C199C2×D412yes—
88C2×C4×C199C2×D411yescharacteristic
89C22×C199—22no—
90C22×C199—22no—
91C22×C199—41no—
92C22×C199—42no—
93C22×C199—42no—
94C22×C199C2×C22⋊C411yescharacteristic
95C22×C199#12012yes—
96C4×C199—24no—
97C4×C199—22no—
98C4×C199—22no—
99C4×C199C2×C22⋊C411yescharacteristic
100C4×C199C2×C22⋊C411yescharacteristic
101C4×C199#12012yes—
102Dic199—82no—
103C2×C199—22no—
104C2×C199—21no—
105C2×C199—42no—
106C2×C199—42no—
107C2×C199#37911yescharacteristic
108C199#179611yescharacteristic, complemented, Sylow
109#1796—1991nomaximal, Sylow
110C2×2−1+4D19911yescharacteristic
111#421—1992no—
112#419—1992no—
113#413—1991no—
114#410—1991no—
1152−1+4—24no—
1162−1+4—24no—
117C2×#41—21no—
118C2×#41—22no—
119C2×#41Dic19912yescomplemented
120C2×#41C2×D19911yescharacteristic
121C2×#41C2×D19911yescharacteristic
122C22×Q8—21no—
123C22×Q8Dic19912yescomplemented
124C22×Q8C2×D19911yescharacteristic
125C2×M16—1992no—
126#117—1992no—
127#104—3984no—
128#103—3984no—
129#101—3981no—
130#100—3981no—
131#99—3982no—
132#41—216no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—22no—
137#41—22no—
138#41—22no—
139#41C22⋊D19914yes—
140C2×Q8—28no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8C2×Dic19911yescharacteristic
147C2×Q8C2×Dic19912yes—
148C2×Q8C22⋊D19914yes—
149C2×Q8C22⋊D19911yescharacteristic
150C2×D4—22no—
151C2×D4—21no—
152C2×D4C2×Dic19911yescharacteristic
153C2×D4C22⋊D19912yes—
154C2×D4C22⋊D19911yescharacteristic
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4C2×Dic19912yes—
160C22×C4C22×D19911yescharacteristic
161M16—3982no—
162M16—3982no—
163C2×C8—3982no—
164C4⋊C4—3982no—
165C22⋊C4—3982no—
166C42—3982no—
167C23—21no—
168C23C22⋊Dic19912yes—
169C23C2×C22⋊D19911yescharacteristic
170Q8—28no—
171Q8—21no—
172Q8—22no—
173Q8—24no—
174Q8—22no—
175Q8C22⋊Dic19912yes—
176Q8C22⋊Dic19914yes—
177D4—28no—
178D4—24no—
179D4—24no—
180D4—21no—
181D4—22no—
182D4C22⋊Dic19912yes—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—7962no—
192C2×C4C22⋊Dic19912yes—
193C2×C4C22×Dic19911yescharacteristic
194C2×C4C2×C22⋊D19912yes—
195C2×C4C2×C22⋊D19912yes—
196C2×C4C2×C22⋊D19911yescharacteristic
197C8—7962no—
198C22—22no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22C2×C22⋊Dic19911yescharacteristic
204C22#2026392412yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—15922no—
209C4C2×C22⋊Dic19911yescharacteristic
210C4C2×C22⋊Dic19911yescharacteristic
211C4#2026392412yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—42no—
216C2#2716387411yescharacteristic
subgroupG/H
centerC2#27163874
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting, prime radicalC2×2−1+4×C199C2
socleC2×C199#379
cosocleC22×C4×C199C23
hypercenterC2×2−1+4D199
hypocenterC199#1796
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #27164035C2semidirect
2. #27164027C2semidirect
3. #27164002C2semidirect
4. #27163992C2semidirect
17. C2×#41×C199C4semidirect
21. C22×Q8×C199C4semidirect
108. C199#1796semidirect
119. C2×#41Dic199semidirect
123. C22×Q8Dic199semidirect

Maximal subgroups

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

subgroupindex
#271640352
#271640272
#271640022
#271639922
C2×2−1+4×C1992
#1796199