Finite Group Databasebeta

← #355×D199 #38770526

Subgroups — #355×D199 #38770526

Order 25472.

260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 57 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#124×D199C211yesmaximal, characteristic
2#101×D199C212yesmaximal, complemented
3#27163815C211yesmaximal, characteristic, complemented
4#27163814C212yesmaximal, complemented
5#355×C199C211yesmaximal, characteristic, complemented
6C2×Q8×D199C2211yescharacteristic
7C2×D4×D199C411yescharacteristic
8M16×D199—22no—
9#20263944C2211yescharacteristic
10#20263943C411yescharacteristic
11#20263942C411yescharacteristic
12#20263941—22no—
13C22⋊C4×D199—22no—
14C42×D199C411yescharacteristic
15#20263874C2212yescomplemented
16#20263873—22no—
17#124×C199C2211yescharacteristic
18#101×C199C2212yescomplemented
19C23×D199—21no—
20C2×Dic398C2×C411yescharacteristic
21C2×Dic398D412yes—
22Q8×D199—41no—
23Q8×D199—41no—
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—21no—
26D4×D199—41no—
27D4×D199—41no—
28C2×C4×D199—21no—
29C2×C4×D199C2×C411yescharacteristic
30C2×C4×D199D412yes—
31#15691870—22no—
32#15691869—42no—
33C8×D199—42no—
34C22⋊Dic199—22no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37C4×Dic199—21no—
38C4×Dic199C2×C411yescharacteristic
39C2×Q8×C199C2311yescharacteristic
40C2×D4×C199C2×C411yescharacteristic
41M16×C199—22no—
42C22⋊C4×C199—22no—
43C42×C199C2×C411yescharacteristic
44C22×D199—21no—
45C22×D199—44no—
46C22×D199—41no—
47C22×D199—41no—
48C22×D199C22⋊C411yescharacteristic
49Dic398—41no—
50Dic398—42no—
51D796—41no—
52C22⋊D199—42no—
53C2×Dic199—21no—
54C2×Dic199C22⋊C411yescharacteristic
55C2×Dic199C22⋊C412yes—
56C4×D199—22no—
57C4×D199—22no—
58C4×D199—21no—
59C4×D199—21no—
60C4×D199—42no—
61C199⋊C8—42no—
62C23×C199—21no—
63Q8×C199—41no—
64D4×C199—41no—
65C2×C4×C199—21no—
66C2×C4×C199C22×C411yescharacteristic
67C2×C4×C199C2×D412yes—
68C8×C199—42no—
69C22×C199—42no—
70C22×C199—41no—
71C22×C199C2×C22⋊C411yescharacteristic
72C2×D199—22no—
73C2×D199—42no—
74C2×D199—41no—
75C2×D199—82no—
76C2×D199—82no—
77C2×D199#9911yescharacteristic
78C2×D199#9911yescharacteristic
79C4×C199—22no—
80C4×C199—21no—
81C4×C199—41no—
82Dic199—21no—
83Dic199—22no—
84Dic199—41no—
85C2×C199—21no—
86C2×C199—81no—
87C2×C199C2×#9911yescharacteristic
88D199—21no—
89D199—81no—
90D199#35512yesdirect product factor
91C199C2×#35511yescharacteristic, complemented, Sylow
92C2×#355—1991nomaximal, Sylow
93C2×#124—1991no—
94C2×#101—1992no—
95#355—1991no—
96#355—1992no—
97#355D19911yescharacteristic, direct product factor
98C22×Q8—1991no—
99C22×D4—1991no—
100C2×M16—3982no—
101#124—3982no—
102#124—1991no—
103#124—1991no—
104#124—1991no—
105#124C2×D19911yescharacteristic
106C2×C22⋊C4—3982no—
107C2×C42—1991no—
108#101—3982no—
109#101—1992no—
110#101C2×D19912yescomplemented
111C24—3981no—
112C2×Q8—7961no—
113C2×Q8—7961no—
114C2×Q8—1991no—
115C2×Q8—1992no—
116C2×Q8C22×D19911yescharacteristic
117C2×D4—7961no—
118C2×D4—7961no—
119C2×D4—3981no—
120C2×D4—1991no—
121C2×D4C4×D19911yescharacteristic
122C22×C4—3981no—
123C22×C4—1991no—
124C22×C4—1992no—
125M16—22no—
126M16—7962no—
127M16—3982no—
128C2×C8—7962no—
129C22⋊C4—22no—
130C22⋊C4—3982no—
131C22⋊C4—3982no—
132C22⋊C4—3982no—
133C42—3981no—
134C42—1991no—
135C42C4×D19911yescharacteristic
136C23—21no—
137C23—7964no—
138C23—7961no—
139C23—7961no—
140C23—3981no—
141C23—1991no—
142Q8—41no—
143Q8—7962no—
144Q8—7961no—
145D4—41no—
146D4—7961no—
147D4—7962no—
148C2×C4—21no—
149C2×C4—7962no—
150C2×C4—3982no—
151C2×C4—3982no—
152C2×C4—3981no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—1991no—
156C2×C4—1992no—
157C2×C4C2×C4×D19911yescharacteristic
158C2×C4D4×D19912yes—
159C8—42no—
160C8—7962no—
161C22—42no—
162C22—41no—
163C22—7962no—
164C22—7961no—
165C22—15922no—
166C22—15922no—
167C22—3982no—
168C22—1991no—
169C22—1991no—
170C22C22⋊C4×D19911yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—7961no—
175C4—3981no—
176C4—3982no—
177C2—21no—
178C2—81no—
179C2—15921no—
180C2—3981no—
181C2—1992no—
182C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×Q8C22×D199
Fitting#355×C199C2
socleC2×C199C2×#99
prime radicalC2×D4×D199C4
cosocleC2×Q8×C199C23
hypercenter#355D199
hypocenterC199C2×#355
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D199C2semidirect
3. #27163815C2semidirect
4. #27163814C2semidirect
5. #355×C199C2semidirect
15. #20263874C22semidirect
18. #101×C199C22semidirect
90. D199#355direct (N × H)
91. C199C2×#355semidirect
97. #355D199direct (N × H)
110. #101C2×D199semidirect

Maximal subgroups

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

subgroupindex
#124×D1992
#101×D1992
#271638152
#271638142
#355×C1992
C2×#355199