Finite Group Databasebeta

← #355×D97 #27143412

Subgroups — #355×D97 #27143412

Order 12416.

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×D97C211yesmaximal, characteristic
2#101×D97C212yesmaximal, complemented
3#20260532C211yesmaximal, characteristic, complemented
4#20260531C212yesmaximal, complemented
5#355×C97C211yesmaximal, characteristic, complemented
6C2×Q8×D97C2211yescharacteristic
7C2×D4×D97C411yescharacteristic
8M16×D97—22no—
9#15690799C2211yescharacteristic
10#15690798C411yescharacteristic
11#15690797C411yescharacteristic
12#15690796—22no—
13C22⋊C4×D97—22no—
14C42×D97C411yescharacteristic
15#15690729C2212yescomplemented
16#15690728—22no—
17#124×C97C2211yescharacteristic
18#101×C97C2212yescomplemented
19C23×D97—21no—
20C2×Dic194C2×C411yescharacteristic
21C2×Dic194D412yes—
22Q8×D97—41no—
23Q8×D97—41no—
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97—21no—
26D4×D97—41no—
27D4×D97—41no—
28C2×C4×D97—21no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D412yes—
31#13095495—22no—
32#13095494—42no—
33C8×D97—42no—
34C22⋊Dic97—22no—
35#13095488—22no—
36#13095488—22no—
37C4×Dic97—21no—
38C4×Dic97C2×C411yescharacteristic
39C2×Q8×C97C2311yescharacteristic
40C2×D4×C97C2×C411yescharacteristic
41M16×C97—22no—
42C22⋊C4×C97—22no—
43C42×C97C2×C411yescharacteristic
44C23×C97—21no—
45C22×D97—21no—
46C22×D97—44no—
47C22×D97—41no—
48C22×D97—41no—
49C22×D97C22⋊C411yescharacteristic
50Q8×C97—41no—
51D4×C97—41no—
52C2×C4×C97—21no—
53C2×C4×C97C22×C411yescharacteristic
54C2×C4×C97C2×D412yes—
55C22⋊D97—42no—
56C2×Dic97—21no—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C412yes—
59D388—41no—
60C4×D97—22no—
61C4×D97—22no—
62C4×D97—21no—
63C4×D97—21no—
64C4×D97—42no—
65Dic194—41no—
66Dic194—42no—
67C8×C97—42no—
68#11715895—42no—
69C22×C97—42no—
70C22×C97—41no—
71C22×C97C2×C22⋊C411yescharacteristic
72C2×D97—22no—
73C2×D97—42no—
74C2×D97—41no—
75C2×D97—82no—
76C2×D97—82no—
77C2×D97#9911yescharacteristic
78C2×D97#9911yescharacteristic
79C4×C97—22no—
80C4×C97—21no—
81C4×C97—41no—
82Dic97—22no—
83Dic97—21no—
84Dic97—41no—
85C2×C97—21no—
86C2×C97—81no—
87C2×C97C2×#9911yescharacteristic
88D97—21no—
89D97—81no—
90D97#35512yesdirect product factor
91C2×#355—971nomaximal, Sylow
92C97C2×#35511yescharacteristic, complemented, Sylow
93C2×#124—971no—
94C2×#101—972no—
95#355—971no—
96#355—972no—
97#355D9711yescharacteristic, direct product factor
98C22×Q8—971no—
99C22×D4—971no—
100C2×M16—1942no—
101#124—971no—
102#124—971no—
103#124—971no—
104#124—1942no—
105#124C2×D9711yescharacteristic
106C2×C22⋊C4—1942no—
107C2×C42—971no—
108#101—972no—
109#101—1942no—
110#101C2×D9712yescomplemented
111C24—1941no—
112C2×Q8—971no—
113C2×Q8—972no—
114C2×Q8—3881no—
115C2×Q8—3881no—
116C2×Q8C22×D9711yescharacteristic
117C2×D4—971no—
118C2×D4—3881no—
119C2×D4—3881no—
120C2×D4—1941no—
121C2×D4C4×D9711yescharacteristic
122C22×C4—971no—
123C22×C4—972no—
124C22×C4—1941no—
125M16—22no—
126M16—3882no—
127M16—1942no—
128C2×C8—3882no—
129C22⋊C4—22no—
130C22⋊C4—1942no—
131C22⋊C4—1942no—
132C22⋊C4—1942no—
133C42—971no—
134C42—1941no—
135C42C4×D9711yescharacteristic
136C23—21no—
137C23—971no—
138C23—3881no—
139C23—3881no—
140C23—3884no—
141C23—1941no—
142Q8—41no—
143Q8—3882no—
144Q8—3881no—
145D4—41no—
146D4—3881no—
147D4—3882no—
148C2×C4—21no—
149C2×C4—971no—
150C2×C4—972no—
151C2×C4—3882no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4C2×C4×D9711yescharacteristic
158C2×C4D4×D9712yes—
159C8—42no—
160C8—3882no—
161C22—42no—
162C22—41no—
163C22—7762no—
164C22—7762no—
165C22—971no—
166C22—971no—
167C22—3882no—
168C22—3881no—
169C22—1942no—
170C22C22⋊C4×D9711yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—3881no—
175C4—1941no—
176C4—1942no—
177C2—21no—
178C2—81no—
179C2—7761no—
180C2—972no—
181C2—1941no—
182C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#355×C97C2
socleC2×C97C2×#99
prime radicalC2×D4×D97C4
cosocleC2×Q8×C97C23
hypercenter#355D97
hypocenterC97C2×#355
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D97C2semidirect
3. #20260532C2semidirect
4. #20260531C2semidirect
5. #355×C97C2semidirect
15. #15690729C22semidirect
18. #101×C97C22semidirect
90. D97#355direct (N × H)
92. C97C2×#355semidirect
97. #355D97direct (N × H)
110. #101C2×D97semidirect

Maximal subgroups

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

subgroupindex
#124×D972
#101×D972
#202605322
#202605312
#355×C972
C2×#35597