Finite Group Databasebeta

← #353×D97 #27143392

Subgroups — #353×D97 #27143392

Order 12416.

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

normal subgroups only

This group has 368 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#127×D97C211yesmaximal, characteristic
2#99×D97C212yesmaximal, complemented
3#20260528C211yesmaximal, characteristic, complemented
4#20260527C212yesmaximal, complemented
5#353×C97C211yesmaximal, characteristic, complemented
6C2×D4×D97—22no—
7C2×D4×D97C411yescharacteristic, complemented
8C2×D4×D97C2211yescharacteristic
9#15690807C411yescharacteristic, complemented
10#15690806—22no—
11#15690806C411yescharacteristic, complemented
12#15690806C2211yescharacteristic
13C22⋊C4×D97—22no—
14C42×D97C411yescharacteristic, complemented
15#15690725C2212yescomplemented
16#15690724—22no—
17#127×C97C2211yescharacteristic
18#99×C97C2212yescomplemented
19C23×D97—21no—
20C23×D97D412yes—
21C2×D388—22no—
22C2×D388C2×C411yescharacteristic
23C2×D388C2×C411yescharacteristic, complemented
24C2×C22⋊D97—22no—
25C2×C22⋊D97—22no—
26C2×C22⋊D97—21no—
27C2×C22⋊D97D412yes—
28D4×D97—41no—
29D4×D97—41no—
30D4×D97—41no—
31D4×D97—41no—
32D4×D97—48no—
33C2×C4×D97—21no—
34C2×C4×D97—42no—
35C2×C4×D97C2×C411yescharacteristic
36C22⋊Dic97—22no—
37#13095488—42no—
38C4×Dic97—21no—
39C4×Dic97C2×C411yescharacteristic, complemented
40C2×D4×C97—22no—
41C2×D4×C97C2×C411yescharacteristic, complemented
42C2×D4×C97C2311yescharacteristic
43C22⋊C4×C97—22no—
44C42×C97C2×C411yescharacteristic, complemented
45C23×C97—21no—
46C23×C97C2×D412yes—
47C22×D97—22no—
48C22×D97—22no—
49C22×D97—21no—
50C22×D97—44no—
51C22×D97—44no—
52C22×D97—41no—
53C22×D97—41no—
54C22×D97C22⋊C411yescharacteristic
55C22×D97C22⋊C412yes—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—44no—
59C2×C4×C97—21no—
60C2×C4×C97—42no—
61C2×C4×C97C22×C411yescharacteristic
62C22⋊D97—42no—
63C22⋊D97—44no—
64C22⋊D97—44no—
65C22⋊D97—42no—
66C2×Dic97—21no—
67C2×Dic97—42no—
68C2×Dic97C22⋊C411yescharacteristic
69D388—41no—
70D388—41no—
71D388—44no—
72C4×D97—24no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—84no—
76C22×C97—22no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C2×D97—22no—
82C2×D97—22no—
83C2×D97—44no—
84C2×D97—42no—
85C2×D97—42no—
86C2×D97—41no—
87C2×D97—82no—
88C2×D97—82no—
89C2×D97—82no—
90C2×D97#9911yescharacteristic
91C2×D97#9911yescharacteristic
92C4×C97—22no—
93C4×C97—21no—
94C4×C97—82no—
95Dic97—22no—
96Dic97—21no—
97Dic97—82no—
98C2×C97—21no—
99C2×C97—42no—
100C2×C97—81no—
101C2×C97C2×#9911yescharacteristic
102D97—21no—
103D97—42no—
104D97—81no—
105D97#35312yesdirect product factor
106C2×#353—971nomaximal, Sylow
107C97C2×#35311yescharacteristic, complemented, Sylow
108C2×#127—971no—
109C2×#99—972no—
110#353—971no—
111#353—972no—
112#353D9711yescharacteristic, direct product factor
113C22×D4—971no—
114C22×D4—971no—
115C22×D4—1942no—
116#127—971no—
117#127—971no—
118#127—971no—
119#127—1942no—
120#127C2×D9711yescharacteristic
121C2×C22⋊C4—1942no—
122C2×C42—971no—
123#99—972no—
124#99—1942no—
125#99C2×D9712yescomplemented
126C24—972no—
127C24—1941no—
128C2×D4—22no—
129C2×D4—972no—
130C2×D4—971no—
131C2×D4—971no—
132C2×D4—3881no—
133C2×D4—3881no—
134C2×D4—3888no—
135C2×D4—3881no—
136C2×D4—3881no—
137C2×D4—1942no—
138C2×D4—1941no—
139C2×D4—1942no—
140C2×D4—1942no—
141C2×D4C4×D9711yescharacteristic, complemented
142C2×D4C22×D9711yescharacteristic
143C22×C4—971no—
144C22×C4—3882no—
145C22×C4—1941no—
146C22⋊C4—22no—
147C22⋊C4—3882no—
148C22⋊C4—1942no—
149C42—971no—
150C42—1941no—
151C42C4×D9711yescharacteristic, complemented
152C23—21no—
153C23—971no—
154C23—972no—
155C23—3881no—
156C23—3881no—
157C23—3884no—
158C23—3884no—
159C23—1942no—
160C23—1942no—
161C23—1941no—
162C23D4×D9712yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—3881no—
167D4—3884no—
168D4—3881no—
169D4—3882no—
170D4—3884no—
171D4—3884no—
172D4—3882no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—7764no—
176C2×C4—971no—
177C2×C4—3882no—
178C2×C4—1941no—
179C2×C4—1941no—
180C2×C4—1944no—
181C2×C4—1941no—
182C2×C4C2×C4×D9711yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—7762no—
188C22—7762no—
189C22—7762no—
190C22—971no—
191C22—971no—
192C22—3882no—
193C22—3882no—
194C22—3884no—
195C22—3881no—
196C22—1942no—
197C22—1942no—
198C22C22⋊C4×D9711yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—7762no—
203C4—1942no—
204C4—1941no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—7761no—
209C2—972no—
210C2—3882no—
211C2—1941no—
212C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#353×C97C2
socleC2×C97C2×#99
prime radical#127×D97C2
cosocleC2×D4×C97C23
hypercenter#353D97
hypocenterC97C2×#353
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D97C2semidirect
3. #20260528C2semidirect
4. #20260527C2semidirect
5. #353×C97C2semidirect
7. C2×D4×D97C4semidirect
9. #15690807C4semidirect
11. #15690806C4semidirect
14. C42×D97C4semidirect
15. #15690725C22semidirect
18. #99×C97C22semidirect
23. C2×D388C2×C4semidirect
39. C4×Dic97C2×C4semidirect
41. C2×D4×C97C2×C4semidirect
44. C42×C97C2×C4semidirect
105. D97#353direct (N × H)
107. C97C2×#353semidirect
112. #353D97direct (N × H)
125. #99C2×D97semidirect
141. C2×D4C4×D97semidirect
151. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
#127×D972
#99×D972
#202605282
#202605272
#353×C972
C2×#35397