Finite Group Databasebeta

← #354×D97 #27143395

Subgroups — #354×D97 #27143395

Order 12416.

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

normal subgroups only

This group has 284 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#99×D97C212yesmaximal
3#20260530C211yesmaximal, characteristic, complemented
4#20260529C212yesmaximal, complemented
5#354×C97C211yesmaximal, characteristic, complemented
6C2×Q8×D97C411yescharacteristic, complemented
7C2×D4×D97C2211yescharacteristic
8#15690799C411yescharacteristic, complemented
9#15690798C411yescharacteristic, complemented
10#15690797C2211yescharacteristic
11#15690796—22no—
12C22⋊C4×D97—22no—
13C22⋊C4×D97—22no—
14C42×D97C411yescharacteristic, complemented
15#15690725C2212yes—
16#15690724—22no—
17#124×C97C2211yescharacteristic
18#99×C97C2212yes—
19C23×D97D412yes—
20C2×Dic194—21no—
21C2×Dic194C2×C411yescharacteristic, complemented
22Q8×D97—41no—
23Q8×D97—41no—
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97D412yes—
26D4×D97—41no—
27D4×D97—41no—
28C2×C4×D97—21no—
29C2×C4×D97—21no—
30C2×C4×D97—42no—
31C2×C4×D97C2×C411yescharacteristic
32C22⋊Dic97—22no—
33C22⋊Dic97—22no—
34#13095488—22no—
35#13095488—22no—
36#13095488—42no—
37C4×Dic97—21no—
38C4×Dic97C2×C411yescharacteristic, complemented
39C2×Q8×C97C2×C411yescharacteristic, complemented
40C2×D4×C97C2311yescharacteristic
41C22⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C42×C97C2×C411yescharacteristic, complemented
44C23×C97C2×D412yes—
45C22×D97—22no—
46C22×D97—22no—
47C22×D97—44no—
48C22×D97C22⋊C411yescharacteristic
49C22×D97C22⋊C412yes—
50Q8×C97—41no—
51D4×C97—41no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—42no—
55C2×C4×C97C22×C411yescharacteristic
56C22⋊D97—42no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—42no—
60C2×Dic97C22⋊C411yescharacteristic
61D388—41no—
62C4×D97—21no—
63C4×D97—21no—
64C4×D97—42no—
65C4×D97—41no—
66C4×D97—41no—
67C4×D97—84no—
68Dic194—42no—
69Dic194—41no—
70C22×C97—22no—
71C22×C97—42no—
72C22×C97C2×C22⋊C411yescharacteristic
73C2×D97—22no—
74C2×D97—22no—
75C2×D97—44no—
76C2×D97—42no—
77C2×D97—82no—
78C2×D97#9911yescharacteristic
79C2×D97#9911yescharacteristic
80C4×C97—21no—
81C4×C97—41no—
82C4×C97—41no—
83C4×C97—82no—
84Dic97—21no—
85Dic97—41no—
86Dic97—41no—
87Dic97—82no—
88C2×C97—21no—
89C2×C97—42no—
90C2×C97C2×#9911yescharacteristic
91D97—21no—
92D97—42no—
93D97#35412yesdirect product factor
94C2×#354—971nomaximal, Sylow
95C97C2×#35411yescharacteristic, complemented, Sylow
96C2×#124—971no—
97C2×#99—972no—
98#354—971no—
99#354—972no—
100#354D9711yescharacteristic, direct product factor
101C22×Q8—971no—
102C22×D4—971no—
103#124—971no—
104#124—971no—
105#124—971no—
106#124—1942no—
107#124C2×D9711yescharacteristic
108C2×C22⋊C4—1942no—
109C2×C22⋊C4—1942no—
110C2×C42—971no—
111#99—972no—
112#99—1942no—
113#99C2×D9712yes—
114C24—972no—
115C2×Q8—971no—
116C2×Q8—3881no—
117C2×Q8—3881no—
118C2×Q8—1941no—
119C2×Q8C4×D9711yescharacteristic, complemented
120C2×D4—972no—
121C2×D4—971no—
122C2×D4—3881no—
123C2×D4—3881no—
124C2×D4C22×D9711yescharacteristic
125C22×C4—971no—
126C22×C4—3882no—
127C22×C4—1941no—
128C22×C4—1941no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—3882no—
132C22⋊C4—1942no—
133C22⋊C4—1942no—
134C22⋊C4—1942no—
135C22⋊C4—1942no—
136C42—971no—
137C42—1941no—
138C42C4×D9711yescharacteristic, complemented
139C23—971no—
140C23—972no—
141C23—3884no—
142C23—1942no—
143C23—1942no—
144C23D4×D9712yes—
145Q8—41no—
146Q8—3881no—
147Q8—3882no—
148D4—41no—
149D4—3881no—
150D4—3882no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—7764no—
155C2×C4—971no—
156C2×C4—3881no—
157C2×C4—3881no—
158C2×C4—3882no—
159C2×C4—3882no—
160C2×C4—1941no—
161C2×C4—1941no—
162C2×C4—1941no—
163C2×C4—1941no—
164C2×C4C2×C4×D9711yescharacteristic
165C22—22no—
166C22—42no—
167C22—7762no—
168C22—971no—
169C22—971no—
170C22—3882no—
171C22—3884no—
172C22—1942no—
173C22—1942no—
174C22C22⋊C4×D9711yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—7762no—
180C4—3881no—
181C4—3881no—
182C4—1941no—
183C2—21no—
184C2—42no—
185C2—972no—
186C2—3882no—
187C2—1941no—
188C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#354×C97C2
socleC2×C97C2×#99
prime radicalC2×D4×D97C22
cosocleC2×D4×C97C23
hypercenter#354D97
hypocenterC97C2×#354
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #20260530C2semidirect
4. #20260529C2semidirect
5. #354×C97C2semidirect
6. C2×Q8×D97C4semidirect
8. #15690799C4semidirect
9. #15690798C4semidirect
14. C42×D97C4semidirect
21. C2×Dic194C2×C4semidirect
38. C4×Dic97C2×C4semidirect
39. C2×Q8×C97C2×C4semidirect
43. C42×C97C2×C4semidirect
93. D97#354direct (N × H)
95. C97C2×#354semidirect
100. #354D97direct (N × H)
119. C2×Q8C4×D97semidirect
138. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
#124×D972
#99×D972
#202605302
#202605292
#354×C972
C2×#35497