Finite Group Databasebeta

← #360×D97 #27143466

Subgroups — #360×D97 #27143466

Order 12416.

232 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 67 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#135×D97C211yesmaximal, characteristic
2M32×D97C211yesmaximal, characteristic, complemented
3#106×D97C211yesmaximal, characteristic, complemented
4#20260546C211yesmaximal, characteristic, complemented
5#20260545C211yesmaximal, characteristic, complemented
6#20260544C211yesmaximal, characteristic, complemented
7#360×C97C211yesmaximal, characteristic, complemented
8#41×D97—21no—
9#15690833C411yescharacteristic, complemented
10#15690832C2211yescharacteristic
11#15690831—21no—
12#15690830—21no—
13#15690829C411yescharacteristic, complemented
14Q16×D97C411yescharacteristic, complemented
15SD16×D97—21no—
16D8×D97C411yescharacteristic, complemented
17C2×C8×D97C2211yescharacteristic
18C4⋊C4×D97—21no—
19#15690750C2211yescharacteristic, complemented
20#15690749—21no—
21C16×D97—21no—
22#15690742C2211yescharacteristic, complemented
23#15690741—21no—
24#135×C97C2211yescharacteristic
25M32×C97C2211yescharacteristic, complemented
26#106×C97C2211yescharacteristic, complemented
27#13095512—21no—
28#13095511—21no—
29#13095511—41no—
30#13095510—21no—
31#13095510—41no—
32Q8×D97—21no—
33D4×D97—21no—
34D4×D97—41no—
35C2×C4×D97—41no—
36C2×C4×D97—41no—
37C2×C4×D97D411yescharacteristic
38Dic388C2×C411yescharacteristic, complemented
39#13095501—21no—
40#13095500—21no—
41#13095499—21no—
42#13095498—21no—
43D776C2×C411yescharacteristic, complemented
44#13095496—21no—
45C2×#11715895D411yescharacteristic
46C8×D97C2×C411yescharacteristic
47C8×D97C2×C411yescharacteristic
48C8×D97D411yescharacteristic
49C8×D97D411yescharacteristic
50C4⋊Dic97—21no—
51#13095490—41no—
52#13095486—21no—
53#41×C97—21no—
54Q16×C97C2×C411yescharacteristic, complemented
55SD16×C97—21no—
56D8×C97C2×C411yescharacteristic, complemented
57C2×C8×C97C2311yescharacteristic
58C4⋊C4×C97—21no—
59C16×C97—21no—
60C22×D97—41no—
61C22×D97D811yescharacteristic
62Q8×C97—21no—
63D4×C97—21no—
64D4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97C2×D411yescharacteristic
68C22⋊D97—41no—
69C22⋊D97—41no—
70C22⋊D97—41no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73C2×Dic97D811yescharacteristic
74D388—21no—
75D388—41no—
76C4×D97—41no—
77C4×D97—41no—
78C4×D97—41no—
79C4×D97—82no—
80C4×D97C22⋊C411yescharacteristic
81C4×D97C22⋊C411yescharacteristic
82C4×D97SD1611yescharacteristic
83C4×D97SD1611yescharacteristic
84Dic194—21no—
85Dic194—41no—
86C8×C97C22×C411yescharacteristic
87C8×C97C2×D411yescharacteristic
88#11715895C22⋊C411yescharacteristic
89#11715895C22⋊C411yescharacteristic
90C22×C97—41no—
91C22×C97C2×D811yescharacteristic
92C2×D97—22no—
93C2×D97—41no—
94C2×D97—82no—
95C2×D97#10211yescharacteristic
96C2×D97#10211yescharacteristic
97C4×C97—41no—
98C4×C97—81no—
99C4×C97C2×C22⋊C411yescharacteristic
100C4×C97C2×SD1611yescharacteristic
101Dic97—41no—
102Dic97—81no—
103Dic97#10211yescharacteristic
104Dic97#10211yescharacteristic
105C2×C97—21no—
106C2×C97—81no—
107C2×C97C2×#10211yescharacteristic
108D97—21no—
109D97—81no—
110D97#36012yesdirect product factor
111C2×#360—971nomaximal, Sylow
112C97C2×#36011yescharacteristic, complemented, Sylow
113C2×#135—971no—
114C2×M32—971no—
115C2×#106—971no—
116#360—971no—
117#360—971no—
118#360—971no—
119#360D9711yescharacteristic, direct product factor
120C2×#41—1941no—
121#135—971no—
122#135—971no—
123#135—971no—
124#135—1941no—
125#135—1941no—
126#135C2×D9711yescharacteristic
127C2×Q16—971no—
128C2×SD16—1941no—
129C2×D8—971no—
130C22×C8—971no—
131C2×C4⋊C4—1941no—
132M32—971no—
133M32—1941no—
134M32C2×D9711yescharacteristic, complemented
135C2×C16—1941no—
136#106—971no—
137#106—1941no—
138#106C2×D9711yescharacteristic, complemented
139#41—21no—
140#41—3881no—
141#41—3881no—
142#41—1941no—
143#41—1941no—
144#41—1941no—
145C2×Q8—1941no—
146C2×D4—3881no—
147C2×D4—1941no—
148C22×C4—971no—
149C22×C4—3881no—
150C22×C4—3881no—
151Q16—971no—
152Q16—1941no—
153Q16C4×D9711yescharacteristic, complemented
154SD16—21no—
155SD16—1941no—
156SD16—1941no—
157SD16—1941no—
158D8—971no—
159D8—1941no—
160D8C4×D9711yescharacteristic, complemented
161C2×C8—971no—
162C2×C8—971no—
163C2×C8—971no—
164C2×C8—971no—
165C2×C8—971no—
166C2×C8C22×D9711yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—3881no—
169C4⋊C4—1941no—
170C16—21no—
171C16—1941no—
172C23—971no—
173C23—3881no—
174Q8—21no—
175Q8—3881no—
176Q8—1941no—
177D4—21no—
178D4—41no—
179D4—3881no—
180D4—3881no—
181D4—3881no—
182D4—3881no—
183D4—1941no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—7762no—
187C2×C4—971no—
188C2×C4—971no—
189C2×C4—971no—
190C2×C4—971no—
191C2×C4—971no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—3881no—
195C2×C4—3881no—
196C2×C4—3881no—
197C2×C4D4×D9711yescharacteristic
198C8—971no—
199C8—971no—
200C8C2×C4×D9711yescharacteristic
201C8D4×D9711yescharacteristic
202C22—41no—
203C22—7762no—
204C22—971no—
205C22—971no—
206C22—3881no—
207C22—1942no—
208C22D8×D9711yescharacteristic
209C4—41no—
210C4—81no—
211C4—7761no—
212C4—971no—
213C4—971no—
214C4—3881no—
215C4C22⋊C4×D9711yescharacteristic
216C4SD16×D9711yescharacteristic
217C2—21no—
218C2—81no—
219C2—7761no—
220C2—972no—
221C2—1941no—
222C2#102×D9711yescharacteristic
subgroupG/H
centerC2#102×D97
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#360×C97C2
socleC2×C97C2×#102
prime radical#135×D97C2
cosocleC2×C8×C97C23
hypercenter#360D97
hypocenterC97C2×#360
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. M32×D97C2semidirect
3. #106×D97C2semidirect
4. #20260546C2semidirect
5. #20260545C2semidirect
6. #20260544C2semidirect
7. #360×C97C2semidirect
9. #15690833C4semidirect
13. #15690829C4semidirect
14. Q16×D97C4semidirect
16. D8×D97C4semidirect
19. #15690750C22semidirect
22. #15690742C22semidirect
25. M32×C97C22semidirect
26. #106×C97C22semidirect
38. Dic388C2×C4semidirect
43. D776C2×C4semidirect
54. Q16×C97C2×C4semidirect
56. D8×C97C2×C4semidirect
110. D97#360direct (N × H)
112. C97C2×#360semidirect
119. #360D97direct (N × H)
134. M32C2×D97semidirect
138. #106C2×D97semidirect
153. Q16C4×D97semidirect
160. D8C4×D97semidirect

Maximal subgroups

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

subgroupindex
#135×D972
M32×D972
#106×D972
#202605462
#202605452
#202605442
#360×C972
C2×#36097