Finite Group Databasebeta

← #27142116

Subgroups — #27142116

Order 12416.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#20261501C211yesmaximal, characteristic, complemented
2#20261281C211yesmaximal, characteristic
3#20260752C211yesmaximal, characteristic, complemented
4#20260708C211yesmaximal, characteristic, complemented
5#20260496C211yesmaximal, characteristic, complemented
6#20260494C211yesmaximal, characteristic, complemented
7#421×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690818—22no—
10#15690816—22no—
11C2×#13095495C2211yescharacteristic
12M16×D97—22no—
13#15690808C411yescharacteristic, complemented
14#15690806C411yescharacteristic, complemented
15#15690799C411yescharacteristic, complemented
16#15690797C411yescharacteristic, complemented
17#15690782—22no—
18C4×C22⋊D97—22no—
19#15690769C2211yescharacteristic
20C2×C4×Dic97C2211yescharacteristic
21#15690737—22no—
22#15690724—22no—
23C2×#41×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25#117×C97C2211yescharacteristic
26#104×C97—22no—
27#13095511—21no—
28#13095511—21no—
29#13095511—21no—
30#13095511—21no—
31C2×Dic194C2×C411yescharacteristic
32C2×D388C2×C411yescharacteristic
33C2×C22⋊D97—22no—
34C2×C22⋊D97D412yes—
35C22×Dic97—21no—
36C2×C4×D97D412yes—
37#13095495—21no—
38#13095495D412yescomplemented
39#13095494—24no—
40C2×#11715895—21no—
41C8×D97—24no—
42C4⋊Dic97—21no—
43#13095490—22no—
44C22⋊Dic97—22no—
45C22⋊Dic97—21no—
46#13095488—22no—
47C4×Dic97—21no—
48C4×Dic97—21no—
49C4×Dic97C2×C411yescharacteristic
50C4×Dic97C2×C411yescharacteristic
51#41×C97—21no—
52#41×C97—21no—
53#41×C97—21no—
54#41×C97—21no—
55C2×Q8×C97C2×C411yescharacteristic
56C2×D4×C97—21no—
57C2×D4×C97C2×C411yescharacteristic
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60M16×C97—21no—
61M16×C97D412yescomplemented
62C2×C8×C97—21no—
63C4⋊C4×C97—21no—
64C22⋊C4×C97—21no—
65C42×C97—21no—
66C23×C97—21no—
67C23×C97C2×D411yescharacteristic
68C22×D97C22⋊C412yes—
69Q8×C97—21no—
70Q8×C97—21no—
71D4×C97—21no—
72D4×C97—21no—
73D4×C97—41no—
74D4×C97—41no—
75C2×C4×C97—21no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97C22×C411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C22⋊D97—22no—
86C22⋊D97—22no—
87C22⋊D97—44no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97—42no—
91C2×Dic97—41no—
92C2×Dic97C22⋊C412yes—
93D388—21no—
94D388—21no—
95C4×D97—22no—
96C4×D97—22no—
97Dic194—21no—
98Dic194—21no—
99C8×C97—22no—
100#11715895—22no—
101C22×C97—41no—
102C22×C97—42no—
103C22×C97—41no—
104C22×C97C2×C22⋊C411yescharacteristic
105C22×C97#12111yescharacteristic
106C22×C97#12111yescharacteristic
107C2×D97—22no—
108C2×D97—42no—
109C4×C97—41no—
110C4×C97—81no—
111C4×C97C4×D411yescharacteristic
112C4×C97C4×D411yescharacteristic
113C4×C97#12011yescharacteristic
114C4×C97#12311yescharacteristic
115Dic97—22no—
116Dic97—22no—
117Dic97—41no—
118Dic97—81no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—21no—
122C2×C97—81no—
123C2×C97#38611yescharacteristic
124D97—42no—
125#1884—971nomaximal, Sylow
126C97#188411yescharacteristic, complemented, Sylow
127#568—971no—
128#532—971no—
129#421—971no—
130#421D9711yescharacteristic, complemented
131#410—971no—
132#339—971no—
133#339—971no—
134C2×#41—971no—
135C2×#41C2×D9711yescharacteristic
136#131—1942no—
137#131—1942no—
138C2×M16—971no—
139C2×M16—1942no—
140C2×M16C2×D9711yescharacteristic
141C4⋊Q8—971no—
142#127—971no—
143#124—971no—
144#124—971no—
145#121—1942no—
146C4×D4—1942no—
147#117—971no—
148#117C2×D9711yescharacteristic
149C2×C42—971no—
150#104—22no—
151#104—1942no—
152#99—1942no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—1941no—
158#41—1941no—
159#41—1941no—
160#41—1941no—
161C2×Q8—971no—
162C2×Q8C4×D9711yescharacteristic
163C2×D4—21no—
164C2×D4—972no—
165C2×D4—971no—
166C2×D4—1942no—
167C2×D4C4×D9711yescharacteristic
168C22×C4—21no—
169C22×C4—972no—
170C22×C4—1941no—
171C22×C4C22×D9711yescharacteristic
172M16—21no—
173M16—972no—
174M16—1944no—
175M16—1941no—
176M16C22⋊D9712yescomplemented
177C2×C8—21no—
178C2×C8—1944no—
179C2×C8—1941no—
180C4⋊C4—21no—
181C4⋊C4—1942no—
182C4⋊C4—1941no—
183C22⋊C4—21no—
184C22⋊C4—1942no—
185C22⋊C4—1942no—
186C22⋊C4—1941no—
187C42—21no—
188C42—971no—
189C42—971no—
190C42—1941no—
191C42—1941no—
192C23—21no—
193C23—972no—
194C23D4×D9711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—1941no—
198Q8—1941no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—3884no—
204D4—1942no—
205D4—1942no—
206D4—1941no—
207D4—1941no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—972no—
213C2×C4—3881no—
214C2×C4—3882no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4—1942no—
218C2×C4—1942no—
219C2×C4C2×C4×D9711yescharacteristic
220C2×C4D4×D9711yescharacteristic
221C2×C4D4×D9711yescharacteristic
222C2×C4C2×C22⋊D9711yescharacteristic
223C2×C4#1309551011yescharacteristic
224C2×C4#1309551111yescharacteristic
225C8—22no—
226C8—1942no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—3882no—
231C22—1942no—
232C22C22⋊C4×D9711yescharacteristic
233C22#1569078111yescharacteristic
234C22#1569078311yescharacteristic
235C4—41no—
236C4—81no—
237C4—7761no—
238C4—3881no—
239C4—1942no—
240C4—1942no—
241C4#1569077011yescharacteristic
242C4C4×C22⋊D9711yescharacteristic
243C4#1569077811yescharacteristic
244C4#1569079211yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—81no—
249C2—3882no—
250C2#2026061111yescharacteristic
subgroupG/H
centerC4#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#421×C97C2
socleC2×C97#386
prime radical#20261281C2
cosocleC22×C4×C97C23
hypercenter#421D97
hypocenterC97#1884
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #20261501C2semidirect
3. #20260752C2semidirect
4. #20260708C2semidirect
5. #20260496C2semidirect
6. #20260494C2semidirect
7. #421×C97C2semidirect
13. #15690808C4semidirect
14. #15690806C4semidirect
15. #15690799C4semidirect
16. #15690797C4semidirect
38. #13095495D4semidirect
61. M16×C97D4semidirect
126. C97#1884semidirect
130. #421D97semidirect
176. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615012
#202612812
#202607522
#202607082
#202604962
#202604942
#421×C972
#188497