Finite Group Databasebeta

← #27142233

Subgroups — #27142233

Order 12416.

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

normal subgroups only

This group has 310 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
1C2×#15690840C211yesmaximal, characteristic
2#20261212C211yesmaximal, characteristic, complemented
3#20260752C211yesmaximal, characteristic
4#20260743C211yesmaximal, characteristic, complemented
5#20260505C211yesmaximal, characteristic, complemented
6#20260491C211yesmaximal, characteristic, complemented
7#410×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9#15690840—22no—
10#15690840—22no—
11C2×#13095501C412yescomplemented
12C2×#13095499C412yescomplemented
13C2×#13095495C2211yescharacteristic
14C2×#13095495C2211yescharacteristic
15C4×Dic194—22no—
16#15690775—22no—
17#15690769C2211yescharacteristic
18#15690766—22no—
19#15690737—22no—
20#15690733—22no—
21C2×#41×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24#99×C97—22no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194C2×C412yes—
29C2×Dic194D412yes—
30C2×Dic194D412yes—
31C22×Dic97—21no—
32#13095501—24no—
33#13095499—24no—
34#13095495—21no—
35#13095495—21no—
36#13095495—21no—
37#13095495—21no—
38C2×#11715895—21no—
39C2×#11715895C2×C412yes—
40C4⋊Dic97—21no—
41#13095490—24no—
42C22⋊Dic97—21no—
43C4×Dic97—24no—
44C4×Dic97D412yes—
45#41×C97—21no—
46#41×C97—21no—
47#41×C97—21no—
48#41×C97—21no—
49C2×Q8×C97C2×C411yescharacteristic
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97—21no—
55C4⋊C4×C97—21no—
56C22⋊C4×C97—21no—
57C22⋊C4×C97D412yes—
58C42×C97—21no—
59C42×C97—21no—
60C23×C97—21no—
61C23×C97C2×D411yescharacteristic
62Q8×C97—21no—
63Q8×C97—21no—
64D4×C97—21no—
65D4×C97—21no—
66D4×C97—41no—
67D4×C97—41no—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97C22×C411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97#4111yescharacteristic
78C2×C4×C97#4111yescharacteristic
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83Dic194—22no—
84Dic194—22no—
85Dic194—22no—
86Dic194C22⋊C414yes—
87#11715895—22no—
88#11715895—41no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—81no—
92C22×C97C4×D411yescharacteristic
93C22×C97#12011yescharacteristic
94C22×C97#12111yescharacteristic
95C4×C97—42no—
96C4×C97—41no—
97C4×C97—81no—
98C4×C97C2×C22⋊C411yescharacteristic
99C4×C97C4×D411yescharacteristic
100C4×C97#12111yescharacteristic
101C4×C97#12311yescharacteristic
102Dic97—24no—
103Dic97—42no—
104C2×C97—21no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97—81no—
108C2×C97#38611yescharacteristic
109#1905—971nomaximal, Sylow
110C97#190511yescharacteristic, complemented, Sylow
111C2×#137—971no—
112#519—971no—
113#421—971no—
114#419—971no—
115#410D9711yescharacteristic
116#344—971no—
117H(3,ℤ/4)—971no—
118C2×#41C2×D9711yescharacteristic
119C22×Q8—971no—
120#137—1942no—
121#137—1942no—
122C2×Q16—972no—
123C2×SD16—972no—
124C2×M16—971no—
125C2×M16—971no—
126C4×Q8—1942no—
127C4×Q8—1942no—
128#117—971no—
129#117—1942no—
130#117C2×D9711yescharacteristic
131#117C2×D9711yescharacteristic
132#104—1942no—
133#103—1942no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—972no—
140C2×Q8—972no—
141C2×Q8—972no—
142C2×Q8—1941no—
143C2×Q8—1941no—
144C2×Q8—1941no—
145C2×Q8C4×D9711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D9711yescharacteristic
148C22×C4—21no—
149C22×C4—1941no—
150C22×C4C22×D9711yescharacteristic
151Q16—1944no—
152SD16—1944no—
153M16—1941no—
154M16—1941no—
155M16—1941no—
156M16—1941no—
157C2×C8—972no—
158C2×C8—1941no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1944no—
162C4⋊C4—1941no—
163C22⋊C4—21no—
164C22⋊C4—1941no—
165C22⋊C4C22⋊D9712yes—
166C42—21no—
167C42—21no—
168C42—972no—
169C42—1944no—
170C23—21no—
171C23C2×C22⋊D9711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—974no—
175Q8—1942no—
176Q8—1942no—
177Q8—1942no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1942no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4D4×D9711yescharacteristic
194C2×C4D4×D9711yescharacteristic
195C2×C4#1309551011yescharacteristic
196C2×C4#1309551111yescharacteristic
197C8—3881no—
198C8—1942no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D9711yescharacteristic
203C22#1569077811yescharacteristic
204C22#1569078111yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—3882no—
209C4—1944no—
210C4C22⋊C4×D9711yescharacteristic
211C4#1569077011yescharacteristic
212C4#1569078311yescharacteristic
213C4#1569079211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2026061111yescharacteristic
subgroupG/H
centerC2#20260611
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#410×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#410D97
hypocenterC97#1905
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20261212C2semidirect
4. #20260743C2semidirect
5. #20260505C2semidirect
6. #20260491C2semidirect
11. C2×#13095501C4semidirect
12. C2×#13095499C4semidirect
110. C97#1905semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
#202612122
#202607522
#202607432
#202605052
#202604912
#410×C972
#190597