Finite Group Databasebeta

← #27142255

Subgroups — #27142255

Order 12416.

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

normal subgroups only

This group has 312 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
2#20261309C211yesmaximal, characteristic
3#20260744C211yesmaximal, characteristic, complemented
4#20260708C211yesmaximal, characteristic
5#20260492C211yesmaximal, characteristic, complemented
6#20260491C211yesmaximal, characteristic, complemented
7#419×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690818—22no—
10#15690816—22no—
11C2×#13095495C2211yescharacteristic
12M16×D97—22no—
13#15690808C412yescomplemented
14#15690799C412yescomplemented
15#15690793—22no—
16#15690786—22no—
17#15690769C2211yescharacteristic
18#15690769C2211yescharacteristic
19#15690734—22no—
20#15690724—22no—
21C22×Q8×C97C2211yescharacteristic
22C2×M16×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24#103×C97—22no—
25#13095511—21no—
26#13095511—21no—
27#13095511—21no—
28#13095511—21no—
29C2×Dic194C2×C411yescharacteristic
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97D412yes—
32C2×C4×D97D412yes—
33#13095495—21no—
34#13095495—21no—
35#13095494—24no—
36C2×#11715895D412yes—
37C8×D97—24no—
38C4⋊Dic97—21no—
39C4⋊Dic97—21no—
40#13095490—24no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43#13095488—24no—
44C4×Dic97—21no—
45C4×Dic97C2×C412yes—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97C2×C412yes—
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97—21no—
56C2×C8×C97D412yes—
57C4⋊C4×C97—21no—
58C22⋊C4×C97—21no—
59C42×C97—21no—
60C23×C97C2×D411yescharacteristic
61C22×D97C22⋊C412yes—
62Q8×C97—22no—
63Q8×C97—22no—
64Q8×C97—41no—
65Q8×C97—41no—
66C2×C4×C97—22no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×C4×C97#4111yescharacteristic
76C22⋊D97—22no—
77C22⋊D97—22no—
78C2×Dic97—22no—
79C2×Dic97—41no—
80C2×Dic97C22⋊C412yes—
81D388—21no—
82D388—21no—
83C4×D97—22no—
84C4×D97—22no—
85Dic194—21no—
86Dic194—21no—
87C8×C97—22no—
88#11715895—22no—
89C22×C97—41no—
90C22×C97C2×C22⋊C411yescharacteristic
91C22×C97#12111yescharacteristic
92C22×C97#12111yescharacteristic
93C2×D97—22no—
94C2×D97—42no—
95C4×C97—42no—
96C4×C97—81no—
97C4×C97C4×D411yescharacteristic
98C4×C97C4×D411yescharacteristic
99C4×C97#12011yescharacteristic
100C4×C97#12311yescharacteristic
101Dic97—22no—
102Dic97—42no—
103Dic97—81no—
104C2×C97—21no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97#38611yescharacteristic
108D97—42no—
109#1909—971nomaximal, Sylow
110C97#190911yescharacteristic, complemented, Sylow
111#568—971no—
112#536—971no—
113#419—971no—
114#419D9711yescharacteristic, complemented
115#410—971no—
116H(3,ℤ/4)—971no—
117H(3,ℤ/4)—971no—
118C2×#41—971no—
119C22×Q8C2×D9711yescharacteristic
120#131—1942no—
121#131—1942no—
122C2×M16—971no—
123C2×M16—1942no—
124C2×M16C2×D9711yescharacteristic
125C4⋊Q8—972no—
126#124—972no—
127#123—1942no—
128C22⋊Q8—1942no—
129#117—971no—
130#117—971no—
131#117C2×D9711yescharacteristic
132#103—22no—
133#103—1942no—
134#99—1942no—
135#41—1941no—
136#41—1941no—
137#41—1941no—
138#41—1941no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—971no—
145C2×Q8C4×D9712yes—
146C2×D4—972no—
147C2×D4—971no—
148C22×C4—21no—
149C22×C4—972no—
150C22×C4C22×D9711yescharacteristic
151M16—21no—
152M16—21no—
153M16—1944no—
154M16—1941no—
155M16—1941no—
156C2×C8—972no—
157C2×C8—1944no—
158C2×C8C22⋊D9712yes—
159C4⋊C4—21no—
160C4⋊C4—1944no—
161C4⋊C4—1941no—
162C4⋊C4—1941no—
163C22⋊C4—21no—
164C22⋊C4—1944no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C42—21no—
168C42—972no—
169C42—1941no—
170C23—972no—
171C23D4×D9711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—1941no—
177Q8—1941no—
178D4—1942no—
179D4—1942no—
180D4—1941no—
181D4—1941no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—972no—
187C2×C4—3881no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4D4×D9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C2×C4#1309551011yescharacteristic
196C2×C4#1309551111yescharacteristic
197C8—22no—
198C8—1942no—
199C22—41no—
200C22—3882no—
201C22—1942no—
202C22C22⋊C4×D9711yescharacteristic
203C22#1569078111yescharacteristic
204C22#1569078311yescharacteristic
205C4—42no—
206C4—81no—
207C4—7761no—
208C4—3882no—
209C4—1942no—
210C4#1569077011yescharacteristic
211C4C4×C22⋊D9711yescharacteristic
212C4#1569077811yescharacteristic
213C4#1569079211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—3882no—
218C2#2026061111yescharacteristic
subgroupG/H
centerC2#20260611
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#419×C97C2
socleC2×C97#386
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenter#419D97
hypocenterC97#1909
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20260744C2semidirect
5. #20260492C2semidirect
6. #20260491C2semidirect
7. #419×C97C2semidirect
13. #15690808C4semidirect
14. #15690799C4semidirect
110. C97#1909semidirect
114. #419D97semidirect

Maximal subgroups

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

subgroupindex
#202615012
#202613092
#202607442
#202607082
#202604922
#202604912
#419×C972
#190997